Download Quantum field theory for matter under extreme conditions

Quantum field theory for matter under extreme
conditions
Prof. Dr. David Blaschke
Institute for Theoretical Physics, University of Wroclaw
Winter semester 2006 - 2007
Abstract
The series of lectures gives an introduction to the modern formulation of quantum
field theories using Feynman path integrals and the hierarchy of Dyson-Schwinger
equations. The formalism is developed for the examples of scalar fields, fermionic
fields and gauge boson fields and extended to the conditions of finite temperatures, densities and strong fields with special emphasis on phase transitions in
QCD, relevant for the physics of ultrarelativistic heavy-ion collisions and the
physics of compact stars. Nonequilibrium quantum field theory is considered for
processes of particle creation in strong, electromagnetic and weak interactions
relevant for the above fields of application and for the high-intensity optical and
X-ray lasers.
Contents
1 Quantum Fields at Zero Temperature
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Minkowski Space Conventions . . . . . . . . . . . . . . . .
1.2.1 Four Vectors . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Dirac Matrices . . . . . . . . . . . . . . . . . . . .
1.3 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Free Particle Solutions . . . . . . . . . . . . . . . .
1.4 Green Functions . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Free-Fermion Propagator . . . . . . . . . . . . . . .
1.4.2 Green Function for the Interacting Theory . . . . .
1.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Path Integral in Quantum Mechanics . . . . . . . . . . . .
1.6 Functional Integral in Quantum Field Theory . . . . . . .
1.6.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . .
1.6.2 Lagrangian Formulation of Quantum Field Theory
1.6.3 Quantum Field Theory for a Free Scalar Field . . .
1.6.4 Scalar Field with Self-Interactions . . . . . . . . . .
1.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Functional Integral for Fermions . . . . . . . . . . . . . . .
1.7.1 Finitely Many Degrees of Freedom . . . . . . . . .
1.7.2 Fermionic Quantum Field . . . . . . . . . . . . . .
1.7.3 Generating Functional for Free Dirac Fields . . . .
1.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Functional Integral for Gauge Field Theories . . . . . . . .
1.8.1 Faddeev-Popov Determinant and Ghosts . . . . . .
1.8.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Dyson-Schwinger Equations . . . . . . . . . . . . . . . . .
1.9.1 Photon Vacuum Polarization . . . . . . . . . . . . .
1.9.2 Fermion Self Energy . . . . . . . . . . . . . . . . .
1.9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Perturbation Theory . . . . . . . . . . . . . . . . . . . . .
1.10.1 Quark Self Energy . . . . . . . . . . . . . . . . . .
1
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1
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50
1.10.2 Dimensional Regularization . .
1.10.3 Regularized Quark Self Energy
1.10.4 Exercises . . . . . . . . . . . . .
1.11 Renormalized Quark Self Energy . . .
1.11.1 Renormalized Lagrangian . . .
1.11.2 Renormalization Schemes . . .
1.11.3 Renormalized Gap Equation . .
1.11.4 Exercises . . . . . . . . . . . . .
1.12 Dynamical Chiral Symmetry Breaking
1.12.1 Euclidean Metric . . . . . . . .
1.12.2 Chiral Symmetry . . . . . . . .
1.12.3 Mass Where There Was None .
References . . . . . . . . . . . . . . .
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2 Quantum Fields at Finite Temperature and Density
2.1 Ensembles and Partition Function . . . . . . . . . . . .
2.2 Bosonic Fields . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Neutral Scalar Field . . . . . . . . . . . . . . .
2.3 Fermionic Fields . . . . . . . . . . . . . . . . . . . . . .
2.4 Gauge Fields . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Quantizing the Electromagnetic Field . . . . . .
2.4.2 Blackbody radiation . . . . . . . . . . . . . . .
2.5 Interactions: Hubbard-Stratonovich Trick . . . . . . . .
2.5.1 Nambu–Jona-Lasinio (NJL) Model . . . . . . .
2.5.2 Mesonic correlations at finite temperature . . .
2.5.3 Matsubara frequency sums . . . . . . . . . . . .
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3 Applications to Heavy-Ion Collisions and Compact Stars
103
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4 Particle Production by Strong Fields
4.1 Quantum kinetic equation for particle production .
4.1.1 Introduction . . . . . . . . . . . . . . . . . .
4.1.2 Dynamics of pair creation . . . . . . . . . .
4.1.3 Discussion of the source term . . . . . . . .
4.1.4 Summary . . . . . . . . . . . . . . . . . . .
4.2 Time-dependent masses: The Inertial Mechanism .
4.2.1 Scalar field . . . . . . . . . . . . . . . . . .
4.2.2 Fermion field . . . . . . . . . . . . . . . . .
4.2.3 Massive vector bosons . . . . . . . . . . . .
4.2.4 Application to conformal cosmology models
4.2.5 The systems with unstable vacuum . . . . .
4.2.6 Summary . . . . . . . . . . . . . . . . . . .
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177
Chapter 1
Quantum Fields at Zero
Temperature
1.1
Introduction
This is the first part of a series of lectures whose aim is to provide the tools for
the completion of a realistic calculation in quantum field theory (QFT) as it is
relevant to Hadron Physics.
Hadron Physics lies at the interface between nuclear and particle (high energy) physics. Its focus is an elucidation of the role played by quarks and gluons
in the structure of, and interactions between, hadrons. This was once particle physics but that has since moved to higher energy in search of a plausible
grand unified theory and extensions of the so-called Standard Model. The only
high-energy physicists still focusing on hadron physics are those performing the
numerical experiments necessary in the application of lattice gauge theory, and
those pushing at the boundaries of applicability of perturbative QCD or trying
to find new kinematic regimes of validity.
There are two types of hadron: baryons and mesons: the proton and neutron are baryons; and the pion and kaon are mesons. Historically the names
distinguished the particle classes by their mass but it is now known that there
are structural differences: hadrons are bound states, and mesons and baryons
are composed differently. Hadron physics is charged with the responsibility of
providing a detailed understanding of the differences.
To appreciate the difficulties inherent in this task it is only necessary to remember that even the study of two-electron atoms is a computational challenge.
This is in spite of the fact that one can employ the Schrödinger equation for this
problem and, since it is not really necessary to quantize the electromagnetic field,
the underlying theory has few complications.
The theory underlying hadron physics is quantum chromodynamics (QCD),
and its properties are such that a simple understanding and simple calculations
1
are possible only for a very small class of problems. Even on the domain for which
a perturbative application of the theory is appropriate, the final (observable)
states in any experiment are always hadrons, and not quarks or gluons, so that
complications arise in the comparison of theory with experiment.
The premier experimental facility for exploring the physics of hadrons is the
Thomas Jefferson National Accelerator Facility (TJNAF), in Newport News, Virginia. Important experiments are also performed at the Fermilab National Accelerator Facility (FermiLab), in Batavia, Illinois, and at the Deutsches Elektronensynchrotron (DESY) in Hamburg. These facilities use high-energy probes
and/or large momentum-transfer processes to explore the transition from the
nonperturbative to the perturbative domain in QCD.
On the basis of the introduction to nonperturbative methods in QFT in the
vacuum, we will develop in the second part of the lecture series the tools for a
generalization to the situation many-particle systems of hadrons at finite temperatures and densities in thermodynamical equilibrium within the Matsubara
formalism.
The third part in the series of lectures is devoted to applications for the
QCD phase transition from hadronic matter to a quark-gluon plasma (QGP)
in relativistic heavy-ion collisions and in the interior of compact stars. Data
are provided from a completed program of experiments at the CERN-SPS and
presently running programs at RHIC Brookhaven. The future of this direction
will soon open a new domain of energy densities (temperatures) at CERN-LHC
(2007) and at the future GSI facility FAIR, where construction shall be completed
in 2015. The CBM experiment will then allow insights into the QCD phase
transition at relatively low temperatures and high baryon densities, a situation
which bears already similarities with the interior of compact stars, formed in
supernova explosions and observed as pulsars in isolation or in binary systems.
Modern astrophysical data have an unprecendented level of accuracy allowing for
new stringent constraints on the behavior of the hadronic equation of state at
high densities.
The final part of lectures enters the domain of nonequilibrium QFT and will
focus on a particular problem which, however, plays a central role: particle production in strong time-dependent external fields. The Schwinger mechanism for
pair production as a strict result of quantum electrodynamics (QED) is still not
experimentally verified. We are in the fortunate situation that developments of
modern laser facilities in the X-ray energy domain with intensities soon reaching
the Schwinger limit for electron- positron pair creation from vacuum will allow
new insights and experimental tests of approaches to nonperturbative QFT in
the strong field situation. These insights will allow generalizations for the other
field theories such as the Standard Model and QCD, where still the puzzles of
initial conditions in heavy-ion collisions and the origin of matter in the Universe
remain to be solved.
2
1.2
Minkowski Space Conventions
In the first part of this lecture series I will use the Minkowski metrics. Later I
will employ a Euclidean metric because that is most useful and appropriate for
nonperturbative calculations.
1.2.1
Four Vectors
Normal spacetime coordinates are denoted by a contravariant four-vector:
xµ := (x0 , x1 , x2 , x3 ) ≡ (t, x, y, z).
(1.1)
Throughout: c = 1 = h̄, and the conversion between length and energy is just:
1fm = 1/(0.197327 GeV) = 5.06773 GeV −1 .
(1.2)
The covariant four-vector is obtained by changing the sign of the spatial components of the contravariant vector:
xµ := (x0 , x1 , x2 , x3 ) ≡ (t, −x, −y, −z) = gµν xν ,
(1.3)
where the metric tensor is




gµν = 
1 0
0
0
0 −1 0
0
0 0 −1 0
0 0
0 −1



.

(1.4)
The contracted product of two four-vectors is (a, b) := gµν aµ bν = aν bν : i.e., a
contracted product of a covariant and a contravariant four-vector. The Poincaréinvariant length of any vector is x2 := (x, x) = t2 − ~x2 .
Momentum vectors are similarly defined:
and
pµ = (E, px , py , pz ) = (E, p~).
(1.5)
(p, k) = pν k ν = Ep Ek − p~~k .
(1.6)
(x, p) = tE − ~xp~ .
(1.7)
∂
∂ 1~
= (i , ∇)
=: i∇µ
∂xµ
∂t i
(1.8)
Likewise,
The momentum operator
pµ := i
3
transforms as a contravariant four-vector, and I denote the four-vector analogue
of the Laplacian as
∂ ∂
.
(1.9)
∂ 2 := −pµ pµ =
∂xµ ∂xµ
The contravariant four-vector associated with the electromagnetic field is
~
Aµ (x) = (φ(x), A(x))
(1.10)
with the electric and magnetic field strengths obtained from
F µν = ∂ ν Aµ − ∂ µ Aν .
(1.11)
For example,
~ = −∇φ
~ − ∂ A.
~
Ei = F 0i ; E
(1.12)
∂t
Similarly, B i = εijk F jk ; j, k = 1, 2, 3. Analogous definitions hold in QCD for the
chromomagnetic field strengths.
1.2.2
Dirac Matrices
The Dirac matrices are indispensable in a manifestly Poincaré covariant description of particles with spin; i.e., intrinsic angular momentum, such as fermions
with spin 12 . The Dirac matrices are defined by the Clifford Algebra
{γ µ , γ ν } = 2g µν
(1.13)
and one common 4 × 4 representation is [each entry represents a 2 × 2 matrix]
γ0 =
"
1 0
0 −1
#
, ~γ =
"
0 −i
i 0
#
0 ~σ
−~σ 0
#
,
(1.14)
where ~σ are the usual Pauli Matrices:
1
σ =
"
0 1
1 0
#
2
,σ =
"
3
,σ =
"
1 0
0 −1
#
,
(1.15)
and 1 = diag[1, 1]. Clearly γ0† = γ0 ; and ~γ † = −~γ . NB. These properties are not
specific to this representation; e.g., γ 1 γ 1 = −1, an analogue of the properties of
a purely imaginary number.
In discussing spin, two combinations of Dirac matrices frequently appear:
and I note that
i
σ µν = [γ µ , γν] , γ 5 = iγ 0 γ 1 γ 2 γ 3 = γ5 ,
2
(1.16)
i
γ 5 σ µν = εµνρσ σρσ ,
2
(1.17)
4
with εµνρσ the completely antisymmetric Lévi-Civita tensor: ε01223 = +1, εµνρσ =
−εµνρσ . In the representation introduced above,
5
γ =
Furthermore,
"
0 1
1 0
#
.
(1.18)
{γ5 , γµ} = 0 , γ5† = γ5 .
(1.19)
~,
γ µ Aµ =: A
/ = γ 0 A0 − ~γ A
γ µ pµ =: /p = γ 0 E − ~γ p~ ,
∂
~ = iγ µ ∂ .
γ µ pµ =: i∇
/ ≡ i∂/ = iγ 0 + i~γ ∇
∂t
∂xµ
(1.20)
(1.21)
γ5 plays a special role in the discussion of parity and chiral symmetry, two key
aspects of the Standard Model.
The “slash” notation is a frequently used shorthand:
(1.22)
(1.23)
The following identities are important in evaluating the cross sections for
decay and scattering processes:
trγ5
tr1
tra
//b
tra
/1 a
/2 a
/3 a
/4
tra
/1 . . . a
/n
trγ5 a
//b
trγ5 a
/1 a
/2 a
/3 a
/4
γµ a
/γ µ
γµ a
//bγ µ
γµ a
//b/cγ µ
=
=
=
=
=
=
=
=
=
=
0,
4,
4(a, b) ,
4[(a1 , a2 )(a3 , a4 ) − (a1 , a3 )(a2 , a4 ) + (a1 , a4 )(a2 , a3 )]
0 , for n odd ,
0,
4iεµνρσ aµ1 aν2 aρ3 aσ4 ,
−2a
/,
4(a, b) ,
−2c
//ba
/.
(1.24)
(1.25)
(1.26)
,(1.27)
(1.28)
(1.29)
(1.30)
(1.31)
(1.32)
(1.33)
(1.34)
They can all be derived using the fact that Dirac matrices satisfy the Clifford
algebra. For example (remember trAB = trBA):
tra
//b = aµ bν trγ µ γ ν
1
= aµ bν tr[γ µ γ ν + γ ν γ µ ]
2
1
= aµ bν tr[2g µν 1]
2
1 µν
= aµ bν 2g 4 = 4(a, b) .
2
(1.35)
(1.36)
(1.37)
(1.38)
(1.39)
5
1.3
Dirac Equation
The unification of special relativity (Poincaré covariance) and quantum mechanics took some time. Even today many questions remain as to a practical implementation of a Hamiltonian formulation of the relativistic quantum mechanics of
interacting systems. The Poincaré group has ten generators: the six associated
with the Lorentz transformations – rotations and boosts – and the four associated
with translations. Quantum mechanics describes the time evolution of a system
with interactions, and that evolution is generated by the Hamiltonian. However,
if the theory is formulated with an interacting Hamiltonian then boosts will almost always fail to commute with the Hamiltonian and thus the state vector
calculated in one momentum frame will not be kinematically related to the state
in another frame. That makes a new calculation necessary in every frame and
hence the discussion of scattering, which takes a state of momentum p to another
state with momentum p0 , is problematic. (See, e.g., Ref. [2]).
The Dirac equation provides the starting point for a Lagrangian formulation
of the quantum field theory for fermions interacting via gauge boson exchange.
For a free fermion
[i∂
/ − m] ψ = 0 ,
(1.40)
where ψ(x) is the fermion’s “spinor” – a four component column vector, while
in the presence of an external electromagnetic field the fermion’s wave function
obeys
[i∂
/ − eA
/ − m] ψ = 0 ,
(1.41)
which is obtained, as usual, via “minimal substitution:” pµ → pµ − eAµ in
Eq. (1.40). These equations have a manifestly covariant appearance but proving
their covariance requires the development of a representation of Lorentz transformations on spinors and Dirac matrices:
ψ 0 (x0 ) = S(Λ) ψ(x) ,
Λνµ γ µ = S −1 (Λ)γ ν S(Λ) ,
i
2
S(Λ) = exp[− σµν ω µν ] ,
(1.42)
(1.43)
(1.44)
where ω µν are the parameters characterising the particular Lorentz transformation. (Details can be found in the early chapters of Refs. [1, 3].)
1.3.1
Free Particle Solutions
As usual, to obtain an explicit form for the free-particle solutions one substitutes
a plane wave and finds a constraint on the wave number. In this case there
are two qualitatively different types of solution, corresponding to positive and
negative energy. (An appreciation of the physical reality of the negative energy
6
solutions led to the prediction of the existence of antiparticles.) One inserts
ψ (+) = e−i(k,x) u(k) , ψ (−) = e+i(k,x) v(k) ,
(1.45)
into Eq. (1.40) and obtains
(k/ − m) u(k) = 0 , (k/ + m) v(k) = 0 .
(1.46)
Assuming that the particle’s mass in nonzero then working in the rest frame
yields
(γ 0 − 1) u(m, ~0) = 0 , (γ 0 + 1) v(m, ~0) = 0 .
(1.47)
There are clearly (remember the form of γ 0 ) two linearly-independent solutions
of each equation:
1
0
0
0
0
1
0
0
0
0
1
0
0






 0 








u(1) (m, ~0) =   , u(2) (m, ~0) =   , v (1) (m, ~0) =   , v (2) (m, ~0) =   .






 0 
1
(1.48)
The solution in an arbitrary frame can be obtained simply via a Lorentz boost
but it is even simpler to observe that







(k/ − m) (k/ + m) = k 2 − m2 = 0 ,
(1.49)
with the last equality valid when the particles are on shell, so that the solutions
for arbitrary k µ are: for positive energy (E > 0),
u
(α)
(k) = q
k/ + m
u
2m(m + E)
(α)
(m, ~0) =
 E + m 1/2


2m

σ·k

 q
2m(m + E)

φα (m, ~0) 

,
 (1.50)
φα (m, ~0) 

with the two-component spinors, obviously to be identified with the fermion’s
spin in the rest frame (the only frame in which spin has its naive meaning)
φ
(1)
=
1
0
!
, φ
(2)
=
0
1
!
;
(1.51)
and, for negative energy,
v
(α)
(k) = q
−k/ + m
2m(m + E)
v
(α)
(m, ~0) =

σ·k
 q

2m(m + E)

 
E + m 1/2
2m
7

χ (m, ~0) 
α
χα (m, ~0)

 (1.52)
,


with χ(α) obvious analogues of φ(α) in Eq. (1.51). (This approach works because
it is clear that there are two, and only two, linearly-independent solutions of
the momentum space free-fermion Dirac equations, Eqs. (1.46), and, for the homogeneous equations, any two covariant solutions with the correct limit in the
rest-frame must give the correct boosted form.)
In quantum field theory, as in quantum mechanics, one needs a conjugate
state to define an inner product. For fermions in Minkowski space that conjugate
is ψ̄(x) = ψ † (x)γ 0 , which satisfies
←
ψ̄(i ∂/ + m) = 0 .
(1.53)
This equation yields the following free particle spinors in momentum space
k/ + m
ū(α) (k) = ū(α) (m, ~0) q
2m(m + E)
v̄ (α) (k) = v̄ (α) (m, ~0) q
−k/ + m
2m(m + E)
(1.54)
,
(1.55)
where I have used γ 0 (γ µ )† γ 0 = γ µ , which relation is easily derived and is particularly important in the discussion of intrinsic parity; i.e., the transformation properties of particles and antiparticles under space reflections (an improper Lorentz
transformation).
The momentum space free-particle spinors are orthonormalised
ū(α) (k) u(β) (k) = δαβ
v̄ (α )(k) v (β) (k) = −δαβ
ū(α )(k) v (β) (k) = 0
.
v̄ (α )(k) u(β) (k) = 0
(1.56)
It is now possible to construct positive and negative energy projection operators. Consider
Λ+ (k) :=
X
α=1,2
u(α) (k) ⊗ ū(α) (k) .
(1.57)
It is plain from the orthonormality relations, Eqs. (1.56), that Λ+ (k) projects
onto positive energy spinors in momentum space; i.e.,
Λ+ (k) u(α) (k) = u(α) (k) , Λ+ (k) v (α) (k) = 0 .
(1.58)
Now, since
X
α=1,2
u
(α)
(m, ~0) ⊗ ū
(α)
(m, ~0) =
1 0
0 0
!
=
1 + γ0
,
2
(1.59)
then
1
1 + γ0
Λ+ (k) =
(k/ + m)
(k/ + m) .
2m(m + E)
2
8
(1.60)
Noting that for k 2 = m2 ; i.e., on shell,
(k/ + m) γ 0 (k/ + m) = 2E (k/ + m) ,
(k/ + m) (k/ + m) = 2m (k/ + m) ,
(1.61)
(1.62)
one finally arrives at the simple closed form:
Λ+ (k) =
k/ + m
.
2m
(1.63)
The negative energy projection operator is
Λ− (k) := −
X
α=1,2
v (α) (k) ⊗ v̄ (α) (k) =
−k/ + m
.
2m
(1.64)
The projection operators have the following important properties:
Λ2± (k) = Λ± (k) ,
tr Λ± (k) = 2 ,
Λ+ (k) + Λ− (k) = 1 .
(1.65)
(1.66)
(1.67)
Such properties are a characteristic of projection operators.
1.4
Green Functions
The Dirac equation is a partial differential equation. A general method for solving
such equations is to use a Green function, which is the inverse of the differential
operator that appears in the equation. The analogy with matrix equations is
obvious and can be exploited heuristically.
Equation (1.41):
[i∂
/x − eA
/(x) − m] ψ(x) = 0 ,
(1.68)
yields the wave function for a fermion in an external electromagnetic field. Consider the operator obtained as a solution of the following equation
[i∂
/x0 − eA
/(x0 ) − m] S(x0 , x) = 1 δ 4 (x0 − x) .
(1.69)
It is immediately apparent that if, at a given spacetime point x, ψ(x) is a solution
of Eq. (1.68), then
Z
ψ(x0 ) :=
d4 x S(x0 , x) ψ(x)
(1.70)
is a solution of
[i∂
/x0 − eA
/(x0 ) − m] ψ(x0 ) = 0 ;
i.e., S(x0 , x) has propagated the solution at x to the point x0 .
9
(1.71)
This effect is equivalent to the application of Huygen’s principle in wave mechanics: if the wave function at x, ψ(x), is known, then the wave function at
x0 is obtained by considering ψ(x) as a source of spherical waves that propagate
outward from x. The amplitude of the wave at x0 is proportional to the amplitude
of the original wave, ψ(x), and the constant of proportionality is the propagator
(Green function), S(x0 , x). The total amplitude of the wave at x0 is the sum over
all the points on the wavefront; i.e., Eq. (1.70).
This approach is practical because all physically reasonable external fields
can only be nonzero on a compact subdomain of spacetime. Therefore the solution of the complete equation is transformed into solving for the Green function,
which can then be used to propagate the free-particle solution, already found,
to arbitrary spacetime points. However, obtaining the exact form of S(x0 , x) is
impossible for all but the simplest cases.
1.4.1
Free-Fermion Propagator
In the absence of an external field Eq. (1.69) becomes
[i∂
/x0 − m] S(x0 , x) = 1 δ 4 (x0 − x) .
(1.72)
Assume a solution of the form:
S0 (x0 , x) = S0 (x0 − x) =
Z
d4 p −i(p,x0 −x)
e
S0 (p) ,
(2π)4
(1.73)
so that substituting yields
(p
/ − m) S0 (p) = 1 ; i.e., S0 (p) =
/p + m
.
p2 − m2
(1.74)
To obtain the result in configuration space one must adopt a prescription for
handling the on-shell singularities in S(p), and that convention is tied to the
boundary conditions applied to Eq. (1.72). An obvious and physically sensible definition of the Green function is that it should propagate positive-energyfermions and -antifermions forward in time but not backwards in time, and vice
versa for negative energy states.
As we have already seen, the wave function for a positive energy free-fermion
is
ψ (+) (x) = u(p) e−i(p,x) .
(1.75)
The wave function for a positive energy antifermion is the charge-conjugate of
the negative-energy fermion solution:
ψc(+) (x) = C γ 0 v(p) ei(p,x)
10
∗
= C v̄(p)T e−i(p,x) ,
(1.76)
where C = iγ 2 γ 0 and (·)T denotes matrix transpose. (This defines the operation
of charge conjugation.) It is thus evident that our physically sensible S0 (x0 − x)
can only contain positive-frequency components for x00 − x0 > 0.
One can ensure this via a small modification of the denominator of Eq. (1.74):
S0 (p) =
/p + m
/p + m
→ 2
,
2
2
p −m
p − m2 + iη
(1.77)
with η → 0+ at the end of all calculations. Inserting this form in Eq. (1.73) is
equivalent to evaluating the p0 integral by employing a contour in the complex-p0
that is below the real-p0 axis for p0 < 0, and above it for p0 > 0. This prescription
defines the Feynman propagator.
To be explicit:
d3 p i~p·(~x0 −~x) 1
e
(2π)3
2 ω(~
p)
"
#
Z ∞
0
dp
/p + m
/p + m
−ip0 (x00 −x0 )
−ip0 (x00 −x0 )
−e
×
e
,
p0 − ω(~
p) + iη
p0 + ω(~
p) − iη
−∞ 2π
(1.78)
√
where the energy ω(~
p) = p~2 + m2 . The integrals are easily evaluated using
standard techniques of complex analysis, in particular, Cauchy’s Theorem.
Focusing on the first term of the sum inside the square brackets, it is apparent
that the integrand has a pole in the fourth quadrant of the complex p0 -plane. For
x00 − x0 > 0 we can evaluate the p0 integral by considering a contour closed by
a semicircle of radius R → ∞ in the lower half of the complex p0 -plane: the
integrand vanishes exponentially along that arc, where p0 = −iy, y > 0, because
(−i) (−iy) (x00 − x0 ) = −y (x00 − x0 ) < 0. The closed contour is oriented clockwise
so that
S0 (x0 − x) =
Z
∞
−∞
Z
dp0 −ip0 (x00 −x0 )
/p + m
−ip0 (x00 −x0 )
e
=
(−)
i
e
(p
/
+
m)
0
0
+
p =ω(~
p)−iη +
2π
p − ω(~
p) + iη
0
= −i e−iω(~p)(x0 −x0 ) (γ 0 ω(~
p) − γ · p~ + m)
−iω(~
p)(x00 −x0 )
= −i e
2m Λ+ (p) .
(1.79)
For x00 − x0 < 0 the contour must be closed in the upper half plane but therein
the integrand is analytic and hence the result is zero. Thus
Z
∞
−∞
/p + m
dp0 −ip0 (x00 −x0 )
0
e
= −i θ(x00 − x0 ) e−iω(~p)(x0 −x) 2m Λ+ (~
p) .
0
+
2π
p − ω(~
p) + iη
Observe that the projection operator is only a function of p~ because p0 = ω(~
p).
Consider now the second term in the brackets. The integrand has a pole in
the second quadrant. For x00 − x0 > 0 the contour must be closed in the lower half
11
plane, the pole is avoided and hence the integral is zero. However, for x00 − x0 < 0
the contour should be closed in the upper half plane, where the integrand is
obviously zero on the arc at infinity. The contour is oriented anticlockwise so
that
Z
∞
−∞
dp0 −ip0 (x00 −x0 )
/p + m
0
−ip0 (x00 −x)
e
=
i
θ(x
−
x
)
e
(p
/
+
m)
0
0
0
p =−ω(~
p)+iη +
2π
p0 + ω(~
p) − iη +
0
= i θ(x0 − x00 ) e+iω(~p)(x0 −x) (−γ 0 ω(~
p) − γ · p~ + m)
0
+iω(~
p)(x00 −x)
= i θ(x0 − x0 ) e
2m Λ− (−~
p) .
(1.80)
Putting these results together, changing variables p~ → −~
p in Eq. (1.80), we
have
0
S0 (x − x) = −i
Z
d3 p m h 0
0
θ(x0 − x0 ) e−i(p̃,x −x) Λ+ (~
p)
3
(2π) ω(~
p)
0
i
+ θ(x0 − x00 ) ei(p̃,x −x) Λ− (~
p) , (1.81)
[(p̃µ ) = (ω(~
p), p~)] which, using Eqs. (1.58) and its obvious analogue for Λ− (~
p),
manifestly propagates positive-energy solutions forward in time and negative energy solutions backward in time, and hence satisfies the physical requirement
stipulated above.
Another useful representation is obtained merely by observing that
Z
∞
−∞
dp0 −ip0 (x00 −x)
/p + m
e
=
2π
p0 − ω(~
p) + iη +
Z
=
Z
/p + m
dp0 −ip0 (x00 −x)
e
=
2π
p0 + ω(~
p) − iη +
Z
∞
−∞
∞
−∞
p) − ~γ · p~ + m
dp0 −ip0 (x00 −x) γ 0 ω(~
e
2π
p0 − ω(~
p) + iη +
0
dp −ip0 (x00 −x)
1
e
2mΛ+ (~
p) 0
,
2π
p − ω(~
p) + iη +
(1.82)
and similarly
Z
∞
−∞
=
Z
∞
−∞
∞
−∞
p) − ~γ · p~ + m
dp0 −ip0 (x00 −x) −γ 0 ω(~
e
2π
p0 + ω(~
p) − iη +
0
1
dp −ip0 (x00 −x)
e
2mΛ− (−~
p) 0
,
2π
p + ω(~
p) − iη +
(1.83)
so that Eq. (1.78) can be rewritten
0
S0 (x − x) =
Z
d4 p −i(p,x0 −x) m
1
1
Λ+ (~
p) 0
.
− Λ− (−~
p) 0
e
4
(2π)
ω(~
p)
p − ω(~
p) + iη
p + ω(~
p) − iη
(1.84)
"
12
#
The utility of this representation is that it provides a single Fourier amplitude
for the free-fermion Green function; i.e., an alternative to Eq. (1.77):
"
#
m
1
1
S0 (p) =
Λ+ (~
p) 0
,
− Λ− (−~
p) 0
ω(~
p)
p − ω(~
p) + iη
p + ω(~
p) − iη
(1.85)
which is indispensable in making a connection between covariant perturbation
theory and time-ordered perturbation theory: the second term generates the Zdiagrams in loop integrals.
1.4.2
Green Function for the Interacting Theory
We now return to Eq. (1.69):
/(x0 ) − m] S(x0 , x) = 1 δ 4 (x0 − x) ,
[i∂
/x0 − eA
(1.86)
which defines the Green function for a fermion in an external electromagnetic
field. As mentioned, a closed form solution of this equation is impossible in all
but the simplest field configurations. Is there, nevertheless, a way to construct a
systematically improvable approximate solution?
To achieve that one rewrites the equation:
[i∂
/x0 − m] S(x0 , x) = 1 δ 4 (x0 − x) + eA
/(x0 ) S(x0 , x) ,
(1.87)
which, it is easily seen by substitution, is solved by
0
0
S(x , x) = S0 (x − x) + e
= S0 (x0 − x) + e
+e
2
Z
4
d y1
Z
Z
Z
d4 y S0 (x0 − y)A
/(y) S(y, x)
(1.88)
d4 y S0 (x0 − y)A
/(y) S0(y − x)
d4 y2 S0 (x0 − y1 )A
/(y1 ) S0 (y1 − y2 )A
/(y2 ) S0 (y2 − x) + . . .
(1.89)
This perturbative expansion of the full propagator in terms of the free propagator
provides an archetype for perturbation theory in quantum field theory. One
obvious application is the scattering of an electron/positron by a Coulomb field,
which is a worked example in Sec. 2.5.3 of Ref. [3]. Equation (1.89) is a first
example of a Dyson-Schwinger equation [4].
This Green function has the following interpretation:
1. It creates a positive energy fermion (antifermion) at spacetime point x;
2. Propagates the fermion to spacetime point x0 ; i.e., forward in time;
3. Annihilates this fermion at x0 .
13
The process can equally well be viewed as
1. The creation of a negative energy antifermion (fermion) at spacetime point
x0 ;
2. Propagation of the antifermion to the spacetime point x; i.e., backward in
time;
3. Annihilation of this antifermion at x.
Other propagators have similar interpretations.
1.4.3
Exercises
1. Prove these relations for on-shell fermions:
(k/ + m) γ 0 (k/ + m) = 2E (k/ + m) ,
(k/ + m) (k/ + m) = 2m (k/ + m) .
2. Obtain the Feynman propagator for the free-field Klein Gordon equation:
(∂x2 + m2 )φ(x) = 0 ,
(1.90)
in forms analogous to Eqs. (1.81), (1.84).
1.5
Path Integral in Quantum Mechanics
Local gauge theories are the keystone of contemporary hadron and high-energy
physics. Such theories are difficult to quantise because the gauge dependence is
an extra non-dynamical degree of freedom that must be dealt with. The modern
approach is to quantise the theories using the method of functional integrals,
attributed to Feynman and Kac. References [3, 5] provide useful overviews of the
technique, which replaces canonical second-quantisation.
It is customary to motivate the functional integral formulation by reviewing
the path integral representation of the transition amplitude in quantum mechanics. Beginning with a state (q1 , q2 , . . . , qN ) at time t, the probability that one
0
will obtain the state (q10 , q20 , . . . , qN
) at time t0 is given by (remember, the time
evolution operator in quantum mechanics is exp[−iHt], where H is the system’s
Hamiltonian):
0
hq10 , q20 , . . . , qN
; t0 |q1 , q2 , . . . , qN ; ti = lim
n→∞

× exp i
n+1
X
j=1
N Z Y
n
Y
α=1
i=1
dqα (ti )
Z n+1
Y
i=1
pα (tj )[qα (tj ) − qα (tj−1 )] − H(p(tj ),
14
dpα (ti )
2π

qtj + qtj −1 
,
)(1.91)
2
where tj = t + j, = (t0 − t)/(n + 1), t0 = t, tn+1 = t0 . A compact notation is
commonly introduced to represent this expression:
hq 0 t0 |qtiJ =
Z
[dq]
Z
( Z
[dp] exp i
t0
t
)
dτ [p(τ )q̇(τ ) − H(τ ) + J(τ )q(τ )] ,
(1.92)
where J(t) is a classical external “source.” NB. The J = 0 exponent is nothing
but the Lagrangian.
Recall that in Heisenberg’s formulation of quantum mechanics it is the operators that evolve in time and not the state vectors, whose values are fixed at a
given initial time. Using the previous formulae it is a simple matter to prove that
the time ordered product of n Heisenberg position operators can be expressed as
hq 0 t0 |T {Q(t1 ) . . . Q(tn )}|qti =
Z
( Z
Z
[dq] [dp] q(t1 ) q(t2 ) . . . q(tn ) exp i
t0
t
)
dτ [p(τ )q̇(τ ) − H(τ )] . (1.93)
NB. The time ordered product ensures that the operators appear in chronological
order, right to left.
Consider a source that “switches on” at ti and “switches off” at tf , t < ti <
tf < t0 , then
0 0
J
hq t |qti =
Z
dqi dqf hq 0 t0 |qf tf i hqf tf |qi ti iJ hqi ti |qti .
(1.94)
Alternatively one can introduce a complete set of energy eigenstates to resolve
the Hamiltonian and write
hq 0 t0 |qti
=
X
n
t0 →−i∞
t → +i∞
=
0
hq 0 |φn i e−iEn (t −t) hφn |qi
0
hq 0 |φ0 i e−iE0 (t −t) hφ0 |qi ;
(1.95)
(1.96)
i.e., in either of these limits the transition amplitude is dominated by the ground
state.
It now follows from Eqs. (1.94), (1.96) that
hq 0 t0 |qti
lim
=
0
−iE0 (t −t) hq 0 |φ i hφ |qi
0
0
t0 →−i∞ e
t → +i∞
Z
dqi dqf hφ0 |qf tf i hqf tf |qi ti iJ hqi ti |φ0 i =: W [J] ;
(1.97)
i.e., the ground-state to ground-state transition amplitude (survival probability)
in the presence of the external source J. From this it will readily be apparent
that, with tf > t1 > t2 > . . . > tm > ti ,
δ m W [J]
= im
δJ(t1 ) . . . δJ(t1 ) J=0
Z
dqi dqf hφ0 |qf tf i hqf tf |T {Q(t1 ) . . . Q(tn )}|qi ti i hqi ti |φ0 i,
(1.98)
15
which is the ground state expectation value of a time ordered product of Heisenberg position operators. The analogues of these expectation values in quantum
field theory are the Green functions.
δ
, is defined analogously to the
The functional derivative introduce here: δJ(t)
derivative of a function. It means to write J(t) → J(t) + (t); expand the
functional in (t); and identify the leading order coefficient in the expansion.
Thus for
Z
Hn [J] = dt0 J(t0 )n
(1.99)
δHn [J] = δ
=
Z
Z
dt0 J(t0 )n =
Z
dt0 [J(t0 ) + (t0 )]n −
dt0 n J(t0 )n−1 (t0 ) + [. . .].
Z
dt0 J(t0 )n
(1.100)
(1.101)
So that taking the limit δJ(t0 ) = (t0 ) → δ(t − t0 ) one obtains
δHn [J]
= n J(t)n−1 .
δJ(t)
(1.102)
The limit procedure just described corresponds to defining
δJ(t)
= δ(t − t0 ) .
δJ(t0 )
(1.103)
This example, which is in fact the “product rule,” makes plain the very close
analogy between functional differentiation and the differentiation of functions so
that, with a little care, the functional differentiation of complicated functions is
straightforward.
Another note is in order. The fact that the limiting values in Eqs. (1.96),
(1.97) are imaginary numbers is a signal that mathematical rigour may be found
more easily in Euclidean space where t → −itE . Alternatively, at least in principle, the argument can be repeated and made rigorous by making the replacement
En → En − iη, with η → 0+ . However, that does not help in practical calculations, which proceed via Monte-Carlo methods; i.e., probability sampling, which
is only effective when the integrand is positive definite.
1.6
Functional Integral in Quantum Field Theory
The Euclidean functional integral is particularly well suited to a direct numerical
evaluation via Monte-Carlo methods because it defines a probability measure.
However, to make direct contact with the perturbation theory of canonical secondquantised quantum field theory, I begin with a discussion of the Minkowski space
formulation.
16
1.6.1
Scalar Field
I will consider a scalar field φ(t, x), which is customary because it reduces the
number of indices that must be carried through the calculation. Suppose that
a large but compact domain of space is divided into N cubes of volume 3 and
label each cube by an integer α. Define the coordinate and momentum via
qα (t) := φα (t) =
1
3
Z
Vα
d3 x φ(t, x) , q̇α (t) := φ̇α (t) =
1
3
Z
Vα
d3 x
∂φ(t, x)
; (1.104)
∂t
i.e., as the spatial averages over the cube denoted by α.
The classical dynamics of the field φ is described by a Lagrangian:
L(t) =
Z
d3 x L(t, x) →
N
X
3 Lα (φ̇α (t), φα (t), φα±s (t)) ,
(1.105)
α=1
where the dependence on φα±s (t); i.e., the coordinates in the neighbouring cells,
is necessary to express spatial derivatives in the Lagrangian density, L(x).
With the canonical conjugate momentum defined as in a classical field theory
∂Lα
∂L
= 3
=: 3 πα (t) ,
∂ φ̇α (t)
∂ φ̇α (t)
pα (t) :=
(1.106)
the Hamiltonian is obtained as
H=
X
α
pα (t) q̇α (t) − L(t) =:
X
3 Hα ,
(1.107)
α
Hα (πα (t), φα (t), φα±s (t)) = πα (t) φ̇α (t) − Lα .
(1.108)
The field theoretical equivalent of the quantum mechanical transition amplitude, Eq. (1.91), can now be written
N Z Y
n
Y
lim
n→∞,→0+ α=1
i=1
dφα (ti )
n+1
X X
dπα (ti )
φα (tj ) − φα (tj−1 )
3
exp i
3 πα (tj )
2π
α
i=1
j=1
=:
Z
( Z
[Dφ] [Dπ] exp i
(

