Download 11 Canonical quantization of classical fields

11
Canonical quantization of classical fields
In Sections 5-10 a formalism was developed which allows to describe quantum
mechanically interactions of relativistic particles. Starting with the second
quantized formulation of quantum mechanics of many-particle systems we
have identified in Section 6 the basic properties one-particle states should
possess, if the theory is to be Poincaré covariant. States describing many
noninteracting particles were constructed as appropriate tensor products of
one-particle states; such many-particle states are in one-to one correspondence (also as far as their transformation properties are concerned) with the
asymptotic in and out states (introduced in Section 7) of interacting particles.
The results of Sections 5 and 6 provided the starting point for for constructing
in Sections 8 and 9 interactions of particles and formulating the perturbative
expansion satisfying the requirements formulated in Section 7, necessary to
obtain relativistically covariant S matrices. The resulting formalism turns
out to have many features of quantum field theory but the basic building
entities remain particles: their set and transformation properties must be
specified from the outset to specify the theory.
Here we present another, complementary approach to the formulation
of theory of interactions of relativistic particles. It is based on the canonical quantization of relativistic fields. Particles emerge in this formalism as
states of quantized fields. The basic entities - at least as far as bosonic degrees of freedom are concerned - are, however, fluctuating quantum fields.
The advantage of the field theory approach is twofold: firstly, it provides
a well defined prescription for constructing generators of the full Poincaré
group (whose structure in terms of the creation and annihilation operators
in the approach of Section 6 can only be guessed at) and, secondly, allows
for an easy discussion of various internal symmetries (providing through the
Noether theorem a recipe for constructing associated conserved charges) and
their spontaneous breaking. Moreover, in the case of fields “whose quanta”1
are bosons (integer spin particles) the picture of fluctuating field values at
different space-time points (complementary to the description in terms of particles) seems more fundamental; it allows for deeper understanding of global
aspects of the theory (such as the role of topologically nontrivial classical
field configurations, symmetry breaking etc.) and definitely proves useful in
diverse modern applications of quantum field theory (e.g. in cosmology).
The picture of fluctuating fields seems however not applicable to quantum fields, “whose quanta” are fermions (half-integer spin particles). While
1
We put the quotation marks bacaus, as will become clear in section 13 the corespondence between the fields entering the Lagrangian and the particle states predicted by a
given theory need not be direct, nor one-to-one.
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it is possible to take as a starting point for such quantum theories the Lagrangian formalism, whose formal aspects are analogous to that of classicla
fields, the basic entities in this case are anticommuting generators of abstract
Grassmann (or Bieriezin) algebra which have no classical counterpart and can
hardly be considered physical. The difference between bosonic and fermionic
fields2 becomes particularly clear in the path integral approach to field quantization (see Section 16). While quantizing in this way bosonic fields one
integrates functionally (over (that is performs summation of contributions
of) all possible classically realisable (and measurable) field configurations,
for fermionic fields one uses for this purposes an operation formally defined
for Grassmann algebra generators only formal aspects of which resemble the
integration. Thus no “classical fermionic fields can classically be measurable
and the only physical picture that in quantum field theory can be associated
with fermions is that provided in Sections 5 and 6. Quantization of classical
fermion fields is only a mathematical trick allowing to easily reproduce the
results of the formalism presented there. In particular it cannot be viewed
(as is sometimes presented in older textbooks) as “quantization of the wave
function satisfying the Dirac equation”.
We begin this Section with a brief review of the classical theory of relativistic fields and their symmetries and then proceed to their quantization.
As the next step we show, that quantum description of noninteracting fields,
whose Lagrangians consist of terms at most bilinear in field variables, leads to
the interpretation of the corresponding Hamiltonian eigenstates in terms of
(noninteracting) particles. In principle, quantization of classical fields is very
similar to the quantization of the system of many coupled harmonic oscillators discussed in Section 5.5: the field value at a given point x in space can
be treated as an oscillator coupled to other oscillators in neighbouring points
(in particular, the common features of both systems: of quantized coupled
anharmonic oscillators and of interacting quantized fields, is the nonconservation of the numbers of phonons and particles, respectively). However,
despite these deep similarities, quantization of some classical fields (including
the electromagnetic one) has some subtle points which require special treatment and we devote separate subsections to these problems. The formalism
allowing to treat fermionic fields on formally equal footing with bosonic ones
is given in section 11.8. After quantizing various types of fields we perform
transition to the so-called interaction picture which for theories of interacting
fields (with more complicated Lagrangians) allows to compute the S-matrix
2
The reason for this sharp difference between bosonic and fermionic fields seems to be
that in the classical ~ → 0 limit bosonic quantum fields go over into true, measurable
(at least in principle) classical fields of forces (like the electromagnetic forces, and various
potential forces) whereas fermionic fields become matter particles described first, at low
energies, by the Schrödinger (or Pauli) equation and ultimately, for ~ → 0, by the classical
mechanics.
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elements using the methods developed in Sections 7 and 9.
11.1
Action and the Noether theorem
Dynamics of a system of classical fields3 φi is determined by the action I[φ]
which is a functional over all differentiable space-time field configurations.
Actions considered in connection with quantum field theory are given in terms
of Lagrangians L obtained as space integrals of local Lagrangian densities
Z t1 Z
Z t1
3
I[φ] =
dt d x L(φ, ∂µ φ) ≡
dt L .
(11.1)
t0
V
t0
The Lagrangian density L is a function of fields φi , i = 1, . . . , N and their
space-time derivatives. In the following we consider Lagrangian densities
depending on first order derivatives only. The volume V may be finite or
infinite; in the latter case one assumes that all the fields φi vanish sufficiently
fast at spatial infinity; in the finite volume V some spatial boundary conditions must be specified. The action I[φ] must be a real and - for relativistic
field theories - Poincaré invariant quantity. In general the fields φi transform
nontrivially (as some, in general reducible, representation of the SO(1, 3)
group or, in the case of fermionic fields to be discussed in section 11.8, of
its universal covering SL(2, C)) under changes of the inertial frame and can
also transform nontrivially under some internal symmetries.
Equations of motion of a system of fields are obtained by requiring that
for the true field configurations the action (11.1) is stationary (i.e. it does
not change to first order in δφi (x)) with respect to substitutions φi (x) →
φi (x) + δφi (x), where the variations δφi (x) are assumed to vanish for t = t1
and t = t2 as well as for |x| → ∞ (if the volume V in (11.1) is finite, δφi (x) are
assumed to vanish at the boundaries). Concisely this requirement is written
as δI = 0. For Lagrangian densities which depend only on fields and their
first derivatives the variation δI of the action due to any variations δφi (x) of
fields (not necessarily subject to some specific boundary conditions) is4
Z
∂L
∂L
4
δI = d x
δφi +
δ(∂ν φi )
(11.2)
∂φi
∂(∂ν φi )
3
The formalism presented in this section carries over unmodified to fermionic fields
discussed in section 11.8 provided all derivatives of the Lagrangian are treated as right
derivatives and the variations stand to the right of these derivatives.
4
Generalization to Lagrangian densities depending on higher field derivatives is
straightforward but r equires assuming appropriate boundary conditions also for derivatives of field variations. However the Hamilton’s formalism presented here, which is important for formulating transition to the quantum theory (without appealing to functional
integrals which will be introduced in Section 16), is adapted to Lagrangian densities depending on first derivatives only. It can be extended to theories whose Lagrangians depend
on higher derivatives using the formalism of constraints presented in section 11.6
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(summation over the index i is understood). In the following we will consider
variations φi (x) → φi (x) + δφi (x) for which δ(∂ν φi ) = ∂ν (δφi ). The variation
δI can be then rewritten as
Z
∂L
∂L
∂L
4
δφi + ∂ν
− ∂ν
δφi .
(11.3)
δI = d x
∂φi
∂(∂ν φi )
∂(∂ν φi )
Due to the boundary conditions imposed on the variations δφi considered
in determination of the true field configurations, the second, surface term
in (11.3) vanishes and the condition δI = 0 leads to the Euler-Lagrange
equations of motions:
∂ν
∂L(φ, ∂φ)
∂L(φ, ∂φ)
= 0.
−
∂(∂ν φi )
∂φi
(11.4)
It is important to remember that Lagrangian densities which differ from each
other by a total derivative of an arbitrary function of fields5
L′ = L + ∂µ X µ (φ) ,
(11.5)
give the same classical equations of motion (due to vanishing of the field
variations on the boundary).
Canonical quantization of a system of fields requires6 going over to the
Hamilton’s formalism. To this end one defines the momenta Πi (t, x) canonically conjugated to the field variables φi (t, x) by
Πi (t, x) =
∂L
δL
≡
,
∂(∂0 φi (t, x))
δ φ̇i (t, x)
and forms the Hamiltonian density
X
H(x) =
Πi (x) φ̇i (x) − L(x) ,
(11.6)
(11.7)
i
and the Hamiltonian
H=
Z
d3 x H(t, x) .