!
φα (tj ) + φα (tj−1 ) φα±s (tj ) + φα±s (tj−1 ) 
πα (tj ),
,

2
2
− Hα
Z

Z Y
n
t0
t
dτ
Z
"
∂φ(τ, ~x)
d x π(τ, ~x)
− H(τ, ~x)
∂τ
3
#)
,
(1.109)
where, as classically, π(t, ~x) = ∂L(t, ~x)/∂ φ̇(t, ~x) and its average over a spacetime
cube is just πα (t). Equation (1.109) is the transition amplitude from an initial
field configuration φα (t0 ) := φα (t) to a final configuration φα (tn+1 ) := φα (t0 ).
17
Generating Functional
In quantum field theory all physical quantities can be obtained from Green functions, which themselves are determined by vacuum-to-vacuum transition amplitudes calculated in the presence of classical external sources. The physical or
interacting vacuum is the analogue of the true ground state in quantum mechanics. And, as in quantum mechanics, the fundamental quantity is the generating
functional:
1
W [J] :=
N
Z
Z
Z
[Dφ] [Dπ] exp i
4
d x [π(x)φ̇(x) − H(x) +
1
iηφ2 (x) + J(x)φ(x)]
2
(1.110)
where N is chosen so that W [0] = 1, and a real time-limit is implemented and
made meaningful through the addition of the η → 0+ term. (NB. This subtracts
a small imaginary piece from the mass.)
It is immediately apparent that
,
1
δ n W [J]
h0̃|T {φ̂(x1 )φ̂(x2 ) . . . φ̂(xn )}|0̃i
=: G(x1 , x2 , . . . , xn ) ,
=
in δJ(x1 )δJ(x2 ) . . . δJ(xn ) J=0
h0̃|0̃i
(1.111)
where |0̃i is the physical vacuum. G(x1 , x2 , . . . , xn ) is the complete n-point Green
function for the scalar quantum field theory; i.e., the vacuum expectation value
of a time-ordered product of n field operators. The appearance of the word
“complete” means that this Green function includes contributions from products
of lower-order Green functions; i.e., disconnected diagrams.
The fact that the Green functions in a quantum field theory may be defined via
Eq. (1.111) was first observed by Schwinger [6] and does not rely on the functional
formula for W [J], Eq. (1.110), for its proof. However, the functional formulism
provides the simplest proof and, in addition, a concrete means of calculating the
generating functional: numerical simulations.
Connected Green Functions
It is useful to have a systematic procedure for the a priori elimination of disconnected parts from a n-point Green function because performing unnecessary
work, as in the recalculation of m < n-point Green functions, is inefficient. A
connected n-point Green function is given by
Gc (x1 , x2 , . . . , xn ) = (−i)
where Z[J], defined via
n−1
δ n Z[J]
,
δJ(x1 )δJ(x2 ) . . . δJ(xn ) J=0
W [J] =: exp{iZ[J]} ,
is the generating functional for connected Green functions.
18
(1.112)
(1.113)
It is instructive to illustrate this for a simple case (recall that I have already
proven the product rule for functional differentiation):
δ 2 Z[J]
δ 2 ln W [J] Gc (x1 , x2 ) = (−i)
=
−
δJ(x1 ) δJ(x2 ) J=0
δJ(x1 ) δJ(x2 ) J=0
"
#
J=0
δW [J] −
δJ(x2 ) δ
1 δW [J]
= −
δJ(x1 ) W [J] δJ(x2 )
1 δW [J]
= + 2
W [J] δJ(x1 )
J=0
h0̃|φ̂(x1 )|0̃i h0̃|φ̂(x2 )|0̃i
h0̃|T {φ̂(x1 )φ̂(x2 )}|0̃i
i
− i2
h0̃|0̃i
h0̃|0̃i
h0̃|0̃i
= −G(x1 ) G(x2 ) + G(x1 , x2 ) .
= i
1.6.2
1
δ 2 W [J] W [J] δJ(x1 ) δJ(x2 ) J=0
(1.114)
Lagrangian Formulation of Quantum Field Theory
The double functional integral employed above is cumbersome, especially since
it involves the field variable’s canonical conjugate momentum. Consider then a
Hamiltonian density of the form
1
~
H(x) = π 2 (x) + f [φ(x), ∇φ(x)]
.
2
(1.115)
In this case Eq. (1.110) involves
Z
1 2
[Dπ] exp i d x [− π (x) + π(x)φ̇(x)]
2
Z
Z
R 4
2
i
{i d x [φ̇(x)] }
4
2
= e
[Dπ] exp −
d x [π(x) − φ̇(x)]
2
R 4
2
Z
= e{i
4
d x [φ̇(x)] }
× N,
(1.116)
where “N” is simply a constant. (This is an example of the only functional
integral than can be evaluated exactly: the Gaussian functional integral.) Hence,
with a Hamiltonian of the form in Eq. (1.115), the generating functional can be
written
Z
Z
N
1
W [J] =
[Dφ] exp i d4 x [L(x) + iηφ2 (x) + J(x)φ(x)] . (1.117)
2
N
Recall now that the classical Lagrangian density for a scalar field is
L(x) = L0 (x) + LI (x) ,
L0 (x) =
1
[∂µ φ(x) ∂ µ φ(x) − m2 φ2 (x)] ,
2
(1.118)
(1.119)
with LI (x) some functional of φ(x) that usually does not depend on any derivatives of the field. Hence the Hamiltonian for such a theory has the form in
Eq. (1.115) and so Eq. (1.117) can be used to define the quantum field theory.
19
1.6.3
Quantum Field Theory for a Free Scalar Field
The interaction Lagrangian vanishes for a free scalar field so that the generating
functional is, formally,
1
W0 [J] =
N̄
Z
Z
[Dφ] exp i
4
dx
1
[∂µ φ(x) ∂ µ φ(x) − m2 φ2 (x) + iηφ2 (x)] + J(x)φ(x)
2
(1.120)
The explicit meaning of Eq. (1.120) is
1
W0 [J] = lim+
→0 N̄
Z Y
α

 X
dφα exp i
4
X
α
β
4
1
2
φα Kαβ φβ +
X
α
4
where α, β label spacetime hypercubes of volume and Kαβ
satisfies
lim+ Kαβ = [−∂ 2 − m2 + iη]δ 4 (x − y) ,


4 Jα φ α  ,
(1.121)
is any matrix that
(1.122)
→0
→0+
→0+
→0+
d4 x; i.e., Kαβ is any matrix whose
where α → x, β → y and α 4 →
continuum limit is the inverse of the Feynman propagator for a free scalar field.
(NB. The fact that there are infinitely many such matrices provides the scope
for “improving” lattice actions since one may choose Kαβ wisely rather than for
simplicity.)
Recall now that for matrices whose real part is positive definite
Z
n
Y
Rn i=1
dxi exp{−
P
n
1 X
2
xi Aij xj +
i,j=1
R
n
X
i=1
b i xi }
n
(2π)n/2
(2π)n/2
1
1 X
= √
exp bt A−1 b .(1.123)
bi (A−1 )ij bj } = √
exp{
2
2
detA
detA
i,j=1
Hence Eq. (1.121) yields
W0 [J] = lim+
→0
1
1
1
1X 4 X 4
√
Jα 8 (K −1 )αβ Jβ } ,
exp{
0
2 α
N̄
i
detA
β
(1.124)
where, obviously, the matrix inverse is defined via
X
Kαγ (K −1 )γβ = δαβ .
(1.125)
γ
Almost as obviously, consistency of limits requires
Z
X
1
4
4
δαβ = δ (x − y) , lim+
lim
= d4 x ,
→0
→0+ 4
α
so that, with
O(x, y) := lim+
→0
20
1
(K −1 )αβ ,
8
(1.126)
(1.127)
.
the continuum limit of Eq. (1.125) can be understood as follows:
lim+
→0
Z
⇒
.
X
4 Kαγ
γ
1
1
(K −1 )γβ = lim+ 4 δαβ
8
→0 d4 w [−∂x2 − m2 + iη]δ 4 (x − w) O(w, y) = δ 4 (x − y)
[−∂x2 − m2 + iη] O(x, y) = δ 4 (x − y) .
··
(1.128)
Hence O(x, y) = ∆0 (x − y); i.e., the Feynman propagator for a free scalar field:
∆0 (x − y) =
Z
d4 p −i(q,x−y)
1
e
.
4
2
(2π)
q − m2 + iη
(1.129)
(NB. This makes plain the fundamental role of the “iη + ” prescription in Eq. (1.77):
it ensures convergence of the expression defining the functional integral.)
Putting this all together, the continuum limit of Eq. (1.124) is
1
i
W [J] =
exp −
2
N̂
1.6.4
Z
4
dx
Z
4
d y J(x) ∆0 (x − y) J(y) .
(1.130)
Scalar Field with Self-Interactions
A nonzero interaction Lagrangian, LI [φ(x)], provides for a self-interacting scalar
field theory (from here on I will usually omit the constant, nondynamical normalisation factor in writing the generating functional):
W [J] =
Z
Z
[Dφ] exp i
" Z
d x LI
" Z
d 4 x LI
= exp i
= exp i
4
d4 x [L0 (x) + LI (x) + J(x)φ(x)]
!# Z
1 δ
[Dφ] exp i d4 x [L0 (x) + J(x)φ(x)] (1.131)
i δJ(x)
!#
Z
Z
1 δ
i
4
4
exp −
d x d y J(x) ∆0 (x − y) J(y) ,
2
i δJ(x)
(1.132)
Z
where
" Z
exp i
4
d x LI
1 δ
i δJ(x)
!#
:=
∞
X
n=0
in
LI
n!
"
1 δ
i δJ(x)
!#n
.
(1.133)
Equation (1.132) is the basis for a perturbative evaluation of all possible
Green functions for the theory. As an example I will work through a first-order
calculation of the complete 2-point Green function in the theory defined by
LI (x) = −
λ 4
φ (x) .
4!
21
(1.134)
The generating functional yields
W [0] =


λ
1−i

4!
Z
δ
d4 x
δJ(x)
!4 

λZ 4
= 1−i
d x 3[i∆0 (0)]2 .
4!

exp −
i
2
Z
d4 u
Z
d4 v J(u) ∆0 (u − v) J(v) J=0
(1.135)
The 2-point function is
λ Z 4
δ 2 W [J]
d x [i∆0 (0)]2 [i∆0 (x1 − x2 )]
= −i∆0 (x1 − x2 ) + i
δJ(x1 ) δJ(x2 )
8
Z
λ
+i
d4 x [i∆0 (0)][i∆0 (x1 − x)][i∆0 (x − x2 )] . (1.136)
2
Using the definition, Eq. (1.111), and restoring the normalisation,
1 1
δ 2 W [J]
(1.137)
i2 W [0] δJ(x1 ) δJ(x2 )
Z
λ
d4 x [i∆0 (0)][i∆0 (x1 − x)][i∆0 (x − x(1.138)
= i∆0 (x1 − x2 ) − i
2 )] ,
2
G(x1 , x2 ) =
where I have used
a + λb
= a + λ(b − ac) + O(λ2 ) .
(1.139)
1 + λc
This complete Green function does not contain any disconnected parts because
the vacuum is trivial in perturbation theory; i.e.,
1 δW [J] h0̃|φ̂(x)|0̃i
=0
:= G(x)|J=0 =
i δJ(x) J=0
h0̃|0̃i
(1.140)
so that the field does not have a nonzero vacuum expectation value. This is
the simplest demonstration of the fact that dynamical symmetry breaking is a
phenomenon inaccessible in perturbation theory.
1.6.5
Exercises
1. Repeat the derivation of Eq. (1.114) for Gc (x1 , x2 , x3 ).
2. Prove Eq. (1.131).
3. Derive Eq. (1.136).
4. Prove Eq. (1.140).
22
1.7
1.7.1
Functional Integral for Fermions
Finitely Many Degrees of Freedom
Fermionic fields do not have a classical analogue: classical physics does not contain anticommuting fields. In order to treat fermions using functional integrals
one must employ Grassmann variables. Reference [7] is the standard source for a
rigorous discussion of Grassmann algebras. Here I will only review some necessary
ideas.
The Grassmann algebra GN is generated by the set of N elements, θ1 , . . . , θN
which satisfy the anticommutation relations
{θi , θj } = 0 , i, j = 1, 2, . . . , N .
(1.141)
It is clear from Eq. (1.141) that θi2 = 0 for i = 1, . . . , N . In addition, the elements
{θi } provide the source for the basis vectors of a 2n -dimensional space, spanned
by the monomials:
1, θ1 , . . . , θN , θ1 θ2 , . . . θN −1 θN , . . . , θ1 θ2 . . . θN ;
(1.142)
i.e., GN is a 2N -dimensional vector space. (NB. One can always choose the pdegree monomial in Eq. (1.142): θi1 θi2 . . . θIN , such that i1 < i2 < . . . < iN .)
Obviously, any element f (θ) ∈ GN can be written
f (θ) = f0 +
X
i1
f1 (i1 ) θi1 +
X
f2 (i1 , i2 ) θi1 θi2 +. . .+
i1 ,i2
X
fN (i1 , i2 , . . . , iN ) θ1 θ2 . . . θN ,
i1 ,i2 ,...,iN
(1.143)
where the coefficients fp (i1 , i2 , . . . , ip ) are unique if they are chosen to be fully
antisymmetric for p ≥ 2.
Both “left” and “right” derivatives can be defined on GN . As usual, they
are linear operators and hence it suffices to specify their operation on the basis
elements:
∂
θi θi θi = δsi1 θi2 . . . θip − δsi2 θi1 . . . θip + . . . + (−)p−1 δsip θi1 θi2 . . . θ(1.144)
ip−1 ,
∂θs 1 2 p
←
θ i1 θ i2 θ ip
∂
= δsip θi1 . . . θip−1 − δsip−1 θi1 . . . θip−2 θip + . . . + (−)p−1 δsi1 θi(1.145)
2 θ ip .
∂θs
The operation on a general element, f (θ) ∈ GN , is easily obtained. It is also
obvious that
∂ ∂
∂ ∂
f (θ) = −
f (θ) .
(1.146)
∂θ1 ∂θ2
∂θ2 ∂θ1
A definition of integration requires the introduction of Grassmannian line
elements: dθi , i = 1, . . . , N . These elements also satisfy Grassmann algebras:
{dθi , dθj } = 0 = {θi , dθj } , i, j = 1, 2, . . . N .
23
(1.147)
The integral calculus is completely defined by the following two identities:
Z
dθi = 0 ,
Z
dθi θi = 1 , i = 1, 2, . . . , N .
(1.148)
For example, it is straightforward to prove, using Eq. (1.143),
Z
dθN . . . dθ1 f (θ) = N ! fN (1, 2, . . . , N ) .
(1.149)
In standard integral calculus a change of integration variables is often used to
simplify an integral. That operation can also be defined in the present context.
Consider a nonsingular matrix (Kij ), i, j = 1, . . . , N , and define new Grassmann
variables ξ1 , . . . , ξN via
θi =
N
X
Kij ξj .
(1.150)
(K −1 )ji dξj
(1.151)
Z
(1.152)
i=j
With the definition
dθi =
N
X
j=1
one guarantees
It follows immediately that
Z
dθi θj = δij =
dξi ξj .
θ1 θ2 . . . θN = (det K) ξ1 ξ2 . . . ξN
dθN dθN −1 . . . dθ1 = (det K −1 ) dξN dξN −1 . . . dξ1 ,
(1.153)
(1.154)
and hence
Z
dθN . . . dθ1 f (θ) = (det K −1 )
Z
dξN . . . dξ1 f (θ(ξ)) .
(1.155)
In analogy with scalar field theory, for fermions one expects to encounter
integrals of the type
I :=
Z
dθN . . . dθ1 exp

N
X

θi Aij θj


,
(1.156)

i,j=1
where (Aij ) is an antisymmetric matrix. NB. Any symmetric part of the matrix,A,
cannot contribute since:
X
θi Aij θj
relable
=
X
θj Aji θi
(1.157)
θj Aij θi
(1.158)
j,i
i,j
use sym.
=
X
i,j
anticom.
=
..·
X
θi Aij θj
=
i,j
24
−
X
θi Aij θj .
(1.159)
0 for A = At .
(1.160)
i,j
Assume for the moment that A is a real matrix. Then there is an orthogonal
matrix S (SS t = I) for which
S t AS =








0
λ1
0
0
−λ1 0
0
0
0
0
0
λ2
0
0 −λ2 0
... ... ... ...
...
...
...
...
...








=: Ã .
(1.161)
PN
Sij ξj and using
Consequently, applying the linear transformation θi =
Eq. (1.155), we obtain
I=
Z
dξN . . . dξ1 exp

Hence
I =
R


R

N
X
ξi Ãij ξj
i,j=1
i=1


.
(1.162)

o
n
dξN . . . dξ1 exp 2 [λ1 ξ1 ξ2 + λ2 ξ3 ξ4 + . . . + λN/2 ξN −1 ξN ] = 2N/2 λ1 λ2 . . . λN , N even
n
o
dξN . . . dξ1 exp 2 [λ1 ξ1 ξ2 + λ2 ξ3 ξ4 + . . . + λ(N −1)/2 ξN −2 ξN −1 ] = 0 , N odd
i.e., since det A = det Ã,
I=
√
det 2A , .
(1.163)
(1.164)
Equation (1.164) is valid for any real matrix, A. Hence, by the analytic
function theorem, it is also valid for any complex matrix A.
The Lagrangian density associated with the Dirac equation involves a field ψ̄,
which plays the role of a conjugate to ψ. If we are express ψ as a vector in GN
then we will need a conjugate space in which ψ̄ is defined. Hence it is necessary
to define θ̄1 , θ̄2 , . . . , θ̄N such that the operation θi ↔ θ̄i is an involution of the
algebra onto itself with the following properties:
i)
(θ̄i ) = θi
ii) (θi θj ) = θ̄j θ̄i
iii) λ θi = λ∗ θ̄i , λ ∈ C .
(1.165)
The elements of the Grassmann algebra with involution are θ1 , θ2 , . . . , θN , θ̄1 , θ̄2 . . . , θ̄N ,
each anticommuting with every other. Defining integration via obvious analogy
with Eq. (1.148) it follows that
Z


dθ̄N dθN . . . dθ1 dθ1 exp −

25
N
X
i,j=1
θ̄i Bij θj



= det B ,
(1.166)
for any matrix B. (NB. This is the origin of the fermion determinant in the
quantum field theory of fermions.) This may be compared with the analogous
result for commuting real numbers, Eq. (1.123):
Z
1.7.2
RN
N
Y
dxi exp{−π
i=1
N
X
i,j=1
xi Aij xj } = √
1
.
det A
(1.167)
Fermionic Quantum Field
To describe a fermionic quantum field the preceding analysis must be generalised
to the case of infinitely many generators. A rigorous discussion can be found in
Ref. [7] but here I will simply motivate the extension via plausible but formal
manipulations.
Suppose the functions {un (x) , n = 0, . . . , ∞} are a complete, orthonormal
set that span a given Hilbert space and consider the Grassmann function
θ(x) :=
∞
X
un (x) θn ,
(1.168)
n=0
where θn are Grassmann variables. Clearly
{θ(x), θ(y)} = 0 .
(1.169)
The elements θ(x) are considered to be the generators of the “Grassman algebra”
G and, in complete analogy with Eq. (1.143), any element of G can be uniquely
written as
f=
∞ Z
X
dx1 dx2 . . . dxN θ(x1 )θ(x2 ) . . . θ(xN ) fn (x1 , x2 , . . . , xN ) ,
(1.170)
n=0
where, for N ≥ 2, the fn (x1 , x2 , . . . , xN ) are fully antisymmetric functions of their
arguments.
In another analogy, the left- and right-functional-derivatives are defined via
their action on the basis vectors:
δ
θ(x1 )θ(x2 ) . . . θ(xn ) =
δθ(x)
δ(x − x1 ) θ(x2 ) . . . θ(xn ) − . . . + (−)n−1 δ(x − xn ) θ(x1 ) . . . θ(xn−1 )(1.171)
,
←
δ
=
δθ(x)
δ(x − xn ) θ(x1 ) . . . θ(xn−1 ) − . . . + (−)n−1 δ(x − x1 ) θ(x2 ) . . . θ(xn )(1.172)
,
θ(x1 )θ(x2 ) . . . θ(xn )
cf. Eqs. (1.144), (1.145).
26
Finally, we can extend the definition of integration. Denoting
[Dθ(x)] := lim dθN . . . dθ2 dθ1 ,
(1.173)
N →∞
consider the “standard” Gaussian integral
I :=
Z
[Dθ(x)] exp
Z
dxdy θ(x)A(x, y)θ(y)
(1.174)
where, clearly, only the antisymmetric part of A(x, y) can contribute to the result.
Define
Z
Aij := dxdy ui (x)A(x, y)uj (y) ,
(1.175)
then
I = lim
N →∞
Z
dθN . . . dθ2 dθ1 e
so that, using Eq. (1.164),
I = lim
N →∞
q
PN
i=1
θi Aij θj
det 2AN ,
(1.176)
(1.177)
where, obviously, AN is the N × N matrix in Eq. (1.175). This provides a
definition for the formal result:
Z
Z
√
(1.178)
dxdy θ(x)A(x, y)θ(y) = Det2A ,
I = [Dθ(x)] exp
where I will subsequently identify functional equivalents of matrix operations
as proper nouns. The result is independent of the the basis vectors since all
such vectors are unitarily equivalent and the determinant is cylic. (This means
that a new basis is always related to another basis via u0 = U u, with U U † =
I. Transforming to a new basis therefore introduces a modified exponent, now
involving the matrix U AU † , but the result is unchanged because det U AU † =
det A.)
In quantum field theory one employs a Grassmann algebra with an involution.
In this case, defining the functional integral via
[D θ̄(x)][Dθ(x)] := lim dθ̄N dθN . . . dθ̄2 dθ2 dθ̄1 dθ1 ,
N →∞
(1.179)
one arrives immediately at a generalisation of Eq. (1.166)
Z
[D θ̄(x)][Dθ(x)] exp −
Z
dxdy θ̄(x) B(x, y) θ(x) = DetB ,
(1.180)
which is also a definition.
The relation
ln det B = tr ln B ,
27
(1.181)
valid for any nonsingular, finite dimensional matrix, has a generalisation that
is often used in analysing quantum field theories with fermions. Its utility is
to make possible a representation of the fermionic determinant as part of the
quantum field theory’s action via
DetB = exp {Tr LnB} .
(1.182)
I note that for an integral operator O(x, y)
Tr O(x, y) :=
Z
d4 x tr O(x, x) ,
(1.183)
which is an obvious analogy to the definition for finite-dimensional matrices.
Furthermore a functional of an operator, whenever it is well-defined, is obtained
via the function’s power series; i.e., if
f (x) = f0 + f1 x + f2 x2 + [. . .] ,
(1.184)
then
4
f [O(x, y)] = f0 δ (x − y) + f1 O(x, y) + f2
1.7.3
Z
d4 w O(x, w)O(w, y) + [. . .] . (1.185)
Generating Functional for Free Dirac Fields
The Lagragian density for the free Dirac field is
Lψ0 (x)
=
Z
d4 x ψ̄(x) (i∂/ − m) ψ(x) .
(1.186)
Consider therefore the functional integral
¯ ξ] =
W [ξ,
Z
Z
[D ψ̄(x)][Dψ(x)] exp i
¯
d4 x ψ̄(x) i∂/ − m + iη + ψ(x) + ψ̄(x)ξ(x) + ξ(x)ψ(x)
h
28
.
(1.187)
Here ψ̄(x), ψ(x) are identified with the generators of G, with the minor additional
complication that the spinor degree-of-freedom is implicit; i.e., to be explicit,
Q
Q
one should write 4r=1 [D ψ̄r (x)] 4s=1 [Dψs (x)]. This only adds a finite matrix
degree-of-freedom to the problem, so that “Det A” will mean both a functional
and a matrix determinant. This effect will be encountered again; e.g., with the
appearance of fermion colour and flavour. In Eq. (1.187) I have also introduced
¯
anticommuting sources: ξ(x),
ξ(x), which are also elements in the Grassmann
algebra, G.
The free-field generating functional involves a Gaussian integral. To evaluate
that integral I write
O(x, y) = (i∂/ − m + iη + )δ 4 (x − y)
i
(1.188)
and observe that the solution of
Z
d4 w O(x, w) P (w, y) = I δ 4 (x − y)
(1.189)
i.e., the inverse of the operator O(x, y) is (see Eq. 1.72) precisely the free-fermion
propagator:
P (x, y) = S0 (x − y) .
(1.190)
Hence I can rewrite Eq. (1.187) in the form
¯ ξ] =
W [ξ,
Z
Z
[D ψ̄(x)][Dψ(x)] exp i
¯
d xd y ψ̄ (x)O(x, y)ψ (y) − ξ(x)S
0 (x − y)ξ(y)
4
h
4
0
0
(1.191)
where
0
ψ̄ (x) := ψ̄(x) +
Z
¯
d w ξ(w)
S0 (w − x) , ψ(x) := ψ(x) +
4
Z
d4 w S0 (x − w) ξ(w) .
(1.192)
Clearly, ψ̄ (x) and ψ (x) are still in G and hence related to the original variables
by a unitary transformation. Thus changing to the “primed” variables introduces
a unit Jacobian and so
0
0
¯ ξ] = exp −i
W [ξ,
×
Z
Z
0
¯ S0 (x − y) ξ(y)
d xd y ξ(x)
4
4
0
Z
[D ψ̄ (x)][Dψ (x)] exp i
Z
= det[−iS0−1 (x − y)] exp −i
=
4
1
exp −i
N0ψ
Z
4
0
0
d xd y ψ̄ (x)O(x, y)ψ (y)
¯ S0 (x − y) ξ(y)
d4 xd4 y ξ(x)
¯ S0 (x − y) ξ(y) ,
d4 xd4 y ξ(x)
(1.193)
(1.194)
(1.195)
where N0ψ := det[iS0 (x − y)]. Clearly.
¯ ξ]
N0ψ W [ξ,
ξ̄=0=ξ
= 1.
(1.196)
The 2 point Green function for the free-fermion quantum field theory is now
easily obtained:
N0ψ
=
ˆ
¯ ξ]
δ 2 W [ξ,
h0|T {ψ̂(x)ψ̄(y)}|0i
=
¯ (−i)δξ(y) h0|0i
iδ ξ(x)
ξ̄=0=ξ
Z
Z
[D ψ̄(x)][Dψ(x)] ψ(x)ψ̄(y) exp i
= i S0 (x − y) ;
4
d x ψ̄(x) i∂/ − m + iη
+
ψ(x)
(1.197)
i.e., the inverse of the Dirac operator, with exactly the Feynman boundary conditions.
29
i
As in the example of a scalar quantum field theory, the generating functional
¯ ξ], defined via:
for connected n-point Green functions is Z[ξ,
¯ ξ] =: exp iZ[ξ,
¯ ξ] .
W [ξ,
n
o
(1.198)
Hitherto I have not illustrated what is meant by “DetO,” where O is an
integral operator. I will now provide a formal example. (Rigour requires a careful
consideration of regularisation and limits.) Consider a translationally invariant
operator
Z
d4 p
O(x, y) = O(x − y) =
O(p) e−i(p,x−y) .
(1.199)
(2π)4
Then, for f as in Eq. (1.185),
o
d4 p n
2
f
+
f
O(p)
+
f
O
(p)
+
[.
.
.]
e−i(p,x−y)
0
1
2
(2π)4
Z
d4 p
f (O(p)) e−i(p,x−y) .
(1.200)
=
(2π)4
f [O(x − y)] =
Z
I now apply this to N0ψ := Det[iS0 (x − y)] and observe that Eq. (1.182) means
one can begin by considering TrLn iS0 (x − y). Writing
"
#
/p
1
S0 (p) = m ∆0 (p ) 1 +
, ∆0 (p2 ) = 2
,
m
p − m2 + iη +
2
(1.201)
the free fermion propagator can be re-expressed as a product of integral operators:
S0 (x − y) =
Z
d4 w m ∆0 (x − w) F (w − y) ,
(1.202)
with ∆0 (x − y) given in Eq. (1.129) and
F (x − y) =
Z
d4 p
/p
e−i(p,x−y) .
1+
4
(2π)
m
#
"
(1.203)
It follows that
n
h
TrLn iS0 (x − y) = Tr Ln i m∆0 (x − y) + Ln δ 4 (x − y) + F (x − y)
io
. (1.204)
Using Eqs. (1.183), (1.200), one can express the second term as
h
TrLn δ 4 (x − y) + F (x − y)
i
d4 p
tr ln [1 + F (p)]
(2π)4
"
#
Z
Z
d4 p
p2
4
=
dx
2 ln 1 − 2
(1.205)
(2π)4
m
=
30
Z
d4 x
Z
and, applying the same equations, the first term is
TrLn im∆0 (x − y) =
Z
4
dx
Z
h
i
d4 p
2 2
2
ln
i
m∆
(p
)
,
0
(2π)4
(1.206)
where in both cases d4 x measures the (infinite) spacetime volume. Combining
these results one obtains
R
Ln N0ψ
= TrLn iS0 (x − y) =
Z
4
dx
Z
d4 p
2 ln ∆0 (p2 ) ,
4
(2π)
(1.207)
where the factor of 2 reflects the spin-degeneracy of the free-fermion’s eigenvalues.
(Including a “colour” degree-of-freedom, this would become “2Nc ,” where Nc is
the number of colours.)
1.7.4
Exercises
1. Verify Eq. (1.149).
2. Verify Eqs. (1.153), (1.154).
3. Verify Eqs. (1.163), (1.164).
4. Verify Eqs. (1.166).
5. Verify Eq. (1.181).
6. Verify Eq. (1.197).
7. Verify Eq. (1.205).
1.8
Functional Integral for Gauge Field Theories
To begin I will consider a non-Abelian gauge field theory in the absence of couplings to matter field, which is described by a Lagrangian density:
1 a
(x)Faµν (x),
L(x) = − Fµν
4
a
Fµν
(x) = ∂µ Bνa − ∂ν Bµa + gfabc Bµb Bνc ,
(1.208)
(1.209)
where g is a coupling constant and fabc are the structure constants of SU (Nc ):
i.e., with {T a : a = 1, . . . , Nc2 − 1} denoting the generators of the group
[Ta , Tb ] = ifabc Tc .
31
(1.210)
a
In the fundamental representation {Ta } ≡ { λ2 }, where {λa } are the generalization
of the eight Gell-Mann matrices, while in the adjoint representation, relevant to
the realization of transformations on the gauge fields,
{T a }bc = −ifabc .
(1.211)
An obvious guess for the form of the generating functional is
W [J] =
Z
[DBµa ] exp
Z
i
4
d x[L(x) +
Jaµ (x)Bµa ]
,
(1.212)
where, as usual, Jaµ (x) is a (classical) external source for the gauge field. It
will immediately be observed that this is a Lagrangian-based expression for the
generating functional, even though the Hamiltonian derived from Eq. (1.208)
is not of the form in Eq. (1.115). It is nevertheless (almost) correct (I will
motivate the modifications that need to be made to make it completely correct)
and provides the foundation for a manifestly Poincaré covariant quantisation of
the field theory. Alternatively, one could work with Coulomb gauge, build the
Hamiltonian and construct W [J] in the canonical fashion, as described in Sec.
??, but then covariance is lost. The Coulomb gauge procedure gives the same Smatrix elements (Green functions) as the (corrected-) Lagrangian formalism and
hence they are completely equivalent. However, manifest covariance is extremely
useful as it often simplifies the allowed form of Green functions and certainly
simplifies the calculation of cross sections. Thus the Lagrangian formulation is
most often used.
The primary fault with Eq. (1.212) is that the free-field part of the Lagrangian
density is singular: i.e., the determinant encountered in evaluating the free-gaugeboson generating functional vanishes, and hence the operator cannot be inverted.
This is easily demonstrated. Observe that
Z
1
d xL0 (x) = −
d4 x[∂µ Bνa (x) − ∂ν Bµa (x)][∂ µ Baν (x) − ∂ ν Baµ (x)](1.213)
4Z
n
o
1
= −
d4 xd4 yBµa (x) [−g µν ∂ 2 + ∂ µ ∂ ν ]δ 4 (x − y) Bνa (y)
(1.214)
2
Z
1
d4 xd4 yBµa (x)K µν (x, y)Bνa (y) .
(1.215)
=:
2
Z
4
The operator K µν (x, y) thus determined can be expressed
µν
K (x − y) =
Z
d4 q µν 2
µ ν
−g
q
+
q
q
e−i(p,x−y)
4
(2π)
(1.216)
from which it is apparent that the Fourier amplitude is a projection operator; i.e.,
a key feature of K µν (x, y) is that it projects onto the space of transverse gauge
field configurations:
qµ (g µν q 2 − q µ q ν ) = 0 = (g µν q 2 − q µ q ν )qν ,
32
(1.217)
where q µ is the four-momentum associated with the gauge field. It follows that
the W0 [J] obtained from Eq. (1.212) has no damping associated with longitudinal
gauge fields and is therefore meaningless. A simple analogy is
Z
∞
−∞
dx
Z
∞
2
−∞
dye−(x−y) ,
(1.218)
in which the integrand does not damp along trajectories in the (x, y)-plane related
via a spatial translation: (x, y) → (x + a, y + a). Hence there is an overall
divergence associated with the translation of the centre of momentum: X =
(x + y)/2. Y = (x − y): X → X + (a, a), Y → Y ,
Z
∞
−∞
dx
Z
∞
−∞
dye−(x−y)
2
=
=
Z
∞
dX
−∞
Z ∞
−∞
Z
dX
∞
−∞
dY e−Y
√
2
π=∞.
(1.219)
(1.220)
The underlying problem, which is signalled by the behavior just identified, is the
R
gauge invariance of the action: L(x); i.e., the action is invariant under local
field transformation
Bµ (x) := igBµa (x)T a → G(x)Bµ (x)G−1 (x) + [∂µ G(x)]G−1 (x) , (1.221)
G(x) = exp{−igT a Θa (x)} .
(1.222)
This means that, given a reference field configuration Bˆµ (x), the integrand in
the generating functional, Eq. (1.212), is constant along the path through the
gauge field manifold traversed by the applying gauge transformations to Bˆµ (x).
Since the parameters characterizing the gauge transformations, Θa , are continuous functions, each such gauge orbit contains an uncountable infinity of gauge
field configurations. It is therefore immediately apparent that the generating
functional, as written, is is undefined: it contains a multiplicative factor proportional to the length (or volume) of the gauge orbit. (NB. While there is, in
addition, an uncountable infinity of distinct reference configurations, the action
changes upon any shift orthogonal to a gauge orbit.)
Returning to the example in Eq. (1.219), the analogy is that the “Lagrangian
density” l(x, y) = (x − y)2 is invariant under translations; i.e., the integrand is
invariant under the operation
(
)
∂
∂
gs (x, y) = exp s
+s
∂x
∂y
0
0
(x, y) → (x , y ) = gs (x, y)(x, y) = (x + s, y + s) .
(1.223)
(1.224)
Hence, given a reference point P = (x0 , y0 ) = (1, 0), the integrand is constant
along the path (x, y) = P0 + (s, s) through the (x, y)-plane. (This is a translation
of the centre-of-mass: X0 = (x0 + y0 )/2 = 1/2 → X0 + s.) Since s∈(−∞, ∞), this
33
path contains an uncountable infinity of points, and at each one the integrand
has precisely the same value. The integral thus contains a multiplicative factor
proportional to the length of the translation path, which clearly produces an
infinite (meaningless) result for the integral. (NB. The value of the integrand
changes upon a translation orthogonal to that just identified.)
1.8.1
Faddeev-Popov Determinant and Ghosts
This problem with the functional integral over gauge fields was identified by
Faddeev and Popov. They proposed to solve the problem by identifying and
extracting the gauge orbit volume factor.
A Simple Model
Before describing the procedure in quantum field theory it can be illustrated
using the simple integral model. One begins by defining a functional of our “field
variable”, (x, y), which intersects the centre-of-mass translation path once, and
only once:
f (x, y) = (x + y)/2 − a = 0 .
(1.225)
Now define a functional ∆f such that
∆f [x, y]
Z
∞
−∞
daδ((x + y)/2 − a) = 1 .
(1.226)
It is clear that ∆f [x, y] is independent of a:
0
0 −1
∆f [x + a , y + a ]
=
ã=a−a0
=
∞
Z
−∞
Z ∞
−∞
daδ((x + a0 + y + a0 )/2 − a)
(1.227)
dãδ((x + y)/2 − ã)
(1.228)
∆f [x, y]−1 .
=
(1.229)
Using ∆f the model generating functional can be rewritten
Z
∞
−∞
dx
Z
∞
−∞
dye
−l(x,y)
=
Z
∞
−∞
da
Z
∞
−∞
dx
Z
∞
−∞
dye−l(x,y) ∆f [x, y]δ((x + y)/2 − a)
(1.230)
and now one performs a centre-of-mass translation: x → x0 = x + a, y → y 0 =
y + a, under which the action is invariant so that the integral becomes
Z
=
∞
−∞
Z ∞
−∞
da
da
Z
∞
−∞
Z ∞
−∞
dx
dx
Z
∞
−∞
Z ∞
−∞
dye−l(x,y) ∆f [x, y]δ((x + y)/2 − a)
(1.231)
dye−l(x,y) ∆f [x + a, y + a]δ((x + y)/2) . (1.232)
34
Now making use of the a−independence of ∆f [x, y], Eq. (1.227),
=
Z
∞
−∞
da
Z Z
∞
−∞
dx
Z
∞
−∞
dye−l(x,y) ∆f [x, y]δ((x + y)/2) .
(1.233)
In this last line the “volume” or “path length” has been explicitely factored out at
the cost of introducing a δ-function, which fixes the centre-of-mass; i.e., a single
point on the path of translationally equivalent configurations, and a functional
∆f , which, we will see, is the analogue in this simple model of the Fadeev-Popov
determinant. Hence the “correct” definition of the generating functional for this
model is
w(~j) =
Z
∞
−∞
dx
Z
∞
−∞
dye−l(x,y)+jx x+jy y ∆f [x, y]δ((x + y)/2) .
35
(1.234)
Gauge Fixing Conditions
To implement this idea for the real case of non-Abelian gauge fields one envisages
an hypersurface, lying in the manifold of all gauge fields, which intersects each
gauge orbit once, and only once. This means that if
fa [Bµa (x); x] = 0 , a = 1, 2, . . . , Nc2 − 1 ,
(1.235)
is the equation describing the hypersurface, then there is a unique element in each
gauge orbit that satisfies one Eq. (1.235), and the set of these unique elements,
none of which cannot be obtained from another by a gauge transformation, forms
a representative class that alone truly characterises the physical configuration of
gauge fields. The gauge-equivalent, and therefore redundant, elements are absent.
Equations (1.235) can also be viewed as defining a set of non-linear equations
for G(x), Eq (??). This in the sense that for a given field configuration, Bµ (x),
it is always possible to find a unique gauge transformation, G1 (x), that yields a
gauge transformed field BµG1 (x), from Bµ (x) via Eq. (??), which is the one and
only solution of fa [Bµa,G1 (x); x] = 0. Equation (1.235) therefore defines a gauge
fixing condition.
In order for Eqs. (1.235) to be useful it must be possible that, when given a
configuration Bµ (x), for which fa [Bµ (x); x] 6= 0, the equation
fa [BµG (x); x] = 0
(1.236)
can be solved for the gauge transformation G(x). To see a consequence of this
requirement consider a gauge configuration Bµb (x) that almost, but not quite,
satisfies Eqs. (1.235). Applying an infinitesimal gauge transformation to this
Bµb (x) the requirement entails that it must be possible to solve
fa [Bµb (x) → Bµb (x) − g fbdc Bµd (x) δΘc (x) − ∂µ δΘb (x); x] = 0
(1.237)
for the infinitesimal gauge transformation parameters δΘa (x). Equation (1.237)
can be written (using the chain rule)
fa [Bµb (x)]
−
Z
i
δfa [Bµb (x); x] h cd
e
δ
∂
+
gf
B
(y)
δΘd (y) = 0 .
dy
ν
ced
ν
c
δBν (y)
4
(1.238)
~ which has a solution for ~θ if, and
This looks like the matrix equation f~ = O θ,
only if, det O 6= 0, and a similar constraint follows from Eq. (1.238): the gauge
fixing conditions can be solved if, and only if,
Det Mf
i
δfa [Bµb (x); x] h cd
e
:= Det −
δ
∂
−
gf
B
(y)
ν
cde
ν
δBνc (y)
(
)
δfa [Bµb (x); x]
=: Det −
[Dν (y)]cd 6= 0 .
δBνc (y)
(
)
36
(1.239)
Equation (1.239) is the so-called admissibility condition for gauge fixing conditions.
A simple illustration is provided by the lightlike (Hamilton) gauges, which are
specified by
nµ Bµa (x) = 0 , n2 > 1 , a = 1, 2, . . . , (N 2 − 1) .
(1.240)
Choosing (nµ ) = (1, 0, 0, 0), the equation for G(x) is, using Eq. (??)
∂
G(t, ~x) = − G(t, ~x) B 0 (t, ~x) ,
∂t
(1.241)
and this nonperturbative equation has the unique solution
G(t, ~x) = T exp −
Z
t
−∞
0
ds B (s, ~x) ,
(1.242)
where T is the time ordering operator. One may compare this with Eq. (1.238),
which only provides a perturbative [in g] solution. While that may be an advantage, Eq. (1.242) is not a Poincaré covariant constraint and that makes it difficult
to employ in explicit calculations.
A number of other commonly used gauge fixing conditions are
∂ µ Bµa (x) = 0 ,
∂ µ Bµa (x) = Aa (x) ,
nµ Bµa (x) = 0 , n2 < 0 ,
nµ Bµa (x) = 0 , n2 = 0 ,
~ · B~a (x) = 0 ,
∇
Lorentz gauge
Generalised Lorentz gauge
Axial gauge
Light-like gauge
Coulomb gauge
(1.243)
and the generalised axial, light-like and Coulomb gauges, where an arbitrary
function, Aa (x), features on the r.h.s.
All of these choices satisfy the admissability condition, Eq. (1.239), for small
gauge field variations but in some cases, such as Lorentz gauge, the uniqueness
condition fails for large variations; i.e., those that are outside the domain of perturbation theory. This means that there are at least two solutions: G1 , G2 , of
Eq. (1.236), and perhaps uncountably many more. Since no nonperturbative solution of any gauge field theory in four spacetime dimensions exists, the actual
number of solutions is unknown. If it is infinite then the Fadde’ev-Popov definition of the generating functional fails in that gauge. These additional solutions
are called Gribov Copies and their existence raises questions about the correct
way to furnish a nonperturbative definition of the generating functional [9], which
are currently unanswered.
Isolating and Eliminating the Gauge Orbit Volume
To proceed one needs a little information about the representation of non-Abelian
groups. Suppose u is an element of the group SU (N ). Every such element can be
37
characterised by (N 2 − 1) real parameters: {Θa , a = 1, . . . , N 2 − 1}. Let G(u) be
the representation of u under which the gauge fields transform; i.e., the adjoint
representation, Eqs. (??), (??). For infinitesimal tranformations
G(u) = I − igT a Θa (x) + O(Θ2 )
(1.244)
where {T a } are the adjoint representation of the Lie algebra of SU (N ), Eq. (??).
Clearly, if u, u0 ∈ SU (N ) then uu0 ∈ SU (n) and G(u)G(u0 ) = G(uu0 ). (These
are basic properties of groups.)
To define the integral over gauge fields we must properly define the gaugefield “line element”. This is the Hurwitz measure on the group space, which is
invariant in the sense that du0 = d(u0 u). In the neighbourhood of the identity
one may always choose
Y
du =
dΘa
(1.245)
a
and the invariance means that since the integration represents a sum over all
possible values of the parameters Θa , relabelling them as Θ̃a cannot matter. It
is now possible to quantise the gauge field; i.e., properly define Eq. (??).
Consider ∆f [Bµa ] defined via, cf. Eq. (??),
∆f [Bµb ]
Z Y
du(x)
x
δ[fa {Bµb,u (x); x}] = 1 ,
Y
x,a
(1.246)
where Bµb,u (x) is given by Eq. (??) with G(x) → u(x). ∆f [Bµa ] is gauge invariant:
Z Y
b,u
∆−1
f [Bµ ] =
du0 (x)
x,a
x
Z Y
=
0
δ[fa {Bµb,u u (x); x}]
Y
d(u0 (x)u(x))
x,a
x
Z Y
=
Y
du00 (x)
x
0
δ[fa {Bµb,u u (x); x}]
00
b
δ[fa {Bµb,u (x); x}] = ∆−1
f [Bµ ] .
Y
x,a
(1.247)
Returning to Eq. (??), one can write
W [0] =
Z Y
Z
du(x) [DBµa ] ∆f [Bµa ]
x
Y
x,b
δ[fb {Bµa,u (x); x}] exp{i
Bµa (x)
Now execute a gauge transformation:
→
invariance of the measure and the action, one has
W [0] = [
Z Y
x
du(x)]
Z
[DBµa ] ∆f [Bµa ]
Y
x,b
38
−1
Bµa,u (x),
δ[fb {Bµa (x); x}]
Z
d4 x L(x)} .
(1.248)
so that, using the
Z
exp{i d4 x L(x)} ,
(1.249)
where now the integrand of the group measure no longer depends on the group
element, u(x) (cf. Eq. (??)). The gauge orbit volume has thus been identified
and can be eliminated so that one may define
Z
W [Jµa ] = [DBµa ] ∆f [Bµa ]
Y
x,b
δ[fb {Bµa (x); x}] exp{i
Z
d4 x [L(x) + Jaµ (x)Bµa (x)]} .
(1.250)
Neglecting for now the possible existence of Gribov copies, Eq. (1.250) is the
foundation we sought for a manifestly Poincaré covariant quantisation of the
gauge field. However, a little more work is needed to mould a practical tool.
Ghost Fields
A first step is an explicit calculation of ∆f [Bµa ]. Since it always appears multiplied by a δ-function it is sufficient to evaluate it for those field configurations
that satisfy Eq. (1.235). Recalling Eqs. (1.238), (1.239), for infinitesimal gauge
transformations
fa [Bµb,u (x); x] = fa [Bµb (x); x] +
= 0+
Z
Z
d4 y (Mf )ac Θc (y)
d4 y (Mf )ac Θc (y) .
(1.251)
Hence, using the definition, Eq. (1.246),
b
∆−1
f [Bµ ]
=
Z Y
a
dΘ (x) δ
x,a
Z
4
d y (Mf )ac Θc (y)
(1.252)
,
(1.253)
and changing variables: Θa → Θ̃a = (Mf )ac Θc (y), this gives
b
∆−1
f [Bµ ] = Det Mf
Z Yn
h
dΘ̃a (x) δ Θ̃a (y)
x,a
io
where the determinant is the Jacobian of the transformation, so that
∆f [Bµb ] = Det Mf .
(1.254)
Now recall Eq. (1.180). This means Eq. (1.254) can be expressed as a functional integral over Grassmann fields: φ̄a , φb , a, b = 1, . . . , (N 2 − 1),
∆f [Bµb ]
Z
= [D φ̄b ][Dφa ] exp −
Z
4
4
d xd y φ̄b (x) (Mf )bc (x, y) φa (y) , .
(1.255)
Consequently, absorbing non-dynamical constants into the normalisation,
W [Jµa ]
=
Z
[DBµa ] [D φ̄b ][Dφa ]
Y
x,b
Z
4
× exp i d x [L(x) +
δ[fb {Bµa (x); x}]
Jaµ (x)Bµa (x)]
39
+i
Z
4
4
d xd y φ̄b (x) (Mf )bc (x, y) φa (y) .
(1.256)
The Grassmann fields {φ̄a , φb } are the Fadde’ev-Popov Ghosts. They are an
essential consequence of gauge fixing.
As one concrete example, consider the Lorentz gauge, Eq. (1.243), for which
h
(ML )ab = δ 4 (x − y) δ ab ∂ 2 − gfabc ∂ µ Bµc (x)
and therefore
W [Jµa ]
Z
=
[DBµa ] [D φ̄b ][Dφa ]
Y
δ[∂
x,a
µ
Bµa (x)]
Z
exp i
4
i
(1.257)
1
4
a
d x − Faµν (x) Fµν
(x)
− ∂µ φ̄a (x) ∂ ν φa (x) + gfabc [∂ µ φ̄a (x)]φb (x)Bµc (x) + Jaµ (x)Bµa (x)(1.258)
.
This expression makes clear that a general consequence of the Fadde’ev-Popov
procedure is to introduce a coupling between the gauge field and the ghosts. Thus
the ghosts, and hence gauge fixing, can have a direct impact on the behaviour of
gauge field Green functions.
As another, consider the axial gauge, for which
i
h
(MA )ab = δ 4 (x − y) δ ab nµ ∂ µ , .
(1.259)
∂ ν Bµa (x) = Aa (x) ,
(1.260)
Expressing the related determinant via ghost fields it is immediately apparent
that with this choice there is no coupling between the ghosts and the gauge field
quanta. Hence the ghosts decouple from the theory and may be discarded as
they play no dynamical role. However, there is a cost, as always: in this gauge
Q
the effect of the delta-function, x,a δ[nµ Bµa (x)], in the functional integral is to
complicate the Green functions by making them depend explicitly on (nµ ). Even
the free-field 2-point function exhibits such a dependence.
It is important to note now that this decoupling of the ghost fields is tied to
an “accidental” elimination of the fabc term in Dµ (y), Eq. (1.239). That term is
always absent in Abelian gauge theories, for which quantum electrodynamics is
the archetype, because all generators commute and analogues of fabc must vanish.
Hence in Abelian gauge theories ghosts decouple in every gauge.
To see how δ[fb {Bµa (x); x}] influences the form of Green functions, consider
the generalised Lorentz gauge:
where {Aa (x)} are arbitrary functions. The Fadde’ev-Popov determinant is the
same in generalised Lorentz gauges as it is in Lorentz gauge and hence
∆GL [Bµa ] = ∆L [Bµa ] ,
(1.261)
where the r.h.s. is given in Eq. (1.257). The generating functional in this gauge
is therefore
W [Jµa ]
=
Z
[DBµa ] [D φ̄b ][Dφa ]
Y
x,a
δ[∂
µ
Bµa (x)
a
Z
− A (x)] exp i
4
1
4
a
d x − Faµν (x) Fµν
(x)
− ∂µ φ̄a (x) ∂ ν φa (x) + gfabc [∂ µ φ̄a (x)]φb (x)Bµc (x) + Jaµ (x)Bµa (x)
40
.
Gauge invariance of the generating functional, Eq. (1.246), means that one can
integrate over the {Aa (x)}, with a weight function to ensure convergence:
Z
[DAa ] exp −
i Z 4 a
d x A (x) Aa (x) ,
2λ
(1.262)
to arrive finally at the generating functional in a covariant Lorentz gauge, specified by the parameter λ:
Z
W [Jµa , ξ¯ga , ξga ] = [DBµa ] [D φ̄b ][Dφa ]
1 µ a
[∂ Bµ (x)] [∂ ν Bνa (x)]
2λ
− ∂µ φ̄a (x) ∂ ν φa (x) + gfabc [∂ µ φ̄a (x)]φb (x)Bµc (x)
Z
exp i
1
4
a
d4 x − Faµν (x) Fµν
(x) −
+Jaµ (x)Bµa (x)
+ ξ¯ga (x)φa (x) + φ̄a (x)ξga (x)
,
(1.263)
where {ξ¯ga , ξga } are anticommuting external sources for the ghost fields. (NB. To
complete the definition one should add convergence terms, iη + , for every field or,
preferably, work in Euclidean space.)
Observe now that the free gauge boson piece of the action in Eq. (1.263) is
:=
Z
1
2
Z
1
2
d4 x d4 y Bµa (x) K µν (x − y; λ) Bνa (y)
1
d4 x d4 y Bµa (x) [g µν ∂ 2 − ∂ µ ∂ ν (1 − ) − ig µν η + ]δ 4 (x − y) Bνa(1.264)
(y)
λ
The operator K µν (x − y; λ) thus defined can be expressed
µν
K (x − y) =
Z
d4 q
(2π)4
1
−g (q + iη ) + q q [1 − ] e−i(q,x−y) ,
λ
µν
2
+
µ ν
(1.265)
cf. Eq. (??), and now
1 ν
q
(1.266)
λ
so that in this case the action does damp variations in the longitudinal components of the gauge field. K µν (x − y; λ) is the inverse of the free gauge boson
propagator; i.e., the free gauge boson 2-point Green function, D µν (x − y), is
obtained via
Z
d4 w Kρµ (x − w) D ρν (w − y) = g µν δ 4 (x − y) ,
(1.267)
qµ K µν = −
and hence
µν
D (x − y) =
Z
d4 q
(2π)4
−g
µν
qµqν
+ (1 − λ) 2
q + iη +
!
q2
1
e−i(q,x−y) .
+
+ iη
(1.268)
The obvious λ dependence is a result of the presence of δ[fb {Bµa (x); x}] in the
generating functional, and this is one example of the δ-function’s direct effect on
the form of Green functions: they are, in general, gauge parameter dependent.
41
1.8.2
Exercises
1. Verify Eq. (1.257).
2. Verify Eq. (1.258).
3. Verify Eq. (1.259).
4. Verify Eq. (1.268).
1.9
Dyson-Schwinger Equations
It has long been known that, from the field equations of quantum field theory, one
can derive a system of coupled integral equations interrelating all of a theory’s
Green functions [6, 10]. This tower of a countable infinity of equations is called
the complex of Dyson-Schwinger equations (DSEs). This intrinsically nonperturbative complex is vitally important in proving the renormalisability of quantum
field theories and at its simplest level provides a generating tool for perturbation theory. In the context of quantum electrodynamics (QED) I will illustrate a
nonperturbative derivation of two equations in this complex. The derivation of
others follows the same pattern.
1.9.1
Photon Vacuum Polarization
Generating Functional for QED
The action for QED with Nf flavours of electomagnetically active fermion, is