(11.8)
We have assumed here that the momenta Πi and the field variables φi are
not subject to any constraints and that the relations (11.6) can be inverted
to give
φ̇i (x) = φ̇i (Π(x), φ(x)) ,
5
(11.9)
For Lagrangian densities depending only on fields and their first field derivatives X µ
can depend also on field derivatives but only in such a way that its variation vanishes for
vanishing variations δφi of fields without imposing any conditions on variations of field
derivatives.
6
In the alternative approach based on path integrals (see Section 16) fields are quantized
using directly the action I[φ].
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where the dependence on φ means possibly also a dependence on ∂i φ(x). As
we will see, in physically interesting cases this assumption is not always true
and we will have to consider appropriate modifications of the canonical formalism. If no such problems arise, the equations of motions in the Hamilton’s
formalism read
φ̇i (t, x) = {φi (t, x), H}PB ,
Π̇i (t, x) = {Πi (t, x), H}PB ,
(11.10)
where the Poisson bracket (PB) of any two functionals F [φ, Π] and G[φ, Π]
is defined as
XZ
δF [φ, Π] δG[φ, Π]
3
F [φ, Π], G[φ, Π]
≡
dx
δφi (t, x) δΠi (t, x)
PB
i
δG[φ, Π] δF [φ, Π]
, (11.11)
−
δφi (t, x) δΠi (t, x)
so that
{φi (t, x), Πj (t, y)}PB = δij δ (3) (x − y) ,
{φi (t, x), φj (t, y)}PB = {Πi (t, x), Πj (t, y)}PB = 0 .
(11.12)
If the canonical variables φi and Πi are not subject to any constraints, the
canonical equations of motion (11.10) are exactly equivalent to the original
Euler-Lagrange equations (11.4).
Formulation of the field dynamics in terms of the action I allows to easily
identify symmetries of the field equations of motion and, via the Noether
theorem, of the corresponding conserved quantities. We begin with the case
of symmetries which do not affect space-time coordinates (these can be ordinary global symmetries or, in the case of theories involving fermionic fields,
rigid supersymmetries). We consider first a change of the field variables
φi (x) → φ′i (x) = φi (x) + δ0 φi (x) .
(11.13)
L′ (φ′ , ∂µ φ′ ) = L(φ, ∂µ φ) .
(11.14)
In the new field variables φ′i (x) the dynamics is described by a new Lagrangian density L′ depending on the fields φ′i , which can be chosen so that7
7
This is just as in classical mechanics in which one is allowed to use any set of dynamical
variables, qi (t) or qi′ (t) = qi′ (q(t), t), to describe a given system. The equations of motion
in the new variables follow then from the new Lagrangian L′
d ∂L′ (q ′ , q̇ ′ , t)
∂L′ (q ′ , q̇ ′ , t)
=
.
dt
∂ q̇i′
∂qi′
Because Lagrangian has physically a well defined interpretation (e.g. L = T − V in
nonrelativistic mechanics), the new Lagrangian L′ (q ′ , q̇ ′ , t) is obtained just by inverting
the relations qi′ = qi′ (q(t), t) and inserting qi (t) = qi (q ′ (t), t) into the original Lagrangian:
L′ (q ′ (q), q̇ ′ (q), t) = L(q(q ′ (t), t), q̇(q ′ (t), t), t).
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In this case
′
I −I ≡
Z
4
′
′
′
d x L (φ , ∂µ φ ) −
Z
d4 x L(φ, ∂µ φ) ≡ 0 ,
(11.15)
i.e. I ′ = I. Consequently, if I is stationary for a configuration φi (x) of
the original fields, that is, if φi (x) satisfy the equations (11.4), then I ′ is
stationary for φ′i (x) related to φi (x), by (11.13), i.e. the configuration φ′i (x)
of the primed fields satisfy the equations
∂ν
∂L′ (φ′ , ∂φ′ )
∂L′ (φ′ , ∂φ′ )
= 0,
−
∂(∂ν φ′i )
∂φ′i
(11.16)
which are in general of different form than the equations (11.4) because
L′ (· , ·) is in general a different function of its arguments than L(· , ·).
The choice of L′ satisfying the condition (11.14) is not the only possibility:
any L′ such that
L′ (φ′ , ∂µ φ′ ) = L(φ, ∂µ φ) + ∂µ X µ (φ) ,
is equally good. In this case, instead of (11.15), one has
Z
′
I = I + d4 x ∂µ X µ (φ) ,
(11.17)
(11.18)
but still, if I is stationary for a configuration
φi (x), then I ′ is such for φ′i (x)
R 4
related to φi (x) by (11.13) because δ d x ∂µ X µ (φ) = 0 for variations δφi
vanishing for t1 and t2 and for |x| → ∞.
It is important to realize, that we are not yet speaking of a symmetry
of the field equations of motion, but rather about the well known fact that
in the Lagrangian formalism a given system can be described by arbitrarily
chosen dynamical variables. A change of variables8 φi → φ′i (φi ) is a symmetry
of the equations of motion if L′ (· , ·) leading to (11.17) is such that
L′ (· , ·) = L(· , ·) + ∂ µ Xµ′ (·) ,
(11.19)
or, using the freedom to redefine L′ by subtracting from it ∂ µ Xµ′ , if for L′ (·, ·)
leading to (11.17) one can just take L(· , ·), because then the equations
8
Mathematically speaking,the fields φi (x), i = 1, . . . , N define mappings from the
space-time into some N -dimensional “target” space T (N ) , called by physicists the internal
space. Therefore φi should be viewed as coordinates of a point in T (N ) onto which the
space-time point characterized by the coordinates xµ is mapped. A change of variables
φi → φ′i can be due to changing the coordinate system of T (N ), in which case we have
to do with a passive transformation, or due to considering a transformed system (active
transformation) - cf. Section 4.1.
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of motion (11.4) satisfied by the fields configuration φi (x) for which I is
stationary have the same form as the equations of motion (11.16) satisfied
by the configuration φ′i (x) for which I ′ is stationary. In other words, φi (x)
and φ′i (x) are both solutions of the same Euler-Lagrange equations. Thus,
the condition that the change φi → φ′i (φi ) is a symmetry reads
L(φ′ , ∂φ′ ) = L(φ, ∂φ) + ∂µ X µ (φ) .
(11.20)
Considering now an infinitesimal symmetry transformation (in the sense
specified above) φi → φi + δ0 φi , we write
L(φ′ , ∂φ′ ) = L(φ + δ0 φ, ∂φ + δ0 ∂φ) = L(φ, ∂φ) + δL(φ, ∂φ) ,
(11.21)
with δL of first order in δ0 φi . Using then (11.3) as well as the condition
(11.20) combined with (11.18) we can write the equality9
Z
0 ≡ d4 x [L(φ′, ∂φ′ ) − L(φ, ∂φ) − ∂µ X µ (φ)]
(11.22)
Z
∂L
∂L
∂L
4
µ
= dx
δ0 φi + ∂µ
− ∂µ
δ0 φi − δX (φ) + . . . .
∂φi
∂(∂µ φi )
∂(∂µ φi )
The first bracket in the above formula vanishes for the fields φi satisfying the
equations of motion (11.4). Thus, for such a fields configuration φi (x) the
quantity
jµ =
∂L
δ0 φi − δXµ (φ) ,
∂(∂ µ φi )
(11.23)
is conserved, i.e. satisfies the equation
∂ µ jµ = 0 ,
(11.24)
because for a symmetry transformation (11.22) is identically zero, and no
special requirements on the values at t = t1 and t = t2 of the field changes
δ0 φi (x) due to the symmetry transformations are imposed (of course, δ0 φi (x)
must vanish at spatial infinity or, in the finite volume, should not change the
boundary conditions satisfied by the fields).10 If the infinitesimal changes
δ0 φi are parametrized by n independent infinitesimal constant parameters θa
9
This reasoning can easily be generalized to Lagrangian densities depending on higher
field derivatives.
10
For this reason the second term on the right hand side of (11.18) is nonvanishing - it
receives contributions from the t = t1 and t = t2 hypersufaces. Notice that the situation is
quite different in the case of theories formulated in the Euclidean space with coordinates
x̄µ : because in this case fields and, therefore, also the symmetry changes δ0 φi are bound
to vanish for |x̄| → ∞ in all directions, from (11.22) one cannot infer the existence of
conserved currents and, hence, there are no conserved charges (11.27).
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(in theories with fields taking values in Grassmann algebras the parameters
θa can be anticommuting Grassmann variables),
δ0 φi = θa δ0a φi ,
(11.25)
there are n conserved Noether currents
jµa =
∂L
δ a φi − Xµa (φ) ,
∂(∂ µ φi ) 0
a = 1, . . . , n,
(11.26)
(for (11.25) δX µ must also be of first order in the transformation parameters,
i.e. δXµ (φ) = θa Xµa (φ)). The Noether charges Qa given by
Z
a
Q = d3 x j0a (t, x) ,
a = 1, . . . , n,
(11.27)
are then time independent (conserved) quantities provided the fields fall off
sufficiently rapidly at spatial infinity (or, in a finite volume, satisfy the appropriate boundary conditions).
In most cases the transformations (11.25) are linear11 in the fields φi
θa δ0a φi = −iθa Tija φj ,
(11.28)
(summations over a and j are understood) with T a being a set of Hermitian
matrices - a matrix representation of the generators of a symmetry group G
- forming a basis of a representation of the dimension n Lie algebra of G (see
Section 4). The matrix generators T a satisfy the commutation rule
a
T , T b = iT c fc ab ,
(11.29)
with some structure constants fc ab and will be assumed to be normalized
by the condition tr(T a T b ) = 21 δ ab . The transformations (11.13) are then
infinitesimal forms of finite field transformations. For such internal symmetries12 Xµa ≡ 0 and it is then easy to see that in the canonical Hamilton’s
formalism the Noether charges Qa generate via Poisson brackets the infinitesimal symmetry transformations (11.28) of field variables. Indeed, the time
component of the current is then simply
j0a (t, x) = −iΠi (t, x) Tija φj (t, x) ,
11
(11.30)
However, symmetries realized on fields nonlinearly also play an important role, mainly
in so-called effective quantum field theories, e.g. in the chiral Lagrangian description of
low energy strong interactions of mesons.
12
More generally, whenever X µ (φ) cannot be removed by a suitable redefinition of L by a
total four-divergence, we have to do with a space-time symmetry (for example, X µ (φ) cannot be removed for supersymmetric transformations); otherwise we speak about internal
symmetries.
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and using (11.12) one gets
θa {Qa , φi (t, x)}PB = iθa Tija φj (t, x) ,
(11.31)
so that φ′i (x) ≈ φi (x) − θa {Qa , φi (x)}PB .
In order to discuss space-time transformations (like translations, rotations
or Lorentz boosts) the formalism has to be generalized. An infinitesimal such
a transformation xµ → x′µ
x′µ = xµ + δxµ (x) = xµ + θa δ a xµ (x) ,
(11.32)
should be accompanied by a corresponding transformation φi (x′ ) → φi (x′ )
of the fields:13
φ′i (x′ ) = φi (x) + δφi (x) = φi (x) + θa δ a φi (x) .
(11.33)
For example, for an infinitesimal transformation xµ → x′µ with
x′µ = xµ + ω µν xν − ǫµ ,
(11.34)
corresponding to a change of the reference frame, the fields φi (x) transform
under some regular (in general reducible) matrix representation (J µν )ij of
the Lorentz group14
φ′i (x′ ) = φi (x) −
i
ωµν (J µν )ij φj (x) .
2
(11.35)
One can also consider transformations xµ → x′µ which correspond e.g. to
infinitesimal conformal transformations etc.
The system of fields can be described in the new space-time coordinates
x in terms of the new field variables φ′ (x′ ) if one uses such a new Lagrangian
density L′ , that for any two fields configurations φi (x) and φ′i (x′ ) related to
each other by (11.32) and (11.33)
Z
Z
′
4 ′ ′
′ ′
′ ′ ′
d x L (φ (x ), ∂µ φ (x )) − d4 x L(φ(x), ∂µ φ(x)) = 0 , (11.36)
I −I =
′
Ω′
Ω
where ∂µ′ denotes the derivative with respect to x′µ and Ω′ is the image of
Ω under the change of variables (11.32). (11.36) is a sufficient condition
for the new fields φ′ (x′ ) obtained via (11.33) from the solutions φ(x) of the
13
This means that the change of the space-time coordinate system entails a related
coordinate change in the internal (target) space.
14
λν µ
The simplest nontrivial is the vector representation with Jvec
given by (B.3)
κ
acting on vector fields φi ≡ V κ , but higher tensor or spinorial representations can also be
considered.
323
equations of motion (11.4) following from L to be solutions of the equations
of motion following from L′ . However as previously, the same conclusion for
φ′i (x′ ) follows if L′ (φ′ (x′ ), ∂µ′ φ′ (x′ )) is chosen such that
Z
′
I = I + d4 x ∂µ X µ (φ) .
(11.37)
Ω
for some four-vector function X µ (φ).
Again, one speaks about a space-time symmetry, if (by an appropriate
choice of the factor Xµ′ (φ′ )) for the new Lagrangian density L′ (·, ·) leading to
(11.37) one can take the original L(·, ·):
L′ (φ′ (x′ ), ∂µ′ φ′ (x′ )) = L(φ′(x′ ), ∂µ′ φ′ (x′ )) ,
(11.38)
because then the equations of motion satisfied by φ′i (x′ ) (in the space-time
coordinates x′µ ) are the same as the equations of motion (in xµ ) satisfied by
φi (x). Taking into account (11.38), the condition (11.37) can be written in
the form
d4 x′ L(φ′ (x′ ), ∂µ′ φ′ (x′ )) = d4 x [L(φ(x), ∂µ φ(x)) + ∂µ X µ (φ(x))] ,
(11.39)
(d4 x′ 6= d4 x, i.e. det(∂x′ /∂x) 6= 1 for a general transformation (11.32)).
In a relativistic field theory, the transformations (11.34) accompanied by
(11.35) should be symmetries in the above sense. For such transformations
det(∂x′ /∂x) = 1 and the condition (11.39) takes the form similar to (11.20),
except for different space-time coordinates on both sides.
Conserved quantities corresponding to symmetry transformations can be
found using the condition (11.37). To this end it is technically convenient to
split the total change of the field φi (x) as follows
δφi (x) ≡ φ′i (x′ ) − φi (x)
= φ′i (x′ ) − φ′i (x) + φ′i (x) − φi (x) ≡ δxµ ∂µ φi (x) + δ0 φi (x) ,
(to the first order in the transformation parameters θa the fields φ′i have been
replaced by φi in the term with δxµ ) and to work with the functional changes
of fields δ0 φi (x) = φ′i (x) − φi (x) for which
δ0 (∂µ φi (x)) = ∂µ (δ0 φi (x)) .
(11.40)
Thus, to the first order in the parameters θa , we write the transformed fields
and their derivatives as15
φ′i (x′ ) = φi (x) + δ0 φi (x) + δxλ ∂λ φi (x) + O(θ2 ) ,
15
From (11.34) it follows that ∂x′λ /∂xκ = δκλ + ∂(δxλ )/∂xκ , and ∂xλ /∂x′µ = δµλ −
∂(δxλ )/∂xµ . Thus, ∂µ′ = ∂µ − (∂µ (δxλ ))∂λ and the terms with derivatives of δxλ cancel
out in (11.41).
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∂µ′ φ′i (x′ ) = ∂µ φi (x) + ∂µ (δ0 φi (x)) + δxλ ∂λ ∂µ φi (x) + O(θ2 ) ,
∂ν′ ∂µ′ φ′i (x′ )
λ
(11.41)
2
= ∂ν ∂µ φi (x) + ∂ν ∂µ (δ0 φi (x)) + δx ∂λ ∂ν ∂µ φi (x) + O(θ ) ,
etc.
This allows to represent L(φ′ (x′ ), ∂µ′ φ′ (x′ )) in the identity (in which x and
x′ are connected by (11.32))
Z
Z
4 ′
′ ′
′ ′ ′
0≡
d x L(φ (x ), ∂µ φ (x )) − d4 x [L(φ(x), ∂µ φ(x)) + ∂µ X µ (φ)] , (11.42)
Ω′
Ω
16
following from (11.37) in the form
L(φ′ (x′ ), ∂µ′ φ′ (x′ )) = L(φ(x), ∂µ φ(x)) + δL ,
(11.43)
∂L
∂L
δ0 φi +
∂µ δ0 φi + δxµ ∂µ L + O(θ2 ) ,
∂φi
∂(∂µ φi )
(11.44)
∂L
∂L
.
+ ∂µ ∂λ φi (x)
δx ∂µ L ≡ δx ∂µ φi (x)
∂φi
∂(∂λ φi )
(11.45)
in which
δL =
where
µ
µ
The Lagrangian density change (11.44) can also be rewritten as
∂L
∂L
∂L
µ
δL = δx ∂µ L +
δ0 φi + ∂µ
− ∂µ
δ0 φi . (11.46)
∂φi
∂(∂µ φi )
∂(∂µ φi )
We then make in the first integral is (11.42) the change of the integration
variables d4 x′ → d4 x (upon which Ω′ → Ω) using the Jacobian17
′
∂xµ
∂δxµ (x)
∂δxµ
ν
= det δµ +
≈ 1 + tr
= 1 + ∂µ δxµ , (11.47)
det
∂xν
∂xν
∂xν
so that finally to first order the integrand of the first term of (11.42) becomes
d4 x′ L(φ′(x′ ), ∂µ′ φ′ (x′ )) = d4 x′ [L(φ(x), ∂µ φ(x)) + δL]
= d4 x L(φ(x), ∂µ φ(x)) + d4 x [δL + L ∂µ δxµ ] . (11.48)
For field configurations φi (x) satisfying the equations of motion (11.4) the
identity (11.42) then gives:
Z
∂L
4
µ
µ
µ
d x L ∂µ δx + δx ∂µ L + ∂µ
δ0 φi − ∂µ δX (φ) + O(θ2 )
∂(∂
φ
)
µ i
Ω
Z
∂L
µ
4
µ
δ0 φi − δX (φ) + O(θ2 ) = 0 . (11.49)
= d x ∂µ L δx +
∂(∂
φ
)
µ i
Ω
16
We again assume, that L depends only on fields and their first derivatives. Conserved
Noether charges corresponding to more complicated Lagrangian densities can be derived
using similar methods.