Nf
X

1
1 µ
/ ψ f − Fµν F µν −
S[Aµ , ψ, ψ̄] = d4 x 
∂ Aµ (x) ∂ ν Aν (x) ,
ψ̄ f i∂/ − mf0 + ef0 A
4
2λ
0
f =1
(1.269)
f
f
where: ψ̄ (x), ψ (x) are the elements of the Grassmann algebra that describe
the fermion degrees of freedom, mf0 are the fermions’ bare masses and ef0 their
charges; and Aµ (x) describes the gauge boson [photon] field, with
Z
Fµν = ∂µ Aν − ∂ν Aµ ,
(1.270)
and λ0 the bare covariant-Lorentz-gauge fixing parameter. (NB. To describe an
electron the physical charge ef < 0.)
The derivation of the generating functional in Eq. (1.263) can be employed
with little change here. In fact, in this context it is actually simpler because
ghost fields decouple. Combining the procedure for fermions and gauge fields,
42
described in Secs. 1.7, ?? respectively, one arrives at
Z
W [Jµ , ξ, ξ̄] = [DAµ ] [Dψ][D ψ̄]
Z
exp i
+
Nf
X
f =1
4
1
4
d x − F µν (x) Fµν (x) −
1 µ
∂ Aµ (x) ∂ ν Aν (x)
2λ0
/ ψ f +J µ (x)Aµ (x) + ξ¯f (x)ψ f (x) + ψ̄ f (x)ξ f (x)
ψ̄ f i∂/ − mf0 + ef0 A
,
where Jµ is an external source for the electromagnetic field, and ξ f , ξ¯f are external sources for the fermion field that, of course, are elements in the Grassmann algebra. (NB. In Abelian gauge theories there are no Gribov copies in the
covariant-Lorentz-gauges.)
Functional Field Equations
As described in Sec. 1.6.1, it is advantageous to work with the generating func¯ ξ] defined via
tional of connected Green functions; i.e., Z[Jµ , ξ,
n
o
W [Jµ , ξ, ξ̄] =: exp iZ[Jµ , ξ, ξ̄] .
(1.271)
The derivation of a DSE now follows simply from the observation that the integral
of a total derivative vanishes, given appropriate boundary conditions. Hence, for
example,
Z
Z
i
h f
δ
0 = [DAµ ] [Dψ][D ψ̄]
exp i S[Aµ , ψ, ψ̄] + d4 x ψ ξ f + ξ¯f ψ f + Aµ J µ
δAµ (x)
=
Z
[DAµ ] [Dψ][D ψ̄]
(
)
Z
i
h f
δS
+ Jµ (x) exp i S[Aµ , ψ, ψ̄] + d4 x ψ ξ f + ξ¯f ψ f + Aµ J µ
δAµ (x)
(
"
#
)
δ
δ
δS
δ
,
=
,−
+ Jµ (x) W [Jµ , ξ, ξ̄] ,
(1.272)
δAµ (x) iδJ iδ ξ¯ iδξ
where the last line has meaning as a functional differential operator on the generating functional.
Differentiating Eq. (1.269) gives
X f f
1
δS
e0 ψ (x)γµ ψ f (x)
= ∂ρ ∂ ρ gµν − 1 −
∂µ ∂ν Aν (x) +
δAµ (x)
λ0
f
(1.273)
so that the explicit meaning of Eq. (1.272) is
−Jµ (x) =
ρ
∂ρ ∂ gµν
X f
δZ
1
− 1−
∂µ ∂ν
+
e0
λ0
δJν (x)
f
43
"
δZ
δ
δ iZ
δZ
γµ ¯f
+ f
γµ ¯f
− f
δξ (x) δ ξ (x) δξ (x)
δ ξ (x)
#!
,
(1.274)
where I have divided through by W [Jµ , ξ, ξ̄]. Equation (1.274) represents a compact form of the nonperturbative equivalent of Maxwell’s equations.
One-Particle-Irreducible Green Functions
The next step is to introduce the generating functional for one-particle-irreducible
(1PI) Green functions: Γ[Aµ , ψ, ψ̄], which is obtained from Z[Jµ , ξ, ξ̄] via a Legendre transformation
Z[Jµ , ξ, ξ̄] = Γ[Aµ , ψ, ψ̄] +
Z
f
d4 x ψ ξ f + ξ¯f ψ f + Aµ J µ .
h
i
(1.275)
A one-particle-irreducible n-point function or “proper vertex” contains no contributions that become disconnected when a single connected m-point Green
function is removed; e.g., via functional differentiation. This is equivalent to the
statement that no diagram representing or contributing to a given proper vertex
separates into two disconnected diagrams if only one connected propagator is cut.
(A detailed explanation is provided in Ref. [3], pp. 289-294.)
A simple generalisation of the analysis in Sec. 1.6.1 yields
δZ
δZ
δZ
= ψ(x) ,
= Aµ (x) , ¯
= −ψ̄(x) ,
µ
δJ (x)
δξ(x)
δ ξ(x)
(1.276)
where here the external sources are nonzero. Hence Γ in Eq. (1.275) must satisfy
δΓ
δΓ
δΓ
= −Jµ (x) ,
= ξ¯f (x) .
= −ξ f (x) ,
µ
f
f
δA (x)
δψ (x)
δ ψ̄ (x)
(1.277)
(NB. Since the sources are not zero then, e.g.,
δA
6= 0 ,
δJ µ (x)
Aµ (x) = Aµ (x; [Jµ , ξ, ξ̄]) ⇒
(1.278)
with analogous statements for the Grassmannian functional derivatives.) It is
easy to see that setting ψ̄ = 0 = ψ after differentiating Γ gives zero unless there
are equal numbers of ψ̄ and ψ derivatives. (This is analogous to the result for
scalar fields in Eq. (1.140).)
Now consider the product (with spinor labels r, s, t)
−
Z
δ2Z
δ2Γ
d4 z f
.
g
¯
h
h
¯
δξr (x)ξt (z) δψt (z)ψ s (y) ξ = ξ = 0
ψ=ψ=0
(1.279)
Using Eqs. (1.276), (1.277), this simplifies as follows:
=
Z
δψth (z) δξsg (y) δξsg (y) dz f
= f
= δrs δ f g δ 4 (x − y) .
¯
h
ξ
=
ξ
=
0
δξr (x) δψt (z)
δξr (x) ψ = ψ = 0
ψ=ψ=0
(1.280)
4
44
Now returning to Eq. (1.274) and setting ξ¯ = 0 = ξ one obtains
i
X f h
1
δΓ ν
f
ρ
A
(x)
−
i
e
tr
γ
S
(x,
x;
[A
])
,
=
∂
∂
g
−
1
−
∂
∂
µ
µ
ρ
µν
µ
ν
0
δAµ (x) ψ=ψ=0
λ0
f
(1.281)
after making the identification
S f (x, y; [Aµ ]) = −
δ2Z
δ2Z
=
(no summation on f ) ,
δξ f (y)ξ¯f (x)
δ ξ¯f (x)ξ f (y)
(1.282)
which is the connected Green function that describes the propagation of a fermion
with flavour f in an external electromagnetic field Aµ (cf. the free fermion Green
function in Eq. (1.197).) I observe that it is a direct consequence of Eq. (1.279)
that
δ2Γ
f
−1
,
(1.283)
S (x, y; [A]) =
δψ f (x)δ ψ̄ f (y) ψ=ψ=0
and it is a general property that such functional derivatives of the generating
functional for 1PI Green functions are related to the associated propagator’s
inverse. Clearly the vacuum fermion propagator or connected fermion 2-point
function is
S f (x, y) := S f (x, y; [Aµ = 0]) .
(1.284)
Such vacuum Green functions are of primary interest in quantum field theory.
To continue, one differentiates Eq. (1.281) with respect to Aν (y) and sets
Jµ (x) = 0, which yields
δ2Γ
δAµ (x)δAν (y) Aµ = 0
ψ=ψ=0
=
∂ρ ∂ ρ gµν − 1 −
−i
X
f

1
∂µ ∂ν δ 4 (x − y)
λ0

−1 
δ2Γ
δ


 
ef0 tr γµ
.
δAν (y) δψ f (x)δ ψ̄ f (x) ψ=ψ=0
(1.285)
The l.h.s. is easily understood. Just as Eqs. (1.283), (1.284) define the inverse
of the fermion propagator, so here is
δ2Γ
(D ) (x, y) :=
µ
ν
δA (x)δA (y) −1 µν
Aµ = 0
ψ=ψ=0
.
(1.286)
The r.h.s., however, must be simplified and interpreted. First observe that

−1
δ
δ2Γ


f
f
δAν (y) δψ (x)δ ψ̄ (x) ψ=ψ=0
=
45
−
Z

−1
δ2Γ

d4 ud4 w  f
f
δψ (x)δ ψ̄ (w) ψ=ψ=0

(1.287)
which is an analogue of the result for finite dimensional matrices:
i
dA−1 (x)
d h
dA(x) −1
A (x) + A(x)
A(x)A−1 (x) = I = 0 =
dx
dx
dx
dA(x)
dA−1 (x)
= −A−1 (x)
A−1 (x) .(1.288)
⇒
dx
dx
Equation (1.287) involves the 1PI 3-point function
ef0 Γfµ (x, y; z) :=
δ2Γ
.
δAν (z) δψ f (x)δ ψ̄ f (y)
δ
(1.289)
This is the proper fermion-gauge-boson vertex. At leading order in perturbation
theory
Γfν (x, y; z) = γν δ 4 (x − z) δ 4 (y − z) ,
(1.290)
a result which can be obtained via the explicit calculation of the functional derivatives in Eq. (1.289).
Now, defining the gauge-boson vacuum polarisation:
Πµν (x, y) = i
X
(ef0 )2
f
Z
h
i
d4 z1 d4 z2 tr γµ S f (x, z1 )Γfν (z1 , z2 ; y)S f (z2 , x) , (1.291)
it is immediately apparent that Eq. (1.285) may be expressed as
−1 µν
ρ
(D ) (x, y) = ∂ρ ∂ gµν
1
− 1−
∂µ ∂ν δ 4 (x − y) + Πµν (x, y) .
λ0
(1.292)
In general, the gauge-boson vacuum polarisation, or “self-energy,” describes the
modification of the gauge-boson’s propagation characteristics due to the presence
of virtual fermion-antifermion pairs in quantum field theory. In particular, the
photon vacuum polarisation is an important element in the description of process
such as ρ0 → e+ e− .
The propagator for a free gauge boson was given in Eq. (1.268). In the
presence of interactions; i.e., for Πµν 6= 0 in Eq. (1.292), this becomes
D µν (q) =
−1
δ2Γ
δ2Γ
δ
 ,

f
f
f
f
δAν (y) δψ (u)δ ψ̄ (w) δψ (w)δ ψ̄ (x) ψ=ψ=0
−g µν + (q µ q ν /[q 2 + iη])
1
qµqν
−
λ
,
0
q 2 + iη
1 + Π(q 2 )
(q 2 + iη)2
(1.293)
where I have used the “Ward-Takahashi identity:”
qµ Πµν (q) = 0 = Πµν (q) qν ,
46
(1.294)
which means that one can write
Πµν (q) = −g µν q 2 + q µ q ν Π(q 2 ) .
(1.295)
Π(q 2 ) is the polarisation scalar and, in QED, it is independent of the gauge parameter, λ0 . (NB. λ0 = 1 is called “Feynman gauge” and it is useful in perturbative
calculations because it obviously simplifies the Π(q 2 ) = 0 gauge boson propagator
enormously. In nonperturbative applications, however, λ0 = 0, “Landau gauge,”
is most useful because it ensures the gauge boson propagator itself is transverse.)
Ward-Takahashi identities (WTIs) are relations satisfied by n-point Green
functions, relations which are an essential consequence of a theory’s local gauge
invariance; i.e., local current conservation. They can be proved directly from the
generating functional and have physical implications. For example, Eq. (1.295)
ensures that the photon remains massless in the presence of charged fermions. (A
discussion of WTIs can be found in Ref. [1], pp. 299-303, and Ref. [3], pp. 407-411;
and their generalisation to non-Abelian theories as “Slavnov-Taylor” identities is
described in Ref. [5], Chap. 2.)
In the absence of external sources for fermions and gauge bosons, Eq. (1.291)
can easily be represented in momentum space, for then the 2- and 3-point functions that appear explicitly must be translationally invariant and hence can be
simply expressed in terms of Fourier amplitudes. This yields
iΠµν (q) = −
X
f
(ef0 )2
Z
d4 `
tr[(iγµ )(iS f (`))(iΓf (`, ` + q))(iS(` + q))] . (1.296)
(2π)d
It is the reduction to a single integral that makes momentum space representations most widely used in continuum calculations.
In QED the vacuum polarisation is directly related to the running coupling
constant. This connection makes its importance obvious. In QCD the connection
is not so direct but, nevertheless, the polarisation scalar is a key component in
the evaluation of the strong running coupling.
In the above analysis we saw that second derivatives of the generating functional, Γ[Aµ , ψ, ψ̄], give the inverse-fermion and -photon propagators and that the
third derivative gave the proper photon-fermion vertex. In general, all derivatives
of this generating functional, higher than two, produce the corresponding proper
Green’s functions, where the number and type of derivatives give the number and
type of proper Green functions that it can serve to connect.
1.9.2
Fermion Self Energy
Equation (1.274) is a nonperturbative generalisation of Maxwell’s equation in
quantum field theory. Its derivation provides the model by which one can obtain
47
an equivalent generalisation of Dirac’s equation. To this end consider that
Z
Z
h g
i
δ
4
g
g g
µ
¯
0 = [DAµ ] [Dψ][D ψ̄] f
exp i S[Aµ , ψ, ψ̄] + d x ψ ξ + ξ ψ + Aµ J
δ ψ̄ (x)
Z
=
[DAµ ] [Dψ][D ψ̄]
)
(
h g
i
δS
f
4
g
¯g ψ g + Aµ J µ
+
ξ
(x)
exp
i
S[A
,
ψ,
ψ̄]
+
d
x
ψ
ξ
+
ξ
µ
δ ψ̄ f (x)
"
#
)
(
δ
δS
δ
δ
f
+ η (x) W [Jµ , ξ, ξ̄]
,−
=
,
δ ψ̄ f (x) iδJ iδ ξ¯ iδξ
"
!
#
δ
δ
f
f
= ξ f (x) + i∂/ − m0 + e0 γ µ
W [Jµ , ξ, ξ̄] .
iδJ µ (x) iδ ξ¯f (x)
Z
(1.297)
(1.298)
This is a nonperturbative, funcational equivalent of Dirac’s equation.
One can proceed further. A functional derivative with respect to ξ f : δ/δξ f (y),
yields
4
δ (x − y)W [Jµ ] − i∂/ −
mf0
+
ef0 γ µ
δ
iδJ µ (x)
!
W [Jµ ] S f (x, y; [Aµ ]) = (1.299)
0,
after setting ξ f = 0 = ξ¯f , where W [Jµ ] := W [Jµ , 0, 0] and S(x, y; [Aµ ]) is defined
in Eq. (1.282). Now, using Eqs. (1.271), (1.277), this can be rewritten
4
δ (x − y) − i∂/ −
mf0
+
ef0 A
/(x; [J])
+
ef0 γ µ
δ
iδJ µ (x)
!
S f (x, y; [Aµ ]) = (1.300)
0,
which defines the nonperturbative connected 2-point fermion Green function
(This is clearly the functional equivalent of Eq. (1.86).)
The electromagentic four-potential vanishes in the absence of an external
source; i.e., Aµ (x; [J = 0]) = 0, so it remains only to exhibit the content of the
remaining functional differentiation in Eq. (1.300), which can be accomplished
using Eq. (1.287):
δ
iδJ µ (x)
S f (x, y; [Aµ ]) =
=
Z

−1
δAν (z)
δ
δ2Γ


d4 z
µ
f
f
iδJ (x) δAν (z) δψ (x)δ ψ̄ (y) ψ=ψ=0
−ef0
= −ef0
Z
Z
d4 z d 4 u d4 w
δAν (z) f
S (x, u) Γν (u, w; z) S(w, y)
iδJµ (x)
d4 z d4 u d4 w iDµν (x − z) S f (x, u) Γν (u, w; z) S(w, y) ,
(1.301)
where, in the last line, I have set J = 0 and used Eq. (1.286). Hence, in the
absence of external sources, Eq. (1.300) is equivalent to
δ 4 (x − y) = i∂/ − mf0 S f (x, y)
− i (ef0 )2
Z
d4 z d4 u d4 w D µν (x, z) γµ S(x, u) Γν (u, w; z) S(w, y) = δ 4 (x (1.302)
− y) .
48
Just as the photon vacuum polarisation was introduced to simplify, or reexpress, the DSE for the gauge boson propagator, Eq. (1.291), one can define a
fermion self-energy:
Z
Σf (x, z) = i(ef0 )2
d4 u d4 w D µν (x, z) γµ S(x, u) Γν (u, w; z) ,
(1.303)
so that Eq. (1.302) assumes the form
Z
d4 z
h
i
i∂/x − mf0 δ 4 (x − z) − Σf (x, z) S(z, y) = δ 4 (x − y) .
(1.304)
Again using the property that Green functions are translationally invariant
in the absence of external sources, the equation for the self-energy can be written
in momentum space:
Σf (p) = i (ef0 )2
Z
d4 `
D µν (p − `) [iγµ ] [iS f (`)] [iΓfν (`, p)] .
(2π)4
(1.305)
In terms of the self-energy, it follows from Eq. (1.304) that the connected fermion
2-point function can be written in momentum space as
S f (p) =
/p −
mf0
1
.
− Σf (p) + iη +
(1.306)
Equation (1.305) is the exact Gap Equation. It describes the manner in which
the propagation characteristics of a fermion moving through the ground state
of QED (the QED vacuum) is altered by the repeated emission and reabsorption of virtual photons. The equation can also describe the real process of
Bremsstrahlung. Furthermore, the solution of the analogous equation in QCD
its solution provides information about dynamical chiral symmetry breaking and
also quark confinement.
1.9.3
Exercises
1. Verify Eq. (1.277).
2. Verify Eq. (1.287).
3. Verify Eq. (1.296).
4. Verify Eq. (1.300).
49
1.10
Perturbation Theory
1.10.1
Quark Self Energy
A key feature of strong interaction physics is dynamical chiral symmetry breaking
(DCSB). In order to understand it one must first come to terms with explicit
chiral symmetry breaking. Consider then the DSE for the quark self-energy in
QCD:
−i Σ(p) =
−g02
Z
d4 `
i
D µν (p − `) λa γµ S(`) Γaν (`, p) ,
4
2
(2π)
(1.307)
where I have suppressed the flavour label. The form is precisely the same as that
in QED, Eq. (1.305), with the only difference being the introduction of the colour
(Gell-Mann) matrices: {λa ; a = 1, . . . , 8} at the fermion-gauge-boson vertex. The
interpretation of the symbols is also analogous: D µν (`) is the connected gluon
2-point function and Γaν (`, `0 ) is the proper quark-gluon vertex.
The one-loop contribution to the quark’s self-energy is obtained by evaluating
the r.h.s. of Eq. (1.307) using the free quark and gluon propagators, and the
quark-gluon vertex:
i
(1.308)
Γaν (0) (`, `0 ) = λa γν ,
2
which appears to be a straightforward task.
To be explicit, the goal is to calculate
(2)
−g02
−i Σ (p) =
Z
i
2
d4 k
(2π)4
× λa γ µ
−g
µν
kµ kν
+ (1 − λ0 ) 2
k + iη +
!
1
i a
λ γµ
k6 + /p − m0 + iη + 2
k2
1
+ iη +
(1.309)
and one may proceed as follows. First observe that Eq. (1.309) can be re-expressed
as
(2)
−i Σ (p) =
−g02
×
C2 (R)
(
Z
d4 k
1
1
2
4
2
2
+
(2π) (k + p) − m0 + iη k + iη +
)
(k, p)6k
γ (6k + /p + m0 ) γµ − (1 − λ0 ) (6k − /p + m0 ) − 2 (1 − λ0 ) 2
k + iη +
(1.310)
µ
where I have used
1 a1 a
λ λ
2
2
= C2 (R) I c ; C2 (R) =
Nc2 − 1
,
2Nc
(1.311)
with Nc the number of colours (Nc = 3 in QCD) and I c , the identity matrix in
colour space. Now note that
2 (k, p) = [(k + p)2 − m20 ] − [k 2 ] − [p2 − m20 ]
50
(1.312)
and hence
(2)
−i Σ (p) =
−g02
(
C2 (R)
Z
d4 k
1
1
2
4
2
+
2
(2π) (k + p) − m0 + iη k + iη +
γ µ (6k + /p + m0 ) γµ + (1 − λ0 ) (p
/ − m0 )
2
+ (1 − λ0 ) (p −
m20 )
)
k6
k6
− (1 − λ0 ) [(k + p)2 − m20 ] 2
.
2
+
k + iη
k + iη +
(1.313)
Focusing on the last term:
d4 k
1
1
k6
[(k + p)2 − m20 ] 2
2
4
2
+
2
+
(2π) (k + p) − m0 + iη k + iη
k + iη +
Z
1
k6
d4 k
=0
(1.314)
=
4
2
+
2
(2π) k + iη k + iη +
Z
because the integrand is odd under k → −k, and so this term in Eq. (1.313)
vanishes, leaving
Z
−i Σ(2) (p) = −g02 C2 (R)
(
1
1
d4 k
2
4
2
+
2
(2π) (k + p) − m0 + iη k + iη +
γ µ (6k + /p + m0 ) γµ + (1 − λ0 ) (p
/ − m0 )
)
k6
+ (1 − λ0 ) (p2 − m20 ) 2
.
k + iη +
(1.315)
Consider now the second term:
(1 − λ0 ) (p
/ − m0 )
Z
d4 k
1
1
.
2
4
2
2
+
(2π) (k + p) − m0 + iη k + iη +
In particular, focus on the behaviour of the integrand at large k 2 :
1
1
1
k 2 →±∞
∼
.
2
2
2
2
+
2
+
(k + p) − m0 + iη k + iη
(k − m0 + iη + ) (k 2 + iη + )
(1.316)
The integrand has poles in the second and fourth quadrant of the k0 -plane but
vanishes on any circle of radius R → ∞. That means one may rotate the contour
anticlockwise to find
Z ∞
1
dk 0 2
2
(k − m0 + iη + ) (k 2 + iη + )
0
Z i∞
1
=
dk 0
2
0
([k 0 ]2 − ~k 2 − m0 + iη + )([k 0 ]2 − ~k 2 + iη + )
Z
∞
1
k 0 →ik4
dk4
=
i
.
(1.317)
0
(−k42 − ~k 2 − m20 ) (−k42 − ~k 2 )
51
Performing a similar analysis of the
Z
R0
−∞
part one obtains the complete result:
d4 k
1
= i
2
4
2
(2π) (k − m0 + iη + ) (k 2 + iη + )
Z
d3 k
(2π)3
Z
1
dk4
.
2
2
−∞ 2π (−~
k − k4 − m20 ) (−~k 2 − k42 )
(1.318)
∞
These two steps constitute what is called a “Wick rotation.”
The integral on the r.h.s. is defined in a four-dimensional Euclidean space;
i.e., k 2 := k12 + k22 + k32 + k42 ≥ 0 . . . k 2 is nonnegative. A general vector in this
space can be written in the form:
(k) = |k| (cos φ sin θ sin β, sin φ sin θ sin β, cos θ sin β, cos β) ,
(1.319)
and clearly k 2 = |k|2 . In this space the four-vector measure factor is
Z
d4E k f (k1 , . . . , k4 )
=
1
2
Z
∞
0
2 2
dk k
Z
π
0
2
dβ sin β
Z
π
0
dθ sin θ
Z
2π
0
dφ f (k, β, θ, φ)
(1.320)
Returning now to Eq. (1.316) the large k behaviour of the integral can be
determined via
Z
Z ∞
d4 k
i
1
1
1
≈
dk 2 2
2
4
2
+
2
+
2
(2π) (k + p) − m0 + iη k + iη
16π 0
(k + m20 )
Z Λ2
i
1
=
lim
dx
2
16π Λ→∞ 0
x + m20
i
2
2
lim
ln
1
+
Λ
/m
→ ∞;
=
0
16π 2 Λ→∞
(1.321)
2
i.e., after all this work, the result is meaningless: the one-loop contribution to
the quark’s self-energy is divergent!
Such “ultraviolet” divergences, and others which are more complicated, appear whenever loops appear in perturbation theory. (The others include “infrared” divergences associated with the gluons’ masslessness; e.g., consider what
would happen in Eq. (1.321) with m0 → 0.) In a renormalisable quantum field
theory there exists a well-defined set of rules that can be used to render perturbation theory sensible. First, however, one must regularise the theory; i.e.,
introduce a cutoff, or some other means, to make finite every integral that appears. Then every step in the calculation of an observable is rigorously sensible.
Renormalisation follows; i.e, the absorption of divergences, and the redefinition
of couplings and masses, so that finally one arrives at S-matrix amplitudes that
are finite and physically meaningful.
The regularisation procedure must preserve the Ward-Takahashi identities
(the Slavnov-Taylor identities in QCD) because they are crucial in proving that
a theory can be sensibly renormalised. A theory is called renormalisable if, and
only if, there are a finite number of different types of divergent integral so that
only a finite number of masses and coupling constants need to be renormalised.
52
1.10.2
Dimensional Regularization
The Pauli-Villars prescription is favoured in QED and that is described, for example, in Ref. [3]. In perturbative QCD, however, “dimensional regularisation”
is the most commonly used procedure and I will introduce that herein.
The key to the method is to give meaning to the divergent integrals by changing the dimension of spacetime. Returning to the exemplar, Eq. (1.316), this
means we consider
T =
Z
dD k
1
1
2
D
2
+
2
(2π) (k + p) − m0 + iη k + iη +
(1.322)
where D is the dimension of spacetime and is not necessarily four.
Observe now that
Γ(α + β) Z 1
1
xα−1 (1 − x)β−1
=
,
dx
a α bβ
Γ(α) Γ(β) 0
[a x + b (1 − x)]α+β
(1.323)
where Γ(x) is the gamma-function: Γ(n + 1) = n!. This is an example of what is
commonly called “Feynman’s parametrisation,” and one can now write
T =
Z
1
0
dx
Z
dD k
1
.(1.324)
2
(2π)D [(k − xp)2 − m0 (1 − x) + p2 x(1 − x) + iη + ]2
(1.325)
The momentum integral is well-defined for D = 1, 2, 3 but, as we have seen, not
for D = 4. One proceeds under the assumption that D is such that the integral
is convergent then a shift of variables is permitted:
T
Z
k→k−xp
=
1
0
dx
Z
dD k
1
,
D
2
2
(2π) [k − a + iη + ]2
(1.326)
where a2 = m20 (1 − x) − p2 x(1 − x).
I will consider a generalisation of the momentum integral:
In =
Z
1
dD k
,
D
2
2
(2π) [k − a + iη + ]n
(1.327)
and perform a Wick rotation to obtain
In =
i
(−1)n
(2π)D
Z
dD k
[k 2
1
.
+ a 2 ]n
(1.328)
The integrand has an O(D) spherical symmetry and therefore the angular integrals can be performed:
SD :=
Z
2π D/2
dΩD =
.
Γ(D/2)
53
(1.329)
Clearly, S4 = 2π 2 , as we saw in Eq. (1.321). Hence
In
(−1)n
1
= i D−1 D/2
2
π
Γ(D/2)
Z
∞
0
dk
k D−1
.
(k 2 + a2 )n
(1.330)
Writing D = 4 + 2 one arrives at
i
In =
(−a2 )2−n
2
(4π)
a2
4π
!
Γ(n − 2 − )
.
Γ(n)
(1.331)
(NB. Every step is rigorously justified as long as 2n > D.) The important point
for continuing with this procedure is that the analytic continuation of Γ(x) is
unique and that means one may use Eq. (1.331) as the definition of In whenever
the integral is ill-defined. I observe that D = 4 is recovered via the limit → 0−
and the divergence of the integral for n = 2 in this case is encoded in
Γ(n − 2 − ) = Γ(−) =
1
− γE + O() ;
−
(1.332)
i.e., in the pole in the gamma-function. (γE is the Euler constant.)
Substituting Eq. (1.331) in Eq. (1.326) and setting n = 2 yields
i Γ(−)
T = (gν )
(4π)2 (4π)
2
Z
1
0
p2
m20
dx
(1 − x) − 2 x(1 − x)
ν2
ν
"
#
,
(1.333)
wherein I have employed a nugatory transformation to introduce the mass-scale
ν. It is the limit → 0− that is of interest, in which case it follows that (x =
exp ln x ≈ 1 + ln x)
1
m20
1
p2
i
dx
ln
−
−
γ
+
ln
4π
−
(1
−
x)
−
x(1 − x)
T = (gν )2
E
(4π)2
ν2
ν2
0
(
! "
#)
i
1
p2
m20
m20
2
= (gν )
− − γE + ln 4π + 2 − ln 2 − 1 − 2 ln 1 − 2
.
(4π)2
ν
p
m0
(1.334)
(
Z
"
#)
It is important to understand the physical content of Eq. (1.334). While it is
only one part of the gluon’s contribution to the quark’s self-energy, many of its
properties hold generally.
1. Observe that T (p2 ) is well-defined for p2 < m20 ; i.e., for all spacelike momenta and for a small domain of timelike momenta. However, at p2 = m20 ,
T (p2 ) exhibits a ln-branch-point and hence T (p2 ) acquires an imaginary
part for p2 > m20 . This imaginary part describes the real, physical process
by which a quark emits a massless gluon; i.e., gluon Bremsstrahlung. In
QCD this is one element in the collection of processes referred to as “quark
fragmentation.”
54
2. The mass-scale, ν, introduced in Eq. (1.333), which is a theoretical artifice,
does not affect the position of the branch-point, which is very good because
that branch-point is associated with observable phenomena. While it may
appear that ν affects the magnitude of physical cross sections because it
modifies the coupling, that is not really so: in going to D = 2n − dimensions the coupling constant, which was dimensionless for D = 4, has
acquired a mass dimension and so the physical, dimensionless coupling constant is α := (gν )2 /(4π). It is this dependence on ν that opens the door
to the generation of a “running coupling constant” and “running masses”
that are a hallmark of quantum field theory.
3. A number of constants have appeared in T (p2 ). These are irrelevant because
they are eliminated in the renormalisation procedure. (NB. So far we have
only regularised the expression. Renormalisation is another step.)
4. It is apparent that dimensional regularisation gives meaning to divergent
integrals without introducing new couplings or new fields. That is a benefit. The cost is that while γ5 = iγ 0 γ 1 γ 2 γ 3 is well-defined with particular
properties for D = 4, a generalisation to D 6= 4 is difficult and hence so is
the study of chiral symmetry.
D-dimensional Dirac Algebra
When one employs dimensional regularisation all the algebraic manipulations
must be performed before the integrals are evaluated, and that includes the Dirac
algebra. The Clifford algebra is unchanged in D-dimensions
{γ µ , γ ν } = 2 g µν ; µ, ν = 1, . . . , D − 1 ,
(1.335)
gµν g µν = D
(1.336)
D 1D ,
(2 − D) γ ν ,
4 g νλ 1D + (D − 4) γ ν γ λ ,
−2 γ ρ γ λ γ ν + (4 − D) γ ν γ λ γ ρ ,
(1.337)
(1.338)
(1.339)
(1.340)
but now
and hence
γµ γ ν
γµ γ ν γ µ
γµ γ ν γ λ γ µ
γµ γ ν γ λ γ ρ γ µ
=
=
=
=
where 1D is the D × D-dimensional unit matrix.
It is also necessary to evaluate traces of products of Dirac matrices. For a
D-dimensional space, with D even, the only irreducible representation of the
Clifford algebra, Eq. (1.335), has dimension f (D) = 2D/2 . In any calculation it
is the (anti-)commutation of Dirac matrices that leads to physically important
55
factors associated with the dimension of spacetime while f (D) always appears as
a common multiplicative factor. Hence one can just set
f (D) ≡ 4
(1.341)
in all calculations. Any other prescription merely leads to constant terms of the
type encountered above; e.g., γE , which are eliminated in renormalisation.
The D-dimensional generalisation of γ5 is a more intricate problem. However,
I will not use it herein and hence will omit that discussion.
Observations on the Appearance of Divergences
Consider a general Lagrangian density:
L(x) = L0 (x) +
X
gi Li (x) ,
(1.342)
i
where L0 is the sum of the free-particle Lagrangian densities and Li (x) represents
the interaction terms with the coupling constants, gi , written explicitly. Assume
that Li (x) has fi fermion fields (fi must be even since fermion fields always appear
in the pairs ψ̄, ψ), bi boson fields and n∂i derivatives. The action
S=
Z
d4 x L(x)
(1.343)
must be a dimensionless scalar and therefore L(x) must have mass-dimension
M 4 . Clearly a derivative operator has dimension M . Hence, looking at the freeparticle Lagrangian densities, it is plain that each fermion field has dimension
M 3/2 and each boson field, dimension M 1 . It follows that a coupling constant
multiplying the interaction Lagrangian density Li (x) must have mass-dimension
3
2
[gi ] = M 4−di , di = fi + bi + n∂i .
(1.344)
It is a fundamentally important fact in quantum field theory that if there is
even one coupling constant for which
[gi ] < 0
(1.345)
then the theory possesses infinitely many different types of divergences and hence
cannot be rendered finite through a finite number of renormalisations. This
defines the term nonrenormalisable.
In the past nonrenormalisable theories were rejected as having no predictive
power: if one has to remove infinitely many infinite constants before one can
define a result then that result cannot be meaningful. However, the modern view
is different. Chiral perturbation theory is a nonrenormalisable “effective theory.”
However, at any finite order in perturbation theory there is only a finite number
56
of undetermined constants: 2 at leading order; 6 more, making 8 in total, when
one-loop effects are considered; and more than 140 new terms when two-loop
effects are admitted. Nevertheless, as long as there is a domain in some physical
external-parameter space on which the one-loop corrected Lagrangian density
can be assumed to be a good approximation, and there is more data that can
be described than there are undetermined constants, then the “effective theory”
can be useful as a tool for correlating observables and elucidating the symmetries
that underly the general pattern of hadronic behaviour.
1.10.3
Regularized Quark Self Energy
I can now return and re-express Eq. (1.315):
(2)
2
−i Σ (p) = −(g0 ν ) C2 (R)
(
Z
dD k 1
1
1
2
4
2
2
+
2
(2π) ν (k + p) − m0 + iη k + iη +
γ µ (6k + /p + m0 ) γµ + (1 − λ0 ) (p
/ − m0 )
2
+ (1 − λ0 ) (p −
m20 )
)
k6
.
k 2 + iη +
(1.346)
It can be separated into a sum of two terms, each proportional to a different
Dirac structure:
Σ(p
/) = ΣV (p2 ) /p + ΣS (p2 ) 1D ,
(1.347)
that can be obtained via trace projection:
ΣV (p2 ) =
1
1 1
2
tr
[p
/
Σ(p
/
)]
,
Σ
(p
)
=
trD [Σ(p
/)] .
D
S
f (D) p2
f (D)
(1.348)
These are the vector and scalar parts of the dressed-quark self-energy, and they
are easily found to be given by
2
dD k 1
(2π)4 ν 2
(
1
[(k + p)2 − m20 + iη + ] [k 2 + iη + ]
#)
"
pµ k µ
2
µ
2
2
2
.
× (2 − D)(p + pµ k ) + (1 − λ0 )p + (1 − λ0 )(p − m0 ) 2
k + iη +
(1.349)
Z
D
d k 1
m0 (D − 1 + λ0 )
ΣS (p2 ) = −i (g0 ν )2 C2 (R)
. (1.350)
(2π)4 ν 2 [(k + p)2 − m20 + iη + ] [k 2 + iη + ]
2
2
p ΣV (p ) = −i (g0 ν ) C2 (R)
Z
(NB. As promised, the factor of f (D) has cancelled.)
57
These equations involve integrals of the general form
dD k
(2π)4
Z
dD k
J µ (α, β; p2 , m2 ) =
(2π)4
2
Z
2
I(α, β; p , m ) =
1
1
,(1.351)
2
2
2
ν [(k + p) − m + iη + ]α [k 2 + iη + ]β
1
kµ
,(1.352)
ν 2 [(k + p)2 − m2 + iη + ]α [k 2 + iη + ]β
the first of which we have already encountered in Sec. 1.10.2. The general results
are (D = 4 + 2)
i
I(α, β; p2 , m2 ) =
(4π)2
p2
× −
4πν 2
!
1
p2
m2
1− 2
p
!α+β−2
!2+−α−β
i
J (α, β; p , m ) = p
(4π)2
µ
2
2
Γ(α + β − 2 − ) Γ(2 + − β)
Γ(α) Γ(2 + )
1
p2
µ
!
m2
p2
1
−
× −
4πν 2
p2
=: pµ J(α, β; p2 , m2 ) ,
2 F1 (α
!α+β−2
!2+−α−β
+ β − 2 − , 2 + − β, 2 + ;
1
),
1 − (m2 /p2 )
(1.353)
Γ(α + β − 2 − ) Γ(3 + − β)
Γ(α) Γ(3 + )
2 F1 (α
+ β − 2 − , 3 + − β, 3 + ;
1
)
1 − (m2 /p2 )
(1.354)
where 2 F1 (a, b, c; z) is the hypergeometric function.
Returning again to Eqs. (1.349), (1.350) it is plain that
ΣV (p2 ) = −i (g0 ν )2 C2 (R)
h
nh
2(1 + ) J(1, 1) − (1 + 2) I(1, 1) − (p2 − m20 ) J(1, 2)
+λ0 (p2 − m20 ) J(1, 2) − I(1, 1)
io
,
i
(1.355)
ΣS (p2 ) = −i (g0 ν )2 C2 (R) m0 (3 + λ0 + 2) I(1, 1; p2, m2 ) ,
(1.356)
where I have omitted the (p2 , m20 ) component in the arguments of I, J.
The integrals explicitly required are
i
p2
1
I(1, 1) =
− + ln 4π − γE − ln − 2 + 2
(4π)2
ν
!
!
!)
2
2
2
m
m
m2
m
,
− 2 ln − 2 − 1 − 2 ln 1 − 2
p
p
p
p
(
!
i 1
1
p2
J(1, 1) =
− + ln 4π − γE − ln − 2 + 2
(4π)2 2
ν
(
2
−
m
p2
!
2
2−
m
p2
!
2
ln −
m
p2
!