17
We use the relation det(1 + A) = exp{tr ln(1 + A)} ≈ 1 + tr(A).
325
Taking into account the arbitrariness of Ω and reexpressing δ0 φi in terms of
δφi ≡ φ′i (x′ ) − φi (x) = δ0 φi + δxµ ∂µ φi we see that conserved is the quantity
∂L
∂L
µ
µ
J = g ρL−
∂ρ φi δxρ +
δφi − δX µ (φ) .
(11.50)
∂(∂µ φi )
∂(∂µ φi )
That is, ∂µ J µ (x) = 0 for fields satisfying the equation (11.4).
We now consider the conserved currents associated with the Poincaré
transformations of the form (11.34). For translations (ωµν = 0 and ǫµ 6= 0)
φ′i (x′ ) = φi (x), so that δφi = 0. The quantity given by (if X µ (φ) = 0, which
is usually the case)
µν
Tcan
=
∂L
∂ νφi − g µν L ,
∂(∂µ φi )
(11.51)
called the canonical energy-momentum tensor is then conserved:
µν
∂µ Tcan
= 0.
The four constants of motion (i.e. time independent quantities)
Z
µ
0µ
P = d3 x Tcan
(t, x) ,
(11.52)
(11.53)
play the role of the total energy P 0 of the system of fields and of its total
νµ
momentum vector P i . It can be shown that if Tcan
is conserved (and only
µ
then), P given by (11.53) transforms as a true four-vector when the reference
frame is changed.
In the case of Lagrangian densities which are of the general form
1
L(φi , ∂µ φi ) = ∂µ φi ∂ µ φi − V (φ) ,
2
(11.54)
where V (φ) is called the field potential, going over to the Hamilton’s formalism, one finds
1
1
00
Tcan
= Πi Πi + ∇φi ·∇φi + V (φ) ≡ H ,
2
2
0k
k
Tcan = Πi ∂ φi .
(11.55)
In this case it is easy to see that the transformations of the field φi (x) corresponding to translations in time and space are generated by Poisson brackets:
{P µ , φi (x)}PB = − ∂ µ φi (x) ,
so that φ′i (x) = φi (x) − {P µ , φi (x)}PB ǫµ .
326
(11.56)
The canonical energy-momentum tensor (11.51) is not always symmetric
µν
in its indices µν. However, if instead of Tcan
one uses another tensor
µν
T µν (x) = Tcan
(x) + H µν (x) ,
(11.57)
where H µν (x) is an arbitrary conserved tensor, ∂µ H µν (x) = 0, such that
Z
d3 x H 0ν (x) = 0 ,
(11.58)
the associated conserved charges P µ obtained from the new tensor T µν are
(for fields vanishing sufficiently fast at spatial infinity) the same as the ones
µν
obtained from Tcan
. This is so if one takes
H µν (x) = ∂ρ H ρµν (x) ,
(11.59)
where H ρµν (x) is antisymmetric in its ρµ indices. Using this freedom one can
µν
µν
νµ
replace Tcan
by a symmetric tensor Tsymm
= Tsymm
. For fields transforming
nontrivially under the Lorentz group of particular interest is the Belinfante
symmetric tensor obtained with
∂L
1
ρµν
(−iJ µν )ij φj
H
=
2 ∂(∂ρ φi )
∂L
∂L
ρν
ρµ
−
(−iJ )ij φj −
(−iJ )ij φj . (11.60)
∂(∂µ φi )
∂(∂ν φi )
It is the Belinfante symmetric energy-momentum tensor which appears as
the right hand side of the Einstein’s equations of General Relativity.
Conserved currents associated with the Lorentz transformations (ǫµ = 0,
6= 0 in (11.34)) are derived in a similar way. Using δxµ = ω µν xν and
δφi = − 2i ωµν (J µν )ij φj (x) following from (11.35) we get (again for X µ = 0)
ω µν
µνκ
µκ
µν
Mcan
(x) = xν Tcan
− xκ Tcan
+
∂L
(−iJ νκ )ij φj .
∂(∂µ φi )
The conserved (i.e. time independent) charges
Z
νκ
0νκ
J = d3 x Mcan
(t, x) ,
(11.61)
(11.62)
µνκ
are antisymmetric J νκ = J κν . Again, it can be shown that if Mcan
is conνκ
served, J transform as a true four-dimensional tensor. The spatial components J ij play the if (11.62) role of the total angular momentum of the
considered system of fields. For Lagrangian densities of the form (11.54) it
327
is straightforward to check that through the Poisson brackets the tensor J νκ
generates Lorentz transformations of fields φi (x).
i
h
(11.63)
{J µν , φi (x)}PB = − (xµ ∂ ν − xν ∂ µ )δij + (−iJ µν )ij φj (x) ,
so that φ′i (x) = φi (x) − 21 ωµν {J µν , φi (x)}PB . It can also be easily shown,
that the tensor (11.61) differs by a total four-divergence from the tensor
µκ
µν
M µνκ = xν Tsymm
− xκ Tsymm
,
(11.64)
µν
in which Tsymm
is the Belinfante symmetric energy-momentum tensor obtained from (11.60), which gives therefore the same conserved Noether charges
J µν (for field configurations satisfying the equations of motion (11.4) and
vanishing at spatial infinity).
11.2
Canonical quantization of a real scalar field
In this subsection we describe in details quantization of the simplest example of a relativistic field - of the real scalar field ϕ(x). No special difficulty
arises in this case. More complicated cases of vector (in particular of the
electromagnetic field) and spinor fields will be discussed in subsequent subsections. We consider first quantization of the noninteracting field ϕ(x) and
then outline modifications introduced by interactions. Finally, the results
are generalized to the case of many fields.
The simplest dynamics of a real field is given (in units in which c = ~ = 1)
by the Poincaré invariant Lagrangian density
1
1
L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 .
2
2
(11.65)
By a rescaling of the field variable ϕ the first term quadratic in first derivatives, called the kinetic term, can always be brought into the canonical form
as above. In the absence in L of higher powers of ϕ (a feature which makes
this theory solvable both classically and quantum mechanically) the negative
sign of the second term ensures the relativistic relation E 2 (k) = k2 + M 2
and boundedness from below of the Hamiltonian spectrum. To stress the
similarity to the system of coupled oscillators we write the corresponding
Lagrangian in the more general, apriori nonlocal, form
Z
Z
Z
1
1
3
2
3
d x ϕ̇ (t, x) −
d x d3 y ϕ(t, x)K(x, y)ϕ(t, y) , (11.66)
L=
2
2
with a symmetric kernel K(x, y) = K(y, x). Formally, the value of ϕ at every
point x can be treated as an independent canonical variable Qx (t) = ϕ(t, x).
328
The Euler-Lagrange equation of motion of the system described by (11.66)
in this formulation read
δL
d δL
−
= 0,
dt δ ϕ̇(t, x) δϕ(t, x)
(11.67)
(where the derivatives are in the functional sense). Applying (11.67) to
(11.66) gives
Z
d2
ϕ(t, x) + d3 y K(x, y)ϕ(t, y) = 0 .
(11.68)
dt2
For the local Lagrangian density (11.65) the kernel K(x, y) takes (restoring
c and ~ for decoration) the form
K(x − y) = −c2 ∇2(x) δ (3) (x − y) +
M 2 c4 (3)
δ (x − y) ,
~2
(11.69)
so that18
1
L=
2
Z
M 2 c4 2
2
2
2
d x ϕ̇ (t, x) − c (∇ϕ(t, x)) −
ϕ (t, x) .
~2
3
(11.70)
The Euler-Lagrange equation (11.67) is in this case equivalent to the field
equation (11.4) and reads
2
M 2 c4
∂
2
2
ϕ(t, x) = 0 .