− 1 − 2
58
2
"
(1.357)
#
2 2
m
m
+ 2
2
p
p

 ln
2
1−
m
p2
!
−

2
m
,
p2 
(1.358)
(
2
2
i
1
m
m
− 2 ln − 2
2
2
(4π) p
p
p
J(1, 2) =
!
2
+
2
m
m
ln 1 − 2
2
p
p
!
+1
)
.
(1.359)
Using these expressions it is straightforward to show that the λ0 -independent
term in Eq. (1.355) is identically zero in D = 4 dimensions; i.e., for − > 0− , and
hence
(gν )2
C2 (R)
ΣV (p2 ) = λ0
(4π)2
m2
−1 − 20 + 1 −
p
(
1
m2
− ln 4π + γE + ln 20
ν
!
!)
4
2
m0
p
ln 1 − 2
.
4
p
m
(1.360)
It is obvious that in Landau gauge: λ0 = 0, in four spacetime dimensions:
ΣV (p2 ) ≡ 0, at this order. The scalar piece of the quark’s self-energy is also
easily found:
(gν )2
ΣS (p ) = m0
C2 (R)
(4π)2
2
m20
1
−(3 + λ0 )
− ln 4π + γE + ln 2
ν
!
!)
2
2
p
m
.
(1.361)
+2(2 + λ0 ) − (3 + λ0 ) 1 − 20 ln 1 − 2
p
m0
(
"
#
Note that in Yennie gauge: λ0 = −3, in four dimensions, the scalar piece of the
self-energy is momentum-independent, at this order. We now have the complete
regularised dressed-quark self-energy at one-loop order in perturbation theory
and its structure is precisely as I described in Sec. 1.10.2. Renormalisation must
follow.
One final observation: the scalar piece of the self-energy is proportional to the
bare current-quark mass, m0 . That is true at every order in perturbation theory.
Clearly then
lim ΣS (p2 , m20 ) = 0
(1.362)
m0 →0
and hence dynamical chiral symmetry breaking is impossible in perturbation
theory.
1.10.4
Exercises
1. Verify Eq. (1.321).
2. Verify Eq. (1.328).
3. Verify Eq. (1.334).
4. Verify Eqs. (1.349) and (1.350).
5. Verify Eqs. (1.360) and (1.361).
59
1.11
Renormalized Quark Self Energy
Hitherto I have illustrated the manner in which dimensional regularisation is employed to give sense to the divergent integrals that appear in the perturbative
calculation of matrix elements in quantum field theory. It is now necessary to
renormalise the theory; i.e., to provide a well-defined prescription for the elimination of all those parts in the calculated matric element that express the divergences and thereby obtain finite results for Green functions in the limit → 0−
(or with the removal of whatever other parameter has been used to regularise the
divergences).
1.11.1
Renormalized Lagrangian
The bare QCD Lagrangian density is
1
2
1
−
2
1
−
4
L(x) = − ∂µ Bνa (x)[∂ µ B νa (x) − ∂ ν B µa (x)] −
1
∂µ Bνa (x)∂µ Bνa (x)
2λ
g fabc [∂ µ B νa (x) − ∂ ν B µa (x)]Bµb (x)Bνc (x)
g 2 fabc fade Bµb (x)Bνc (x)B µd (x)B νe (x)
−∂µ φ̄a (x)∂ µ φa (x) + g fabc ∂µ φ̄a (x) φb (x)B µc (x)
+q̄ f (x)i∂/q f (x) − mf q̄ f (x)q f (x) +
1
2
g q̄ f (x)λaB
/ a (x)q f (x) ,(1.363)
where: Bµa (x) are the gluon fields, with the colour label a = 1, . . . , 8; φ̄a (x), φa (x)
are the (Grassmannian) ghost fields; q̄ f (x), q f (x) are the (Grassmannian) quark
fields, with the flavour label f = u, d, s, c, b, t; and g, mf , λ are, respectively, the
coupling, mass and gauge fixing parameter. (NB. The QED Lagrangian density is
immediately obtained by setting fabc ≡ 0. It is clearly the non-Abelian nature of
the gauge group, SU (Nc ), that generates the gluon self-couplings, the triple-gluon
and four-gluon vertices, and the ghost-gluon interaction.)
The elimination of the divergent parts in the expression for a Green function
can be achieved by adding “counterterms” to the bare QCD Lagrangian density,
one for each different type of divergence in the theory; i.e., one considers the
renormalised Lagrangian density
LR (x) := L(x) + Lc (x) ,
(1.364)
with
Lc (x) = C3Y M
1
2
+C1Y M
1
4
∂µ Bνa (x)[∂ µ B νa (x) − ∂ ν B µa (x)] + C6
1
2
1
∂µ Bνa (x)∂µ Bνa (x)
2λ
g fabc [∂ µ B νa (x) − ∂ ν B µa (x)]Bµb (x)Bνc (x)
+C5 g 2 fabc fade Bµb (x)Bνc (x)B µd (x)B νe (x)
60
+C̃3 ∂µ φ̄a (x)∂ µ φa (x) − C̃1 g fabc ∂µ φ̄a (x) φb (x)B µc (x)
−C2F q̄ f (x)i∂/q f (x) + C4 mf q̄ f (x)q f (x) − C1F
1
2
g q̄ f (x)λaB
/ a (x)q f (x) .
(1.365)
To prove the renormalisability of QCD one must establish that the coefficients, Ci ,
each understood as a power series in g 2 , are the only additional terms necessary
to remove all the ultraviolet divergences in the theory at every order in the
perturbative expansion.
In the example of Sec. 1.10.2 I illustrated that the divergent terms in the regularised self-energy are proportional to g 2 . This is a general property and hence
all of the Ci begin with a g 2 term. The Ci -dependent terms can be treated just as
the terms in the original Lagrangian density and yield corrections to the expressions we have already derived that begin with an order-g 2 term. Returning to the
example of the dressed-quark self-energy this means that we have an additional
contribution:
∆Σ(2) (p
/) = C2F /p − C4 m ,
(1.366)
and one can choose C2F , C4 such that the total self-energy is finite.
The renormalisation constants are introduced as follows:
Zi := 1 − Ci
(1.367)
so that Eq. (1.365) becomes
Z3Y M
Z
∂µ Bνa (x)[∂ µ B νa (x) − ∂ ν B µa (x)] − 6 ∂µ Bνa (x)∂µ Bνa (x)
2
2λ
Z1Y M
µ νa
ν µa
b
g fabc [∂ B (x) − ∂ B (x)]Bµ (x)Bνc (x)
−
2
Z
− 5 g 2 fabc fade Bµb (x)Bνc (x)B µd (x)B νe (x)
4
LR (x) = −
−Z̃3 ∂µ φ̄a (x)∂ µ φa (x) + Z̃1 g fabc ∂µ φ̄a (x) φb (x)B µc (x)
+Z2F q̄ f (x)i∂/q f (x) − Z4 mf q̄ f (x)q f (x) +
Z1Y M
2
g q̄ f (x)λaB
/ a (x)q f (x) ,
(1.368)
NB. I have implicitly assumed that the renormalisation counterterms, and hence
the renormalisation constants, are flavour independent. It is always possible to
choose prescriptions such that this is so.
I will now introduce the bare fields, coupling constants, masses and gauge
61
fixing parameter:
1/2
1/2
B0µa (x) := Z3Y M B µa (x) ,
q0f (x) := Z2F q f (x) ,
1/2
φ̄a0 (x) := Z̃3 φ̄a (x) ,
1/2
φa0 (x) := Z̃3 φa (x) ,
−3/2
g0Y M := Z1Y M Z3Y M g ,
−1/2
−1/2
g̃0 := Z̃1 Z̃3−1 Z3Y M ,
(1.369)
1/2
−1
g0F := Z1F Z3Y M Z2F
g,
−1
g05 := Z5 Z3Y
M g
−1
mf0 := Z4 Z2F
mf ,
λ0 := Z6−1 Z3Y M λ .
The fields and couplings on the r.h.s. of these definitions are called renormalised,
and the couplings are finite and the fields produce Green functions that are finite
even in D = 4 dimensions. ( NB. All these quantities are defined in D = 4 + 2dimensional space. Hence one has for the Lagrangian density: [L(x)] = M D , and
the field and coupling dimensions are
[q(x)] = [q̄(x)] = M 3/2+ ,
[B µ (x)] = M 1+ ,
[φ(x)] = [φ̄(x)] = M 1+ ,
[g] = M − ,
[λ] = M 0 ,
[m] = M 1 .)
(1.370)
The renormalised Lagrangian density can be rewritten in terms of the bare
quantities:
1
2
1
−
2
1
−
4
a
LR (x) = − ∂µ B0ν
(x)[∂ µ B0νa (x) − ∂ ν B0µa (x)] −
1
a
a
∂µ B0ν
(x)∂µ B0ν
(x)
2λ0
a
a
g0Y M fabc [∂ µ B0νa (x) − ∂ ν B0µa (x)]B0µ
(x)B0ν
(x)
2
b
c
g05
fabc fade B0µ
(x)B0ν
(x)B0µd (x)B0νe (x)
−∂µ φ̄a0 (x)∂ µ φa0 (x) + g̃0 fabc ∂µ φ̄a0 (x) φb0 (x)B0µc (x)
+q̄0f (x)i∂/q0f (x) − mf0 q̄0f (x)q0f (x) +
1
2
g0F q̄0f (x)λaB
/ a0 (x)q0f (x) .
(1.371)
It is apparent that now the couplings are different and hence LR (x) is not invariant
under local gauge transformations (more properly BRST transformations) unless
g0Y M = g̃0 = g0F = g05 = g0 .
(1.372)
Therefore, if the renormalisation procedure is to preserve the character of the
gauge theory, the renormalisation constants cannot be completely arbitrary but
must satisfy the following “Slavnov-Taylor” identities:
g0Y M = g̃0
g0Y M = g0F
g0Y M = g05
Z̃3
Z3Y M
=
,
Z1Y M
Z̃1
Z2F
Z3Y M
=
,
⇒
Z1Y M
Z1F
Z2
⇒ Z5 = 1Y M .
Z3Y M
⇒
62
(1.373)
In QED the second of these equations becomes the Ward-Takahashi identity:
Z1F = Z2F .
1.11.2
Renormalization Schemes
At this point we can immediately write an expression for the renormalised dressedquark self-energy:
(2)
ΣR (p
/) = ΣV (p2 ) + C2F /p + ΣS (p2 ) − C4 m .
(1.374)
The subtraction constants are not yet determined and there are many ways one
may choose them in order to eliminate the divergent parts of bare Green functions.
Minimal Subtraction
In the minimal subtraction (MS) scheme one defines a dimensionless coupling
α :=
(gν )2
4π
(1.375)
and considers each counterterm as a power series in α with the form
Ci =
j
∞ X
X
(2j)
Ci,k
j=1 k=1
1
k
α
π
j
,
(1.376)
where the coefficients in the expansion may, at most, depend on the gauge parameter, λ.
Using Eqs. (1.360), (1.361) and (1.374) we have
(2)
ΣR (p
/)
m2
α 1
1
λ C2 (R)
− ln 4π + γE + ln 2
(1.377)
= /p
π 4
ν
!
!#
)
m2
m4
p2
−1 − 2 + 1 − 4 ln 1 − 2
+ C2F
(1.378)
p
p
m
(
"
#
α1
1
m2
+m
C2 (R) −(3 + λ)
− ln 4π + γE + ln 2
(1.379)
π 4
ν
!
!
)
p2
m2
+2(2 + λ) − (3 + λ) 1 − 2 ln 1 − 2 − C4 . (1.380)
p
m
(
"
Now one chooses C2F , C4 such that they cancel the 1/ terms in this equation
and therefore, at one-loop level,
α 1
1
λ C2 (R) ,
π 4
α
1
1
= 1 + (3 + λ) C2 (R) ,
4
π
Z2F = 1 − C2F = 1 +
Z4 = 1 − C 4
63
(1.381)
(1.382)
and hence Eq. (1.380) becomes
(2)
ΣR (p
/)
m2
α 1
λ C2 (R) − ln 4π + γE + ln 2
= /p
π 4
ν
!
!#
)
2
4
2
m
m
p
−1 − 2 + 1 − 4 ln 1 − 2
+ C2F
p
p
m
(
!
m2
α1
C2 (R) −(3 + λ) − ln 4π + γE + ln 2
+m
π 4
ν
!
!
)
2
2
m
p
+2(2 + λ) − (3 + λ) 1 − 2 ln 1 − 2 − C4
p
m
"
(
!
(1.383)
(1.384)
(1.385)
, (1.386)
which is the desired, finite result for the dressed-quark self-energy.
It is not common to work explicitly with the counterterms. More often one
uses Eq. (1.371) and the definition of the connected 2-point quark Green function
(an obvious analogue of Eq. (1.282)):
iSRf (x, y; m, λ, α) = −
δ 2 ZR [Jµa , ξ, ξ̄]
−1
= h0|q f (x) q̄ f (y)|0i = Z2F
h0|q0f (x) q̄0f (y)|0i
f
f
¯
δξ (y)δ ξ (x)
(1.387)
to write
n
o
−1
SR (p
/; m, λ, α) = lim Z2F
S0 (p
/; m0 , λ0 , α0 ; ) ,
→0
(1.388)
where in the r.h.s. m0 , λ0 , α0 have to be substituted by their expressions in terms
of the renormalised quantities and the limit taken order by order in α.
To illustrate this using our concrete example, Eq. (1.388) yields
n
o
n
1 − ΣV R (p2 ; m, λ, α) /p − m 1 + ΣSR (p2 ; m, λ, α)/m
=
lim− Z2F
→0
nh
i
h
o
1 − ΣV (p2 ; m0 , λ0 , α0 ) /p − m0 1 + ΣS (p2 ; m0 , λ0 , α0 )/m0
io
(1.389)
and taking into account that m = Z4−1 Z2F m0 then
1 − ΣV R (p2 ; m, λ, α) =
1 + ΣSR (p2 ; m, λ, α)/m =
lim− Z2F
→0
n
n
o
1 − ΣV (p2 ; m0 , λ0 , α0 ) ,
(1.390)
o
lim− Z4 1 + ΣS (p2 ; m0 , λ0 , α0 )/m0 .(1.391)
→
Now the renormalisation constants, Z2F , Z4 , are chosen so as to exactly cancel
the 1/ poles in the r.h.s. of Eqs. (1.390), (1.391). Using Eqs. (1.360), (1.361),
Eqs. (1.381), (1.382) are immediately reproduced.
Modified Minimal Subtraction
The modified minimal substraction scheme MS is also often used in QCD. It takes
advantage of the fact that the 1/ pole obtained using dimensional regularisation
64
always appears in the combination
1
− ln 4π + γE
(1.392)
so that the renormalisation constants are defined so as to eliminate this combination, in its entirety, from the Green functions. At one-loop order the renormalisation constants in the MS scheme are trivially related to those in the MS
scheme. At higher orders there are different ways of defining the scheme and the
relation between the renormalisation constants is not so simple.
Momentum Subtraction
In the momentum subtraction scheme (µ-scheme) a given renormalised Green
function, GR , is obtained from its regularised counterpart, G, by subtracting
from G its value at some arbitrarily chosen momentum scale. In QCD that scale
is always chosen to be a Euclidean momentum: p2 = −µ2 . Returning to our
example of the dressed-quark self-energy, in this scheme
ΣAR (p2 ; µ2 ) := ΣA (p2 ; ) − ΣA (p2 ; ) ; A = V, S ,
(1.393)
and so
!
(
α(µ)
1
1
1
=
λ(µ) C2 (R) −m2 (µ) 2 + 2
4
π
p
µ
!)
!
!
!
4
2
m (µ)
p
m4 (µ)
µ2
+ 1−
,
ln 1 −
− 1−
ln 1 + 2
p4
m(µ)2
µ4
m (µ)
(1.394)
α(µ) 1
(2)
C2 (R) {−[3 + λ(µ)]
ΣSR (p2 ; µ2 ) = m(µ)
4
π
"
!
!
!
!#)
m2 (µ)
p2
m2 (µ)
µ2
× 1−
ln 1 − 2
− 1+
ln 1 + 2
p2
m (µ)
µ2
m (µ)
(1.395)
(2)
ΣV R (p2 ; µ2 )
where the renormalised quantities depend on the point at which the renormalisation has been conducted. Clearly, from Eqs. (1.390), (1.391), the renormalisation
constants in this scheme are
(2)
Z2F
(2)
Z4
(2)
= 1 + ΣV (p2 = −µ2 ; )
= 1−
(2)
ΣS (p2
2
= −µ ; )/m(µ) .
(1.396)
(1.397)
It is apparent that in this scheme there is at least one point, the renormalisation mass-scale, µ, at which there are no higher order corrections to any of the
Green functions: the corrections are all absorbed into the redefinitions of the coupling constant, masses and gauge parameter. This is valuable if the coefficients of
65
the higher order corrections, calculated with the parameters defined through the
momentum space subtraction, are small so that the procedure converges rapidly
on a large momentum domain. In this sense the µ-scheme is easier to understand
and more intuitive than the MS or MS schemes. Another advantage is the manifest applicability of the “decoupling theorem,” Refs. [11], which states that quark
flavours whose masses are larger than the scale chosen for µ are irrelevant.
This last feature, however, also emphasises that the renormalisation constants
are flavour dependent and that can be a nuisance. Nevertheless, the µ-scheme is
extremely useful in nonperturbative analyses of DSEs, especially since the flavour
dependence of the renormalisation constants is minimal for light quarks when the
Euclidean substraction point, µ, is chosen to be very large; i.e., much larger than
their current-masses.
1.11.3
Renormalized Gap Equation
Equation (1.307) is the unrenormalised QCD gap equation, which can be rewritten as
−iS0−1 (p)
= −i(p
/ − m0 ) +
g02
Z
d4 `
i
D0µν (p − `) λa γµ S0 (`) Γa0ν (`, p) . (1.398)
4
2
(2π)
The renormalised equation can be derived directly from the generating functional
defined using the renormalised Lagrangian density, Eq. (1.368), simply by repeating the steps described in Sec. 1.9.2. Alternatively, one can use Eqs. (1.371) to
derive an array of relations similar to Eq. (1.387):
µν
−1 a
ΓRν (k, p) ,
D0µν (k) = Z3Y M DR
(k) , Γa0ν (k, p) = Z1F
(1.399)
and others that I will not use here. (NB. It is a general feature that propagators
are multiplied by the renormalisation constant and proper vertices by the inverse
of renormalisation constants.) Now one can replace the unrenormalised couplings,
masses and Green functions by their renormalised forms:
−1
−iZ2−1 SR−1 (p) = −ip
/ + iZ2F
Z 4 mR
Z
4
d`
i
µν
2
−1
−2 2
−1 a
+Z1F
Z3Y
Z3Y M DR
(p − `) λa γµ Z2F SR (`) Z1F
ΓRν (`, p) ,
M Z2F gR
4
2
(2π)
(1.400)
which simplifies to
−iΣR (p) = i(Z2F − 1) /p − i(Z4 − 1) mR
Z
d4 `
i
−Z1F gR2
D µν (p − `) λa γµ SR (`) ΓaRν (`, p)
2
(2π)4 R
=: i(Z2F − 1) /p − i(Z4 − 1) mR − iΣ0 (p) ,
(1.401)
66
where Σ0 (p) is the regularised self-energy.
In the simplest application of the µ-scheme one would choose a large Euclidean
mass-scale, µ2 , and define the renormalisation constants such that
ΣR (/
p + µ = 0) = 0
(1.402)
which entails
Z2F = 1 + Σ0V (p
/ + µ = 0) , Z4 = 1 − Σ0S (p
/ + µ = 0)/mR (µ) ,
(1.403)
where I have used Σ0 (p) = Σ0 (p) /p + Σ0S (p). (cf. Eqs. (1.396), (1.397).) This is
simple to implement, even nonperturbatively, and is always appropriate in QCD
because confinement ensures that dressed-quarks do not have a mass-shell.
On-shell Renormalization
If one is treating fermions that do have a mass-shell; e.g., electrons, then an
on-shell renormalisation scheme may be more appropriate. One fixes the renormalisation constants such that
SR−1 (p
/)
p
/=mR
= /p − mR ,
(1.404)
which is interpreted as a constraint on the pole position and the residue at the
pole; i.e., since
S
−1
(p
/) = [/
p − mR − ΣR (p
/)]|p/=mR
"
d
− [/
p − mR ]
ΣR (p
/)
d/
p
then Eq. (1.404) entails
ΣR (p
/)|p/=mR
#
+...
(1.405)
p
/=mR
d
= 0,
ΣR (p
/)
= 0.
d/
p
p
/=mR
(1.406)
The second of these equations requires
Z2F
d
d 0 2 (1.407)
= 1 + Σ0V (m2R ) + 2 m2R 2 Σ0V (p2 ) 2 2 + 2 mR
Σ (p )
2 2
p =mR
dp
dp2 S
p =m
R
and the first:
Z4 = Z2F − Σ0V (m2R ) − Σ0S (m2R ) .
67
(1.408)
1.11.4
Exercises
1. Using Eqs. (1.360), (1.361) derive Eqs. (1.381), (1.382).
2. Verify Eqs. (1.396), (1.397).
3. Verify Eq. (1.401).
4. Verify Eq. (1.403).
5. Beginning with Eq. (1.404), derive Eqs. (1.407), (1.408). NB. /p/p = p2 and
d
d
hence dp
f (p2 ) = dp
f (p
//p) = 2 /p dpd2 f (p2 ).
/
/
1.12
Dynamical Chiral Symmetry Breaking
This phenomenon profoundly affects the character of the hadron spectrum. However, it is intrinsically nonperturbative and therefore its understanding is best
sought via a Euclidean formulation of quantum field theory, which I will now
summarise.
1.12.1
Euclidean Metric
I will formally describe the construction of the Euclidean counterpart to a given
Minkowski space field theory using the free fermion as an example. The Lagrangian density for free Dirac fields is given in Eq. (1.186):
Lψ0 (x)
=
Z
∞
−∞
dt
Z
(1.409)
dtE f (−itE )
(1.410)
d3 x ψ̄(x) i∂/ − m + iη + ψ(x) .
Recall now the observations at the end of Sec. 1.5 regarding the role of the iη +
in this equation: it was introduced to provide a damping factor in the generating
functional. The alternative proposed was to change variables and introduce a
Euclidean time: t → −itE . As usual
Z
∞
−∞
dt f (t) =
Z
∞
−∞
d(−itE ) f (−itE ) = −i
Z
∞
−∞
if f (t) vanishes on the curve at infinity in the second and fourth quadrants of
the complex-t plane and is analytic therein. To complete this Wick rotation,
however, we must also determine its affect on i∂/, since that is part of the “f (t);”
i.e., the integrand:
i∂/ = iγ 0
∂
∂
+ iγ i i
∂t
∂x
M →E
→
=
=:
68
∂
∂
+ iγ i i
E
∂(−it )
∂x
∂
∂
−γ 0 E − (−iγ i ) i
∂t
∂x
∂
−γµE E ,
∂xµ
iγ 0
(1.411)
(1.412)
where (xE
x, x4 := −it) and the Euclidean Dirac matrices are
µ ) = (~
γ4E = γ 0 γiE = −iγ i .
(1.413)
These matrices are Hermitian and satisfy the algebra
{γµE , γνE } = 2δµν ; µ, ν = 1, . . . , 4,
(1.414)
where δµν is the four-dimensional Kroncker delta. Henceforth I will adopt the
notation
a E · bE =
4
X
E
aE
µ bν .
(1.415)
µ=1
Now, assuming that the integrand is analytic where necessary, one arrives
formally at
Z
∞
−∞
= −i
Z
dt
∞
Z
−∞
d3 x ψ̄(x) i∂/ − m + iη + ψ(x)
dtE
Z
d3 x ψ̄(xE ) −γ E · ∂ E − m + iη + ψ(xE ) .
(1.416)
Intepreted naively, however, the action on the r.h.s. poses problems: it is not real
if one inteprets ψ̄(xE ) = ψ † (xE ) γ4 .
Let us study its involution
AE =
Z
d4 xE ψ̄(xE ) (−γ E · ∂ E − m + iη + ) ψ(xE )
(1.417)
where ψ̄r (xE ) and ψs (xE ) are intepreted as members of a Grassmann algebra with
involution, as described in Sec. 1.7.1:
AE =
Z
=
Z
E ] − m δ + iη + δ ) ψ (xE )
d4 xE ψ̄r (xE ) (−[γ E · ∂→
rs
rs
rs
s
4 E
E
d x ψ̄s (x )
−[γ E
·
E]
∂←
rs
∗
+
!
− m δrs − iη δrs ψr (xE ) . (1.418)
We know that the mass is real: m∗ = m, and that as an operator the gradient is
anti-Hermitian; i.e.,
E = −∂ E
∂←
→
(1.419)
[γµE ]rs = [(γµE )T ]sr .
(1.420)
(γµE )T := −CE γµE CE† ,
(1.421)
but that leaves us with
If we define
69
where CE = γ2 γ4 is the Euclidean charge conjugation matrix then
AE = A E
(1.422)
when ψ̄ and ψ are members of a Grassmann algebra with involution and we take
the limit η + = 0, which we will soon see is permissable.
Return now to the generating functional of Eq. (1.187):
¯ ξ] =
W [ξ,
Z
Z
[D ψ̄(x)][Dψ(x)] exp i
¯
d4 x ψ̄(x) i∂/ − m + iη + ψ(x) + ψ̄(x)ξ(x) + ξ(x)ψ(x)
h
Z
h
d4 xE ψ̄(xE ) γ · ∂ + m − iη + ψ(xE )
i
(1.424)
so that the Euclidean generating functional
¯ ξ] :=
W E [ξ,
Z
× exp −
[D ψ̄(xE )][Dψ(xE )]
Z
¯ E )ψ(xE )
d x ψ̄(x ) (γ · ∂ + m) ψ(x ) + ψ̄(x )ξ(x ) + ξ(x
4
h
E
E
E
E
i
(1.425)
involves a positive-definite measure because the free-fermion action is real, as I
have just shown. Hence the “+iη + ” convergence factor is unnecessary here.
Euclidean Formulation as Definitive
Working in Euclidean space is more than simply a pragmatic artifice: it is possible
to view the Euclidean formulation of a field theory as definitive (see, for example,
Refs. [12, 13, 14, 15]). In addition, the discrete lattice formulation in Euclidean
space has allowed some progress to be made in attempting to answer existence
questions for interacting gauge field theories [15].
The moments of the Euclidean measure defined by an interacting quantum
field theory are the Schwinger functions:
S n (xE,1 , . . . , xE,n )
(1.426)
which can be obtained as usual via functional differentiation of the analogue of
Eq. (1.425) and subsequently setting the sources to zero. The Schwinger functions
are sometimes called Euclidean space Green functions.
70
.
(1.423)
A Wick rotation of the action transforms the source-independent part of the
integrand, called the measure, into
exp −
i
Given a measure and given that it satisfies certain conditions [i.e., the Wightman and Haag-Kastler axioms], then it can be shown that the Wightman functions, W n (x1 , . . . , xn ), can be obtained from the Schwinger functions by analytic
continuation in each of the time coordinates:
W n (x1 , . . . , xn ) = lim
S n ([~x1 , x14 + ix01 ], . . . , [~xn , xn4 + ix0n ])
i
x4 →0
(1.427)
with x01 < x02 < . . . < x0n . These Wightman functions are simply the vacuum expectation values of products of field operators from which the Green functions
[i.e., the Minkowski space propagators] are obtained through the inclusion of step[θ-] functions in order to obtain the appropriate time ordering. (This is described
in some detail in Refs. [13, 14, 15].) Thus the Schwinger functions contain all of
the information necessary to calculate physical observables.
This notion is used directly in obtaining masses and charge radii in lattice
simulations of QCD, and it can also be employed in the DSE approach since
the Euclidean space DSEs can all be derived from the appropriate Euclidean
generating functional using the methods of Sec. 1.9 and the solutions of these
equations are the Schwinger functions.
All this provides a good reason to employ a Euclidean formulation. Another is
a desire to maintain contact with perturbation theory where the renormalisation
group equations for QCD and their solutions are best understood [16].
Collected Formulae for Minkowski ↔ Euclidean Transcription
To make clear my conventions (henceforth I will omit the supescript E used to
denote Euclidean four-vectors): for 4-vectors a, b:
a · b := aµ bν δµν :=
4
X
a i bi ,
(1.428)
i=1
so that a spacelike vector, Qµ , has Q2 > 0; the Dirac matrices are Hermitian and
defined by the algebra
{γµ , γν } = 2 δµν ;
(1.429)
and I use
γ5 := − γ1 γ2 γ3 γ4
(1.430)
tr [γ5 γµ γν γρ γσ ] = −4 εµνρσ , ε1234 = 1 .
(1.431)
so that
The Dirac-like representation of these matrices is:
~γ =
0 −i~τ
i~τ
0
!
, γ4 =
71
τ0 0
0 −τ 0
!
,
(1.432)
where the 2 × 2 Pauli matrices are:
1 0
0 1
0
τ =
!
0 1
1 0
1
, τ =
!
0 −i
i 0
2
, τ =
!
3
, τ =
1 0
0 −1
!
. (1.433)
Using these conventions the [unrenormalised] Euclidean QCD action is
S[B, q, q̄] =
Z

1
1 a a
Fµν + ∂ · B a ∂ · B a +
d4 x  Fµν
4
2λ
Nf
X
f =1
1
2
q̄f γ · ∂ + mf + ig λa γ · B a qf  ,
(1.434)
a
where Fµν
= ∂µ Bνa − ∂ν Bµa − gf abc Bµb Bνc . The generating functional follows:
W [J, ξ, ξ̄] =
Z
dµ(q̄, q, B, ω̄, ω) exp
Z
i
h
d4 x q̄ ξ + ξ¯ q + Jµa Aaµ ,
(1.435)
with sources: η̄, η, J, and a functional integral measure
dµ(q̄, q, B, ω̄, ω) :=
(1.436)
Y
Y
YY
a
g
a
a
Dq̄φ (x)Dqφ (x) Dω̄ (x)Dω (x) DBµ (x) exp(−S[B, q, q̄] − S [B, ω, ω̄]) ,
x
µ
a
φ
where φ represents both the flavour and colour index of the quark field, and ω̄
and ω are the scalar, Grassmann [ghost] fields. The normalisation
W [η̄ = 0, η = 0, J = 0] = 1
(1.437)
is implicit in the measure. As we saw in Sec. 1.8.1, the ghosts only couple directly
to the gauge field:
Sg [B, ω, ω̄] =
Z
h
i
d4 x −∂µ ω̄ a ∂µ ω a − gf abc ∂µ ω̄ a ω b Bµc ,
(1.438)
and restore unitarity in the subspace of transverse [physical] gauge fields. I note
that, practically, the normalisation means that ghost fields are unnecessary in the
calculation of gauge invariant observables using lattice-regularised QCD because
the gauge-orbit volume-divergence in the generating functional, associated with
the uncountable infinity of gauge-equivalent gluon field configurations in the continuum, is rendered finite by the simple expedient of only summing over a finite
number of configurations.
It is possible to derive every equation introduced above, assuming certain
analytic properties of the integrands. However, the derivations can be sidestepped
using the following transcription rules:
72