(11.71)
−c ∇ +
∂t2
~2
To quantize the system we have first to set up the canonical (Hamilton’s) formalism. We define the momenta Px (t) ≡ Π(t, x) conjugated to the
canonical variables Qx (t) = ϕ(t, x):
Π(t, x) =
δL
= ϕ̇(t, x) ,
δ ϕ̇(t, x)
and construct the Hamiltonian
Z
H = d3 x Π(t, x)ϕ̇(t, x) − L ,
(11.72)
(11.73)
in which ϕ̇(t, x) has to be expressed in terms of Π(t, x) and ϕ(t, x):
Z
Z
Z
1
1
3
2
3
d x Π (t, x) +
d x d3 y ϕ(t, x)K(x − y) ϕ(t, y) . (11.74)
H=
2
2
18
R
In units [M ], [T ] and [L] (mass, time and length) the action I = dtL has dimension
of ~, that is, [M ][L]2 [T ]−1 . It follows that the field ϕ has dimension [M ]1/2 [L]−1/2 . Furthermore, since Π = ϕ̇, it has dimension [M ]1/2 [L]−1/2 [T ]−1 and the Hamiltonian (11.74)
has the right dimension [M ][L]2 [T ]−2 of energy. See also Appendix F.
329
It is easy to check, that the classical Hamilton’s equations
δH
d
ϕ(t, x) = {ϕ(t, x), H}PB =
,
dt
δΠ(t, x)
d
δH
Π(t, x) = {Π(t, x), H}PB = −
,
dt
δϕ(t, x)
(11.75)
are fully equivalent to the Euler-Lagrange equation (11.4), i.e. to (11.71).
Quantization in the Schrödinger picture of a classical system means promoting its canonical variables Qi and Pj taken at one particular instant,
usually t = 0, to time independent operators Q̂i and P̂j (acting in some as
yet unspecified Hilbert space) satisfying the commutation rules19
h
i
Q̂i , P̂j = i~ {Qi , Pj }PB = i~ δij
(11.76)
(and [Q̂i , Q̂j ] = [P̂i , P̂j ] = 0).
Following this prescription, in the case of the real scalar field we impose
h
i
ϕ̂(x), Π̂(y) = i~δ (3) (x − y) ,
h
i
[ϕ̂(x), ϕ̂(y)] = Π̂(x), Π̂(y) = 0 .
(11.77)
Upon quantization the Hamiltonian (11.74) also becomes an operator.
In principle, in full analogy with ordinary quantum mechanics in position
space, in which the system is described by a wave function ψ(Qi , t) (the
probability amplitude of finding the system in the classical state characterized
by the values Qi of its canonical variables) on which Q̂i ’s act by multiplication
by Qi and P̂j ’s act as −i~∂/∂Qj , one can introduce the “wave functional”
Ψ[ϕ, t] ≡ hϕ(x)|Ψ(t)i which represents the probability amplitude that the
field takes on the configuration ϕ(x). On Ψ[ϕ, t] the operators ϕ̂(x) and
Π̂(x) act as
ϕ̂(x)Ψ[ϕ, t] = ϕ(x)Ψ[ϕ, t] ,
Π̂(x)Ψ[ϕ, t] = −i~
δΨ[ϕ, t]
.
δϕ(x)
(11.78)
19
A stronger requirement, that for any pair of classical observables F (Q, P ) and G(Q, P )
such that {F (Q, P ), G(Q, P )}PB = A(Q, P ) in the quantum theory one has
h
i
F (Q̂, P̂ ), G(Q̂, P̂ ) = i~A(Q̂, P̂ ) ,
cannot in general be imposed because of problems with ordering of operators; it can
hold only for observables F (Q, P ) and G(Q, P ) which are at most linear in the canonical
variables Q and P .
330
While this approach offers some useful insight (we will return to it in the
section devoted to path integrals), it does not lead immediately to the interpretation of field states in terms of particles. Such an interpretation emerges
naturally if we find a representation of the ϕ(x) and Π(x) operators (from
now on we omit “hats”) in terms of some other operators, so that the Hamiltonian becomes similar to the one of uncoupled harmonic oscillators. To this
end we define first new operators (Fourier transforms) ϕ̃(k) and Π̃(k) by
Z
Z
3
−ik·x
ϕ̃(k) = d x ϕ(x) e
,
Π̃(k) = d3 x Π(x) e−ik·x .
(11.79)
Since upon quantization real classical field ϕ and real Π go over into Hermitian operators ϕ† (x) = ϕ(x) and Π† (x) = Π(x), their Fourier transforms
satisfy
ϕ̃† (k) = ϕ̃(−k) ,
Π̃† (k) = Π̃(−k) .
From the commutation relations (11.77) it then follows that
h
i
′
ϕ̃(k), Π̃(k ) = i~(2π)3 δ (3) (k + k′ ) ,
h
i
[ϕ̃(k), ϕ̃(k′ )] = Π̃(k), Π̃(k′ ) = 0 .
(11.80)
(11.81)
Below we consider only translationally invariant kernels K(x, y) = K(x−y).
It is then convenient to introduce the quantity ω 2 (k) by
Z
Z 3
dk 2
2
3
−ik·x
ω (k) eik·x . (11.82)
ω (k) = d x K(x) e
,
K(x) =
(2π)3
For a real kernel K(x) one has ω 2 (−k) = ω 2∗ (k), and because K(x) =
K(−x), ω 2(−k) = ω 2(k). Hence, ω 2 (k) is real. We assume it is positive
(this restricts possible forms of the kernel K(x)). For the local K(x) of the
form (11.69) corresponding to the Klein-Gordon Lagrangian density (11.65)
ω 2 (k) = c2 |k|2 +
M 2 c4
.
~2
(11.83)
Expressing the Hamiltonian (11.74) in terms of ϕ̃(k), Π̃(k) and ω 2 (k) we
obtain
Z 3 h
i
1
dk
2
H=
Π̃(k)Π̃(−k) + ω (k) ϕ̃(k)ϕ̃(−k) .
(11.84)
2 (2π)3
By analogy with the harmonic oscillator case we define now the operators
r
ω(k)
i
ϕ̃(k) +
Π̃(k) ,
a(k) =
2~
ω(k)
r
ω(k)
i
†
a (k) =
ϕ̃(−k) −
Π̃(−k) ,
(11.85)
2~
ω(k)
331
whose commutation rules follow directly from (11.81):
a(k), a† (k′ ) = (2π)3 δ (3) (k − k′ ) ,
[a(k), a(k′ )] = a† (k), a† (k′ ) = 0 .
In terms of a(k) and a† (k) one finds
s
~ a(k) + a† (−k) ,
ϕ̃(k) =
2ω(k)
1
Π̃(k) =
i
r
(11.86)
~ω(k) a(k) − a† (−k) ,
2
and inserting these expressions in (11.84) we get
Z 3
†
1
d k
†
H=
~ω(k)
a
(k)a(k)
+
a(k)a
(k)
2 (2π)3
Z 3
dk
1
†
3 (3)
=
E(k) a (k)a(k) + (2π) δ (0) ,
(2π)3
2
(11.87)
where E(k) = ~ω(k). The delta function (2π)3 δ (3) (0) in the second term can
be interpreted as the volume factor. The term
Z 3
Z 3
d k
dk
1
1
3 (3)
E(k) (2π) δ (0) → V ×
E(k) ,
(11.88)
3
2 (2π)
2 (2π)3
can be therefore identified with the contribution to the field energy contained
in the volume V of zero point oscillations of infinitely many field modes
(numbered by the wave vectors k). This infinite constant contribution can
be discarded as long as we do not couple the quantized field to gravity and
as long as we do not compare energies of vacuum states of quantized fields
subject to different boundary conditions (see Section 11.3).
Expressed in terms of the operators a† (k) and a(k), the Schrödinger picture (time independent) field operators ϕ(x) and Π(x) take the form
Z 3 s
dk
~
ik·x
†
−ik·x
ϕ(x) =
a(k)
e
+
a
(k)
e
,
(11.89)
(2π)3 2ω(k)
1
Π(x) =
i
Z
d3 k
(2π)3
r
~ω(k)
a(k) eik·x − a† (k) e−ik·x ,
2
(11.90)
One can then easily check, that with (11.86) the basic commutation rules
(11.77) are satisfied.
The commutation rules (11.86) of a(k) and a† (k) differ from the rules
obtained for their counterparts in the case of the harmonic oscillators in that
332
a(k) and a† (k′ ) commute to the delta function instead of the Kronecker delta.
Therefore the standard algebraic arguments cannot be directly used to determine the spectrum of the Hamiltonian (11.87). There are two alternative
methods of dealing with this difficulty. Firstly, one can take an arbitrary
complete countable set of normalizable functions fα (k) such that
Z 3
X
dk ∗
f
(k)f
(k)
=
δ
,
fα (k)fα∗ (k′ ) = (2π)3 δ (3) (k − k′ ) ,
β
αβ
α
3
(2π)
α
and define the new operators20
Z 3
d k ∗
aα =
f (k) a(k) ,
(2π)3 α
a†α
=
Z
d3 k
fα (k) a† (k) ,
(2π)3
(11.91)
such that
i
h
†
aα , aβ = δαβ .
It is then possible to use the same argument as for ordinary harmonic oscillators that there must exist a vector |0, . . . , 0, . . .i ≡ |Ω0 i annihilated by all
aα ’s and to construct the states
(a† )nα
(a†1 )n1 (a†2 )n2
√
. . . √α
. . . |Ω0 i ,
|n1 , n2 , . . . , nα , . . .i = √
n1 ! n2 !
nα !
(11.92)
which span the whole Hilbert space and represent quantum excitations of the
field ϕ, which can be given a particle interpretation: the state |n1 , n2 , . . .i is
interpreted as the state in which n1 particles are in the first one-particle state
characterized by the momentum space wave function f1 (k), n2 particles are in
the second one-particle state characterized byf2 (k), etc. Particles can occupy
infinitely many one-particle states (in the case of N coupled oscillators there
were only N types of phonons corresponding to N possible 1-phonon states).
However, except for the state |Ω0 i, the states (11.92) are not eigenstates of
the Hamiltonian (although, for the appropriately chosen functions fα (k) they
can be made arbitrarily close to the true eigenstates).
The second alternative is to enclose the classical system in a box of finite
volume V = L3 and to impose periodic boundary conditions on the allowed
field configurations ϕ(x). The wave vectors k form then a discrete set: k =
(2π/L)n, where n = (nx , ny , nz ) are vectors whose components are integers
and the integrals over d3 k are replaced by the sums over discrete k, that is,
20
Similarly as ϕ(x) and Π(x), a† (k) and a(k) are said to be operator-valued distributions; this wise terminology does not shed however much light on physical problems.
333
over (nx , ny , nz ). Periodic boundary conditions21 for the operators ϕ(x) and
Π(x) are ensured by the expansions
1 X
1 X
ϕ̃k eik·x ,
Π(x) = √
Π̃k eik·x ,
ϕ(x) = √
V k
V k
Z
Z
1
1
3
−ik·x
d x ϕ(x) e
,
Π̃k = √
d3 x Π(x) e−ik·x . (11.93)
ϕ̃k = √
V
V
Imposing the relations
h
i
ϕ̃k , Π̃k′ = i~ δ−k,k′
(11.94)
etc. ensures that the commutation rules (11.77) are satisfied. The Hamiltonian (11.74) takes then the form
X
1 X
1
†
2
H=
~ωk ak ak +
Π̃k Π̃−k + ωk ϕ̃k ϕ̃−k =
.
(11.95)
2 k
2
k
The fields can again be written in terms of the creation and annihilation
operators
s
X
~
ak eik·x + a†k e−ik·x ,
ϕ(x) =
2V ω(k)
k
r
1 X ~ω(k) Π(x) =
(11.96)
ak eik·x − a†k e−ik·x ,
i k
2V
satisfying the rules
i
h
ak , a†k′ = δk,k′ ,
h
i
[ak , ak′ ] = a†k , a†k′ = 0 .
(11.97)
One can therefore use the standard reasoning to show that there is a state
|Ω0 i annihilated by all ak ’s and that the Hilbert space is spanned by the
vectors22
1
|nk1 , nk2 , . . .i ≡ p
|k1 , . . . , k1 , k2 , . . . , k2 , . . .i
nk1 ! nk2 ! . . .
nk nk
1
2
a†k1
a†k2
p
= p
. . . |Ω0 i ,
(11.98)
nk1 !
nk2 !
21
The necessary orthogonality and completeness relations read
Z
X
′
d3 x eix·(k−k ) = V δk,k′ ,
eik·(x−y) = V δ (3) (x − y) .
V
k
22
Since the Hamiltonian (11.95) is a sum of Hamiltonians of independent harmonic
oscillators (one per each k), one can, using the analogy with the nonrelativistic quantum
p̂2
+ 21 mω 2 q 2 and the wave function of the ground state
mechanics, in which for H = 2m
334
which are interpreted as representing nk1 bosons of momenta k1 , nk2 bosons
of momenta k2 , etc.
The interpretation of the H eigenstates in terms of noninteracting particles (bosons) given above23 is supported by the statistical properties of the
quantized field: To show this let us consider the quantum field enclosed in
the box of volume V = L3 in contact with a heat bath of temperature T , and
compute the statistical sum Zstat = e−βF , where24 β = 1/kB T and F is the
Helmholtz free energy of the Gibbs Canonical Ensemble. This reduces to
−βH
Zstat = Tr e
=
∞
∞
X
X
nk1 =0 nk2 =0
=
Y
k
"
. . . hnk1 , nk2 , . . . |e−βH |nk1 , nk2 , . . .i
e−~ωk /2kB T
∞
X
e−~ωk /kB
nk =0
T nk
#
.
(11.99)
The geometric series can easily be summed and one finds
F (T, V ) = −kB T ln Zstat =
X
1X
~ωk + kB T
ln 1 − e−~ωk /kB T , (11.100)
2 k
k
or, going over to the continuous normalization,
Z 3 d k 1
−~ωk /kB T
~ωk + kB T ln 1 − e
,
(11.101)
F (T, V ) = V
(2π)3 2
p
with ωk = c2 k2 + c4 M 2 /~2 . Apart from the first term which represents the
contribution of the zero point oscillations (which could have been subtracted
from the beginning) the temperature dependent part of F is precisely the
free energy of a system of nointeracting relativistic bosons (which do not
carry any conserved number and whose total number - similarly to the total
number of photons - is therefore determined only by the condition of thermal
is ψ0 (q) = (mω/π~)1/4 exp −mωq 2 /2~ , represent the vacuum functional of the ground
state of the field in the form
!
Y ωk 1/4
1 X
exp −
hϕk |Ω0 i ≡ ΨΩ0 [ϕ] =
ωk ϕk ϕ∗k .
π~
2~
k
k
The operators ϕk act in this representation as multiplication whereas the operators Πk act
as derivatives. Wave functionals of particle states can in principle be obtained by acting
with a†k in this representation on the vacuum ΨΩ0 [ϕ]. The analogy with ordinary quantum
mechanics has to be exploited cautiously, however, because the variables ϕk and Πk are
complex here.
23
Notice that having quantized the field in a finite box we cannot appeal to the Poincaré
transformation properties of the states.
24
kB = 8.6 × 10−5 eV/K is the Boltzmann constant.
335
equilibrium). Thus, one of the arguments that allow us to interpret states of
quantized fields as particles is essentially identical to the statistical argument
used by Einstein in 1905 to argue that electromagnetic radiation behaves as
a collection of noninteracting particles (photons).
Calculation of the partition function of an interacting quantum field is
most easily formulated in the path integral approach. Still, presenting that
approach in Section 16 we will need the result (11.101) to fix an additive
constant in F which in the functional approach is difficult to compute.
Fields quantized in the infinite space can be treated similarly. We assume
that there is a ground state |Ω0 i which has zero energy (after subtracting
the infinite contribution of infinitely many zero mode oscillations) and the
other states spanning the Hilbert space are obtained by acting on it with the
creation operators a† (k)
|ki = a† (k)|Ω0 i ,
|k1 , k2 i = a† (k1 ) a† (k2 )|Ω0 i ,
|k1 , k2 , k3 i = a† (k1 ) a† (k2 ) a† (k3 )|Ω0 i ,
...
(11.102)
where all momenta k, k1 , k2 , etc. are arbitrary. (These states are not properly normalized when two or more momenta coincide; this has to be taken
care of in the completeness relation as it was done in (5.14)). Although nonnormalizable (if we assume that the state |Ω0 i has the norm equal to 1) these
are true eigenstates of the Hamiltonian H (11.87). Their particle interpretation follows now from their Poincaré transfomation properties which we
discuss at the end of this subsection.
If the Lagrangian defining the theory is not quadratic in the field ϕ but
instead of (11.65) takes the more general form (we set c = 1 but for decoration
keep ~ in some formulae below)
1
1
1
L = ∂µ ϕ∂ µ ϕ − V (ϕ) = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − Hint (ϕ) ,
2
2
2
(11.103)
where Hint (ϕ), and therefore also the whole field potential V (ϕ), is usually
a polynomial of fields,25 e.g. Hint (ϕ) = (λ/4!)ϕ4 , the quantization in the
Schrödinger picture proceeds in the same way as described above: the commutation rules (11.77) of ϕ and Π remain unchanged. The Hamiltonian takes
then the form
Z
H = H0 + Vint = H0 + d3 x Hint (ϕ) ,
(11.104)
25
We will see that in the presence of Hint (ϕ) the sign of the term quadratic in ϕ can
also be positive (negative M 2 ).
336
H0 =
Z
1 2 2
1 2 1
2
d x Π + (∇ϕ) + M ϕ .
2
2
2
3
The form (11.104) of H remains valid for Hint (ϕ), or even V (ϕ) explicitly
depending on time (e.g. if the quantized field ϕ interacts with some external
agents). As in the case of the free field, one can expand the Schrödinger
picture operators ϕ(x) and Π(x) as in (11.89) and (11.90) into the operators
a(k) and a† (k) with the same commutation rules as previously but the Hamiltonian (11.104) will now contain terms (a(k))4 , (a† (k))4 , etc. and the states
(11.102) will not be its eigenstates.26 Still, if the coefficient M 2 does not
depend on time, they are eigenstates of the free part H0 of the Hamiltonian
(11.104) and will play an important role in the formulation of the S-matrix
approach to the scattering theory. This will be discussed in Section 11.9.