Configuration Space
1.
Z
M
d4 xM → −i
Momentum Space
Z
E
d 4 xE
1.
Z
M
d4 k M → i
Z
E
2. ∂
/ → iγ E · ∂ E
2. k
/ → −iγ E · k E
3. A
/ → −iγ E · AE
3. A
/ → −iγ E · AE
4. Aµ B µ → −AE · B E
4. kµ q µ → −k E · q E
5. xµ ∂µ → xE · ∂ E
5. kµ xµ → −k E · xE
d4 k E
These rules are valid in perturbation theory; i.e., the correct Minkowski space
integral for a given diagram will be obtained by applying these rules to the
Euclidean integral: they take account of the change of variables and rotation
of the contour. However, for the diagrams that represent DSEs, which involve
dressed n-point functions whose analytic structure is not known as priori, the
Minkowski space equation obtained using this prescription will have the right
appearance but it’s solutions may bear no relation to the analytic continuation
of the solution of the Euclidean equation. The differences will be nonperturbative
in origin.
1.12.2
Chiral Symmetry
Gauge theories with massless fermions have a chiral symmetry. Its effect can be
visualised by considering the helicity: λ ∝ J · p, the projection of the fermion’s
spin onto its direction of motion. λ is a Poincaré invariant spin observable that
takes a value of ±1. The chirality operator can be realised as a 4 × 4-matrix, γ 5 ,
and a chiral transformation is then represented as a rotation of the 4 × 1-matrix
quark spinor field
q(x) → eiγ5 θ q(x) .
(1.439)
A chiral rotation through θ = π/2 has no effect on a λ = +1 quark, qλ= + → qλ= + ,
but changes the sign of a λ = −1 quark field, qλ= − → − qλ= − . In composite
hadrons this is manifest as a flip in their parity: J P = + ↔ J P = − ; i.e., a θ = π/2
chiral rotation is equivalent to a parity transformation. Exact chiral symmetry
therefore entails that degenerate parity multiplets must be present in the spectrum of the theory.
For many reasons, the masses of the u- and d-quarks are expected to be very
small; i.e., mu ∼ md ΛQCD . Therefore chiral symmetry should only be weakly
broken, with the strong interaction spectrum exhibiting nearly degenerate parity
partners. The experimental comparison is presented in Eq. (1.440):
+
N ( 12 , 938)
−
N ( 21 , 1535)
π(0− , 140)
a0 (0+ , 980)
73
ρ(1− , 770)
.
a1 (1+ , 1260)
(1.440)
Clearly the expectation is very badly violated, with the splitting much too large
to be described by the small current-quark masses. What is wrong?
Chiral symmetry can be related to properties of the quark propagator, S(p).
For a free quark (remember, I am now using Euclidean conventions)
m − iγ · p
(1.441)
S0 (p) =
m2 + p 2
and as a matrix
m
−iγ · p
+ e2iγ5 θ 2
(1.442)
S0 (p) → eiγ5 θ S0 (p)eiγ5 θ = 2
2
p +m
p +m2
under a chiral transformation. As anticipated, for m = 0, S0 (p) → S0 (p); i.e., the
symmetry breaking term is proportional to the current-quark mass and it can be
measured by the “quark condensate”
Z
Z
d4 p
m
d4 p
tr
[S(p)]
∝
,
(1.443)
−hq̄qi :=
4
4
2
(2π)
(2π) p + m2
which is the “Cooper-pair” density in QCD. For a free quark the condensate
vanishes if m = 0 but what is the effect of interactions?
As we have seen, interactions dress the quark propagator so that it takes the
form
−iγ · pA(p2 ) + B(p2 )
1
= 2 2 2
,
(1.444)
S(p) :=
iγ · p + Σ(p)
p A (p ) + B 2 (p2 )
where Σ(p) is the self energy, expressed in terms of the scalar functions: A and
B, which are p2 -dependent because the interaction is momentum-dependent. On
the valid domain; i.e., for weak-coupling, they can be calculated in perturbation
theory and at one-loop order, Eq. (1.395) with p2 → −(pE )2 µ2 , m2 (µ),
αS
p2
B(p ) = m 1 −
ln
π
m2
"
2
#!
,
(1.445)
which is ∝ m. This result persists: at every order in perturbation theory every
mass-like correction to S(p) is ∝ m so that m is apparently the only source of
chiral symmetry breaking and hq̄qi ∝ m → 0 as m → 0. The current-quark
masses are the only explicit chiral symmetry breaking terms in QCD.
However, symmetries can be “dynamically” broken. Consider a point-particle
in a rotationally invariant potential V (σ, π) = (σ 2 + π 2 − 1)2 , where (σ, π) are the
particle’s coordinates. In the state depicted in Fig. 1.1, the particle is stationary
at an extremum of the action that is rotationally invariant but unstable. In the
ground state of the system, the particle is stationary at any point (σ, π) in the
trough of the potential, for which σ 2 + π 2 = 1. There are an uncountable infinity
of such vacua, |θi, which are related one to another by rotations in the (σ, π)plane. The vacua are degenerate but not rotationally invariant and hence, in
general, hθ|σ|θi 6= 0. In this case the rotational invariance of the Hamiltonian is
not exhibited in any single ground state: the symmetry is dynamically broken
with interactions being responsible for hθ|σ|θi 6= 0.
74
•
1.5
1
1
0.5
0.5
0
0
-1
-0.5
-0.5
0
0.5
-1
1
Figure 1.1: A rotationally invariant but unstable extremum of the Hamiltonian
obtained with the potential V (σ, π) = (σ 2 + π 2 − 1)2 .
1.12.3
Mass Where There Was None
The analogue in QCD is hq̄qi 6= 0 when m = 0. At any finite order in perturbation
theory that is impossible. However, using the Dyson-Schwinger equation [DSE]
for the quark self energy [the QCD “gap equation”]:
iγ · p A(p2 ) + B(p2 ) = Z2 iγ · p + Z4 m
Z Λ
d4 ` 2
λa
1
+Z1
g
D
(p
−
`)
γ
Γa (`, p) , (1.446)
µν
µ
4
2
(2π)
2 iγ · `A(` ) + B(`2 ) ν
depicted in Fig. 1.2, it is possible to sum infinitely many contributions. That
allows one to expose effects in QCD which are inaccessible in perturbation theR
ory. [NB. In Eq. (1.446), m is the Λ-dependent current-quark bare mass and Λ
represents mnemonically a translationally-invariant regularisation of the integral,
with Λ the regularisation mass-scale. The final stage of any calculation is to remove the regularisation by taking the limit Λ → ∞. The quark-gluon-vertex and
quark wave function renormalisation constants, Z1 (ζ 2 , Λ2 ) and Z2 (ζ 2 , Λ2 ), depend
on the renormalisation point, ζ, and the regularisation mass-scale, as does the
mass renormalisation constant Zm (ζ 2 , Λ2 ) := Z2 (ζ 2 , Λ2 )−1 Z4 (ζ 2 , Λ2 ).]
The quark DSE is a nonlinear integral equation for A and B, and it is the
nonlinearity that makes possible a generation of nonperturbative effects. The
75
Σ
D
=
γ
S
Γ
Figure 1.2: DSE for the dressed-quark self-energy. The kernel of this equation
is constructed from the dressed-gluon propagator (D - spring) and the dressedquark-gluon vertex (Γ - open circle). One of the vertices is bare (labelled by γ)
as required to avoid over-counting.
kernel of the equation is composed of the dressed-gluon propagator:
g 2 Dµν (k) = δµν
kµ kν
− 2
k
!
g2
G(k 2 )
2
,
G(k
)
:=
,
k2
[1 + Π(k 2 )]
(1.447)
where Π(k 2 ) is the vacuum polarisation, which contains all the dynamical information about gluon propagation, and the dressed-quark-gluon vertex: Γaµ (k, p).
The bare (undressed) vertex is
Γaµ (k, p)bare = γµ
λa
.
2
(1.448)
Once Dµν and Γaµ are known, Eq. (1.446) is straightforward to solve by iteration.
One chooses an initial seed for the solution functions: 0A and 0B, and evaluates
the integral on the right-hand-side (r.h.s.). The bare propagator values: 0A = 1
and 0B = m are often adequate. This first iteration yields new functions: 1A and
1B, which are reintroduced on the r.h.s. to yield 2A and 2B, etc. The procedure
is repeated until nA = n+1A and nB = n+1B to the desired accuracy.
It is now easy to illustrate DCSB, and I will use three simple examples.
Nambu–Jona-Lasinio Model
The Nambu–Jona-Lasinio model [17] has been popularised as a model of lowenergy QCD. The commonly used gap equation is obtained from Eq. (1.446) via
the substitution
1
g 2 Dµν (p − `) → δµν 2 θ(Λ2 − `2 )
(1.449)
mG
in combination with Eq. (1.448). The step-function in Eq. (1.449) provides a
momentum-space cutoff. That is necessary to define the model, which is not
76
renormalisable and hence can be regularised but not renormalised. Λ therefore
persists as a model-parameter, an external mass-scale that cannot be eliminated.
The gap equation is
iγ · p A(p2 ) + B(p2 )
= iγ · p + m +
4
3
1
m2G
Z
d4 `
−iγ · `A(`2 ) + B(`2 )
2
2
θ(Λ
−
`
)
γ
γ(1.450)
µ
µ,
(2π)4
`2 A2 (`2 ) + B 2 (`2 )
where I have set the renormalisation constants equal to one in order to complete
the defintion of the model. Multiplying Eq. (1.450) by (−iγ · p) and tracing over
Dirac indices one obtains
2
2
2
p A(p ) = p +
8
3
1
m2G
Z
d4 `
A(`2 )
2
2
θ(Λ − ` ) p · ` 2 2 2
,
(2π)4
` A (` ) + B 2 (`2 )
(1.451)
from which it is immediately apparent that
A(p2 ) ≡ 1 .
(1.452)
This property owes itself to the the fact that the NJL model is defined by a fourfermion contact interaction in configuration space, which entails the momentumindependence of the interaction in momentum space.
Simply tracing over Dirac indices and using Eq. (1.452) one obtains
B(p2 ) = m +
16
3
1 Z d4 `
B(`2 )
2
2
,
θ(Λ
−
`
)
m2G
(2π)4
`2 + B 2 (`2 )
(1.453)
from which it is plain that B(p2 ) = constant = M is the only solution. This, too,
is a result of the momentum-independence of the model’s interaction. Evaluating
the angular integrals, Eq. (1.453) becomes
1 1 Z Λ2
M
1 1
M =m+ 2 2
,
dx x
= m + M 2 2 C(M 2 , Λ2 )(1.454)
2
3π mG 0
x+M
3π mG
h
C(M 2 , Λ2 ) = Λ2 − M 2 ln 1 + Λ2 /M 2
i
(1.455)
Λ defines the model’s mass-scale and I will henceforth set it equal to one so
that all other dimensioned quantities are given in units of this scale, in which
case the gap equation can be written
M =m+M
1 1
C(M 2 , 1) .
3π 2 m2G
(1.456)
Irrespective of the value of mG , this equation always admits a solution M 6= 0
when the current-quark mass m 6= 0.
Consider now the chiral-limit, m = 0, wherein the gap equation is
M =M
1 1
C(M 2 , 1) .
2
2
3π mG
77
(1.457)
This equation admits a solution M ≡ 0, which corresponds to the perturbative
case considered above: when the bare mass of the fermion is zero in the beginning
then no mass is generated via interactions. It follows from Eq. (1.443) that the
condensate is also zero. This situation can be described as that of a theory
without a mass gap: the negative energy Dirac sea is populated all the way up
to E = 0.
Suppose however that M 6= 0 in Eq. (1.457), then the equation becomes
1=
1 1
C(M 2 , 1) .
3π 2 m2G
(1.458)
It is easy to see that C(M 2 , 1) is a monotonically decreasing function of M with
a maximum value at M = 0: C(0, 1) = 1. Consequently Eq. (1.458) has a M 6= 0
solution if, and only if,
1 1
> 1;
(1.459)
3π 2 m2G
i.e., if and only if
Λ2
' (0.2 GeV )2
(1.460)
3π 2
for a typical value of Λ ∼ 1 GeV. Thus, even when the bare mass is zero, the NJL
model admits a dynamically generated mass for the fermion when the coupling
exceeds a given minimum value, which is called the critical coupling: the chiral
symmetry is dynamically broken! (In this presentation the critical coupling is
expressed via a maximum value of the dynamical gluon mass, mG .) At these
strong couplings the theory exhibits a nonperturbatively generated gap: the initially massless fermions and antifermions become massive via interaction with
their own “gluon” field. Now the negative energy Dirac sea is only filled up to
E = −M , with the positive energy states beginning at E = +M ; i.e., the theory
has a dynamically-generated, nonperturbative mass-gap ∆ = 2M . In addition
the quark condensate, which was zero when evaluated perturbatively because
m = 0, is now nonzero.
Importantly, the nature of the solution of Eq. (1.456) also changes qualitatively when mG is allowed to fall below it’s critical value. It is in this way that
dynamical chiral symmetry breaking (DCSB) continues to affect the hadronic
spectrum even when the quarks have a small but nonzero current-mass.
m2G <
Munczek-Nemirovsky Model
The gap equation for a model proposed more recently [18], which is able to
represent a greater variety of the features of QCD while retaining much of the
simplicity of the NJL model, is obtained from Eq. (1.446) by using
G(k 2 )
= (2π)4 G δ 4 (k)
k2
78
(1.461)
in Eq. (1.447) with the bare vertex, Eq. (1.448). Here G defines the model’s
mass-scale.
The gap equation is
iγ · p A(p2 ) + B(p2 ) = iγ · p + m + G γµ
−iγ · p A(p2 ) + B(p2 )
γµ (,1.462)
p2 A2 (p2 ) + B 2 (p2 )
where again the renormalisation constants have been set equal to one but in this
case because the model is ultraviolet finite; i.e., there are no infinities that must
be regularised and subtracted. The gap equation yields the following two coupled
equations:
A(p2 )
p2 A2 (p2 ) + B 2 (p2 )
B(p2 )
B(p2 ) = m + 4 2 2 2
,
p A (p ) + B 2 (p2 )
A(p2 ) = 1 + 2
(1.463)
(1.464)
where I have set the mass-scale G = 1.
Consider the chiral limit equation for B(p2 ):
B(p2 ) = 4
B(p2 )
.
p2 A2 (p2 ) + B 2 (p2 )
(1.465)
The existence of a B 6≡ 0 solution; i.e., a solution that dynamically breaks chiral
symmetry, requires
p2 A2 (p2 ) + B 2 (p2 ) = 4 ,
(1.466)
measured in units of G. Substituting this identity into equation Eq. (1.463) one
finds
1
A(p2 ) − 1 = A(p2 ) ⇒ A(p2 ) ≡ 2 ,
(1.467)
2
which in turn entails
2
q
B(p ) = 2 1 − p2 .
(1.468)
The physical requirement that the quark self energy be real in the spacelike
region means that this solution is only acceptable for p2 ≤ 1. For p2 > 1 one
must choose the B ≡ 0 solution of Eq. (1.465), and on this domain we then find
from Eq. (1.463) that
q
2
1
2
⇒
A(p
)
=
A(p ) = 1 + 2
1
+
1 + 8/p2 .
2
p A(p2 )
2
(1.469)
Putting this all together, the Munczek-Nemirovsky model exhibits the dynamical
chiral symmetry breaking solution:
A(p2 ) =
B(p2 ) =


2;
q
1

1 + 1 + 8/p2
2
( √
1 − p2 ;
0;
79
p2 ≤ 1
p2 > 1 .
;
p2 ≤ 1
p2 > 1
(1.470)
(1.471)
which yields a nonzero quark condensate. Note that the dressed-quark self-energy
is momentum dependent, as is the case in QCD.
It is important to observe that this solution is continuous and defined for all
2
p , even p2 < 0 which corresponds to timelike momenta, and furthermore
p2 A2 (p2 ) + B 2 (p2 ) > 0 , ∀ p2 .
(1.472)
This last fact means that the quark described by this model is confined: the propagator does not exhibit a mass pole! I also note that there is no critical coupling
in this model; i.e., the nontrivial solution for B always exists. This exemplifies a
contemporary conjecture that theories with confinement always exhibit DCSB.
In the chirally asymmetric case the gap equation yields
2 B(p2 )
,
m + B(p2 )
4 [m + B(p2 )]2
B(p2 ) = m +
.
B(p2 )([m + B(p2 )]2 + 4p2 )
A(p2 ) =
(1.473)
(1.474)
The second is a quartic equation for B(p2 ) that can be solved algebraically with
four solutions, available in a closed form, of which only one has the correct p2 →
∞ limit: B(p2 ) → m. Note that the equations and their solutions always have a
smooth m → 0 limit, a result owing to the persistence of the DCSB solution.
Renormalization-Group-Improved Model
Finally I have used the bare vertex, Eq. (1.448) and G(Q) depicted in Fig. 1.3, in
solving the quark DSE in the chiral limit. If G(Q = 0) < 1 then B(p2 ) ≡ 0 is the
only solution. However, when G(Q = 0) ≥ 1 the equation admits an energetically
favoured B(p2 ) 6≡ 0 solution; i.e., if the coupling is large enough then even in the
absence of a current-quark mass, contrary to Eq. (1.445), the quark acquires a
mass dynamically and hence
hq̄qi ∝
Z
d4 p
B(p2 )
6= 0 for m = 0 .
(2π)4 p2 A(p2 )2 + B(p2 )2
(1.475)
These examples identify a mechanism for DCSB in quantum field theory. The
nonzero condensate provides a new, dynamically generated mass-scale and if its
magnitude is large enough [−hq̄qi1/3 need only be one order-of-magnitude larger
than mu ∼ md ] it can explain the mass splitting between parity partners, and
many other surprising phenomena in QCD. The models illustrate that DCSB is
linked to the long-range behaviour of the fermion-fermion interaction. The same
is true of confinement.
The question is then: How does the quark-quark interaction behave at large
distances in QCD? It remains unanswered.
80
1.0
0.8
G(Q)
0.6
0.4
0.2
0.0
0
1
10
100
Q (GeV)
Figure 1.3: Illustrative forms of G(Q): the behaviour of each agrees with perturbation theory for Q > 1 GeV. Three possibilities are canvassed in Sec.1.12.3:
G(Q = 0) < 1; G(Q = 0) = 1; and G(Q = 0) > 1.
81
Bibliography
[1] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill,
New York, 1964).
[2] B.D. Keister and W.N. Polyzou, “Relativistic Hamiltonian dynamics in nuclear and particle physics,” Adv. Nucl. Phys. 20, 225 (1991).
[3] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New
York, 1980).
[4] C.D. Roberts and A.G. Williams, “Dyson-Schwinger Equations and their
Application to Hadronic Physics,” Prog. Part. Nucl. Phys. 33, 477 (1994).
[5] P. Pascual and R. Tarrach, Lecture Notes in Physics, Vol. 194, QCD: Renormalization for the Practitioner (Springer-Verlag, Berlin, 1984).
[6] J.S. Schwinger, “On The Green’s Functions Of Quantized Fields: 1 and 2,”
Proc. Nat. Acad. Sci. 37 (1951) 452; ibid 455.
[7] F.A. Berezin, The Method of Second Quantization (Academic Press, New
York, 1966).
[8] L.D. Faddeev and V.N. Popov, “Feynman Diagrams For The Yang-Mills
Field,” Phys. Lett. B 25 (1967) 29.
[9] V.N. Gribov, “Quantization Of Nonabelian Gauge Theories,” Nucl. Phys. B
139 (1978) 1.
[10] F.J. Dyson, “The S Matrix In Quantum Electrodynamics,” Phys. Rev. 75
(1949) 1736.
[11] K. Symanzik, “Infrared Singularities And Small Distance Behavior Analysis,” Commun. Math. Phys. 34 (1973) 7; T. Appelquist and J. Carazzone,
“Infrared Singularities And Massive Fields,” Phys. Rev. D 11 (1975) 2856.
[12] K. Symanzik in Local Quantum Theory (Academic, New York, 1969) edited
by R. Jost.
82
[13] R.F. Streater and A.S. Wightman, A.S., PCT, Spin and Statistics, and All
That, 3rd edition (Addison-Wesley, Reading, Mass, 1980).
[14] J. Glimm and A. Jaffee, Quantum Physics. A Functional Point of View
(Springer-Verlag, New York, 1981).
[15] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Theory
and Statistical Mechanics (Springer-Verlag, New York, 1982).
[16] D.J. Gross, “Applications Of The Renormalization Group To High-Energy
Physics,” in Proc. of Les Houches 1975, Methods In Field Theory (North
Holland, Amsterdam, 1976) pp. 141-250.
[17] Y. Nambu and G. Jona-Lasinio, “Dynamical Model Of Elementary Particles
Based On An Analogy With Superconductivity. I,II” Phys. Rev. 122 (1961)
345, 246
[18] H.J. Munczek and A.M. Nemirovsky, Phys. Rev. D 28 (1983) 181.
83
Chapter 2
Quantum Fields at Finite
Temperature and Density
2.1
Ensembles and Partition Function
In equilibrium statistical mechanice, one normally encounters three types of ensemble. The microcanonical ensemble is used to describe an isolated system
which has a fixed energy E, a fixed particle number N , and a fixed volume V .
The canonical ensemble is used to describe a system in contact with a heat reservoir at temperature T . The system can freely exchange energy with the reservoir.
Thus T , N , and V are fixed. In the grand canonical ensemble, the system can
exchange particles as well as energy with a reservoir. In this ensemble, T , V , and
the chemical potential µ are fixed variables.
In the latter two ensembles T −1 = β may be thought of as a Lagrange multiplier which determines the mean energy of the system. Similarly, µ may be
thought of as a Lagrange multiplier which determines the mean number of particles in the system. In a relativistic quantum system where particles can be
created and destroyed, it is most straightforward to compute observables in the
grand canonical ensemble. So we therefore use this ensemble. This is without
loss of generality since one can pass over to either of the other ensembles by
performing a Laplace transform on the variable µ and/or the variable β.
Consider a system described by a Hamiltonian Ĥ and a set of conserved
number operators N̂i . In relativistic QED, for example, the number of electrons
minus the number of positrons is a conserved quantity, not the electron number or
the positron number separately. These number operators must be Hermitian and
must commute with H as well as with each other. Also, the number operators
must be extensive in order that the usual macroscopic thermodynamic limit can
be taken. The statistical density matrix is
h
i
ρ̂ = exp −β(Ĥ − µi N̂i ) ,
84
(2.1)
where a summation over i is implied. The ensemble average of an operator  is
Trρ̂Â
.
Trρ̂
A=
(2.2)
The grand canonical partition function is
Z = Trρ̂ .
(2.3)
The function Z = Z(T, V, µ1 , µ2 , . . .) is the single most important function in
thermodynamics. From it all other standard thermodynamic properties may be
determined. For example the equations of state for, e.g., the pressure P , the
particle number N , the entropy S, and the energy E are, in the infinite volume
limit,
∂ ln Z
,
∂V
∂ ln Z
,
Ni = T
∂µi
∂(T ln Z)
,
P =
∂T
E = −P V + T S + µi Ni .
P = T
(2.4)
(2.5)
(2.6)
(2.7)
Note that the notion of ensembles as introduced here for the situation in thermodynamical equilibrium can be extended to the nonequilibrium situation, where a
generalized Gibbs ensemble can be introduced for systems which are in a nonequilibrium situation that can be characterized by further observables such as currents
or reaction variables. These additional observables shall be accounted for by an
enlarged set of Lagrange multipliers thus arriving at a statistical operator of the
nonequilibrium state, also called relevant statistical operator within the Zubarev
formalism.
In order to calculate the partition using methods of quantum field theory, we
recall that in quantum statistics
Z = Tr e−β(Ĥ−µi N̂i ) =
Z
dφa hφa |e−β(Ĥ−µi N̂i ) |φa i ,
(2.8)
where the sum runs over all (eigen-)states. This has an appearance very similar
to the transition amplitude discussed in the previous chapter. In fact, we can
express the partition function Z as a functional integral over fields and their
conjugate momenta. Before doing so, we must switch to an imaginary time
variable τ = i t and limit the integration over τ to the region between 0 and β.
The trace operation means that we have to integrate over all fields φa . Finally, if
the system admits some conserved charge, then we must make the replacement
H(π, φ) → K(pi, φ) = H(π, φ) − µN (π, φ) ,
85
(2.9)
where N (π, φ) is the conserved charge density. Thus, we finally arrive at the
fundamental formula
Z=
Z
Dπ
Z
periodic
Dφ exp
(Z
β
0
Z
∂φ
dx iπ
− H(π, φ) + µN (π, φ)
∂τ
3
!)
. (2.10)
The term “periodic” means that the integration over the field is constrained so
that φ(~x, 0) = φ(~x, β). This is a consequence of the trace operation, setting
φa (~x) = φ(~x, 0) = φ(~x, β). There is no restriction on the π integration. The
generalization of (2.10) to an arbitrary number of fields and conserved charges is
obvious.
2.2
2.2.1
Bosonic Fields
Neutral Scalar Field
The most general renormalizable Lagrangian for a neutral scalar field is
1
1
L = ∂µ φ∂ µ φ − m2 φ2 − U (φ) ,
2
2
(2.11)
U (φ) = gφ3 + λφ4 ,
(2.12)
where the potential is
and λ ≥ 0 for stability of the vacuum. The momentum conjugate to the field is
π=
∂L
∂φ
=
,
∂(∂0 φ)
∂t
(2.13)
and the Hamiltonian is
H=π
∂φ
1
1
1
− L = π 2 + (∇φ)2 + m2 φ2 + U (φ) .
∂t
2
2
2
(2.14)
There is no conserved charge.
The first step in evaluating the partition function is to return to the discretized
version
Z =
lim
N →∞
exp
ΠN
i=1

N Z
X

j=1
Z
∞
−∞
!
dπi Z
dφi
2π periodic
d3 x iπj (φj+1 − φj ) − ∆τ
(2.15)
1 2 1
1
πj + (∇φj )2 + m2 φ2j + U (φj )
2
2
2


The momentum integrations can be done immediately since they are just products
of Gaussian integrals. Divide position space into M 3 little cubes with V = L3 ,
L = aM , a → 0, M → ∞, M an integer.
86

.
For convenience, and to ensure that Z remains explicitely dimensionless at
each step in the calculation, we write πj = Aj /(a3 ∆τ )1/2 and integrate Aj from
−∞ to +∞. For each cube we obtain
Z
∞
−∞

1
a3
exp − A2j + i
2
∆τ
dAj
2π
!1/2

(φj+1 − φj )Aj 
−a3 (φj+1 − φj )2
= (2π)−1/2 exp
2∆τ
"
#
.
(2.16)
Thus far we have
Z =
lim (2π)
M,N →∞


exp ∆τ

−M 3 N/2
N Z
X
j=1
Z h
ΠN
i=1 dφi

1
d 3 x −
2
i
(φj+1 − φj )
∆τ
!2
Returning to the continuum limit, we obtain
Z = N0
Z
periodic
Z
Dφ exp
β
0


1
1
) .
− (∇φj )2 − m2 φ2j − U (φj(2.17)

2
2
dτ
Z
d3 xL
!
.
(2.18)
The Lagrangian is expressed as a functional of φ and its first derivatives. The
formula (2.17) expresses Z as a functional integral over φ of the exponential of
the action in imaginary time. The normalization constant is irrelevant, since
multiplication of Z by any constant does not change the thermodynamics.
Next, we turn to the case of noninteracting fields with U (φ) = 0. Interactions
will be discussed separately. Define
S=
Z
β
0
dτ
Z
1Z β Z 3
d xL = −
dτ d x
2 0
3
"
∂φ
+ (∇φ)2 + m2 φ2
∂τ
!#
.
(2.19)
Integrating by parts and taking note of the periodicity of φ, we obtain
1
S=−
2
Z
β
0
dτ
Z
∂2
d xφ − 2 − ∇2 + m2 φ .
∂τ
!
3
(2.20)
The field can be decomposed into a Fourier series according to
φ(~x, τ ) =
β
V
!1/2
∞ X
X
ei(~p~x+ωn τ ) φn (~
p) ,
(2.21)
n=−∞ p
~
where ωn = 2πnT , due to the constraint of periodicity that φ(~x, β) = φ(~x, 0) for
all ~x. The normalization of (2.21) is chosen conveniently so that each Fourier
87
amplitude is dimensionless. Substituting (2.21) into (2.20), and noting that
φ−n (−~
p) = φ∗n (~
p) as required by the reality of φn (~
p), we find
1 XX 2
S = − β2
(ωn + ω 2 )φn (~
p)φ∗n (~
p)
2
n p
~
(2.22)
√
with ω = p~2 + m2 . The integrand depends only on the amplitude of φ and not
its phase. The phases can be integrated out to get
Z = N 0 Πn Πp~
Z
h
1
dAn (~
p) exp − β 2 (ωn2 + ω 2 )A2n (~
p)
2
−∞
∞
= N 0 Πn Πp~ 2π/(β 2 (ωn2 + ω 2 ))
i1/2
.
(2.23)
Ignoring an overall multiplicative factor independent of β and V , which does not
affect the thermodynamics, we arrive at
h
Z = Πn Πp~ β 2 (ωn2 + ω 2 )
i−1/2
.
(2.24)
More formally one can arrive at this result by using the general rules for Gaussian
functional integrals over commuting (bosonic) variables, derived before in the
QFT chapter, since (2.18) and (2.20) can be expressed as
Z = N0
Z
1
Dφ exp − (φ, Dφ) = N 0 constant(det D)−1/2 ,
2
(2.25)
where D = β 2 (ωn2 + ω 2 ) in (~
p, ωn ) space and (φ, Dφ) denotes the inner product
on the function space.
Thus far we have
ln Z = −
i
1 XX h 2 2
ln β (ωn + ω 2 ) .
2 n p~
(2.26)
Note that the sum over n is divergent. This unfortunate feature stems from
our careless handling of the integration measure Dφ. A more rigorous treatment
using the proper definition of Dφ gives a finite result. In order to handle (2.26),
we make use of
h
ln (2πn)2 + β 2 ω 2 )
i
Z
=
β 2 ω2
1
h
i
dΘ2
2
+
ln
1
+
(2πn)
,
Θ2 + (2πn)2
(2.27)
where the last term is β− independent and thus can be ignored. Furthermore,
∞
X
1
2π 2
2
=
1
+
2
2
Θ
eΘ − 1
−∞ n + (Θ/2π)
hence
ln Z = −
XZ
p
~
βω
1
1
1
dΘ
+ Θ
2 e −1
88
.
,
(2.28)
(2.29)
Carrying out the Θ integral, and throwing away a β− independent piece, we
finally arrive at
ln Z = V
Z
d3 p
1
− βω − ln(1 − e−βω ) ,
3
(2π)
2
(2.30)
from which we obtain immediately the well-known expression for the ideal Bose
gas (µ = 0), once we subtract the divergent expressions for the zero-point energy
E0 = −
∂
ln Z0 = V
∂β
Z
d3 p ω
,
(2π)3 2
(2.31)
and for the zero-point pressure
P0 = T
∂
E0
ln Z0 = −
,
∂V
V
(2.32)
which are typical for the quantum field-theoretical treatment. With this subtraction the vacuum is defined as the state with zero energy and pressure.
2.3
Fermionic Fields
Dirac fermions are descibed by a four-spinor field ψ with a Lagrangian density
L = ψ̄(i∂/ − m)ψ
!
0 ∂
† 0
~ −m ψ .
= ψ γ iγ
+ i~γ · ∇
∂t
(2.33)
The momentum conjugate to this field is
Π=
∂L
= iψ † ,
∂(∂ψ/∂t)
(2.34)
because γ 0 γ 0 = 1. Thus, somewhat paradoxically, ψ and ψ † must be treated as
independent entities in the Hamiltonian formulation. The Hamiltonian density
is found by standard procedures,
!
∂ψ
∂
~ + m)ψ ,
H=Π
− L = ψ† i
ψ − L = ψ̄(−i~γ · ∇
∂t
∂t
(2.35)
and the partition function is
Z = Tr e−β(Ĥ−µQ̂) ,
with the conserved charge Q =
reads
Z=
Z
†
Dψ Dψ exp
"Z
β
0
dτ
R
Z
(2.36)
d3 xψ † ψ. The functional integral representation
3
d xψ
†
!
∂
~ − m + µγ 0 ψ
−γ
+ i~γ · ∇
∂τ
0
89
#
(2.37)
As with bosons, it is most convenient to work in (~
p, ωn ) space instead of (~x, τ )
space, i.e.,
!1/2 ∞
X X
β
ψα (~x, τ ) =
ei(~p~x+ωn τ ) ψ̃α;n (~
p) ,
(2.38)
V
n=−∞ p
~
where now ωn = (2n + 1)πT due to the antiperiodicity of the (Grassmannian)
Fermion field at the borders of the fundamental strip 0 ≤ τ ≤ β in the imaginary
time, ψ(~x, 0) = ψ(~x, β).
Now we are ready to evaluate the fermionic partition function (2.37),
Z =
Πn Πp~ Πα
XX
S =
n
Z
†
idψ̃α;n
(~
p)dψ̃α;n (~
p)
†
iψα;n
(~
p)Dαρ ψρ;n (~
p) ,
eS ,
p
~
h
i
D = −iβ (−iωn + µ) − γ 0~γ · p~ − mγ 0 ,
(2.39)
using our knowledge about Grassmannian integration of Gaussian functional integrals, resulting in
Z = det D .
(2.40)
Employing the identity
ln det D = Tr ln D ,
(2.41)
and evaluating the determinant in Dirac space explicitly (Exercise !), one finds
ln Z = 2
XX
n
p
~
h
n
ln β 2 (ωn + iµ)2 + ω 2
io
.
(2.42)
Since both positive and negative frequencies have to be summed over, the latter
expression can be put in a form analogous to the above expression in the bosonic
case,
ln Z =
XXn
n
p
~
h
ln β 2 ωn2 + (ω − µ)2
i
h
+ ln β 2 ωn2 + (ω + µ)2
io
.
(2.43)
In the further evaluation we can go similar steps as in the bosonic case, with two
exceptions: (1) the presence of a chemical potential, splitting the contributions of
particles and antiparticles; (2) the Matsubara frequencies are now odd multiples
of πT , so that the infinite sum to be exploited reads
∞
X
1
1 1
1
=
−
2 2
2
Θ 2 eΘ + 1
n=−∞ (2n + 1) π + Θ
.
(2.44)
Integrating over the auxiliary variable Θ, and dropping terms independent of β
and µ, we finally obtain
ln Z = 2V
Z
i
d3 p h
−β(ω−µ)
−β(ω+µ)
βω
+
ln(1
+
e
)
+
ln(1
+
e
)
.
(2π)3
90
(2.45)
Notice that the factor 2 crresponding to the spin- 21 nature of the fermions comes
out automatically. Separate contributions from particles (µ) and antiparticles
(-µ) are evident. Finally, the zero-point energy of the vacuum also appears in
this formula.
91
2.4
2.4.1
Gauge Fields
Quantizing the Electromagnetic Field
In this chapter we would like to understand how the Faddeev-Popov ghosts as
introduced to eliminate divergences of the gauge freedom will act in the pure
gauge theory to eliminate unphysical degrees of freedom and allow to derive the
blackbody radiation law with the correct number of physical degrees of freedom.
We start with recalling the path integral formulation of QED for the photon
field Aµ (x) with the field strength tensor
Fµν = ∂µ Aν − ∂ν Aµ
(2.46)
and the free action functional
S=−
1
4
Z
d4 xFµν F µν .
(2.47)
This action is invariant under gauge transformations
Aµ (x) → Aµ (x) = A0µ (x) + ∂µ ω(x) ,
(2.48)
where ω(x) is a scalar function which parametrizes the gauge transformations.
The momenta conjugate to the space components of Ai (x) are, up to a sign, the
components Ei (x) = Ei (x) of the electric field
πi = −Ei = −F0i ,
(2.49)
while the magnetic field B(x) is
Bi = εijk ∂j Ak .
(2.50)
We work in an axial gauge, A3 = 0 to be specific. The momenta π1 and π2 are
independent variables; E3 is not an independnet variable, but it is a function of π1
and π2 , which may be computed from Gauss’s law ∇ · E = 0. There are thus two
dynamical variables A1 and A2 with conjugate momenta π1 and π2 . We define
π3 = −E3 (π1 , π2 , but π3 is not to be interpreted as a conjugate momentum. The
partition function is written as a Hamiltonian path integral
Z=
Z
D(π1 , π2 )
Z
Ai (0)=Ai (β)
1
2
D(A , A ) exp
"Z
β
0
4
1
2
#
d x(iπ1 ∂τ A + iπ2 ∂τ A − H) (2.51)
,
where we have used the notation
Z
β
0
d4 x =
Z
β
0
92
dτ
Z
d3 x ,
(2.52)
while the Hamiltonian density H is
1 2
1
π1 + π22 + E23 (π1 , π2 ) + B2
H = (E2 + B2 ) =
2
2
(2.53)
Equation (2.51) is then transformed by using
Z
1=
Dπ3 δ(π3 + E3 (π1 , π2 )) ,
(2.54)
and
∂(∇ · π)
δ(π3 + E3 (π1 , π2 )) = δ(∇ · π) det
∂π3
h
!
i
= det ∂3 δ 3 (x − y) δ(∇ · π) .
(2.55)
In the following step, one inserts an integral representation of δ(∇ · π)
δ(∇ · π) =
Z
" Z
DA4 exp i
β
0
#
4
d xA4 (∇ · π) ,
(2.56)
where A4 = iA0 , and we work in Euclidean space now: xµ = (x, x4 ) = (x, τ ),
A4 = (A, A4 ). Performing the π− integration, we are left with
"Z
#
1
1
d x (i∂τ A − i∇A4 )2 − B2
Z = D(A1 , A2 , A4 ) det ∂3 δ (x − y) exp
,
2
2
0
(2.57)
where A = (A1 , A2 , 0). Note that the argument of the exponential is
Z
h
i
3
β
4
1 2 1 2
E − B =L.
2
2
(2.58)
The A− integration is rendered more aesthetic by inserting
1=
Z
DA3 δ(A3 ) ,
(2.59)
and the partition function assumes the form
Z=
Z
µ
h
3
i
DA δ(A3 ) det ∂3 δ (x − y) exp
Z
β
0
4
d xL
!
.
(2.60)
The axial gauge A3 = 0 is not a particularly convenient gauge to use for practical
computations. Furthermore, it is not immediately apparent that (2.60) is a gauge
invariant expression for Z.
Take an arbitrary gauge specified by F = 0, where F is some function of Aµ
and its derivatives. For the gauge above, F = A3 . For this gauge, (2.60) is given
by
!
!
Z
Z β
∂F
µ
4
Z = DA δ(F ) det
exp
d xL .
(2.61)
∂α
0
93
Equation (2.61) is manifestly gauge invariant: L is invariant, the gauge fixing
factor times the Jacobian of the transformation δ(F ) det(∂F/∂α) is invariant,
and the integration is over all four components of the vector potential. Equation
(2.61) reduces to (2.60) in the case of the axial gauge A3 = 0. We know this is
correct since it was derived from first principles in the Hamiltonian formulation
of the gauge theory, Z = Tr e−β Ĥ
2.4.2
Blackbody radiation
It is important to verify that (2.61) describes blackbody radiation with two polarization degrees of freedom. We will do this here in the axial gauge A3 = 0, the
Feynman gauge is left as an exercise.
In the axial gauge, we rewrite (2.57) as
Z =
S0
Z
1
=
2
D(A0 , A1 , A2 ) det(∂3 )eS0
Z
dτ
Z
d3 x(A0 , A1 , A2 )
∂
−∂1 ∂τ
∂22 + ∂32 +
−∂1 ∂2
∇2

∂
×  −∂1 ∂τ
∂
−∂2 ∂τ

∂
−∂2 ∂τ
∂1 ∂2
∂12 + ∂32 +
∂2
∂τ 2
∂2
∂τ 2


A0


  A1  .
A2
(2.62)
We can express the determinant of ∂3 as a functional integral over a complex
ghost field C: that is, a Grassmann field with spin-0,
det(∂3 ) =
Z
D C̄DC exp
Z
β
0
dτ
Z
3
d xC̄∂3 C
!
.
(2.63)
These ghost fields C̄ and C are not physical fields since they do not appear
in the Hamiltonian. Furthermore, since they are anticommuting scalar fields
they violate the connection between spin and statistics. It is simply a convenient
functional integral representation of the determinant of an operator. The greatest
applicability of these ficticious ghost fields will be to non-Abelian gauge theories,
see also Sect. 5.2.1. of Le Bellac [2].
In frequency-momentum space the partition function is expressed as
ln Z = ln det(βp3 ) −
D =


β2 
p2
−ωn p1
−ωn p1
1
ln det(D) ,
2

−ωn p1
−ωn p2

ωn2 + p22 + p23
−p1 p2
 .
−p1 p2
ωn2 + p21 + p23
Carrying out the determinantal operation,
ln Z =
i
h
1
1
Tr ln(β 2 p23 ) − Tr ln β 6 p23 (ωn2 + p2 )2
2
2
94
h
= ln Πn Πp β 2 (ωn2 + p2 )
= 2V
Z
i−1 1
d3 p
− βω − ln(1 − e−βω ) .
3
(2π)
2
(2.64)
Here, ω = |p|. Comparison with the result for the scalar field case shows that
(2.64) describes massless bosons with two spin degrees of freedom in thermal
equilibrium; in other words, blackbody radiation.
2.5
Interactions: Hubbard-Stratonovich Trick
So far we have dealt with free quantum fields in the absence of interactions and
have obtained nice closed expressions for the thermodynamic potential, i.e., therefore also for the generating functionals of the thermodynamic Green functions.
However, once we switch on the interactions in our model field theories, there is
only a very limited class of soluble models, in general we have to apply approximations. The most common technique is based on perturbation theory, which
requires a small parameter. For strong interactions at low momentum transfer
(the infrared region), the coupling is nonperturbatively strong and alternative,
nonperturbative methods have to be invoked. One of the strategies, which is especially suitable for the treatment of quantum field theories within the path integral
formulation is based on the introduction of collective variables (auxialiary fields)
by an exact integral transformation due to Stratonovich and Hubbard which allows to eliminate (integrate out) the elementary degrees of freedom. Generally,
the (dual) coupling of the auxialiary fields is weak so that perturbative expansions
of the nonlinear effective action make sense and provide useful results already at
low orders of this expansion.
A general class of interactions for which the Hubbard-Stratonovich (HS) transformation is immediately applicable, are four-fermion couplings of the currentcurrent type
Lint = G(ψ̄ψ)2 .
(2.65)
A Fermi gas with this typ of interaction serves as a model for electronic superconductivity (Bardeen-Cooper-Schrieffer (BCS) model, 1957) or for chiral symmetry
breaking in quark matter (Nambu–Jona-Lasinio (NJL) model, 1961).
The HS-transformation for (2.65) reads
h
exp G(ψ̄ψ)
2
i
=N
Z
σ2
+ ψ̄ψσ
Dσ exp
4G
"
#
(2.66)
and allows to bring the functional integral over fermionic fields into a quadratic
(Gaussian) form so that fermions can be integrated out. This is also called
Bosonization procedure.
95
2.5.1
Nambu–Jona-Lasinio (NJL) Model
Here we will present an application of the HS technique to the NJL model for
quark matter at finite densities and temperatures. This is possible since the
interaction of this model is of the current-current form and therefore the HS trick
for the bosonization of 4-fermion interactions applies. The Lagrangian density is
given by
L = q̄iα (i∂/δij δαβ − Mij0 δαβ + µij,αβ γ 0 )qjβ
+ GS
8 h
X
(q̄τfa q)2 + (q̄iγ5 τfa q)2
a=0
+ GD
Xh
i
C
(q̄iα ijk αβγ qjβ
)(q̄iC0 α0 i0 j 0 k α0 β 0 γ qj 0 β 0 )
k,γ
i
C
+ (q̄iα iγ5 ijk αβγ qjβ
)(q̄iC0 α0 iγ5 i0 j 0 k α0 β 0 γ qj 0 β 0 ) ,
(2.67)
where from here on the quark spinor is qiα , with the flavor index i = u, d, s and
α = r, g, b stands for the color degree of freedom. Mij0 = diag(m0u , m0d , m0s ) is
the current quark mass matrix in flavor space and µij,αβ is the chemical potential
matrix in color and flavor space. The grand cononical thermodynamical potential
Ω = T ln Z is known once we manage to evaluate the partition function in a reasonable approximation. A closed solution, even for the simple model Lagrangion
(2.67), is not possible. The bosonization of the partition function proceeds as
follows
Z
Z
Z = Dq̄Dq exp d4 xL
(2.68)
96
2.5.2
Mesonic correlations at finite temperature
In the previous section we have seen how the concept of order parameters can be
introduced in quantum field theory in the mean-field approximation. By analysis
of the gap equations describing the minima of the thermodynamical potential in
the space of order parameters we have we could investigate the phenomenon of
spontaneous symmetry breaking, indicated by a nonvanishing value of the order
parameter (gap). Important examples being: chiral symmetry breaking (mass
gap) and superconductivity (energy gap). At finite temperature and density the
values of these gaps change and their vanishing indicates the restoration of a
symmetry. As the symmetries prevailing under given thermodynamical conditions of temperature and chemical potential (density) characterize a phase of the
system, we have thus acquainted ourselves with a powerful quantum field theoretical method of analysing phase transitions. The results of such an analysis
are summarized in phase diagrams. A prominent example is the phase diagram
of QCD in the temperature- density plane, shown schematically in Fig. ??. It
exhibits two major domains: Hadronic matter (confined quarks and gluons) and
the Quark-gluon plasma (QGP), separated by the phenomenon of quark (and
gluon) deconfinement under investigation in heavy-ion collisions, but also in the
physics of compact stars and in simulations of Lattice-gauge QCD on modern Teraflop computers. The hadronic phase is subdivided into a nuclear matter phase
at low temperatures with a gas-liquid transition (similar to the van-der-Waals
treatment of real gases) and a ’plasma phase’ of a hot hadron gas with a multitude of hadronic resonances. The QGP phase is subdivided into a quark matter
phase at low temperatures where most likely gluons are still condensed (confined)
and the strongly correlated Fermi-liquid of quarks shoild exhibit the phenomenon
of superconductivity/ superfludity with a Bose condensate of Cooper pairs of .
diquarks. At asymptotic temperatures and densities one expects a system of
free quarks and gluons (due to asymptotic freedom of QCD). In any real situation in terrestrial experiments or in astrophysics, one expects to be quite far
from this ideal gas state. As the nature of the confinement-deconfinement transition is not yet clarified, we must speculate what to expect in the vicinity of the
conjectured deconfinement phase transition. In the lattice-QCD simulations one
has found evidence for strong correlations even above the critical temperature
Tc ∼ 170 MeV, which is defined by an increase of the effective number of degrees
of freedom ε(T )/T 4 by about one order of magnitude in a close vicinity of Tc .
Indications for strong correlations in the QGP one has also found in recent RHIC
experiments, where from particle production and flow one has found fast thermalization and an extremely low viscosity (’perfect fluid’). One speaks about an
< 2 Tc .
’sQGP phase’ at temperatures Tc <
∼T ∼
In order to investigate the phase diagram experimentally, one has to correlate
the observables with the regions. As the detectors are situated in the ’vacuum’
at zero temperature and chemical potential where confinement prevails, no direct
97
Big Bang
Quark-Gluon-Plasma
1.5
0.1
DECONFINEMENT
CERN-SPS
FAIR (Project)
AGS Brookhaven
Hadron gas
CONFINEMENT
QCD - Lattice Gauge Theory
Temperature [TH =140 MeV]
RHIC, LHC (construction)
s
SIS Darmstadt
on
yI
SuperNovae
Nuclear matter
1
ion
is
oll
C
Quark Matter
av
He COLOR SUPERCONDUCTIVITY
Neutron / Quark Stars
Baryon Density
3
-3
[nο =0.16 fm ]
Figure 2.1: The conjectured phase diagram of QCD and places where to investigate it.
detection of quarks and gluons from the plasma phase is possible and one has
to conjecture about the properties of the sQGP phase(s) from the traces which
they might leave in the hadronic and electromagnetic spectra emitted by the
hadronizing fireball formed in a heavy-ion collision or by the neutrino-emitting
cooling processes of compact stars.
As a prerequisite for discussing hadronic spectra and their modifications, we
will study the mesonic spectral function(s), and for their analysis within effective
models like the NJL or MN models, we need to evaluate loop diagrams at finite
temperature. A basic ingredient are Matsubara frequency sums.
2.5.3
Matsubara frequency sums
In computing Feynman diagrams with internal fermion lines we shall encounter
frequency sums, and we have to learn how to evaluate them. Let us denote the
quantity (ωn2 + Ep2 )−1 , which appears in the free Fermion propagator
SF (iωn , p) = −
Z
∞
−∞
dp0 ρF (p0 , p)
m−p
/
= 2
2π iωn − p0
ωn − E2p
98
(2.69)
˜ n , Ep ), and its mixed representation by ∆(τ,
˜ Ep ), suppressing the subby ∆(iω
script F for simplicity
˜ Ep ) = T
∆(τ,
X
˜ n , Ep ) .
e−iωn τ ∆(iω
(2.70)
n
It can easily be checked (Exercise !) that
i
1 h
(1 − ñ(Ep ))e−Ep τ − ñ(Ep )eEp τ
2Ep
X s
=
(1 − f˜(sEp ))e−sEp τ ,
s=±1 2Ep
˜ Ep ) =
∆(τ,
(2.71)
where the Fermi-Dirac distribution ñ(p0 ) is
ñ(p0 ) =
1
eβ|p0 | + 1
.
(2.72)
One should note the absolute value of p0 in (2.72), in contrast with the definition
of f˜(p0 ) = 1/[exp(βp0 ) + 1]. Note also that
f˜(E) = ñ(E) , f˜(−E) = 1 − ñ(E) ,
1 − f˜(E) − f˜(−E) = 0 .
(2.73)
In frequency space, the formula corresponding to (2.71) is
X
1
˜ s (iωn , Ep
=
∆
2
− Ep
s=±1
X
s
1
−
.
=
2Ep iωn − sEp
s=±1
˜ n , Ep ) =
∆(iω
ωn2
(2.74)
The frequency sums are performed by following the methods of analytic continuation iωn → k0 and contour integration with a function having simple poles at the
discrete frequencies k0 = iωn with unit residuum and convergence for |k0 | → ∞.
Let us give two general examples for Matsubara sums of one-loop diagrams with
two external (amputated) legs.
• Fermion-boson case, see Fig. 2.5.3:
s1 s2 1 + f (s1 E1 ) − f˜(s2 E2 )
4E1 E2 iω − s1 E1 − s2 E2
n
(2.75)
˜
X
˜ s (i(ω − ωn ), E2 ) = − is2 1 + f (s1 E1 ) − f (s2 E2 )
T
ωn ∆s1 (iωn , E1 )∆
2
4E2 iω − s1 E1 − s2 E2
n
(2.76)
T
X
˜ s2 (i(ω − ωn ), E2 ) = −
∆s1 (iωn , E1 )∆
99
φ
1
k
q
φ
φ
2
q−k
Figure 2.2: One-loop diagram for the fermion-antifermion polarization function.
φ
q
k
φ
1
φ
2
q−k
Figure 2.3: One-loop diagram for the fermion self-energy.
100
• Fermion-antifermion case, see Fig. 2.5.3:
˜
˜
˜ s1 (iωn , E1 )∆
˜ s2 (i(ω − ωn ), E2 ) = − s1 s2 1 − f (s1 E1 ) − f (s2 E2 )
∆
4E1 E2 iω − s1 E1 − s2 E2
n
(2.77)
˜
˜
X
˜ s1 (iωn , E1 )∆
˜ s2 (i(ω − ωn ), E2 ) = − is2 1 − f (s1 E1 ) − f (s2 E2 )
ωn ∆
T
4E2 iω − s1 E1 − s2 E2
n
(2.78)
T
X
Note that you can obtain unify these formula using the rule: f (sE) → −f˜(sE)
when replacing a bosonic by a fermionic line.
Exercise: Verify these expressions!
101
Bibliography
[1] J.I. Kapusta, Finite-temperature Field Theory, Cambridge University Press,
1989
[2] M. Le Bellac, Thermal Field Theory, Cambridge University Press, 1996
[3] D. Blaschke, et al., Phys. Rev. D 72 (2005)
XXXXXXXXXXXXXXXXXXXXXXXXXX
102
Chapter 3
Applications to Heavy-Ion
Collisions and Compact Stars
3.1
Introduction
103
A typical frequency sum, which we shall encounter in the calculation of loop
diagrams is
X
S(iωn ) = T
∆(i(ωm − ωn ))∆0 (iωn ) ,
(3.1)
n
where the oscillator frequencies of ∆ and ∆0 are ω and ω 0 , respectively. Using
the Fourier representation of ∆(iωn )
∆(iωn ) =
Z
∞
0
dτ eiωn τ ∆(τ )
(3.2)
and the relation
T
X
n
where p = 0, ±1, ±2, . . ., yields
eiωn τ =
X
p
δ(τ − pβ) ,
S(iωn ) =
124
(3.3)
(3.4)
Chapter 4
Particle Production by Strong
Fields
4.1
4.1.1
Quantum kinetic equation for particle production
Introduction
Particle production in ultrarelativistic heavy-ion collisions raises a number of
challenging problems. One of these interesting questions is how to incorporate the
mechanism of particle creation into a kinetic theory[1, 2, 3, 4, 5, 6, 7, 8, 9]. Within
the framework of a flux tube model[4, 10, 11, 12], a lot of promising research has
been carried out during the last years. In the scenario where a chromo-electric
field is generated by a nucleus-nucleus collision, the production of parton pairs
can be described by the Schwinger mechanism[13, 15, 15]. The produced charged
particles generate a field which, in turn, modifies the initial electric field and may
cause plasma oscillations. The interesting question of the back reaction of this
field has been analyzed within a field theoretical approach[16, 17, 18]. The results
of a simple phenomenological consideration based on kinetic equations and the
field-theoretical treatment[5, 19, 20, 21] agree with each other. The source term
which occurs in such a modified Boltzmann equation was derived phenomenologically in Ref.[22]. However, the systematic derivation of this source term in
relativistic transport theory is not yet fully carried out. For example, recently it
was pointed out that the source term may have non-Markovian character[6, 23]
even for the case of a constant electric field.
In the present lecture, a kinetic equation is derived in a consistent field theoretical treatment for the time evolution of the pair creation in a time-dependent
and spatially homogeneous electric field. This derivation is based on the Bogoliubov transformation for field operators between the asymptotic in-state and the
instantaneous state. In contrast with phenomenological approaches, but in agree125
ment with[6, 23], the source term of the particle production is of non-Markovian
character. The kinetic equation derived reproduces the Schwinger result in the
low density approximation in the weak field limit.
4.1.2
Dynamics of pair creation
Creation of fermion pairs
In this section we demonstrate the derivation of a kinetic equation with a source
term for fermion-antifermion production. As an illustrative example we consider
electron-positron creation, however the generalization to quark-antiquark pair
creation in a chromoelectric field in Abelian approximation is straightforward.
For the description of e+ e− production in an electric field we start from the QED
lagrangian
1
L = ψ̄iγ µ (∂µ + ieAµ )ψ − mψ̄ψ − Fµν F µν ,
4
(4.1)
where F µν is the field strength, the metric is taken as g µν = diag(1, −1, −1, −1)
and for the γ– matrices we use the conventional definition [24]. In the following
we consider the electromagnetic field as classical and quantize only the matter
field. Then the Dirac equation reads
(iγ µ ∂µ − eγ µ Aµ − m)ψ(x) = 0 .
(4.2)
We use a simple field-theoretical model to treat charged fermions in an external
electric field charactarized by the vector potential Aµ = (0, 0, 0, A(t)) with A(t) =
A3 (t). The electric field
E(t) = E3 (t) = −Ȧ(t) = −dA(t)/dt
(4.3)
is assumed to be time-dependent but homogeneous in space (E1 = E2 = 0). This
quasi-classical electric field interacts with a spinor field ψ of fermions. We look
for solutions of the Dirac equation where eigenstates are represented in the form
(±)
ψp̄r (x) =
iγ 0 ∂0 + γ k pk − eγ 3 A(t) + m χ(±) (p̄, t) Rr eip̄x̄ ,
(4.4)
where k = 1, 2, 3 and the superscript (±) denotes eigenstates with the positive
and negative frequencies. Herein Rr (r = 1, 2) is an eigenvector of the matrix
γ0γ3




1
0
 0 
 1 




 ,
(4.5)
 ,
R2 = 
R1 = 
 −1 
 0 
0
−1
126
so that Rr+ Rs = 2δrs . The functions χ(±) (p̄, t) are related to the oscillator-type
equation
χ̈
(±)
2
(p̄, t) = − ω (p̄, t) + ieȦ(t) χ(±) (p̄, t) ,
(4.6)
where we define the total energy ω 2 (p̄, t) = ε2⊥ + Pk2 (t), the transverse energy
ε2⊥ = m2 + p̄2⊥ and Pk (t) = pk − eA(t). The solutions χ(±) (p̄, t) of Eq. (4.6)
for positive and negative frequencies are defined by their asymptotic behavior at
t0 = t → −∞, where A(t0 ) = 0. We obtain
χ(±) (p̄, t) ∼ exp ( ± iω0 (p̄) t) ,
(4.7)
where the total energy in asymptotic limit is given as ω0 (p̄) = ω0 (p̄, t0 ) =
lim ω(p̄, t). Note that the system of the spinor functions (4.106) is complete
t→−∞
and orthonormalized. The field operators ψ(x) and ψ̄(x) can be decomposed in
the spinor functions (4.106) as follows:
ψ(x) =
X
r,p̄
(−)
(+)
ψp̄r (x) bp̄r (t0 ) + ψp̄r (x) d+
−p̄r (t0 ) .
(4.8)
+
+
The operators bp̄r (t0 ), bp̄r
(t0 ) and dp̄r (t0 ), dp̄r
(t0 ) describe the creation and annihilation of electrons and positrons in the in-state |0in > at t = t0 , and satisfy the
anti-commutation relations [24]
{bp̄r (t0 ), bp̄+0 r0 (t0 )} = {dp̄r (t0 ), dp̄+0 r0 (t0 )} = δrr0 δp̄p̄0 .
(4.9)
The evolution affects the vacuum state and mixes states with positive and negative energies resulting in non-diagonal terms that are responsible for pair creation.
The diagonalization of the hamiltonian corresponding to a Dirac-particle (Eq.
(4.2)) in the homogeneous electric field (4.3) is achieved by a time-dependent
Bogoliubov transformation
bp̄r (t) = αp̄ (t) bp̄r (t0 )
+ βp̄ (t) d+
−p̄r (t0 ) ,
dp̄r (t) = α−p̄ (t) dp̄r (t0 ) − β−p̄ (t)
(4.10)
b+
−p̄r (t0 )
with the condition
|αp̄ (t)|2 + |βp̄ (t)|2 = 1 .
(4.11)
Here, the operators bp̄r (t) and dp̄r (t) describe the creation and annihilation of
quasiparticles at the time t with the instantaneous vacuum |0t >. Clearly, the
operator system b(t0 ), b+ (t0 ); d(t0 ), d+ (t0 ) is unitary non-equivalent to the system
b(t), b+ (t); d(t), d+ (t). The substitution of Eqs. (4.10) into Eq. (4.8) leads to the
new representation of the field operators
ψ(x) =
X
(−)
Ψp̄r (x)
bp̄r (t) +
r,p̄
127
(+)
Ψp̄r (x)
d+
−p̄r (t)
.
(4.12)
(±)
The link between the new Ψp̄r (x) and the former (4.106) basis functions is given
by a canonical transformation
(−)
(−)
(+)
(+)
(−)
Ψp̄r (x)
ψp̄r (x) = αp̄ (t) Ψp̄r (x) − βp̄∗ (t) Ψp̄r (x) ,
(+)
ψp̄r (x)
αp̄∗ (t)
=
Ψ
(x)p̄r + βp̄ (t)
(4.13)
.
(±)
Therefore it is justified to assume that the functions Ψp̄r have a spin structure
(+)
similar to that of ψp̄r in Eq. (4.106),
(±)
(±)
Ψp̄r (x) = iγ 0 ∂0 + γ k pk − eγ 3 A(t) + m φp̄ (x) Rr e±iΘ(t) eip̄x̄ ,
(4.14)
where the dynamical phase is defined as
Θ(p̄, t) =
Z
t
t0
dt0 ω(p̄, t0 ) .
(4.15)
(±)
The functions φp̄ are yet unknown. The substitution of Eq. (4.14) into Eqs.
(4.13) leads to the relations
(−)
(+)
χ(−) (p̄, t) = αp̄ (t) φp̄ (t) e−iΘ(p̄,t) − βp̄∗ (t) φp̄ (t) eiΘ(p̄,t) ,
χ
(+)
(p̄, t) =
αp̄∗ (t)
(+)
φp̄ (t)
e
iΘ(p̄,t)
+ βp̄ (t)
(−)
φp̄ (t)
e
−iΘ(p̄,t)
(4.16)
.
Now we are able to find explicit expressions for the coefficients αp̄ (t) and βp̄ (t).
Taking into account that the functions χ(±) (p̄, t) are defined by Eq. (4.6), we
introduce additional conditions to Eqs. (4.16) according to the Lagrange method
χ̇
(−)
(p̄, t) = −iω(p̄, t) αp̄ (t)
χ̇(+) (p̄, t) =
(−)
φp̄ (t)
e
−iΘ(p̄,t)
(+)
iω(p̄, t) αp̄∗ (t) φp̄ (t) eiΘ(p̄,t)
+
βp̄∗ (t)
(+)
φp̄ (t)
(−)
e
iΘ(p̄,t)
,
(4.17)
− βp̄ (t) φp̄ (t) e−iΘ(p̄,t) .
Differentiating these equations with respect to time, using Eqs. (4.6) and (4.16)
and then choosing as an Ansatz
(±)
φp̄ (t) =
v
u
u ω(p̄, t) ± Pk (t)
t
ω(p̄, t)
,
(4.18)
we obtain the following differential equations for the coefficients
α̇p̄ (t) =
eE(t)ε⊥ ∗
β (t) e2iΘ(p̄,t) ,
2ω 2 (p̄, t) p̄
(4.19)
β̇p̄∗ (t) = −
eE(t)ε⊥
αp̄ (t) e−2iΘ(p̄,t) .
2ω 2 (p̄, t)
128
As the result of the Bogoliubov transformation we obtained the new coefficients
of the instantaneous state at the time t. The relations between them read
1
(−)
(−)
ω(p̄, t) χ (p̄, t) + i χ̇ (p̄, t) eiΘ(p̄,t) ,
αp̄ (t) = q
2 ω(p̄, t) (ω(p̄, t) − Pk (t))
βp̄∗ (t)
1
= − q
ω(p̄, t) χ
2 ω(p̄, t) (ω(p̄, t) − Pk (t))
(−)
(p̄, t) − i χ̇
(−)
(p̄, t) e
−iΘ(p̄,t)
It is convenient to introduce new operators which absorb the dynamical phase
Bp̄r (t) = bp̄r (t) e−iΘ(p̄,t) ,
Dp̄r (t) = dp̄r (t) e−iΘ(p̄,t)
(4.21)
(4.22)
satisfying the anti-commutation relations:
{Bp̄r (t), Bp̄+0 r0 (t)} = {Dp̄r (t), Dp̄+0 r0 (t)} = δrr0 δp̄p̄0 .
(4.23)
It is easy to show that these operators satisfy the Heisenberg-like equations of
motion
dBp̄r (t)
eE(t)ε⊥ +
=− 2
D (t) + i [H(t), Bp̄r (t)] ,
dt
2ω (p̄, t) −p̄r
(4.24)
eE(t)ε⊥ +
B (t) + i [H(t), Dp̄r (t)] ,
2ω 2 (p̄, t) −p̄r
dDp̄r (t)
=
dt
where H(t) is the hamiltonian of the quasiparticle system
H(t) =
X
ω(p̄, t)
r,p̄
+
Bp̄r
(t)
Bp̄r (t) − D−p̄r (t)
+
D−p̄r
(t)
.
(4.25)
The first term on the r.h.s. of Eqs. (4.24) is caused by the unitary non-equivalence
of the in-representation and the quasiparticle one.
Now we explore the evolution of the distribution function of electrons with
the momentum p̄ and spin r defined as
+
+
fr (p̄, t) =< 0in |bp̄r
(t) bp̄r (t)|0in >=< 0in |Bp̄r
(t) Bp̄r (t)|0in > .
(4.26)
According to the charge conservation the distribution functions for electrons and
positrons are equal fr (p̄, t) = f¯r (p̄, t), where
+
f¯r (p̄, t) = < 0in |d+
−p̄r (t) d−p̄r (t)|0in > = < 0in |D−p̄r (t) D−p̄r (t)|0in > .
(4.27)
The distribution functions (4.26) and (4.27) are normalized to the total number
of pairs N (t) of the system at a given time t
X
r,p̄
fr (p̄, t) =
X
f¯r (p̄, t) = N (t) .
r,p̄
129
(4.28)
(4.20)
.
Time differentiation of Eq. (4.26) leads to the following equation
eE(t)ε⊥
dfr (p̄, t)
=− 2
Re{Φr (p̄, t)} .
dt
ω (p̄, t)
(4.29)
Herein, we have used the equation of motion (4.24) and evaluated the occurring
commutator. The function Φr (p̄, t) in Eq. (4.29) describes the creation and
annihilation of an electron-positron pair in the external electric field E(t) and is
given as
Φr (p̄, t) =< 0in |D−p̄r (t) Bp̄r (t)|0in > .
(4.30)
It is straightforward to evaluate the derivative of this function. Applying the
equations of motion (4.24), we obtain
dΦr (p̄, t)
eE(t)ε⊥
2fr (p̄, t) − 1 − 2iω(p̄, t) Φr (p̄, t) ,
=
dt
2ω 2 (p̄, t)
(4.31)
where because of charge neutrality of the system, the relation fr (p̄, t) = f¯r (p̄, t)
is used. The solution of Eq. (4.31) may be written in the following integral form
Φr (p̄, t) =
ε⊥
2
Z
t
t0
dt0
eE(t0 )
0
2fr (p̄, t0 ) − 1 e2i[Θ(p̄,t )−Θ(p̄,t)] .
2
0
ω (p̄, t )
(4.32)
The functions Θ(p̄, t) and Θ(p̄, t0 ) in Eq. (4.32) can be taken at t0 (see the
definition (4.15)). Hence with A(t0 ) = 0, the function Φr (p̄, t) vanishes at t0 .
Inserting Eq. (4.32) into the r.h.s of Eq. (4.29) we obtain
eE(t)ε⊥
dfr (p̄, t)
=
dt
2ω 2 (p̄, t)
Z
0
0 eE(t )ε⊥
dt 2
1
ω (p̄, t0 )
−∞
t
0
0
− 2fr (p̄, t ) cos 2[Θ(p̄, t) − Θ(p̄, t )] .
(4.33)
Since the distribution function obviously does not depend on spin (4.33), we can
define: fr = f . With the substitution f (p̄, t) → F (P̄ , t), where the 3-momentum
is now defined as P̄ (p⊥ , Pk (t)), the kinetic equation (4.33) is reduced to its final
form:
dF (P̄ , t)
∂F (P̄ , t)
∂F (P̄ , t)
=
+ eE(t)
= S(−) (P̄ , t) ,
(4.34)
dt
∂t
∂Pk (t)
with the Schwinger source term
0
eE(t)ε⊥ Z t
0 eE(t )ε⊥
dt
1 − 2F (P̄ , t0 ) cos 2[Θ(p̄, t) − Θ(p̄, t0 )] .
2ω 2 (p̄, t) −∞
ω 2 (p̄, t0 )
(4.35)
Recently in Ref.[6], a kinetic equation similar to (4.34) has been derived within a
projection operator formalism for the case of a time-independent electric field
where it was first noted that this source term has non-Markovian character.
As well as in this method the multiple pair creation is not considered. The
S(−) (P̄ , t) =
130
presence of the Pauli blocking factor [1 − 2F (P̄ , t)] in the source term has been
obtained earlier in Ref.[18]. We would like to emphasize the closed form of the
kinetic equation in the present work where the source term does not include
the anomalous distribution functions (4.30) for fermion-antifermion pair creation
(annihilation).
Creation of boson pairs
In this subsection we derive the kinetic equation with source term for the case of
pairs of charged bosons in a strong electric field.
The Klein-Gordon equation reads
µ
µ
(∂ + ieA )(∂µ + ieAµ ) + m
2
φ(x) = 0 .
(4.36)
The solution of the Klein-Gordon equation in the presence of the electric field
defined by the vector potential Aµ = (0, 0, 0, A(t)) is taken in the form [24]
(±)
φp̄ (x) = [2ω(p)]−1/2 eix̄p̄ g (±) (p̄, t) ,
(4.37)
where the functions g (±) (p̄, t) satisfy the oscillator-type equation with a variable
frequency
g̈ (±) (p̄, t) + ω 2 (p̄, t) g (±) (p̄, t) = 0 .
(4.38)
Solutions of Eq. (4.38) for positive and negative frequencies are defined by their
asymptotic behavior at t0 = t → −∞ similarly to Eq. (4.7).
The field operator in the in-state is defined as
φ(x) =
Z
(−)
(+)
d3 p [ φp̄ (x) ap̄ (t0 ) + φp̄ (x) b+
vp (t0 ) ] .
(4.39)
The diagonalization of the hamiltonian corresponding to the instantaneous stateis
achieved by the transition to the quasiparticle representation. The Bogoliubov
transformation for creation and annihilation operators of quasiparticles has the
following form
ap̄ (t)
= αp̄ (t) ap̄ (t0 )
+ βp̄ (t) b+
−p̄ (t0 ) ,
(4.40)
b−p̄ (t) = α−p̄ (t) bp̄ (t0 ) + β−p̄ (t) a+
−p̄ (t0 )
with the condition
|αp̄ (t)|2 − |βp̄ (t)|2 = 1 .
(4.41)
The derivation of the Bogoliubov coefficients α and β is similar to that given in
the previous subsection. We obtain the equations of motion for the coefficients
of the canonical transformation (4.40) as follows
ω̇(p̄, t) ∗
βp̄ (t) e2iΘ(p̄,t) ,
2ω(p̄, t)
ω̇(p̄, t) ∗
αp̄ (t) e2iΘ(p̄,t) .
β̇p̄ (t) =
2ω(p̄, t)
α̇p̄ (t) =
131
(4.42)
(4.43)
Following the derivation for the case of fermion production, we arrive at the
final result for the source term in the bosonic case
0
eE(t)ε⊥ Z t
0
0
0 eE(t )ε⊥
1 + 2F (P̄ , t ) cos 2[Θ(p̄, t) − Θ(p̄, t )] ,
dt 2
S(+) (P̄ , t) =
2ω 2 (p̄, t) −∞
ω (p̄, t0 )
(4.44)
which differs from the fermion case just by the sign in front of the distribution
function due to the different statistics of the produced particles.
4.1.3
Discussion of the source term
Properties of the source term
We can combine the results for fermions (4.35) and for bosons (4.44) into a single
kinetic equation
dF(±) (P̄ , t)
∂F(±) (P̄ , t)
∂F(±) (P̄ , t)
= S(±) (P̄ , t) .
=
+ eE(t)
dt
∂t
∂P3
(4.45)
Here, the upper (lower) sign corresponds to the Bose-Einstein (Fermi-Dirac)
statistics. Based on microscopic dynamics, these kinetic equations are exact
within the approximation of a time-dependent homogeneous electric field and
the neglect of collisions. The Schwinger source terms (4.35) and (4.44) are characterized by the following features:
1. The kinetic equations (4.45) are of non-Markovian type due to the explicit
dependence of the source terms on the whole pre-history via the statistical
factor 1 ± 2F (P̄ , t) for fermions or bosons, respectively. The memory effect
is expected to lead to a modification of particle pair creation as compared
to the (Markovian) low-density limit, where the statistical factor is absent.
2. The difference of the dynamical phases, Θ(p̄, t)−Θ(p̄, t0 ), under the integrals
(4.35) and (4.44) generates high frequency oscillations.
3. The appearance of such a source term leads to entropy production due
to pair creation (see also Rau[6]) and therefore the time reversal symmetry
should be violated, but it does not result in any monotonic entropy increase
(in absence of collisions).
4. The source term and the distribution functions have a non-trivial momentum dependence resulting in the fact that particles are produced not only
at rest as assumed in previous studies, e.g. Ref.[19].
5. In the low density limit and in the simple case of a constant electric field
we reproduce the pair production rate given by Schwinger’s formula
cl
−3
S = lim (2π) g
t→+∞
Z
e2 E 2
d P S(P̄ , t) =
exp
4π 3
3
132
πm2
−
|eE|
.
(4.46)
As noted above, Rau[6] found that the source term has a non-Markovian
behaviour by deriving the production rate within a projector method. In the
limit of a constant field our results agree with Ref.[6]. In our approach the electric
field is treated as a general time dependent field and hence there is no a priori
limitation to constant fields. However our result allows to explore the influence
of any time-dependent electric field on the pair creation process. It is important
to note that in general this time dependence should be given by a selfconsistent
solution of the coupled field equations, namely the Dirac (Klein-Gordon) equation
and the Maxwell equation. This would incorporate back reactions as mentioned in
the introduction. However, the solution of such a system of equations is beyond
the scope of this work. Herein we will restrict ourselves to the study of some
features of the new source term.
Finally we remark that the source term is characterized by two time scales:
the memory time
ε⊥
(4.47)
τmem ∼
eE
and the production interval
τprod = 1/ < S(±) > ,
(4.48)
with < S(±) > denoting the time averaged production rate. As long as E <<
m2 /e < ε2⊥ /e, the particle creation process is Markovian: τmem << τprod . This
results for constant fields agree with those of Rau[6].
30.0
20.0
S/ε
10.0
0.0
-10.0
1.0
S/ε
0.0
-1.0
-2.0
-3.0
-1.0
1.0
3.0
p|| /ε
Figure 4.1: The pair production rate as a function of parallel momentum for
a constant, strong field (upper plot: Ẽ0 = 1.5) and a weak field (lower plot:
Ẽ0 = 0.5) at t̃ = 0.
133
Numerical results
In order to study the new source term, we consider two different cases, namely a
constant field and a time dependent electric field. For the numerical evaluation
we start with Eq. (4.35) assuming a dilute system, F = 0, and
0
S(p̃k , t̃)
Ẽ(t̃) Z t̃
0 Ẽ(t̃ )
d
t̃
cos 2[Θ(p̃k , t̃) − Θ(p̃k , t̃0 )] ,
=
ε⊥
2ω̃ 2 (p̃k , t̃) −∞ ω̃ 2 (p̃k , t̃0 )
(4.49)
where we have introduced dimensionless variables
S̃(p̃k , t̃) =
Ẽ(t̃) = eE(t̃)/ε2⊥ ,
p̃k = pk /ε⊥ ,
t̃ = tε⊥ ,
(4.50)
ω̃ = ω/ε⊥ .
(4.51)
This notation is convenient to distinguish the weak field (Ẽ < 1) and strong field
(Ẽ > 1) limits. A particularly simple result is obtained if we assume a constant
field,
Ã(t̃) = A(t̃)/ε⊥ = t̃Ẽ0 /e ,
(4.52)
where the electric field does not depend on time and the energy is given as
ω̃02 (p̃k , t̃) = 1 + (p̃k − Ẽ0 t̃)2 .
(4.53)
For the source term we obtain
S̃(p̃k , t̃) =
Ẽ02
2ω̃02 (p̃k , t̃)
Z
t̃
−∞
dt̃0
1
cos 2
ω̃02 (p̃k , t̃0 )
Z
t̃
t̃0
dt̃00 ω̃0 (p̃k , t̃00 ) .
(4.54)
In Fig. 4.1 we plot the particle production rate as function of the parallel
momentum pk for a weak constant field and a strong field, respectively. The
rates are normalized to be of the order of one for Ẽ0 = 0.5 at zero values of both
momentum and time. The production rate is positive when particles are produced
with positive momenta. Pairs with negative momenta are moving against the
field and hence get annihilated, clearly to be seen in the negative production
rate. These results mainly agree with those of Ref.[6], since no time dependence
was considered. Note that using the prefactor and field strengths chosen by Rau,
we reproduce exactly the results given in Ref.[6].
In considering the time dependence of the source term, we go beyond the
analysis of Ref.[6]. In Fig. 4.2, we display the time dependence of the production
rate at zero momentum. The maximum of the production rate is concentrated
around zero and shows an oscillating behaviour for large times. Indeed it is
possible to write Eq. (4.54) in terms of the Airy function because the constant
field provides an analytical solution for the dynamical phase difference using the
energy given in Eq. (4.53). The production of pairs for strong fields is larger
134
30.0
20.0
S/ε
10.0
0.0
-10.0
1.0
S/ε
0.0
-1.0
-2.0
-3.0
2.0
7.0
tε
Figure 4.2: The pair production rate as a function of time for a constant, strong
field (upper plot: Ẽ0 = 1.5) and a weak field (lower plot: Ẽ0 = 0.5) at zero
parallel momentum.
Figure 4.3: The pair production rate, S̃(p̃k , t̃), as a function of time and parallel
momentum for a time dependent weak electric field charactarized by Ẽ0 = 0.5,
σ̃ = 1 and τ = 0 . All plotted values are dimensionless as described in the text.
135
Figure 4.4: The pair production rate, S̃(p̃k , t̃), as a function of time and parallel
momentum for a time dependent strong electric field charactarized by Ẽ0 = 1.5,
σ̃ = 1 and τ = 0 . All plotted values are dimensionless as described in the text.
than that of weak fields, and the typical time period of the oscillations is smaller.
The situation changes if we allow the electric field to be time dependent. We
assume a Gaussian field at the dimensionsless time τ with the width σ̃ = σε⊥ ,
Ẽ(t̃) = Ẽ0 e−(t̃−τ )
2 /σ̃ 2
(4.55)
and obtain
√ π
Ã(t̃) = −Ẽ0 σ
Erf[(t̃ − τ )/σ̃] − Erf[(−τ − t0 )/σ̃] .
2
The occuring error function is defined as
Z z
2
2
Erf(z) = √
e−x dx .
π 0
(4.56)
(4.57)
Using this Ansatz for the field strength in Eq. (4.49) we obtain the numerical
results plotted in Figs. 4.3 and 4.4. Therein all occuring values are dimensionsless. The electric field is non zero within a certain width σ̃ = 1 for τ = 0 around
t̃ = 0. Therefore the oscillations for times beyond the time where the electrical
field is finite are damped out. The pair production rate is peaked around small
momenta for both weak and strong fields. For strong fields the distribution is
shifted to positive momenta but remains still close to small parallel momenta. It
is important to point out that the production of particles happens not only at
rest (pk = 0) what is assumed in many works also addressing the back reaction
problem, e.g.[19]. In contrary we find a non-trivial momentum dependence of the
pair creation rate depending on the field strength and on time.
136
4.1.4
Summary
We have derived a quantum kinetic equation within a consistent field theoretical
treatment which contains the creation of particle-antiparticle pairs in a timedependent homogeneous electric field. For both fermions and bosons we obtain a
source term providing a kinetic equation of non-Markovian character. The source
term is characterized by a pair production rate that contains a time integration
over the evolution of the distribution function and therefore involves memory
effects.
For the simple case of a constant electric field in low density limit and Markovian approximation, we analytically and numerically reproduce the results of
earlier works[6]. Since our approach is not restricted to constant fields, we have
explored the dependence of the source term on a time dependent (model) electric
field with a Gaussian shape. Within these two different Ansätze for the field,
we have performed investigations of the time structure of the source term. The
production of pairs does not happen at rest only. We observe a non-trivial momentum dependence of the source term depending on the field strength and on
time.
The particle production source is dominated by two time scales: the memory
time and the production time. The numerical results mainly show oscillations
due to the dynamical phase and urge the need to include the Maxwell equation
to determine the electric field by physical boundary conditions (back reactions),
work in this direction is in progress. Furthermore, it would be of great interest to
extend this approach to the QCD case to explore the impact of a non-Markovian
source term on the pre-equilibrium physics in ultrarelativistic heavy-ion collisions.
137
Bibliography
[1] N.B. Naroshni and A.I. Nikishov, Yad. Fiz. 11 (1970) 1072 (Sov. J. Nucl.
Phys. 11 (1970) 596); V.S. Popov and M.S. Marinov, Yad. Fiz. 16 (1972)
809 (Sov. J. Nucl. Phys. 16 (1974) 449); V.S. Popov, Zh. Eksp. Teor. Fiz.
62 (1972) 1248 (Sov. Phys. JETP 35 (1972) 659); M.S. Marinov and V.S.
Popov, Fort. d. Phys. 25 (1977) 373.
[2] A. Bialas and W. Czyż, Phys. Rev. D 30 (1984) 2371; 31 (1985) 198;
Z. Phys. C 28 (1985) 255; Nucl. Phys. B 267 (1985) 242; Acta Phys. Pol. B17
(1986) 635.
[3] K. Kajantie and T. Matsui, Phys. Lett. B 164 (1985) 373.
[4] G. Gatoff, A.K. Kerman, and T. Matsui, Phys. Rev. D 36 (1987) 114.
[5] F. Cooper, J.M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky,
Phys. Rev. D 48 (1993) 190.
[6] J. Rau, Phys.Rev. D 50 (1994) 6911.
[7] R.S. Bhalerao and V. Ravishankar, Phys.Lett. B 409 (1997) 38.
[8] W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong
Fields (Springer-Verlag, Berlin, 1985).
[9] A.A. Grib, S.G. Mamaev, and V.M. Mostepanenko, Vacuum quantum effects
in strong external fields, (Atomizdat, Moscow, 1988).
[10] S. Nussinov, Phys. Rev. Lett. 34 (1975) 1286.
[11] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys. Rep. 97
(1993) 31.
[12] N.K. Glendenning and T. Matsui, Phys. Rev. D 28 (1983) 2890.
[13] F. Sauter, Z. Phys. 69 (1931) 742.
[14] W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714.
138
[15] J. Schwinger, Phys. Rev. 82 (1951) 664.
[16] F. Cooper and E. Mottola, Phys. Rev. D 40 (1989) 456.
[17] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,
Phys. Rev. Lett. 67 (1991) 2427.
[18] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,
Phys. Rev. D 45 (1992) 4659.
[19] Y. Kluger, J.M. Eisenberg, and B. Svetitsky, Int. J. Mod. Phys. E 2 (1993)
333 (here further references may be found).
[20] F. Cooper, S. Habib, Y. Kluger, E. Mottola, J.P. Paz, and P.R. Anderson,
Phys. Rev. D 50 (1994) 2848.
[21] J.M. Eisenberg, Phys. Rev. D 51 (1995) 1938.
[22] Ch. Best and J.M. Eisenberg, Phys. Rev. D 47 (1993) 4639.
[23] J. Rau and B. Müller, Phys. Rep. 272 (1996) 1.
[24] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill,
New York, 1964).
139
4.2
Time-dependent masses: The Inertial Mechanism
This lecture is devoted to construct the kinetic theory for vacuum creation of
particles with the masses depending on time. We shall name for short such
mechanism as inertial. A microscopic foundations of that dependence can be
different. The Higgs mechanism leads to the most popular models of such class,
when corresponding mean fields depend on time. It can consider rather general
quantum field models with non-polynomial interactions, where the separation of
non-stationary mean fields forms the time depending mass [1]. A well known example of this kind is the Witten – Di Veccia – Veneziano model [2, 3], where the
mean field conception was analyzed in the work [4]. The model of the NambuJona-Lasinio [5] and σ-model [6] are the other examples of such kind, where the
hadron masses are defined by intensity of a quark condensate. Its evolution can
be described at the hydrodynamical [7] or kinetic level [8]. The particle mass can
depend on the many-particle interaction in hot and dense nonstationary matter
[9, 10, 11]. The field dependence on the mass is a general factor determining its
time evolution in all these cases (F-class models). The conformal invariance of
the scalar-tensor gravitational theory establishes a time dependence of the particle mass by means of the conformal multiplier [12, 13]. Such theories should
be referred to other class (C-class). In the F-class theories, the vacuum particle
creation admits a well known interpretation based on the simplified vacuum tunneling model in an external field [14, 15, 16]. The C-class models are not admit
the similar interpretation. On the phenomenological level, however, both classes
have the uniform mathematical description. It will be shown in Sec. 4.2.1-4.2.3
for the scalar, fermion and massive vector boson QFT, correspondingly.
The kinetic theory construction will be based on the oscillator (holomorphic)
representation (OR) [17], which is the most economical method for the nonpertrubative description (in comparison with the Bogoliubov’s method of the
canonical transformation [18]) for the vacuum particle creation under the action
of a time-dependent strong fields. This approach leads at once to the quasiparticle representation (QPR) with diagonal operator forms in the momentum space
for the dynamical variable set and allows to get rather easily the Heisenberg-type
equations of motion for the creation and annihilation operators. An important
feature of the time dependent Fock representation is the necessity of agreement
of commutation (anti-commutation) relations with the equations of motion. This
circumstance can bring to the non-canonical quantization rules ( such example
will be considered in Sect.4.1).
On this OR basis it is possible to derive directly the corresponding kinetic
equations (KE) using the well known methods [19]. Some particular results are
published in the works [1, 20]. The kinetic theory for the scalar, spinor and
massive vector fields is constructed in the Sect. 2,3 and 4, respectively. The main
140
attention will be spared to the particle creation in the conformal cosmological
models (Sec. 4.2.4). It was shown there, that the choice of the equation of state
(EoS) of the Universe allows to obtain, in principle, the observed number densities
of matter and photons and, possibly, dark matter. The basic problem here is the
description of vacuum particle creation which must be self-consistent with EoS.
This problem remains outside of the framework of the present article.
Finally, in the Sect.6 it is considered an other class of scalar QFT systems with
non-monotonic self-interaction potentials admitting decomposition of the field
amplitude into the quasiclassical space homogeneous time-dependent background
field and the fluctuation part. In this case, the particle mass is defined by intensity
of quasiclassical field. As an example, it is analyzed here the self-interaction
potentials of the simplest polynomial type and a non-trivial case (the pseudoscalar
sector of the Witten – Di Veccia – Veneziano model). It is showed, that the
relevant definition of vacuum states allows to avoid of the tachyonic modes rise.
The main purpose of this work is the summarizing of all known relevant results
and to separate unsolved problems which will be listed shortly in Sec. 4.2.6.
We use the metric g µν = diag(1, −1, −1, −1) and the natural system h̄ = c =
1.
4.2.1
Scalar field
Oscillator and quasiparticle representations
Let consider the simplest case of the scalar real field with the time dependent
mass m(t) and the equation of motion
[∂µ ∂ µ + m2 (t)]ϕ(x) = 0.
(4.58)
The corresponding Lagrange function is
1
1
L = ∂µ φ ∂ µ φ − m2 (t)φ2 .
2
2
(4.59)
The assumption about the space homogeneity of the system allows to look for