In older formulations one used to define the quantum theory by taking for
the Hamiltonian operator not the expression (11.104), but its normal odered
counterpart Ĥ = : H(11.104) :. Normal ordered form : OS : of a Schrödinger
picture operator OS is the form, in which all annihilation operators are placed
to the right with respect to creation operators; for example
(−)
(−)
(−)
(+)
: ϕi (x)ϕj (x) : = ϕi (x)ϕj (x) + ϕi (x)ϕj (x)
(−)
(+)
(+)
(+)
+ϕj (x)ϕi (x) + ϕi (x)ϕj (x) ,
for bosonic field operators (i.e. ones satisfying commutation relations) and
(−)
(+)
: ψα (x)ψβ (x) : = ψα(−) (x)ψβ (x) + ψα(−) (x)ψβ (x)
(−)
(+)
−ψβ (x)ψα(+) (x) + ψα(+) (x)ψβ (x) ,
for fermionic operators (i.e. ones satisfying anticommutation relations -see
section 11.8), where ϕ(−) , (ϕ(+) ) is the part of (11.89) with the creation (annihilation) operator. Operators ordered in this way have all (except for the
unit operator) zero expectation value in the Fock state |Ω0 i annihilated (by
definition) by all the annihilation operators. This prescription for Ĥ removes
some of the infinities encountered in practical calculations, in particular it
removes the addtive infinite part (11.88) in the expectation value of the free
field Hamiltonian (11.87) in the state |Ω0 i. Normal ordering, ubiquitus on
older approaches to quantum field theory, has been now largely abandoned.
First of all, it does not remove all divergences which must be renormalized
anyway (see Section 14) and there is no point to invoke two different prescription for removing divergences having a common ultraviolet origin. Furthermore, normal ordering of operators defined in terms of the creation and
annihilation operators diagonalizing H0 (i.e. normal ordering with respect
26
Moreover, if Hint depends explicitly on time, the Hamiltonian H does not, strictly
speaking, possess time-independent eigenstates.
337
to the Fock state |Ω0 i) does not imply that hΩ− | : OS : |Ω+ i = 0 - matrix elements can have nonperturbative contributions. Finally, and probably most
importantly, in the modern functional approach (see Section 16) quantum
field theory is viewed as a formalism allowing to take into account real quantum fluctuations of fields; from this point of view every quantum field theory
is defined with Fourier momenta bounded by an ultraviolet cutoff Λ which
should have a real physical meaning. All contributions to amplitudes, expectation values, etc. one computes using the quantum field theory formalism
are then equally physical and should not be subtracted using an arbitrarily
defined prescription. The divegences (arising for Λ → ∞) disappear however
if the computed quantities are expressed in terms of some other measurable
quantities; subtracting divergences is then merely done for computational
convenience and has no fundamental meaning; hence defining the Hamiltonian as a normal ordered operator becomes then a completely useless and
obsolete prescription.27
We have quantized the classical real scalar field ϕ in the Schrödinger picture in which the operators ϕ(x) and Π(x) satisfying the rules (11.77) do not
depend on time. Transition to the Heisenberg picture, in which the states of
the system do not vary with time whereas the operators do, is usually convenient, particularly in relativistic field theories. In general (see Section 1), this
is achieved by choosing the moment t0 at which the two pictures coincide and
defining Heisenberg picture operators OH (t, x) corresponding to Schrödinger
picture operators OS (t, x) = O(t, ϕ(x), Π(x)) by the formula
OH (t, x) = U † (t, t0 )OS (t0 , x)U(t, t0 ) = O(t, ϕH (t, x), ΠH (t, x)) , (11.105)
in which
i
U(t, t0 ) = T exp −
~
Z
t
t0
′
′
dt H(t ) ,
(11.106)
is the Schrödinger picture evolution operator. By construction the operators
OH (t, x) satisfy then the Heisenberg equation (see Sec. 1.1)
1
∂O
d
QH (t, x) =
[OH (t, x), HH (t)] +
dt
i~
∂t H
1 †
∂O
≡
, (11.107)
U (t, t0 ) [OS (t0 , x), H(t0 )] U(t, t0 ) +
i~
∂t H
in which HH (t) ≡ U † (t, t0 )H(t0 )U(t, t0 ). (Most of the Schrödinger picture
operators OS do not depend on time and the second term in (11.107) is
27
Normal ordering is therefore only relevant for the Wick theorem (13.12) used to set
the perturbative expansion.
338
absent; one notable exception is the boost generators given by space integrals
(11.62) of M 00i given by (11.61)).
It is easy to check that the Heisenberg picture operators ϕH (t, x) and
ΠH (t, x), corresponding to the canonical variables and obtained from the
prescription (11.105), satisfy the canonical equal time commutation rules
[ϕH (t, x), ΠH (t, y)] = iδ (3) (x − y) ,
[ϕH (t, x), ϕH (t, y)] = [ΠH (t, x), ΠH (t, y)] = 0 ,
(11.108)
for arbitrary time t. These rules together with the Hamiltonian H (following
from the action I) are in fact the basic relations defining the quantum version
of the classical field theory model. Furthermore, for H of the form (11.104)
(even if V (ϕ) in the Lagrangian density (11.103) depends explicitly on time,
i.e. for a system interacting with external agents) the Heisenberg equations
(11.107) satisfied by ϕH and ΠH ,
d
ϕH (t, x) =
dt
d
ΠH (t, x) =
dt
1
[ϕH (t, x), HH (t)] ,
i~
1
[ΠH (t, x), HH (t)] ,
i~
(11.109)
′
yield ϕ̇H = ΠH and Π̇H = c2 ∇2 ϕH − M 2 ϕH − Hint
(ϕH ). As a result, the
Heisenberg operator ϕH satisfies the “classical” field equation of motion
2
∂
2
′
2
(ϕH (t, x)) ,
(11.110)
− ∇ + M ϕH (t, x) = −Hint
2
∂t
while the operator ΠH satifies the equation
2
∂
2
2
′′
− ∇ + M + Hint (ϕH (t, x)) ΠH (t, x) = 0 .
∂t2
(11.111)
Together with the equal time commutation rules (11.108) these two equations
can also constitute the complete specification of the theory.
Of course, for closed systems, which the system of fields usually are, H is
independent of time. The evolution operator U(t, t0 ) (11.106) reduces then to
the ordinary exponent, HH (t) = H, so that in the considered theory (11.103)
it takes the form
Z
1 2
1
3
2
H = d x ΠH (t, x) + (∇ϕH ) (t, x) + V (ϕH (t, x)) , (11.112)
2
2
formally the same as in the classical theory, and choosing t0 = 0 as it is
customary, one has
ϕH (t, x) = eiHt/~ ϕ(x) e−iHt/~ ,
ΠH (t, x) = eiHt/~ Π(x) e−iHt/~ .
339
(11.113)
For the free field whose Hamiltonian H (11.74) can be represented in
the form (11.87), the equations (11.109) can be easily solved. ϕH (t, x) and
ΠH (t, x) are then given by the expressions (11.89) and (11.90) but with a(k)
and a† (k) replaced by aH (t, k) and a†H (t, k) satisfying the equations
1
d
aH (t, k) =
[aH (t, k), H] ,
dt
i~
d †
1 †
aH (t, k) =
[a (t, k), H] ,
dt
i~ H
(11.114)
with the initial conditions aH (0, k) = a(k), a†H (0, k) = a† (k). One then finds
[aH (t, k), H] = ~ω(k)aH (t, k) and [a†H (t, k), H] = −~ω(k)a†H (t, k). The
equations (11.114) can be then easily integrated28 to give
aH (t, k) = e−iω(k)t a(k) ,
a†H (t, k) = eiω(k)t a† (k) ,
(11.115)
so that
ϕH (t, x) =
Z
1
ΠH (t, x) =
i
Z
d3 k
(2π)3
s
d3 k
(2π)3
~ a(k) e−iω(k)t+ik·x + a† (k) eiω(k)t−ik·x , (11.116)
2ω(k)
r
~ω(k) a(k) e−iω(k)t+ik·x − a† (k) eiω(k)t−ik·x ,(11.117)
2
that is, the exponents in the Heisenberg operators depend on the Lorentz
invariant products xµ kµ , where xµ = (ct, x) and k µ = (ω(k)/c, k). Both field
operators, ϕH and ΠH , satisfy in this case the free Klein-Gordon equation.