the solution of Eq. (4.58) in a discrete momentum space in the following form
1 X ipx
e ϕ(p, t),
ϕ(x) = √
V p
(4.60)
where V = L3 and pi = (2π/L)ni with the integer ni for each i = 1, 2, 3.
The thermodynamical limit can be fulfilled in the resulting equations. Then the
oscillator-type equation of motion follows from the Eqs. (4.58) and decomposition
(4.60)
ϕ̈(±) + ω 2 (p, t)ϕ(±) (p, t) = 0,
(4.61)
141
where ω(p, t) is the time-dependent one-particle energy,
ω(p, t) =
q
m2 (t) + p2 .
(4.62)
The symbols (±) correspond to the positive and negative frequency solutions of
Eq. (4.60) defined by its free asymptotics in the infinite past (future) [18],
ϕ(±) (p, t → ∓∞) ∼ e±iω∓ t ,
where ω∓ =
q
(4.63)
m2∓ + p2 are defined by asymptotics of mass
m∓ = lim m(t).
t→∓∞
(4.64)
The asymptotics (4.63) correspond to the in(out) - states and are necessary for
definition in(out) - vacuums. This requirement, however, can be broken in cosmology [21]. We suppose here that such asymptotics exist and the relevant vacuum states will be denoted by |0i without indices ”in” or ”out” , which can be
defined from context. In the considered class of the problems, the classification
of the states on frequency sign turns out impossible for arbitrary time moment.
According to general analysis of the work [72], it leads to instability of vacuum
state at the action period of the inertial mechanism and vacuum particle creation. Then it is possible to talk about quasiparticle excitations at the period of
evolution of the system (ṁ(t) 6= 0) (QR) and to describe their vacuum creation
and annihilation over the vacuum state |0i.
The conception of ’quasiparticle’ plays the central role in QFT in the presence
of strong time dependent quasiclassical external fields [18, 72]. In the considered
conditions, it is the nearest realization of the articleconception in the standard
Fock representation of QFT, where an external field can be taken into account
on the basis of perturbative approach. Thus, QPR corresponds to a possibility to
write in diagonal form the row of commutative operators of physical (observable)
values (the complete QPR) or, as minimum, the Hamiltonian only (the uncomplete QPR in the case of several observables). It is natural, the question is about
the operators having the quadratic form in the Fock representation. Hence, interaction between the field constituents and self-interaction are not taken into
account here. It corresponds to the non-dissipative approximation in the kinetic
theory [73].
The transition to QPR can be realized by different ways. The traditional
method is based on time dependent canonical Bogoliubov’s transformation [18].
The alternative approach is based on the OR, which leads at once to QRP [17]. In
the considered case the transition to OR is fulfilled by means of the substitution
m± → m(t) in the dispersion law of the corresponding free field decompositions
(4.60)
n
o
1
1 X
q
ϕ(x) = √
eipx a(−) (p, t) + a(+) (−p, t)
(4.65)
V p 2ω(p, t)
142
and the corresponds generalized momenta
i X
π(x) = − √
V p
s
o
ω(p, t) ipx n (−)
a (p, t) − a(+) (−p, t) ,
e
2
(4.66)
where a(±) (p, t) are the positive and negative frequency amplitudes corresponding
to the asymptotic (4.63). The decompositions (4.65) and (4.66) are postulated
essentially and they are the basis of a nonperturbative dynamics, which will be
discussed below.
The substitution of the decompositions (4.65) and (4.66) into the Hamiltonian
H(t) =
1
2
Z
n
d3 x π 2 (x) + [∇ϕ(x)]2 + m2 (t)ϕ2 (x)
o
(4.67)
leads simultaneously to a diagonal form, which corresponds to the QPR,
H(t) =
n
o
1X
ω(p, t) a(+) (p, t)a(−) (p, t) + a(−) (p, t)a(+) (p, t) .
2 p
(4.68)
The equations of motion for the field amplitudes a(±) (p, t) can be obtained
now from the minimal action principle [17] or on the basis of the Hamilton equations
∂H
∂H
φ̇ =
= π,
π̇ = −
= 4φ − m2 (t)φ.
(4.69)
∂π
∂φ
The relevant equations of motion can be obtained from here with taking into
account of the decompositions (8) and (9),
ȧ(±) (p, t) =
1
∆(p, t)a(∓) (−p, t) ± iω(p, t)a(±) (p, t),
2
(4.70)
where the overdot denotes the time derivative and
∆(p, t) =
ω̇(p, t)
m(t)ṁ(t)
= 2
ω(p, t)
ω (p, t)
(4.71)
is the transition amplitude between states with positive and negative energies.
Then the canonical commutation relation
[ϕ(x), π(x0 )]t=t0 = iδ(x − x0 )
(4.72)
with the decompositions (4.65) and (4.66) form the standard commutation relations for the time dependent creation and annihilation operators,
h
i
a(−) (p, t), a(+) (p0 , t) = δpp0
(4.73)
etc. That relation is consistent obviously with the equations of motion (4.70).
143
The corresponding Hamilton operator (4.68) will be equal
H(t) =
X
ω(p, t) a
p
(+)
(p, t)a
(−)
1
(p, t) +
2
= H0 (t) + hν (t),
(4.74)
where
H0 (t) =
X
ω(p, t)a(+) (p, t)a(−) (p, t)
(4.75)
p
and
hν (t) =
1X
ω(p, t).
2 p
(4.76)
Thus, vacuum energy of zero oscillations (”zitterbewegung” ) depends on time.
The equation of motion (4.70) can be rewritten as the Heisenberg-type equation
h
i
1
(4.77)
ȧ(±) (p, t) = ∆(p, t)a(∓) (−p, t) + i H(t), a(±) (p, t) .
2
These equations (or Eqs. (4.70)) in the instantaneous QPR are the basis for
nonperturbative derivation of KE describing the processes of the scalar particles
creation and annihilation under action of the inertial mechanism.
Kinetic equation
The basic object of the kinetic theory is the quasiparticle distribution function
which is for the space homogeneous case
L3 f (p, t) = h0|a(+) (p, t)a(−) (p, t)|0i,
(4.78)
where |0i = |0iin is the initial (t → −∞) vacuum state. Differentiating the
distribution function (4.78) with respect to time, we obtain using the equations
of motion (4.70)
n
o
1
f˙(p, t) = ∆(p, t) f (+) (p, t) + f (−) (p, t) ,
2
(4.79)
where the auxiliary one-particle correlation functions are introduced
L3 f (+) (p, t) = h0|a(+) (p, t)a(+) (−p, t)|0i,
L3 f (−) (p, t) = h0|a(−) (−p, t)a(−) (p, t)|0i.
(4.80)
The equation of motion for the functions (4.80) can be obtained by analogy with
the equation (4.79). We write it out in the integral form
f
(±)
(p, t) =
Zt
0
dt0 ∆(p, t0 ) [1 + 2f (p, t0 )] e±2iθ(p;t,t ) ,
t0
144
(4.81)
where the initial conditions have been used f (±) (p, t0 ) = 0. According to Eq.
(4.79), it means the statement of the Cauchy problem f (p, t0 ) = 0. This condition should be considered together with the asymptotic relations (4.63) and
(4.64). Finally, the dynamical phase in Eq.(4.81) is equal
0
θ(p; t, t ) =
Zt
dτ ω(p, τ ).
(4.82)
t0
The substitution of Eqs.(4.81) in Eq.(4.79) brings to the resulting KE (we
write this equation in the thermodynamical limit V → ∞ at the fixed particle
density)
1
f˙(p, t) = ∆(p, t)
2
Zt
dt0 ∆(p, t) [1 + 2f (p, t0 )] cos [2θ(p; t, t0 )] ,
(4.83)
t0
where the source term in right side describing a variation of particle number with
given momentum as the result of vacuum creation and annihilation processes under the action of the inertial mechanism, ∆(p, t) is defined by Eq.(4.71). The
non-Markovian KE (4.83) has the same form that KE for the Schwinger mechanism of pair creation in an electric field [19]. This equation had been investigated
in detail in the works [22, 24] in application to description of pre-equilibrium
evolution of quark-gluon plasma generated at collisions of ultrarelativistic heavy
ions. The case of the scalar QED has considered in [17] for the electric field of
the arbitrary polarization.
As it follows from Eq.(4.71), in the framework of the inertial mechanism the
particle production rate is defined by the velocity of mass change
ξ(t) =
1 dm(t)
,
m(t) d(m0 t)
(4.84)
where m0 is the characteristic mass fixing the time scale. In particular, the
adiabatic process (ξ(t) 1) is not accompanied by vacuum particle creation.
Let us underline that KE (4.83) is valid in the framework of two basis suppositions:
a) the particles (or antiparticles) are absent in the in-state;
b) the collisionless approximation is used (i.e. the corresponding dissipative processes are not taken into account).
In the low density approximation f (p, t) 1 KE (4.83) brings to the solution
[22, 27]
2
t
Z
1
f (p, t) = dt0 ∆(p, t0 ) exp [2iθ(p; t, t0 )] ≥ 0.
4
t0
145
(4.85)
The KE (4.83) can be transformed to a system of ordinary differential equations with zero initial conditions
1
f˙ = ∆u,
2
u̇ = ∆(1 + 2f ) − 2ωv,
v̇ = 2ωu,
(4.86)
which is convenient for numerical analysis. This equation system has the first
integral
(1 + 2f )2 − u2 − v 2 = 1,
(4.87)
according to it the phase trajectories are located on two-cavity hyperboloid with
top coordinates f = u = v = 0 (physical branch) and f = −1, u = v = 0
(nonphysical one).
Excluding the function f from Eqs. (4.86), we obtain the two dimensional
dynamical system with the equations of motion
√
u̇ = ∆ 1 + u2 + v 2 − 2ωv,
v̇ = 2ωu,
(4.88)
having non-Hamilton structure.
The functions f (p, t) and u(p, t) have the direct physical meaning (the last
function describes the vacuum polarization effects, see below Sect.2.3) and are
invariants relatively of the time inversion t → −t while the auxiliary function
v(p, t) changes its sign. Thus, KE (4.83) is invariant at the time inversion.
Observable variables and regularization
The KE (4.83) describes the vacuum quasiparticle excitations existing during
an external influence (ṁ(t) 6= 0, in the considered case). After switch off of this
action, some density of real (residual) particles and antiparticles are remained. In
absence of an interaction between the system constituents, the real particles are
”on-shell” one with the dispersion laws ω± of the free particles with the masses
m± (4.64) while the quasiparticles are ”off-shell” one with the dispersion law
(4.62). In general case, the on-shell condition
m(t) − m 0
m0
1
(4.89)
(m0 is some characteristic mass, e.g., m0 = m± ), is not connected directly with
< 1, where ξ(t)
the condition of the effectiveness of vacuum particle creation, ξ(t) ∼
is defined by Eq.(4.84). Differently, the presence of high frequencies in the function m(t) is necessary for vacuum creation and is not contradict to the on-shell
condition (4.89). In principle, the KEs of such type are intended for description
of evolution both real particles and quasiparticles. In particular, the distribution
function of the residual particles is fout (p) = lim f (p, t). The simple formula
t→∞
146
for fout (p) follows from Eq. (4.85) in the low density approximation. However,
the presence of the fast oscillated multiplier in the source term in r.h.s. of KE
(4.83) brings to huge volume of numerical calculations which make impossible
the research of the system evolution for rather great scales after the initial moment of switch on an external forces. The corresponding large scaling methods
of calculations based on the KE (4.83) are not worked out at presence. The estimation of some properties of residual particle - antiparticle plasma as the result
of a limited pulse of an external field action can be fulfilled on the basis of the
imaginary time method [68].
The distribution function is the basic observable characteristic of system. The
densities of other observable variables are some integrals in the momentum space
containing the distribution function and the auxiliary functions u(p, t), which
describe the effects of vacuum polarization. The simplest variable of such type
is the total density of particles and antiparticles considered as the quasiparticles.
We have in the thermodynamical limit L → ∞
ntot (t) =
Z
[dp]f (p, t),
(4.90)
where [dp] = (2π)−3 d3 p.
The other important characteristics of the system are the energy density and
pressure, which can be obtained as the average value of the energy-momentum
tensor corresponding to the Lagrange density (4.127),
Tµν = ∂µ φ∂ν φ − gµν L.
(4.91)
As the result we have [18]
ε = h0|T00 |0i =
3p = ε −
Z
Z
[dp] ωf,
(4.92)
1
m2
[dp]
(f + u) + ωu .
ω
2
(4.93)
The last two terms in the Eq.(4.93) represent the contribution of vacuum polarization. These equations produce the time-dependent p-ε EoS of the system.
Finally, the entropy density can be introduced
S(t) = −
Z
[dp][f ln f − (1 + f ) ln (1 + f )].
(4.94)
It does not conserve (Ṡ(t) 6= 0) even in the considered non-dissipative approximation because the system is open (the mass change is defined by external causes).
The direct proof of the convergence of (4.90), (4.92), (4.93), (4.94) integrals
is complicated here because of an absence of an explicit form of these functions.
Therefore, the method of the asymptotic expansions in power series of the inverse
momentum p−n (N-wave regularization technique) [25] is used usually (the other
147
approach is based on WKB approximation [26]). The applicability of this method
in the considered case can raise the doubts because of the researched functions are
quickly oscillating at p → ∞. Let us show it by means of the explicit asymptotic
solutions of the system (4.86) by |p| m. It is assumed that the integral (4.90)
is convergent at any time. Then the function f (p, t) must decrease at p → ∞
and hence f (p, t) 1 in that region. This inequality corresponds to the low
density approximation (4.85), where the KE solution can be written in explicit
form
Zt
2
1 0
0
0
0 ∞
(4.95)
f (p, t) =
dt m(t )ṁ(t ) exp [2ip (t − t )] ,
4
4p
t0
p = |p| → ∞ and
1
u (p, t) + iv (p, t) = 2
p
∞
∞
Zt
t0
dt0 m(t0 )ṁ(t0 ) exp [2ip (t − t0 )],
(4.96)
because δ ∞ (p, t) = m(t)ṁ(t)/p2 according to Eq. (4.71). Thus, asymptotic
solution is indeed some quickly oscillating function (this fact was observed first
in the work [20] in the case of massive vector bosons, see Sect. 4.2.3). Such
behaviour corresponds to quasiparticle interpretation of vacuum exitations under
action of the inertial mechanism. The real (observed) particles are the result of
the evolution to the moment, when ṁ = 0 and the out-vacuum state is realized.
The asymptotics (4.95) and (4.96) can be changed for other (non-inertial) mechanisms of vacuum particle creation ( e.g., in the case of harmonic ”laser” electric
field [28, 29].
The regularizing procedure of the Pauli-Villars type is based on the substraction of the corresponding counterterms into the integrals of the type (4.90), (4.92),
(4.93), (4.94) . These counterterms can be obtained by the change p2 → p2 + M 2
in the denominators of the asymptotics (4.95) and (4.96),
fR = f − f M ,
uR = u − u M .
(4.97)
If the regularizing mass M m(t) can be chosen rather large, M Λ (Λ is
”the computer cut-off parameter”), influence of the conuterterms on the results
of numerical calculations is negligible small.
4.2.2
Fermion field
Quasiparticle representation
The equations of motion of the fermion field with variable mass have the form
[iγ µ ∂µ − m(t)]ψ(x) = 0,
ψ̄(x)[iγ µ ∂µ + m(t)] = 0,
148
(4.98)
where ψ̄ = ψ + γ 0 . The corresponding Hamiltonian is (k = 1, 2, 3)
H(t) = i
Z
3
+
d x ψ ψ̇ =
Z
d3 x ψ̄{−iγ k ∂k + m(t)}ψ.
(4.99)
In the considered case, the system is space homogeneous and non-stationary.
Therefore the transition to Fock space can be realized on the basis functions ϕ =
exp(±ikx) and creation and annihilation operators become the time dependent
ones. So, we have the following decompositions of the field functions in discrete
momentum space
1 X X ipx
(α)
ψ(x) = √
{e aα (p, t)u(α) (p, t) + e−ipx b+
(p, t)},
α (p, t)v
V p α=1,2
1 X X −ipx +
{e
aα (p, t)ū(α) (p, t) + eipx bα (p, t)v̄(α) (p, t)}. (4.100)
ψ̄(x) = √
V p α=1,2
The OR is intended for a derivation of the motion equations for the creation
and annihilation operators on the basis of the primary equations (4.98) and the
free u, v - spinors as the basic functions with the natural substitution m → m(t).
The following free-like equations for the spinors are postulated in OR as
[γp − m(t)] u(p, t) = 0,
[γp + m(t)] v(p, t) = 0,
(4.101)
where p0 = ω(p, t). These definitions create the set of the standard orthonormality conditions [30]
m(t)
δαβ ,
v̄ α (p, t)
ω(p, t)
m(t)
δαβ ,
v (β) (p, t) = −
ω(p, t)
u(α)+ (p, t)u(β) (p, t) = v (α)+ (−p, t)v(β) (−p, t) = δαβ ,
ūα (p, t)v(β) (p, t) = ū(α) (p, t)γ 0 v(β) (−p, t) = 0.
ūα (p, t)u(β) (p, t) =
(4.102)
The decompositions (4.100) and the relations (4.102) lead to the diagonal
form of the Hamiltonian (4.99) at once (before second quantization)
H(t) =
X
p,α
+
ω(p, t)[a+
α (p, t)aα (p, t) − bα (−p, t)bα (−p, t)].
(4.103)
Such form of the Hamiltonian is necessary for interpretation of the time dependent operators a+ , a (and b+ , b) as creation and annihilation operators of
quasiparticles (anti-quasiparticles). Thus, this way results to QPR. We will not
take an interest in diagonalization of the spin operator. Therefore, such QPR
can be named as incomplete one (see below Sec. 4.2.3).
149
Now, in order to get the equation of motion for the creation and annihilation
operators in the OR, let us substitute the decomposition (4.100) in Eqs.(4.98)
and use the relations (4.102). Then we obtain as the intermediate result the
following closed system of equations of motion in the matrix form
ȧ(p, t) + U(1) (p, t)a(p, t) + U(2) (p, t)b+ (−p, t) = −iω(p, t)a(p, t),
+
+
ḃ(−p, t) + a+ (p, t)V(1)
(p, t) − b(−p, t)V(2)
(p, t) = −iω(p, t)b(−p,(4.104)
t).
The spinor constructions were introduced here
αβ
U(1)
= u+(α) (p, t)u̇(β) (p, t),
αβ
V(1)
= v +(α) (−p, t)u̇(β) (p, t),
αβ
U(2)
= u+(α) (p, t)v̇(β) (−p, t),
αβ
V(2)
= v +(α) (−p, t)u̇(β) (−p, t) .
(4.105)
The matrixes U(2) and V(1) describe transitions between states with the positive
and negative energies and different spin while the matrixes U(1) and V(2) show the
spin rotations only.
The equations (4.104) are compatible with the standard anti-commutation
relations
0
+
0
{aα (k, t), a+
β (k , t)} = {bα (k, t), bβ (k , t)} = δkk0 δαβ ,
(4.106)
+
that can be shown with the help of the relation U(1)
= −U(1) .
The material part of the Hamilton operator will be equal then:
H0 (t) =
X
+
ω(p, t)[a+
α (p, t)aα (p, t) + bα (p, t)bα (p, t)] .
(4.107)
p,α
Let us write now the u, v - spinors in the explicit form, using the corresponding
free spinors [31]:
u+(1) (p, t)
u+(2) (p, t)
v +(1) (−p, t)
v +(2) (−p, t)
=
=
=
=
A(p) [ω+ , 0, p3 , p− ] ,
A(p) [0, ω+ , p+ , −p3 ] ,
A(p) [−p3 , −p− , ω+ , 0] ,
A(p) [−p+ , p3 , 0, ω+ ] ,
(4.108)
where p± = p1 ± ip2 , ω+ = ω + m and A(p) = [2ωω+ ]−1/2 .
The spin rotations matrixes (4.105) in the representation (4.108) are equal to
zero,
U(1) = V(2) = 0.
(4.109)
150
The rest matrixes (4.105) are equal
11
22
U(2)
(p, t) = −U(2)
(p, t) =
12
U(2)
(p, t) =
ṁ(t)p−
,
2ω 2 (p, t)
ṁ(t)p3
,
2ω 2 (p, t)
21
U(2)
(p, t) =
ṁ(t)p+
2ω 2 (p, t)
(4.110)
and in addition V(1) = −U(2) .
Kinetic equation
The equations of motion (4.104) are not contain the spin rotation matrices (4.109)
and they are similar to Eqs.(4.70), therefore the KE derivation has not problems
now. Let us introduce one-particle correlation function of electrons and positrons
gαβ (p, t) = < 0|a+
β (p, t)aα (p, t)|0 >,
g̃αβ (p, t) = < 0|bβ (−p, t)b+
α (−p, t)|0 > .
(4.111)
Differentiation the first of them over time leads to equation (we use the equations
of motion (4.104) and the short notation U(2) = U )
ġ(p, t) = −U(p, t)g(+) (p, t) − g(−) (p, t)U(p, t),
(4.112)
where the auxiliary correlation functions were introduced
(+)
+
gαβ (p, t) = < 0|a+
β (p, t)bα (−p, t)|0 >,
(−)
gαβ (p, t) = < 0|bβ (−p, t)aα (p, t)|0 > .
(4.113)
The subsequent transformations are distinguished from the KE derivation in the
case of scalar particles (Sec. 4.2.1) by replacement of the commutator (4.73) in
the anticommutator (4.106) and taking into account of matrix character of the
equations of motion (4.104). As the result, we have
ġ(p, t) =
Zt
t0
0
dt0 {U(p, t)[g̃(p, t0 )U(p, t0 ) − U(p, t0 )g(p, t0 )]e−2iθ(p,t ,t) + h.c.}.
(4.114)
The KE for the all component of the correlation function fαβ (p, t) (4.111) is not
of physical interest by virtue of isotropy of the system. Hence, gαβ = f δαβ , where
the distribution function of the fermi particles is introduced,
f (p, t) =
1 X
gαα (p, t).
2 α=1,2
151
(4.115)
Further, it is easy to calculate T r U (p, t)U(p, t0 ) with help of the equalities
(4.110). The corresponding KE for this distribution function has the standard
form [19, 22]:
f˙(p, t) = 2λ(p, t)
Zt
t0
dt0 λ(p, t0 )[1 − 2f (p, t0 )] cos[2θ(t, t0 )],
(4.116)
where the amplitude λ(p, t) is defined by the relation (p = |p|)
ω̇p
.
ω(p, t)m(t)
λ(p, t) =
(4.117)
The KEs (4.83) and (4.116) have the analogical form but differ by the statistical factors 1 ± 2f (the Boze enhancement or the fermi suppression) and by
structure of the vacuum transition amplitudes (4.71) and (4.117).
The corresponding system of ordinary differential equations has the following
form:
1
f˙ = λu,
u̇ = λ[1 − 2f ] − 2ωv,
v̇ = 2ωu.
(4.118)
2
This system possesses one first integral of motion (e.g., [33])
(1 − 2f )2 + v 2 + u2 = 1.
(4.119)
This relation represent the ellipsoid in the phase space of (f, u, v) variables. After
excluding function f from (4.118), we obtain the dynamical system in d = 2 + 1
space
√
v̇ = ±λ 1 − v 2 − u2 − 2ωu,
u̇ = 2ωv.
(4.120)
This system is non-Hamiltonian as a consequence of non-unitary transition from
the initial to the quasiparticale representation (see, Sect.2.3.).
As it can proof easily, KE (4.116) is invariant in respect to time inversion.
The equations analogous to (4.86) and (4.118) was obtained in the work [32] (see
also [18]) for the conformal flat space-time.
Observables and regularization
The observable variables of spinor field are defined by following expressions:
ntot (t) = 2
Z
[dp]f (p, t),
ε = h0|T00 |0i = 2
p(t) =
1
(ε + m2
3 Z
S(t) = −2
Z
Z
(4.121)
[dp]ωf,
[dp]u),
[dp][f ln f + (1 − f ) ln (1 − f )].
152
(4.122)
(4.123)
(4.124)
The same regularization procedure for calculation of the divergence integrals
showed in Sect.2.3 for the scalar bosons can be realized here. The Eq.(4.85) for
the distribution function in the low density approximation is valid here also after
the replacement ∆(p, t) → λ(p, t). The asymptotic of the amplitude (4.117) is
equal λ∞ (p, t) = ṁ(t)/p. Using Eq.(4.85) and the rules of Sect 2.3, it is easily
to obtain the following expressions for the counterterms for the case of fermion
fields:
t
Z
1 2
(p + M 2 )| dt0 ṁ(t0 ) exp [2ip(t − t0 )]|2 ,
fM (p, t) =
4
(4.125)
t0
uM (p, t) = √
p2
1
+ M2
Zt
t0
dt0 ṁ(t0 ) cos [2p(t − t0 )].
(4.126)
However these counterterms can be ignored by computer calculations.
4.2.3
Massive vector bosons
Complete QPR
The simplest version of quantum field theory of the massive vector bosons is
based on the Lagrangian density [30]
L(§) = −∂µ uν ∂ µ uν + m∈ (t)uν uν ,
(4.127)
which corresponds to the equation of motion
(∂µ ∂ µ + m2 (t))uν = 0
(4.128)
with the additional exterior constraint
∂µ uµ = 0.
(4.129)
The alternative way is based on the Lagrangian [34]
∞
∞
L(§) = − F µν Fµν + m∈ (t)uν uν ,
4
∈
(4.130)
where Fµν = ∂ν uµ −∂µ uν . The constraint (4.129) is here a consequence of dynamical equations. The Lagrangians (4.127) and (4.130) bring to different energymomentum tensors, that can be of great importance for EoS.
The transition to the QPR is carried out on the basis of the standard decomposition of the free field operators and momenta in the discrete momentum space
with the replacement m → m(t) in the dispersion law (see Sect.4.2.1),
uµ (x)
=
P
√1
V p
πµ (x) = − √iV
√
Pq
p
1
2ω(p,t)
eip x
n
o
aµ(−) (p, t) + a(+)
µ (−p, t) ,
n
o
(−)
2ω(p, t) e−ip x a(+)
µ (p, t) − aµ (−p, t) .
153
(4.131)
The substitution of the field operators (4.131) into the Hamiltonian
H=−
Z
dx πµ π µ + ∇uµ ∇uµ + m2 (t)uµ uµ
(4.132)
brings it at once to a diagonal form in Fock space, which corresponds to the QPR,
H=−
X
ω(p, t)
p
(−)µ
a(+)
(p, t)
µ (p, t)a
+
aµ(−) (−p, t)a(+)µ (−p, t)
.
(4.133)
However, this quadratic form is not positively defined. In order to exclude the
µ = 0 component with the help of the additional condition (4.129), it is necessary
to derive the equations for the amplitudes a± .
Substituting the decompositions (4.131) into the equation of motion (4.128),
we find the Heisenberg equations of motion for the time-dependent creation and
annihilation amplitudes
1
∆(p, t)aµ(∓) (−p, t) ± iω(p, t)aµ(±) (p, t),
(4.134)
2
where ∆(p, t) is defined by Eq.(4.71).
The additional condition (4.129) may be transformed now with the help of
Eqs. (4.134) to the following form (i = 1, 2, 3)
ȧµ(±) (p, t) =
(±)
(±)
ω(p, t)a0 (p, t) = pi ai (p, t).
(4.135)
These equations allow to exclude the µ = 0 component in the Hamiltonian (4.133)
with the result
H=
X
(+)
(−)
(−)
(+)
ω(p, t) ai (p, t)ai (p, t) + ai (−p, t)ai (−p, t)
p,i,k
1
(−)
(−)
(+)
(+)
t) .
pi ai (p, t)pk ak (p, t) + pi ai (−p, t)pk ak (−p,(4.136)
−
2
ω (p, t)
The next step is the additional diagonalization of the quadratic form (4.136)
by means of the linear transformations [30]
ω
(±)
(±)
(±)
a(±) (p, t) = Eα(±) (p, t) ≡ e1 α1 (p, t) + e2 α2 (p, t) + e3
α3 (p, t),
m(t)
(4.137)
where {e1 (p), e2 (p), e3 (p)} is the local basis built on the vector e3 = p/|p|.
These real unit vectors form the triad,
eik ejk = eki ekj = δij ,
eik = (ei )k .
(4.138)
The presence of the factor ω/m in the non-unitary matrix E in Eq. (4.137) leads
to a violation of the unitary equivalence between the a and α - representations.
The transformation (4.137) establishes the positively defined Hamiltonian
H=
X
p
ω(p, t)
(+)
(−)
αi (p, t)αi (p, t)
+
154
(−)
(+)
αi (−p, t)αi (−p, t)
.
(4.139)
The equations of motion for these new amplitudes follow from a combination
of Eqs. (4.134) and (4.137):
1
(∓)
(±)
(±)
∆(p, t)αi (−p, t) ± iω(p, t)αi (p, t) + ηij (p, t)αj (p, t).
2
(4.140)
The spin rotation matrix ηij is defined as
(±)
α̇i (p, t) =
ηik (p, t) = −∆m δi3 δk3 ,
(4.141)
where ∆m = −ṁ/m + ∆. This relation shows the distinguished role of the third
component.
Together with the Hamiltonian (4.133), the total momentum operator takes
also the diagonal form. However, the spin operator
Si = εijk
Z
dx uk πj + πj uk − uj πk − πk uj
(4.142)
has non-diagonal form in spin space in terms of the operator α(±)
Sk = iεijk
X
(+)
(−)
αi (p, t)αj (p, t)
p
−
(−)
(+)
αi (−p, t)αj (−p, t)
,
(4.143)
εijk is the unit antisymmetric tensor. In particular, the spin projection to the
p3 -axis is
S3 = i
X
p
+
(+)
(−)
(+)
(−)
α1 (p, t)α2 (p, t) − α2 (p, t)α1 (p, t)
(−)
(+)
α2 (−p, t)α1 (−p, t)
−
(−)
(+)
α1 (−p, t)α2 (−p, t)
.
(4.144)
Thus, this representation can be named an incomplete quasiparticle one with
non-fixed spin projection. The operator (4.144) can be diagonalized with a linear
transformation to the new basis [30]
(±)
(±)
(±)
ci (p, t) = Rik αk (p, t),
(4.145)
1 ∓i 0
1 

±
R = √  ±i 1 √0  .
2
2
0 0
(4.146)
with the unitary matrix


As a result, the new amplitudes c(±) correspond to the creation and annihilation
operators of vector quasiparticles with the total energy, 3-momentum and spin
155
projection on the chosen direction,
H(t) =
X
ω(p, t)
p
Π(t) =
X p
(+)
(−)
ci (p, t)ci (p, t)
(+)
(−)
ci (p, t)ci (p, t)
p
S3 (t) =
X
p
(+)
−
(−)
+
(−)
(+)
ci (−p, t)ci (−p, t)
(−)
(+)
ci (−p, t)ci (−p, t)
(−)
,
,(4.147)
(4.148)
(+)
c1 (p, t)c1 (p, t) − c1 (−p, t)c1 (−p, t)
(−)
(+)
(+)
(−)
+ c2 (−p, t)c2 (−p, t) − c2 (p, t)c2 (p, t) .
(4.149)
This representation can be named the complete quasiparticle one. The equations
of motion for these amplitudes follow from Eqs. (4.140),(4.145)
1
(∓)
(±)
(±)
(±)
∆(p, t)ci (−p, t) ± iω(p, t)ci (p, t) + ηij (p, t)cj (p, t),
2
(4.150)
where the matrix ηij is fixed by Eq.(4.141).
The transition to this representation from the initial one is defined by the
combination of the transformations (4.137) and (4.145),
(±)
ċi (p, t) =
c(±) (p, t) = U(±) (p, t)a(±) (p, t),
√
with non-unitary operator (e(±) = (e1 ± ie2 )/ 2)
(∓)
e1
U (±) (p, t) = R(±) · E−1 (p, t) = e1(±)
m
e
ω 31
(∓)
e2
(±)
e2
m
e
ω 32
(4.151)
(∓)
e3
(±)
e3 .
m
e
ω 33
(4.152)
The quantization problem is solved while taking into account the equation of
motion. The commutation relation has the non-canonical form
(−)
[ci
(+)
(p, t), cj (p0 , t)] = Qik (p, t)Qjk (p, t)δpp0 ,
(4.153)
where the matrices Qil (p, t) are defined by the equations
Q̇ij (p, t) = ηik (p, t)Qkj (p, t)
(4.154)
with the initial conditions
lim Qij (p, t) = δij ,
t→−∞
(4.155)
i.e. the commutation relation transform to the canonical form only in the asymptotic limit t → −∞. The relation (4.153) provides the definition of positive energy quasiparticle excitations with some time dependent energy reservoir of the
vacuum.
156
Kinetic equations
The standard procedure to derive the KE [19] is based on the Heisenberg-type
equations of motion (4.134) or (4.150). Let us introduce the one-particle correlation functions of vector particles and antiparticles in the initial (a)-representation
(−)
Fµν (p, t) = h0|a(+)
µ (p, t)aν (p, t)|0i,
F̃µν (p, t) = h0|aµ(−) (−p, t)a(+)
ν (−p, t)|0i,
(4.156)
where the averaging procedure is performed over the in-vacuum state [18]. Differentiating the first one with respect to time, we obtain
n
o
1
(−)
∆(p, t) F(+)
(p,
t)
+
F
(p,
t)
,
µν
µν
2
where the auxiliary correlation functions are introduced as
(4.157)
Ḟµν (p, t) =
(±)
Fµν
(p, t) = h0|aµ(±) (±p, t)aν(±) (±p, t)|0i.
(4.158)
The equations of motion for these functions can be obtained by analogy with Eq.
(4.157). We write them out in the integral form
(±)
Fµν
(p, t)
1
=
2
Zt
h
i
0
dt0 ∆(p, t0 ) Fµν (p, t0 ) + F̃µν (p, t0 ) e∓2iθ(p;t,t ) .
−∞
(4.159)
(±)
The asymptotic condition Fµν
(p, −∞) = 0 (the absence of quasi-particles in the
initial time) has been introduced here. The substitution of Eq. (4.159) into Eq.
(4.157) leads to the resulting KE
1
Ḟµν (p, t) = ∆(p, t)
2
Zt
dt0 ∆(p, t0 )[Fµν (p, t0 ) + F̃µν (p, t0 )] cos[2 θ(p; t, t0 )].
−∞
(4.160)
This KE is an almost natural generalization of the corresponding KE for scalar
particles (Sec. 4.2.1).
On this stage, however, there is a number of problems that are specific for the
theory of massive bosons: the energy is not positively defined, the spin operator
has a non-diagonal form in the space of spin states etc., see above. This circumstance hampers the physical interpretation of the distribution function (4.156).
In order to overcome this difficulty, it is necessary to pass on to the complete
QPR in which the system has well-defined values of energy, spin etc. The simplest way of derivation of the KE in this QPR is based on the application of the
transformations (4.151) directly onto the KE (4.160).
By analogy with the definitions (4.156), we introduce the correlation functions
of vector particles and antiparticles in the complete QPR
(+)
(−)
fik (p, t) = h0in |ci (p, t)ck (p, t)|0in i,
(+)
(−)
f˜ik (p, t) = h0in |ci (−p, t)ck (−p, t)|0in i.
157
(4.161)
They are connected with the primordial correlation functions (4.156) by relations
of the type
−
fik (p, t) = U+
(4.162)
in (p, t)Ukm (p, t)Fnm (p, t),
where Fnm (p, t) is the ”spatial” part of the tensor function Fµν (p, t) (4.156)
(m, n = 1, 2, 3). As a result, the KE (4.160) takes the following form:
1
f˙ik (p, t) =
∆(p, t)
2
Zt
dt0 ∆(p, t0 )Mikjl (p, t, t0 )[fjl (p, t0 ) + f̃jl (p, t0 )] cos 2θ(p; t, t0 )
−∞
−∆m (p, t)[δi3 f3k (p, t) + δk3 fi3 (p, t)],
(4.163)
where
ω(t0 ) m(t)
ω(t0 ) m(t) h
⊥
⊥
δi3 δk3 δj3 δl3
δ
δ
δ
+
δ
δ
δ
+
Mikjl (t, t ) =
+
i3
j3
k3
l3
kl
ij
ω(t) m(t0 )
ω(t) m(t0 )
(4.164)
⊥
and δik = δik − δi3 δk3 .
As to was expected, the distribution functions fαβ (p, t) satisfy the same KE
(4.160) for α = 1, 2. The feature of the complete QPR becomes apparent only in
the components of tensor distribution function fik (p, t) that contain the preferred
values of spin index i and (or) k = 3. Let us select the KE for the diagonal
components of the correlation functions (4.161) having a direct physical meaning
as the distribution functions of the transversal (i = 1, 2)
0
#
⊥
δij⊥ δkl
1
f˙i (p, t) = ∆(p, t)
2
Zt
dt0 ∆(p, t0 )[1 + 2fi (p, t0 )] cos 2θ(p; t, t0 ),
(4.165)
−∞
and longitudinal components
t
1
m2 (t) Z
ω 2 (t0 )
˙
f3 (p, t) =
∆(p, t) 2
dt0 ∆(p, t0 ) 2 0 [2f3 (p, t0 ) + Q(p, t0 )] cos 2θ(p; t, t0 )
2
ω (t)
m (t )
−∞
−2∆m (p, t)f3 (p, t) .
(4.166)
Here the shorthand notation fii = fi has been introduced for the diagonal components of the matrix correlation functions (4.161), and
∆=
mṁ
,
ω2
∆m = −∆
p2
.
m2
(4.167)
The longitudinal and transversal distribution functions are connected by the relation
f3 (p, t) = Q(p, t)f1 (p, t),
(4.168)
158
where Q(p, t) is the function occurring in the commutator of the creation and
annihilation operators for the longitudinal bosons,
(−)
(+)
[c3 (p, t), c3 (p0 , t)] = Q(p, t) δpp0 ,
"
m(t) ω(t0 )
Q(p, t) = exp −2 ∆m (t0 )dt0 =
m(t0 ) ω(t)
t0
Z
t
#2
,(4.169)
where m(t0 ) and ω(t0 ) are corresponding initial values.
Owing to Eq. (4.168), it is sufficient to solve the one equation (4.165). KE for
transversal bosons is reduced from the integro-differential form to the system of
ordinary differential equations as well as the previous cases (Sects. 2.2 and 3.2):
1
f˙k = ∆uk ,
2
u̇k = ∆(1 + 2fk ) − 2ωvk ,
v̇k = 2ωuk .
(4.170)
The general initial condition for all diagonal components of the distribution function
lim fk (t) = lim uk (t) = lim vk (t) = 0
(4.171)
t→−∞
t→−∞
t→−∞
establishes the following requirement
lim m(t) = m0 ,
t→−∞
or
lim ṁ(t) = 0.
t→−∞
(4.172)
The main characteristic of the vacuum creation process is the total number
density of vector bosons,
1
ntot (t) = 2
π
Z∞
0
1
p dp [2f1 (p, t) + f3 (p, t)] = 2
π
2
Z∞
p2 dp f1 (p, t)[2 + Q(p, t)], (4.173)
0
where isotropy of the system was taken into account, p = |p|. As it will be shown
in Sec. 4.2.3, the integral (4.173) is convergent.
We will consider below the case when the time dependence of vector boson
mass is defined by the conformal evolution of the Universe.
The equation set similar to the (4.170) was be obtained first in the work [50]
in the framework of the alternative model Lagrangian (4.130) in FRW space-time.
The quantization procedure is ordinary in this approach. It would be useful to
compare the predictions of these two models in detail.
EoS for the isotropic case
The relations for the energy density and pressure can be obtained from the energymomentum tensor corresponding to the Lagrangian (4.127)
Tµν = −∂µ uα ∂ν uα − ∂ν uα ∂µ uα − gµν L.
159
(4.174)
Using the decompositions (4.131) and the relations, which are a consequence of
the spatial homogeneity of the system (in the discrete momentum representation),
(−)
0
h0|a(+)
µ (p, t)aν (p , t)|0i = δpp0 Fµν (p, t),
0
h0|aµ(−) (−p, t)a(+)
ν (−p , t)|0i = δpp0 F̃µν (p, t),
(±)
h0|aµ(±) (p, t)aν(±) (−p0 , t)|0i = δpp0 Fµν
(p, t),
(4.175)
we obtain
ε(t) =
< T00 > = −
p(t) =
1
< Tii > =
3
2
+ (2p + 3m
2
Z
[dp] ω(p, t)[Fµµ (p, t) + F̃µµ (p, t)],
[dp]
−p2 [Fµµ (p, t) + F̃µµ (p, t)]
ω(p, t)
Z
)[F(+)µ
(p, t)
µ
+
Fµ(−)µ (p, t)]
.
(4.176)
We perform now the series of the consecutive transformations of the functions
Fµµ (p, t), F̃µµ (p, t) and Fµ(±)µ (p, t): the exclusion of the µ = 0 component (according to Eq. (4.135)) and the transition to the complete QR representation
with the help of Eqs. (4.152). Thus, we arrive at
Fµµ (p, t) = −
Fµ(±)µ (p, t) = −
3
X
l=1
3
X
l=1
fl (p, t), F̃µµ (p, t) = −
(±)
fl
(p, t) − 2
3
X
f̃l (p, t),
l=1
p2 (±)
f (p, t),
m2 3
(4.177)
where fl (p, t), f˜l (p, t) and f (±) (p, t) are the corresponding functions in the complete QR representation. The substitution of these relations into Eqs. (4.176)
and (4.176) leads to the following expressions:
ε(t) = 2
1
p(t) =
3
3 Z
X
l=1
3 Z
X
l=1
[dp]ω(p, t)fl(p, t),
[dp]
2p2 fl (p, t) − [2ω 2 (p, t) + m2 ]
ω(p, t)
!
i h
i 2p2 h (+)
(−)
(+)
(−)
×
f (p, t) + f3 (p, t) + fl (p, t) + fl (p, t) (4.178)
.
3m2 3
Taking into account the isotropy of the system, we obtain the EoS for the massive
vector boson gas
Z
ε(t) = 2 [dp]ω(2 + Q)f1 ,
p(t) =
i
2 Z [dp] h 2
2p (2 + Q)f1 + δpvac (t),
3
ω
160
(4.179)
where δpvac (t) is the contribution in the pressure induced by the vacuum polarization,
2
δpvac (t) = −
3
Z
[dp]
2ω 2 + m2
ω
"
1 p2
1+Q
+
2 m2
!#
u1 .
(4.180)
In order to prove the convergence of the integrals (4.179), (4.180) we investigate the asymptotic behavior of the solution to the equations system (4.170).
This system can be solved exactly in the asymptotic limit p m for the case
α = 1/2 (the parameter α is defined in the Sec. 5)
m2 1
f˙ = H 2 u,
4tH p
u̇ =
m2H 1
(1 + 2f ) − 2pv,
2tH p2
v̇ = 2pu .
(4.181)
The asymptotic solution of Eqs. (4.181) with the initial conditions (4.171) is
f (p, t) =
v(p, t)
sin2 p(t − t0 )
∼
,
(2p/m0 )3
16(p/m0 )6
u(p, t) ≈
sin 2p(t − t0 )
,
4(p/m0 )3
(4.182)
2/3 −1/3
where m0 = m(t0 ) = mH tH and t0 = 1/m0 . The numerical investigation of
the general Eqs. (4.170) shows that the basic features of the solutions (4.182)
for α = 1/2 are conserved also for other α > 0. It corresponds to the results of
Sect.2.3.
According to Eq. (4.182), the particle number and energy densities (Eqs.
(4.173) and (4.179), respectively) are convergent, but the vacuum polarization
contribution to the pressure (4.180) is divergent. Moreover, irrelevant fast vacuum oscillations of the pressure are observed here. Such a behavior of the pressure
for a plasma created from vacuum is not a special feature of the present theory
but is characteristic also for the models where an electron - positron plasma is
created in strong, time-dependent electric fields as investigated in [35]. The standard regularization procedure of similar integrals with some unknown functions
satisfying ordinary differential equations is based on the investigation of asymptotic decompositions of these functions in power series of the inverse momentum,
1/pn (see Sect.4.2.1). In the considered case, such a procedure is not effective
because the solutions (4.181) has quickly oscillating factors (”zitterbewegung” ),
the asymptotic decompositions of which lead to the secular terms. Therefore we
regularize the pressure by a momentum cut-off at p = 10m0 for the numerical
calculation and separate its stable part by the time averaging
1
< p >=
(t − t0 )
Zt
p(t)dt.
(4.183)
t0
The such ”coarse graining” procedure was proposed in [36] in order to exclude the
”zitterbewegung” from the description of vacuum particle creation. In reality,
161
0,30
n1
1,5
Density [ cm
-3
]
Distribution function [ 10
-4
]
n3
0,24
0,18
0,12
0,06
α = 1/2
0,00
1,2
f1
f3
0,9
0,6
0,3
0,0
1,0x10
-11
3,0x10
-11
5,0x10
-11
0
1
2
3
4
p / m0
Time [ s ]
Figure 4.5: Time evolution of number density of vector bosons with the initial
condition m0 · t0 = 1 for α = 1/2 (left) and the corresponding momentum distribution at the time t t0 (right).
the smoothing of these fast oscillations occurs due to dissipative processes that
are not taken into account here (the first attempt of a derivation of the collision
integral for the scalar quasiparticle gas in a strong electric field was made in the
work [37]).
4.2.4
Application to conformal cosmology models
The description of the vacuum creation of particles in the time dependent gravitational fields of cosmological models goes back to Refs. [38]- [41] and has been
reviewed, e.g., in the monographs [21, 42, 57]. The specifics of our work consists in the consideration of vacuum generation of particles in the conditions of
the early Universe in the framework of a conformal-invariant cosmological model
[43], thus assuming, that the space-time is conformally flat and that the expansion of the Universe in the Einstein frame (with metric g̃µν ) with constant masses
m̃ can be replaced by the change of masses in the Jordan frame (with metric
gµν ) due to the evolution of the cosmological (scalar) dilaton background field
[12, 13, 44]. This mass change is defined by the conformal factor Ω(x) of the
conformal transformation
g̃µν (x) = Ω2 (x)gµν .
(4.184)
As mass terms generally violate conformal invariance, a space-time dependent
mass term
1
m̃
(4.185)
m(x) =
Ω(x)
has been introduced which formally keeps the conformal invariance of the theory
[13, 21]. In the important particular case of the isotropic FRW space-time, the
conformal factor is equal to the scale factor, Ω(x) = a(t̃), and hence m(t̃) =
162
16
10
6
10
n1
]
n3
-3
2
10
0
10
-2
10
α = 1/2
-4
12
10
10
10
8
10
α = 1/3
6
10
10
-6
10
n1
14
10
n3
Final density [ cm
Final density [ cm
-3
]
4
10
4
-18
10
-16
10
-14
10
-12
10
-10
10
10
-8
10
-22
10
-20
10
Initial time [ s ]
-18
-16
10
10
Initial time [ s ]
-14
10
-12
10
Figure 4.6: The dependence of final number density of vector bosons on the initial
time t0 : α = 1/2 (left), α = 1/3 (right).
a(t̃)mobs , where t̃ is ”the Einstein time” and mobs is the observable present-day
mass. Such a dependence was used, e.g., in Ref. [18] for the FRW metric. On
the other hand, the scaling factor a(t̃) is defined by the cosmic equation of state.
For a barotropic fluid, this EoS has the form
pph = (γ − 1)εph = c2s εph ,
(4.186)
where pph and εph are phenomenological pressure and energy density (in the
distinction from ”dynamical” p and ε, see Sect. 4.2.3), γ is the barotropic parameter, cs is the sound velocity. The solution of the Friedman equation for such
EoS leads to the following scaling factor
a(t̃) ∼ t̃
2/3γ
.
(4.187)
The kinetics of vacuum creation of massive vector bosons (Sect. 4.2.3) was
constructed in the flat Jordan frame with the proper conformal time t, which is
necessary to introduce now in the Eq. (4.187). The transition to the conformal
time is defined by relation dt = dt̃/a(t̃). From this relation and Eq. (4.187)
follows
! #3γ/(3γ−2)
"
2
.
(4.188)
t
t̃ ∼ 1 −
3γ
The substitution of this relation into Eq. (4.187) establishes the mass evolution
law in the terms of the conformal time
m(t) = (t/tH )α mW ,
α=
2
,
3γ − 2
(4.189)
where tH = [(1 + α)H]−1 is the scaling factor (the age of the Universe), H is
the Hubble constant and mW = 80GeV is the W-boson mass. The values of the
163
6
4
]
-3
[ Gev * cm
]
-3
[ Gev * cm
4
energy density
pressure
mean pressure
2
3
2
energy density
mean pressure
1
0
0
-2
-1
-4
1,82x10
-10
1,84x10
-10
1,86x10
-10
1,88x10
-10
1,90x10
-2
-10
3,0x10
-10
6,0x10
-10
9,0x10
-10
1,2x10
-9
Time [ s ]
Time [ s ]
Figure 4.7: Time dependence of energy density and pressure at t t0 with initial
condition mi · ti = 1 for α = 1/2.
parameter α for some popular EoS are: γ = 2, α = 1/2 (stiff fluids); γ = 4/3,
α = 1 (radiation); γ = 1, α = 2 (dust).
Due to back reactions and dynamical mass generation during the cosmic evolution the detailed mass history remains to be worked out. The central question,
however, is whether the number density of produced W-bosons could be of the
same order as that of the CMB photons, nCMB ∼ 465 cm−3 . If this question
could be answered positively, the vacuum pair creation of W-bosons from a timedependent scalar field (mass term) could be suggested as a mechanism for the
generation of matter and radiation in the early Universe. The non-Abelian nature
of the W-bosons could even imply consequences for the generation of the baryon
(and lepton) asymmetry due to topological effects [45].
The numerical analysis of the Eqs. (4.170) for massive vector bosons is performed by a standard Runge-Kutta method on a one-dimensional momentum
grid. As one can see from Fig. 4.5, the creation process ends very quickly and
the particle density saturates at some final value. The momentum distribution
of particles is formed also very early when m(t) ≈ m0 and frozen in such form so
that later on for times t t0 most of the particles have very small momentum
p m(t). The spectrum of created bosons is essentially non-equilibrium, hence
we should continue further the analysis of relevant dissipative mechanisms and
other observable manifestations of the non-equilibrium state (e.g., CMB photons;
in this connection, see, for example, [46, 47]).
The dependence of the corresponding final value of density on initial time is
shown in Fig. 4.6. The final density n1 of the transversal vector bosons with
spin projection ±1 reaches a maximum, when we let the initial time go to very
early times, close to the birth of the Universe. However, in the same limit, the
density f3 of the longitudinal particles with spin projection zero grows beyond
all bounds. The choice of the EoS changes drastically the quantity of the created
164
0.2
t = 5 . 10 s
-11
1.5
f(k)
Density [10
44
-3
cm ]
2.0
0.1
1.0
0.5
0
2
4
t [1/m0]
6
8
0
Figure 4.8: The time dependence of
fermion pair density for a mass evolution given by α = 1/2 and mq = 170
GeV.
10
k [m0]
20
Figure 4.9: The momentum distribution of fermions created by the inertial
mechanism at t = 5 · 10−11 s.
particles, thus giving values which are too small (α = 1/2) or too large (α = 1/3)
in comparison with the observed CMB photon densities. In order to improve this
model, we should use an improved EoS, assuming that the barotropic parameter
γ characterizing the evolution of the particle masses can change during the time
evolution. Such a time-dependence could be induced by the back-reaction of the
created particles on the scalar field. Furthermore, we could use another spacetime model, e.g., the Kasner space-time [48] instead of the conformally flat de
Sitter one. The main achievement relative to the earlier work [44] is that in the
present approach, there is no divergence in the distribution function, thus we do
not need to introduce some ambiguous regularization procedure.
Fig. 4.7 shows that the mean pressure remains negative and its magnitude
becomes negligible in comparison with the energy density. On large times the
energy density grows but the pressure stays very small, p ' 0. The energy growth
with the condition p ' 0 results in the conclusion, that the massive vector bosonanti-boson gas created from the vacuum is a cold one (see also Fig. 1). It can
be seen directly from Eq. (4.179), that at large times ε(t) ' m(t)ntot (t), because
of ω(t) ' m(t). Such EoS of the massive vector boson gas(ε 6= 0 and p ' 0)
corresponds to dust-like matter [49], which would characterize the evolution of
the Universe during those stages when the vector boson gas is the dominant
component of its matter (energy) content. At a qualitative level, this conclusion
is valid independently of the concrete choice of an EoS and, in particular, in
the case of the dust-like EoS. It would be interesting to obtain a formula of the
type (4.189) as a result of the solution of the Friedman equation with the EoS
(4.179)-(4.180) (such a procedure represents the back reaction problem) and to
investigate self-consistently the production of vector bosons in the Universe. Let
165
]
35
10
30
10
25
10
20
10
15
10
10
Density [ cm
-3
10
10
5
10
0
10
-15
10
-13
10
-11
10
-9
10
-7
10
-5
10
-3
10
-1
10
1
∆m /m
Figure 4.10: The final density of created particles for the conformal time
dependence of mass (4.189) as function of the initial time t0 for α = 1/2.
Figure 4.11: The final density of created particles at the linear change of
the mass in time of order 1/m0 by
m0 = 0.2 Gev.
us remark, that vacuum creation of massive vector bosons in the FRW metric
was be considered at first in the work [50].
In order to investigate the features of the fermions vacuum creation in the
conformal model of the Universe, let us use the corresponding basic KE (4.116)
and chose as an example the system of the heavy top quarks with mass mq =
170Gev (with the change mw → mq in Eq. (4.189). The values of the parameter
α are the same as were taken for the vector boson.
As one can see from Fig. 4.8, the creation process ends very quickly and the
particle density saturates at some final value. According to Eq. (4.117), the creation of fermion pairs with zero momentum is forbidden, Fig. 4.9, in contrast to
the pair creation in the electric field, where the p = 0 corresponds to the maximum of the distribution function [18]. The similar ”spectrum inversion” has the
place for the massive vector bosons. This feature can be used as an observational
tool to discriminate between the two different mechanisms of vacuum particle
creation.
Finally, Fig. 4.10 shows the dependence of the final density value on the initial
time t0 for m(t) given by Eq. (4.189). The intensity of fermion creation at fast
change of their mass is presented in Fig. 4.11.
On the qualitative level, the same picture will have place for the vacuum
creation of neutralino, which can be main component of dark matter (e.g., [51]).
Some analogous problem in early cosmology was be considered in the work [52] on
the basis of the original approach close to the Bialynicki-Birula-Gornicki-Rafelsky
method of the covariant Wigner functions [58, 59].
166
4.2.5
The systems with unstable vacuum
Wording of the problem
Here it will be considered the rather general mechanism of mass formation under
action of some space homogeneous time dependent quasiclassical field.
Separating of the quasiclassical background field is a standard procedure of
different non-perturbative approaches in QFT [18, 21, 60, 71]. In the framework
of this procedure the quantum fluctuations can be described by perturbation
theory.
There is a class of problems, in which the strong background field produces
particles, that in turn influence on the background field (the back reaction problem). It is worthy to mention such problems as decay of disoriented chiral condensate [61], the resonant decay of CP-odd metastable states [4, 62], QGP preequilibrium evolution [19, 22, 23, 24], phase transition in systems with broken
symmetry [63] etc.
The construction of general kinetic theory of such kind for various potential
types is shown in Sect. 4.2.5. We will derive the closed system of equations
for the self-consistent description of back reaction (BR) problem, including the
KE with non-perturbative source term describing particle creation in the quasiclassical background field and equation of motion for this background field. We
use the OR to derive the KE. As an illustrative examples, in Subsect. 4.2.5
one-component scalar theory with φ4 and double-well potential is considered. In
these examples, we study some features of proposed approach. In particular,
the problem of stable vacuum state definition and possibility to emerge tachyonic
regimes is discussed. As an example of less trivial case, the pseudo-scalar sector of
the Witten-De Veccia-Veneziano model will be considered. The similar analysis
was done for some other models of such kind (e.g., [4, 62, 64, 65, 66]).
In this section we follow to the work [1].
The System of Basic Equation
Let us consider now the scalar field Lagrangian with a self-interaction potential
V [Φ]:
L[⊕] =
1
1
∂µ Φ∂ µ Φ − m20 Φ2 − V [Φ],
2
2
(4.190)
where m0 is the bare mass. In general case, the potential V [Φ] is an arbitrary
continuous non-monotonous function with at least one minimum Vmin [Φ] (this
is necessary for correct definition of vacuum state). It is assumed that the field
Φ can be decomposed in the neighborhood of some Vmin into the quasi-classical
space homogeneous time dependent background field φ0 (t) and fluctuation part
φ(x)
Φ(x) = φ0 (t) + φ(x).
(4.191)
167
In accordance with definition of fluctuations, we have hφi = 0 and hΦi = φ0 ,
where symbol h. . .i denotes some averaging procedure. The background field
φ0 (t) can be treated as quasi-classical one at the condition [67]:
√
h̄c
,
(4.192)
|φ̇0 | (c∆t)2
where ∆t is the characteristic time of the field averaging.
We consider the case of rather small fluctuations in the neighborhood of the
background field. Therefore, the potential energy expansion in powers of φ(x)
can be performed
1
V [Φ] = V [φ0 ] + R1 φ + R2 φ2 + Vr [φ0 , φ],
2
(4.193)
where V [φ0 ] Vmin
R1 = R1 [φ0 ] =
d2 V [φ0 ]
dV [φ0 ]
, R2 = R2 [φ0 ] =
dφ0
dφ20
(4.194)
and Vr [φ0 , φ] is a residual term containing the high orders to be neglected in
current article (non-dissipative approximation). The decomposition (4.193) can
be finite (for polynomial theories) or infinite. After field decomposition (4.191)
the equation of motion
dV [Φ]
=0
dΦ
(4.195)
(−∂µ ∂ µ − m2 )φ = Q[φ0 , φ],
(4.196)
m2 (t) = m2 [φ0 ] = m20 + R2 [φ0 ]
(4.197)
∂µ ∂ µ Φ + m20 Φ +
can be rewritten in the following form:
where
is the time-dependent in-medium mass and
Q[φ0 , φ] = φ̈0 + m20 φ0 + R1 [φ0 ] + Q2 [φ0 , φ],
1 dR2 [φ0 ] 2
φ .
Q2 [φ0 , φ] =
2 dφ0
(4.198)
(4.199)
As a result of averaging Eq.(4.196), the equation of motion for background
field is obtained
φ̈0 + m20 φ0 + R1 [φ0 ]+ < Q2 [φ0 , φ] >= 0.
(4.200)
The time independence of the averaging procedure is taken into account.
168
The assumption about space-homogeneity means that the function < Q2 [φ0 , φ] >
in Eq. (4.200) can only depend on time. As it follows from Eqs. (4.199) and
(4.200), the source term in the right side of Eq. (4.196) is exclusively defined by
the fluctuations,
Q[φ0 , φ] = Q2 [φ0 , φ]− < Q2 [φ0 , φ] > .
(4.201)
On the other hand, the field function φ(x) in nonstationary situation allows
the decomposition:
φ(x) =
Z
[dk]{φ(+) (k, t)e−ıkx + φ(−) (k, t)eıkx },
(4.202)
[dk] = (2π)(−3/2) d3 k, φ(±) (k, t) are the positive and negative frequency solutions
of the equation of motion
φ̈(±) (k, t) + ω 2 (k, t)φ(±) (k, t) = −Q[φ0 , φ| ± k],
(4.203)
ω 2 (k, t) = m2 (t) + k2
(4.204)
where
and Q[φ0 , φ|k] is the Fourier image of the function Q[φ0 , φ],
Q[φ0 , φ] =
Z
[dk]Q[φ0 , φ|k]e−ıkx .
(4.205)
The function Q[φ0 , φ|k] contains non-linear contribution to the Eq. (4.203).
(±)
We suppose a finite limit lim φ(±) (k, t) = φ− (k) in the infinite past and ast→−∞
sume that the solutions φ(±) (k, t) become asymptotically free φ(±) (k, t) → e±iω− t ,
where ω− (k, t) = lim ω(k, t). The existence of the last limit is based on adiat→−∞
batic hypothesis about switching off of self-interaction in the Eq.(4.197).
After the decompositions (4.191) and (4.193) the Hamiltonian density is
H[Φ] = H[φ0 ] + H1 [φ0 , φ] + H2 [φ0 , φ] + Vr [φ0 , φ],
(4.206)
where H0 [φ0 ] is the background field Hamiltonian and H1 and H2 are the Hamiltonian functions of the first and second order with respect to the fluctuation
field,
1
1
H[φ0 ] = H0 [φ0 ] + V [φ0 ] = φ̇20 + m20 φ20 + V [φ0 ],
2
2
2 2
H1 [φ0 , φ] = φ̇0 φ̇ + (m0 φ0 + R1 [φ0 ])φ,
1 2 1
1
H2 [φ0 , φ] =
φ̇ + (∇φ)2 + m2 φ2 .
2
2
2
(4.207)
(4.208)
(4.209)
One can built the Hilbert space on the system of basic functions φ(±) (k, t)
defined by Eq. (4.203) and proceed to the Fock representation. After the decomposition (4.193), H2 (4.209) has non-diagonal form with respect to creation and
169
annihilation operators and hence does not allow quasi-particle interpretation [18]
(Sect.2.1). In order to diagonalize Hamiltonian it is necessary to apply either the
Bogoliubov transformation [18] or OR [17].
Following the idea of OR, let us write the free field decomposition of φ(x) and
φ̇(x) in the absence of background field φ0 with the substitution the free particle
energy ω(k) = (m20 + k2 )1/2 to the time dependent one (4.204) (see also Sect.
4.2.1)
φ(x) =
[dk]
Z
q
Z
2ω(k, t)
π(x) = ı [dk]
q
n
2ω(k, t) n
2
o
a(+) (k, t)e−ıkx + a(−) (k, t)eıkx ,
(4.210)
o
a(+) (k, t)e−ıkx − a(−) (k, t)eıkx ,
(4.211)
where π(x) plays a role of the canonical momentum. The remarkable feature of
the decomposition (4.210) and (4.211) is fulfillment of the standard commutative
relations
[π(x), φ(x)]|t=t0 = −ıδ(x̄ − x̄0 ),
(4.212)
self-considered with the commutative relations for time dependent creation and
annihilation operators
[a(−) (k, t), a(+) (k, t)] = δ(k − k0 )
(4.213)
(the rest commutators are equal to zero). It is not less important that the Hamiltonian (4.209) after the replacement φ̇ → π has the diagonal form,
H2 [φ0 , φ] =
Z
d3 k ω(k, t)a(+) (k, t)a(−) (k, t)
(4.214)
and hence OR is the quasiparticle representation [17].
In order to obtain the equations of motion for the operators a(±) (k, t), let us
write corresponding action with the Hamiltonian (4.207) – (4.209) (this way is
alternative one for the Hamilton approach of the Sect. 4.2.1)
S[φ] =
Z
d4 x {π φ̇ − H1 − H2 − Vr }.
(4.215)
After substitution here of decompositions (4.210) and (4.211), we get
S[φ] =
ı
dt d3 k { [a(+) (k, t)ȧ(−) (k, t) − a(−) (k, t)ȧ(+) (k, t)]
(4.216)
2
ω̇(k, t) (+)
−
[a (k, t)a(+) (−k, t) − a(−) (−k, t)a(+) (k, t)]
(4.217)
2ω(k, t)
1
− ω(k, t)[a(+) (k, t)a(−) (k, t) + a(−) (k, t)a(+) (k, t)] − Vrk [φ0 , φ]} + (4.218)
S1 [φ] ,
2
Z
170
where S1 [φ] is the part of the action corresponding to the Hamiltonian (4.208)
and Vk [φ0 , φ] is the Fourier image of the residual potential term. Variation
with respect to a(±) (k, t) and subsequent transition to the occupation numbers
representation lead to the Heisenberg-type equations of motion (k 6= 0)
ȧ(±) (k, t) = W(k, t)a(∓) (k, t) + ı[H2 + Vr , a(±) (k, t)],
(4.219)
where
W (k, t) =
Ṙ2 [φ0 ]
ω̇(k, t)
=
2ω(k, t)
4ω 2
(4.220)
is the vacuum transition amplitude of a particle creation. In the Eq. (4.219), the
condensate contribution generated by the action part S1 [φ] is omitted because it
corresponds to k = 0 (the connection mechanism of the condensate state with
k = 0 and excitations is absent in the present model).
For the first time, equations of the type (4.219) was obtained in the work [68]
in QED in the framework of Bogoliubov time-dependent canonical transformations approach. They are the basis for nonperturbative derivation of KE. We will
use now this procedure in the present statement of problem. Let us introduce the
distribution function of quasiparticles (it is convenient to do in discrete momentum representation and subsequent transition to infinite volume of the system
in the resulting KE, L3 → ∞)
L3 f (k, t) =< in | a(+) (k, t)a(−) (k, t) | in >,
(4.221)
where the averaging procedure is carried out over the in-vacuum state. We will
consider nondissipative approximation Vr [φ0 , φ] → 0. Using the method of the
Sect. 4.2.1 and basic equation (4.219) it is not difficult to get the KE
df (k, t)
= 2W (k, t)
dt
Z
t
−∞
0
0
Z
dt W(k, t )[1 + 2f (k, t)]s 2
t
t0
dτ ω(k, τ )
(4.222)
where ω(k, t) is defined by Eq. (4.204). The KE (4.222) can be transformed
to a system of ordinary differential equations, which is convenient for numerical
analysis,
f˙ = W u,
u̇ = 2W (1 + 2f ) − 2ωv,
v̇ = 2ωu.
(4.223)
To rewrite Eq. (4.200) for background field in nondissipative approximation,
one has to calculate the averaging value < in | φ2 (x) | in >. Using Eq. (4.210)
and the relations (in the limit L3 → ∞)
< in | a(+) (k, t)a(−) (k0 , t) | in > = (2π)3 f (k, t)δ(k − k0 ),
171
(4.224)
in the space homogeneous case one can obtain
< in | φ2 (x) | in >=
1 Z d3 k
[1 + 2f (k, t) + u(k, t)].
2 ω(k, t)
(4.225)
Then Eq. (4.200) is
φ̈0 +
m20 φ0
1 dR2
+ R1 [φ0 ] +
2 dφ0
Z
1
d3 k
f (k, t) + u(k, t) = 0
ω(k, t)
2
(4.226)
(the vacuum unit is omitted in the integral).
The KE (4.222) (or the equivalent system of equations (4.223)) and Eq.
(4.226) form the closed system of nonlinear equations describing of the BR problem. In the case of v[Φ0 , Φ] = 0, this system of equations is a direct nonperturbative consequence of the dynamics and the assumption (4.191). For the description
of particle creation we will use the particle density (4.90) as well as background
field energy Ecl and energy of created quasiparticles Eq
1 2 1 2 2
φ̇ + m φ + V (φ0 ),
2 0 2 0 0
Z
d3 p
=
ω(k, t)f (p, t).
(2π)3
Ecl =
(4.227)
Eq
(4.228)
The conservation of the full energy of the system can be shown analitically. The
proof based on the differentiation of total energy Ecl + Eq with respect to time
and taking into account (4.226) and relations (4.194),(4.197) and (4.220).
The constructed formalism allows the consideration of the following initial
problems at the time t = t0 : some initial excitation level of background field
φ0 (t0 ) and φ̇0 (t0 ) are given with the additional condition either f (p, t0 ) = 0 or
f (p, t0 ) 6= 0, where f (p, t0 ) is some initial plasma distribution.
Examples
Φ4 potential The separation of background field (4.191) in the potential
1
V [Φ] = λΦ4 ,
4
λ>0
(4.229)
R2 [φ0 ] = 3λφ20 ,
(4.230)
leads to the follows decomposition coefficients
R1 [φ0 ] = λφ30 ,
background field potential
1
V [φ0 ] = λφ40
4
(4.231)
Vr [φ0 , φ] = λ(φ0 + φ/4)φ3 .
(4.232)
and residual potential
172
Thus, the time dependent quasiparticle mass of fluctuating field (4.197) is equal
to
m2 (t) = m20 + 3λφ20 .
(4.233)
If λ < 0 and the excitation level is large enough, it is possible that the
tachyonic mode will arise, that corresponds to unstable state [69]. The mass
(4.233) determines the vacuum transition amplitude (4.220)
W (k, t) = λ
2φ0 φ̇0
.
ω 2 (k, t)
(4.234)
KE (4.222) with this amplitude is correct in non-dissipative approximation which
corresponds to the neglecting of the residual potential (4.232).
Let us write also the equation of motion for the background field (4.226) in
the same approximation:
φ̈0 + M 2 (t)φ0 + λφ30 = 0
(4.235)
with the corresponding mass equal to
2
M (t) =
m20
+ 3λ
Z
1
d3 k
f (k, t) + u(k) ,
ω(k, t)
2
(4.236)
i.e. the mass of the condensate excitations is defined both the distribution of
quasiparticles and vacuum polarization.
In numerical calculations we apply zero initial conditions for the distribution
function and nonzero one for the background field φ0 (t0 ) = 1.2. The choice
of parameters (λ = 1 and m0 = 1) as well as initial conditions here and in the
Sec. 4.2.5 is motivated by the desire to make a qualitative comparison between
our work and [63], where authors offered the alternative method for description
of quantum systems under strong background field action (so called CornwallJackiw-Tomboulis method [70]).
As it can be seen on Fig.8 at the early evolution stage all energy is mainly
concentrated in the field oscillation. For t < 50 we have a slow growing of
number density. However it drastically increases at t ∼ 50 and after this time
the quantum energy dominates over classical one.
The case λ < 0 (absolutely unstable potential) is associated with tachyonic
regime, that is realized for enough high excitation level, when the initial amplitude
φ0 (t0 ) satisfies with condition m20 + 3λφ20 (t0 ) ≤ 0. The low excitation level m20 +
3λφ20 (t0 ) > 0 corresponds to pre-tachyonic regime. The vacuum particle creation
in the pre-tachyonic region is characterized by developing instability, i.e. by
(c)
unrestricted growth at tendency φ0 (t) to the critical value φ0 = m0 (3|λ|)−1/2 .
173
Double well potential
1
1
V [Φ] = λΦ4 − µ2 Φ2 ,
4
2
λ>0
(4.237)
leads to the same equation (4.235) for background field with new mass (we set
here m0 = 0 and µ2 > 0)
2
2
M (t) = −µ + 3λ
Z
1
d3 k
f (k, t) + u(k, t) .
ω(t)
2
(4.238)
The vacuum transition amplitude (4.220) is equal here
W (k, t) = λ
3φ0 φ̇0
,
2ω 2 (k, t)
(4.239)
where now the quasiparticle frequency (4.204) contains the mass
m2 (t) = −µ2 + 3λφ20 .
(4.240)
The neighborhood of the central point φ0 (t) = 0 is the instability region. In
this region the group velocity vg = dω(k)/dk = k/ω(k) is either super-luminous
(vg > 1 and Imω = 0 for k > kc , where kc is the root of the equation ω(k, t) = 0)
or indefinite (Reω(k) = 0 for k < kc ). Thus, it is tachyonic region.
√
Let us mark the minimum points of the potential (4.237) as Ψ± = ±µ/ λ =
±Ψ0 and to put new origin of frame of reference in one from these points,
Φ = Ψ± + Ψ. We separate now background component φ0 from the field Ψ,
i.e. Ψ = φ0 + φ. We omit the marks (±), which identify the belonging of the
fields Ψ, φ0 andφ to the corresponding branch Ψ± . The rather small level of excitations in the vicinity
of the stable
√
√ points Ψ± will be considered below only,
|φ(x)|, |φ(x)0 | << 2Ψ0 , where ± 2Ψ0 are the roots of the equation V [Ψ] = 0.
Using the Eq.(4.195) and the methods of the Sect.6.2, it is easy to obtain the
following system of equations of motion:
φ̈0 + 2µ2 φ0 + 3λΨ± φ20 + λφ30 + 3λ(Ψ± + φ0 )hφ2 i + λhφ3 i = 0,
−[∂µ ∂ µ + m2± (t)]φ + 3λ(Ψ± + φ0 )[hφ2 i − φ2 ] + λ[hφ3 i − φ3 ] = 0,
(4.241)
(4.242)
where
m2± (t) = 2µ2 + 3λφ0 (φ0 + 2Ψ± ).
(4.243)
In the accepted nondissipative approximation, we must keep in the Eq.(4.242)
the linear terms only. That leads to KE (4.222) with the amplitude defined by
time dependent mass (4.243)
W± (k, t) =
ω̇(k, t)
3λφ̇0 (φ0 + Ψ± )
=
.
2ω(k, t)
2ω 2 (k, t)
174
(4.244)
The mean values hφ2 i and hφ3 i are calculated either in minimal order of
perturbative theory (for λ 1) or in RPA. We obtain the result (4.225) for hφ2 i
and hφ3 i = 0. Thus, we have
d3 k
[2f (k, t) + v(k, t)] = 0.
2 ω(k, t)
(4.245)
Fig.9 present the numerical results for parameters set m = 1, λ = 1, φ0 (0) =
0.58, near to border of a tachyonic mode. The stationary regime is achieved
faster, than in case of symmetric potential.
Other formalism for the description of strong field problem for the quantum
field system with the potential (4.237) is developed by J. Baacke et al ([63],
and works cited therein). The general kinetic approach for the description of
an arbitrary strong excited nonequilibrium states in the scalar QFT with selfinteraction admitting the existence of unstable vacuum states was developed
in the present work. We restrict ourselves by the collisionless (non-dissipative)
approximation. However the attempts to go beyond this approximation was done
[37]. As the concrete example, φ4 and double-well potentials were investigated.
It would be interested to research some other properties of the considered
model, such as the features of excitation transitions between states with different
vacuums (in the same space-time point). Apparently,
it is possible both under
√
rather high initial excitation level |φ0 (0) ≤ 2Ψ(0) and as a result of tunneling
process through the central barrier. The last problem is interested especially in
the generalized double-well potential model with non-degenerated vacuum states.
φ̈0 + λφ0 (2 + 3φ0 )Ψ± +
φ20
+ 3λ(Ψ± + φ0 )
Z
η-meson system We will use now the cumulative experience in order to consider more complicated quantum field system - η-meson sector of the WittenDiVecchia-Veneziano model [2]:
i
a
fπ2
[θ − tr(ln U − ln U + )]2 ,
tr(∂µ U ∂µ U + ) + tr(M U + M U + ) −
Lef f =
4
Nc
2
(4.246)
which describes the low-energy dynamics of the nonet or the pseudoscalar mesons
in the large Nc limit of QCD. We focus here our attention on the singlet state of
this model with the following Lagrangian:
η
1
a
L = ∂µ η∂ µ η + f 2 µ2 cos
(4.247)
− η2,
2
f
2
q
where f = 32 fπ , fπ = 92 MeV is the semi-leptonic pion decay constant, µ2 =
1
(m2π + 2m2K ) ' 0.171 GeV2 , mπ and mK are π- and K - meson masses, a =
3
m2η + m2η0 − 2m2K ' 0.726 GeV2 for zero temperature. The corresponding total
Hamiltonian density is given by
1
1
1
H = H0 + Vin = η̇ 2 + (∇η)2 + aη 2 + Vin ,
(4.248)
2
2
2
175
Vin = 2f 2 µ2 sin2
η
,
2f
(4.249)
where the constant addend f 2 µ2 was discarded.
The Hamiltonian density (4.248), (4.249) can be reconstructed to the form [2]
1
H = H0 + Vin = η̇ 2 +
2

η
Hin = 2f 2 µ2 sin2
−
2f
1
1
(∇η)2 + m20 η 2 + Hin ,
2
2
!2 
η 
,
2f
(4.250)
(4.251)
where m20 = a + µ2 . The effective potential Hin is selected here so that its
formal decomposition with respect to field function η(x) does not contain the
corresponding square contribution that is associated with the mass term. However, the redefinition (4.250), (4.251) leads to absolutely unstable potential (Sec.
4.2.5) with corresponding tachyonic modes [2]. In our opinion, they have artificial character and can be eliminated by return to the original Hamilton density
(4.248), (4.249), on the basis of which will be fulfilled below the investigation of
excitations in the η- meson system.
By analogy with Eq. (4.191), let us select the quasi-classical field η0 (t) and
quantum fluctuation part φ(x),
η(x) = η0 (t) + φ(x).
(4.252)
Now we can use the general formalism of Sec. 4.2.5 to η-meson system with
Hamiltonian density (4.248), (4.249). The master equation (4.226) takes the
form
η0
η̈0 + aη0 + f µ sin
f
2
!"
1
1− 2
2f
Z
1
d3 k
f+ v
ω
2
#
=0
(4.253)
where ω(k, t) is quasi-particle energy (4.204) with the mass (4.197)
m2 (t) = a + µ2 cos
η0
.
f
(4.254)
KE (4.222) or equivalent to it the equation system (4.223) is defined via the
vacuum transition amplitude (4.220) that has now the form
W (k, t) = −
η0
µ2
η̇
sin
.
0
4f ω 2 (k, t)
f
(4.255)
In principle, the mass formula (4.254) contains a possibility of tachyonic modes
appearance. However, the concrete values of the parameters a and µ2 in this
model are such that m2 (t) > 0 for any amplitudes of quasi-classical field η0 (t)
and, consequently, the pre-tachyonic regime is realized only (Sec. (4.2.5)).
176
4.2.6
Summary
This last chapter of the lecture series had the aim to summarize the information
about the kinetic description of vacuum particle creation as a result of inertial
mechanism action stipulated on the phenomenological level by some time dependencies of particle masses. Three basic quantum field models were considered
here: massive scalar, vector and spinor fields. The constructed kinetic theory was
applied here (Sect. 4.2.4) upon to a conformal cosmology model for investigation
of matter creating from vacuum in an early period of the Universe evolution.
In particular, it was showed that the density of the produced vector bosons is
sufficient for the explanation of the present density of CMB photons.
The obtained results can be used for the subsequent investigation of the matter
dynamics created from vacuum in the early Universe (the equation of state, the
long wave-length acoustic excitations, the back-reaction problem etc.). We have
considered here the single mechanism of mass change induced by the conformal
expansion of the Universe with switching on the mass m0 (t0 ) at some arbitrary
initial time t0 . For the construction of a more consistent theory, one should
eventually take into account the inflation mechanism of mass generation acting
during an earlier period the Universe evolution [53]-[56].
It would be interesting to consider the generation of particles of different
masses and statistics. The equation (4.189) is valid for all particles independent
of their inner symmetry. The chemical composition of created matter and the EoS
of the Universe can be fixed rather precisely and can be subject to experimental
verification.
The interesting perspectives are opened also the results of the Sect. 4.2.5,
where it was investigated some simplest quantum field models of scalar fields
with different kinds of selfinteractions and the corresponding quasiclassical nonstationary condensate field (the problem of phase transition under restoration
of a broken symmetry, particles tunneling between states with different vacuum
etc).
Of course, the considered examples are not exhaust of all variety of the systems, where the vacuum generation of particles is stipulated by the inertial mechanism. in particular, the large class of experimentally controlled models of meson
and quark subsystems evolved in the vicinity of the relativistic phase transition
had remained beyond our scope.
Acknowledgement
D.B. was supported by the Virtual Institute of the Helmholtz association under
grant No. VH-VI-041. We are grateful to Prof. M.P. Da̧browski for discussion of
some cosmological aspects of the present work.
177
1,5
1,0
1,0
0,8
0,5
n(t)
φ0(t)
0,6
0,0
0,4
-0,5
0,2
-1,0
-1,5
0
20
40
60
80
0,0
100
0
20
40
t
60
80
100
t
1,2
8000
t = 40
t = 50
6000
0,8
f(p,t)
E(t)
E tot
E cl
E qu
4000
0,4
2000
0,0
0
20
40
60
80
100
0,00
t
0,05
0,10
0,15
0,20
0,25
p
Figure 4.12: Time evolution for the symmetric Φ4 potential. Parameters are:
m0 = 1, φ0 (0) = 1.2, λ = 1. From left to right: (a) evolution of the mean field;
(b) evolution of the particle density; (c) evolution of the energy; (d) momentum
spectrum of particles at times t = 40 and t = 50.
178
1,3
0,07
1,2
0,06
n(t)
1,1
0,05
φ 0(t)
1,0
0,04
0,9
0,03
0,8
0,02
0,7
0,01
0,6
0,5
0,00
0
20
40
60
80
100
0
20
t
40
60
t
140
0,08
120
0,00
f(p)
0,04
E qu
E cl
-0,04
t=40
t=10
100
80
E tot
-0,08
60
-0,12
40
-0,16
20
-0,20
-0,24
0
20
40
0
0,00
60
t
0,25
0,50
0,75
1,00
p
Figure 4.13: Time evolution for the bistable Φ4 potential (4.237). Parameters are:
µ2 = 1, λ = 1, φ0 (0) = 0.58. From left to right: (a) evolution of the mean field;
(b) evolution of the particle density; (c) evolution of the energy; (d) momentum
spectrum of particles at time t = 10 and t = 40.
179
Bibliography
[1] S. A. Smolyansky, V. V. Skokov and A. V. Prozorkevich, PEPAN Lett. 2, 50
(2005); hep-th/0310073.
[2] G. Veneziano, Nucl. Phys. B. 159, 213 (1979); P. Di Veccia and G. Veneziano,
Nucl. Phys. B. 171, (1980) P. Di Veccia et al., Nucl. Phys. B. 181, 318
(1981).
[3] E. Witten, Ann. Phys. 128, 363 (1980); Phys. Rev. Lett. 81, 2862 (1998).
[4] D. B. Blaschke et al., Phys. Rev. D. 65, 054039 (2002).
[5] M. K. Volkov, Ann. Phys. 157, 282 (1984); Sov. J. Part. Nucl. 17, (1986).
[6] V. Koch Int. J. Mod. Phys. E. 6, 203 (1997); nucl-th/9706075.
[7] R. Baier, M. Dirks and K. Redlich. Phys. Rev. D. 55, 4344 (1997); hepph/9711213.
[8] W. Florkowski, Acta Phys. Polon. B. 28, 2079 (1997); W. Florkowski et al.
Ann. Phys. 245, 445 (1996).
[9] R. Pisarski R., Phys. Lett. B. 110, 155 (1982).
[10] R. J. Furnstahl, T. Hatsuda, and S. H. Loe, Phys. Rev. D. 42, 1744 (1990).
[11] F. Cooper and E. Mottola, Phys. Rev. D. 40, 456 (1989).
[12] Y. Fujii and K.-I. Maeda, The Scalar-Tensor Theory of Gravitation, (Cambrige Univ. Press, 2004).
[13] M. P. Dabrowski, D. Benke, and D. Blaschke, Conformal Relativity and Its
Cosmological Implementation, in preparation.
[14] F. Sauter Z. Phys. 69, 742 (1931).
[15] W. Heisenberg and H. Euler Z. Phys. 98, 714 (1936).
[16] J. Schwinger, Phys. Rev. 82, 664 (1951).
180
[17] V. N. Pervushin et al., Int. J. Mod. Phys. A. 20, 5689 (2005); hepph/0307200.
[18] A. A. Grib, S. G. Mamaev and V. M. Mostepanenko, Vacuum Quantum Effects in Strong External Fields, (Friedman Lab. Publ., St.-Peterburg, 1994).
[19] S. M. Schmidt et al., Int. J. Mod. Phys. E. 7, 709 (1998).
[20] D. B. Blaschke et al., Phys. Atom. Nucl. 68, 1087 (2005); hep-ph/0411383.
[21] N. D. Birrell and P. C. M. Davies, Quantum Fields in Curved Spaces, (Cambridge Univ. Press., Cambridge, 1982).
[22] S. M. Schmidt et al., Phys. Rev. D. 59, 094005 (1999).
[23] J. C. R. Bloch et al., Phys. Rev. D. 60, 116011 (1999).
[24] D. V. Vinnik et al., Eur. Phys. J. C. 22, 341 (2001).
[25] Ya. B. Zeldovich and A. A. Starobinsky, Sov. Phys. JETP. 34, 1159 (1972).
[26] A. Casher, J. Kogut and L. Susskind, Phys. Rev. D. 10, 732 (1974).
[27] S. A. Smolyansky, A. V. Prozorkevich and A. V. Tarakanov, in: Voprosy Prikladnoi Fiziki, N11, (Saratov State University, 2004) p.124 (2004).
[28] A. V. Prozorkevich et al., Proceeding of SPIE, 5476, 68 (2004); D. B.
Blaschke et al., Pulsations of the Electron-Positron Plasma in the Field of
Optical Lasers, physics/0410114.
[29] D. B. Blaschke et al., Proceedings of the International Conference on
Strongly Coupled Coulomb Systems, June 22-25, 2005, Moscow, Russia. J.
Phys. A. (in press).
[30] N. N. Bogolubov and D. V. Shirkov, Introduction to the Theory of Quantum
Fields, (3rd eq., Wiley, 1980).
[31] L. H. Rayder, Quantum Field Theory, (Cambrige Univ. Pres., Cambrige,
1985).
[32] S. G. Mamaev, V. M. Mostepanenko and V. M. Frolov, Yad. Fiz. 23, 1118
(1976).
[33] D. V. Vinnik et al., Phys. Atom. Nucl. 64, 775 (2001).
[34] G. Wentzel, Quantum Theory of Fields, (N.-Y., 1949).
[35] S. G. Mamaev and N. N. Trunov, Yad. Phys. 30, 1301 (1979).
181
[36] J. Rau and B. Müller, Phys. Rep. 272, 1 (1996).
[37] S. A. Smolyansky et al., Proc. of Conf. ”Progress in Nonequilibrium Green’s
Functions II”. M. Bonitz and D. Semkat eds., (World Scientific Publ., Singapore, 2003) p.384.
[38] L. Parker, Phys Rev. 183, 1057 (1969); L. Parker, Phys. Rev. D. 7, 976
(1973).
[39] A. A. Grib and S. G. Mamaev, Yad. Fiz. 10, 1276 (1969).
[40] R. U. Sexl and H. K. Urbantke, Phys. Rev. 179, 1247 (1969).
[41] Ya. B. Zeldovich, Pisma Zh. Exp. Teor. Fiz. 12, 443 (1970); Ya.B. Zeldovich
and A.A. Starobinsky, Zh. Exp. Teor. Fiz. 61, 1261 (1971).
[42] Ya. B. Zeldovich and I. D. Novikov, Structure and Evolution of the Universe,
(Univ. Chicago Press., Chicago, 1983).
[43] V. N. Pervushin, D. V. Proskurin and A. A. Gusev, Grav. Cosmology. 8, 181
(2002).
[44] D. B. Blaschke et al., Phys. Atom. Nucl. 67, 1074 (2004).
[45] D. B. Blaschke and M. P. Dabrowski, Conformal Relativity Versus BransDicke and Superstring Theories, hep-th/0407078.
[46] E. Komatsu and D. N. Spergel, Phys.Rev. D. 63, 063002 (2001).
[47] F. Vernizzi, A. Melchiorri and R. Durrer, Phys. Rev. D. 63, 063501 (2001).
[48] S. A. Smolyansky et al., in: Proc. of Int. Workshop ”Hot Point in Astrophysics”, (Dubna, 2000), p. 364.
[49] M. P. Dabrowski and J. Stelmach, Ap. J. 97, 978 (1989).
[50] A. N. Veryaskin, V. G. Lapchinsky and V. A. Rubakov, Preprint P-0198, Institute of Nuclear Research, Moscow, 1981;
A. V. Veryaskin, V. G. Lapchinsky, and V. A. Rubakov, Theor. Math. Phys.
45, 407 (1980).
[51] M. I. Vysotskii and R. B. Nevzorov, Physics-Uspekhi, 44, 919 (2001).
[52] B. Garbrecht, T. Prokopec, and M. G. Schmidt, Eur. Phys. J. C. 38, 135
(2004).
[53] A. Dolgov, in: ”Multiple Facets of Quantization and Supersymmetry”. M.
Olshavetsky, A. Vainshtein eds., (World Scientific, Singapore, 2002), p. 104.
182
[54] P. B. Green and L. Kofman, Phys.Lett. B. 448, 6 (1999).
[55] Proceedings of the 18-th IAP Astrophys. Colloquium ”On the Nature of Dark
Energy”. July 1-5. 2002. Eds: Ph. Brax, J. Martin, J.-Ph. Uzan, (Frontier
Group, Paris, 2002).
[56] J. A. Casas, J. Garcia-Bellido, and M. Quiros, Class. Quant. Grav. 9, 1371
(1992); G. W. Anderson, S. M. Carroll, Dark Matter with Time-Dependent
Mass, astro-ph/9711288.
[57] A. Linde, Particle Physics and Inflationary Cosmology, (Harwood Academic
Publ., Switzerland, 1990).
[58] I. Bialynicki-Birula, P. Górnicki, and J. Rafelski, Phys. Rev. D. 44, 1825
(1991).
[59] V. G. Morozov, G. Röpke, and A. Höll, Theor. Math. Phys. 131, 812 (2002);
V. G. Morozov, G. Röpke, and A. Höll, Theor. Math. Phys. 132, 1026 (2002);
A. Höll, V. G. Morozov, and G. Röpke, Covariant Linear Response Theory
of Relativistic QED Plasmas, quant-ph/0208083.
[60] F. Cooper et al., Phys. Rev. D 48, 190 (1993).
[61] D. I. Kaiser, Phys. Rev. D. 59, 117901 (1999).
[62] D. Ahrensmeier, R. Baier and D. Dirk, Phys. Lett. B. 484, 58 (2000).
[63] J. Baacke and A. Heinen, Phys. Rev. D. 67, 105020 (2003).
[64] R. Friedberg and T.D. Lee, Phys. Rev. D. 16, 1096 (1977).
[65] S. Loh et al., Nucl. Phys. A. 619, 321 (1997).
[66] T. Hamazaki et al., Phys. Rev. D. 53, 2045 (1996).
[67] V. B. Berestetsky, E. M. Lifshits, and L. P. Pitaevsky, Quantum Electrodynamics, (Moscow, ”Nauka”, 1980).
[68] V. S. Popov and M. S. Marinov, Yad. Fiz. 16, 809 (1972).
[69] A. Yu. Andreev and D. V. Kirshnitz, Physics-Uspekhi, 39, 1071 (1996).
[70] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D. 10, 2428 (1974).
[71] E. V. Shuryak, The QCD Vacuum, Hadrons and the Superdense Matter, (
World Scientific Publ.Co., 1988).
[72] E.S. Fradkin and D.M. Gitman, Phys. Rev. D. 53, 7162 (1996).
183
[73] L.P. Kadanoff and G. Baym, Quantum Statistical Mechanics, (Benjamin,
New York, 1962)
[74] F. Gelis and R. Venugopalan, Particle Production in Field Theories Coupled
to Strong External Sources, hep-ph/0601209.
184