It is also good to remark that in certain cases (the most notable being the
one of the free electromagnetic field quantized using the approach of Gupta
and Bleuler discussed in section 11.11 with the gauge fixing parameter ξ 6= 1)
even if the Hamiltonian is quadratic in field variables and conjugated momenta their expansions analogous to (11.89), (11.90) are not easy to find and
do not render the Hamiltonian diagonal; the time dependendent operators
(11.113) are then not simply given by replacing in (11.89), (11.90) e±ik·x by
e∓ikx as in (11.116), (11.117). Still, the explicit form of the time dependent
operators and their momenta can be found by directly solving (though not
so easily) the canonical equations (11.109).
28
Notice, that if the Lagrangian contains terms with ϕ in powers higher than the second,
the Heisenberg equations resulting from (11.114) are more complicated and their solution
cannot be found in a closed form. Hence, the Heisenberg picture operators ϕH (x) and
ΠH (x) corresponding to the interacting field cannot be written in the forms (11.116) and
(11.117); only ϕH (0, x) ≡ ϕ(0, x) and ΠH (0, x) ≡ Π(0, x) (recall, we have chosen to equate
the Schrödinger and Heisenberg pictures at t0 = 0) can be written as in (11.89) and (11.90).
340
At this point, in order to simplify the notation, we set also ~ = 1 and drop
the subscript H on Heisenberg picture operators. In addition, it is convenient
to change thep
normalization of the thepcreation and annihilation operators
a(k) → a(k)/ 2E(k), a† (k) → a† (k)/ 2E(k), where E(k) = ~ωk , so that
their commutator becomes
a(k), a† (k′ ) = 2E(k)(2π)3 δ (3) (k − k′ ) ≡ δΓ (k − k′ ),
(11.118)
and the (Heisenberg picture) free field operators (11.116), (11.117) take the
simple form
Z
ϕ(x) = dΓk a(k) e−ik·x + a† (k) eik·x ,
(11.119)
1
Π(x) =
i
Z
dΓk Ek a(k) e−ik·x − a† (k) eik·x ,
(11.120)
where the measure dΓk ≡ d3 k/(2π)3 2E(k) is separately Lorentz invariant.
In theories of interacting fields (when in the full Hamiltonian there are
terms with powers fields higher than the second) the time dependent operators termed the interaction picture operators obtained by solving the equations (11.109) but with time independent H0 replacing the full HH (t) and
hence given in closed form by (11.113) again with H0 instead of H, play
an important role in formulating the perturbation expansion for computing
S-matrix elements. This will be discussed in section 11.9.
We can now discuss the Poincaré transformations. Using the rules given in
Section 11.1 one finds the canonical energy-momentum tensor corresponding
to the Lagrangian density (11.103)
1 2 2
µν
µ
ν
µν 1 λ
Tcan = ∂ ϕ∂ ϕ − g
∂ ϕ∂λ ϕ − M ϕ − Hint (ϕ) .
(11.121)
2
2
It is in this case automatically symmetric.29 In the quantum theory ∂ 0 ϕ must
of course be expressed in terms of the canonical variables ϕ and Π. One then
has (we drop the subscript H)
1 2
Π (t, x) + ∇ϕ(t, x) ·∇ϕ(t, x) + M 2 ϕ2 (t, x) + Hint (ϕ(t, x)) ,
2
0i
Tcan
(t, x) = Π(t, x) ∂ i ϕ(t, x) .
(11.122)
00
Tcan
(t, x) =
29
µν
Still, as it turns out, even in this simple case Tcan
must be modified by adding to it
µν
ρµν
a tensor H = ∂ρ H
as in (11.57), in order to ensure finiteness of matrix elements of
the energy-momentum tensor (treated as an operator) in the theory of interacting fields.
341
00
P 0 given by the integral of Tcan
over d3 x is just the Hamiltonian (11.104)
written in terms of the Heisenberg field operators whereas the momentum
operator is given by
Z
Z
i
3
0i
P = d x Tcan (t, x) = d3 x Π(t, x)∂ i ϕ(t, x) .
(11.123)
µν
From Tcan
the generators of the Lorentz transformations are obtained according to the formulae (11.61) and (11.62):
Z
µν
0ν
0µ
J = d3 x xµ Tcan
− xν Tcan
.
(11.124)
Using the canonical commutation rules (11.77) these generators can be shown
to satisfy the commutation rules of the Poincaré algebra (6.19) or (6.21).
Moreover, P µ and J µν generate transformations of the field operator ϕ(x)
related to changes of the reference frame
µ
µ
eiaµ P ϕ(x) e−iaµ P = ϕ(x + a) ,
i
µν
i
e 2 ωµν J ϕ(x) e− 2 ωµν J
µν
= ϕ(Λ−1 (ω) ·x) ,
(11.125)
or, in the infinitesimal form,
i [P µ , ϕ(x)] = ∂ µ ϕ(x) ,
i [J µν , ϕ(x)] = (xµ ∂ ν − xν ∂ µ )ϕ(x) .
(11.126)
It is important to stress that the operators P i , H, J i and K i satisfy the commutation rules of the Poincaré algebra and generate the field transformations
as in (11.125) only by virtue of the canonical commutation relations (11.108),
independently of the form of Hint (ϕ) in (11.103), i.e. also in the quantum
theory of the interacting field ϕ. The Poincaré symmetry generators P i , H,
J i and K i are by construction independent of time. This can be checked
by using the Heisenberg equation (11.107). As such they can be computed
for any time t. A particularly convenient (for checking their commutation
relations) is the choice of t = 0, for which the Heisenberg picture operators
ϕH and ΠH can be expanded into the creation and annihilation operators.
For V (ϕ) = 21 M 2 ϕ2 , i.e. in the quantum theory of the free field (with
Hint = 0), H takes in terms of the creation and annihilation operators the
form (11.87) and the generators P i and J ij and K i = J 0i are given by
Z
i
P = dΓk k i a† (k)a(k) ,
Z
∂
ij
†
i ∂
a(k) ,
(11.127)
− kj
J = i dΓk a (k) k
∂kj
∂ki
Z
∂
K i = i dΓk a† (k) Ek
a(k) .
∂ki
342
In this case, the states (11.102) are the eigenstates of H and P with the
eigenvalues (Ek , ~k), (Ek1√+ Ek2 , ~k1 + ~k2 ), etc., where Ek is given by the
relativistic formula Ek = ~2 k2 c2 + M 2 c4 = ~ω(k). One can also check that
the one particle states |ki transform properly under the Lorentz transformations,
U(Λ)|ki = U(Λ) a† (k) U −1 (Λ) U(Λ)|Ω0 i = a† (kΛ )|Ω0 i ,
(11.128)
i.e. that they transform in the way appropriate for states of a spin 0 particle
(see Section 6.2) provided the vacuum state |Ω0 i is assumed to be Lorentz
invariant U(Λ)|Ω0 i = |Ω0 i. In the interacting field case, if ϕH (0, x) and
ΠH (0, x) are expanded into creation and annihilation operators, the operators
P i and J ij take usually30 the same form as in (11.127), whereas the boost
generators K i are modified by the interaction, as argued in Section 7.3.
Quantization of a single real scalar field described above can immediately
be generalized to the case of many real scalar fields ϕi (x) where i = 1, . . . , N,
whose classical dynamics is governed by the Lagrangian density (11.54). Furthermore, a system of N classical complex fields can always be represented
as a system of 2N real fields and quantized using the same prescriptions.
For several fields ϕi the Lagrangian density may be invariant (or invariant up to a total divergence - see the formula (11.20)) under a group of
continuous transformations which in the infinitesimal form can be written
as ϕi → ϕ′i = ϕi + θa δ a ϕi . In this case, upon replacing classical fields ϕi
and their derivatives by the corresponding field operators (as ∂ 0 ϕi one has
of course to use the expressions ϕ̇i = ϕ̇i (Π, ϕ) given by the canonical formalism), the classical expressions for Noether currents jµa become current
operators and the quantities
Z
a
Q̂ = d3 x ja0 (t, x) ,
(11.129)
become the symmetry generators acting in the Hilbert space.31 By using the
canonical commutation rules (11.108) one can show that independently of
the precise form of L (provided it is symmetric) the time-like components of
the Noether currents satisfy the commutation relations
0
(11.130)
ja (t, y), jb0 (t, x) = jc0 (t, x) ifc ab δ (3) (x − y) ,
30
Exception are theories quantized around a nontrivial classical background ϕcl (x) in
which the field operator takes the form ϕH (t, x) = ϕcl (x) 1̂ + χH (t, x).
31
Of course, the operators Q̂a in (11.129)
are well defined if their matrix elements
R
between physical states: hΦ|Q̂a |Ψi = d3 x hΦ|ja0 (t, x)|Ψi are well defined; it may then
happen that the right hand side is not integrable due to the presence in the H spectrum
of massless particles which mediate long range forces. This is so whenever a classical
continuous symmetry of the Lagrangian is spontaneously broken by the vacuum state (see
Section 22).
343
which ensure that the Noether charges (11.129) satisfy the symmetry algebra
relations
[Q̂a , Q̂b ] = Q̂c ifc ab .
(11.131)
Moreover, by virtue of the canonical commutation rules (11.77) the Noether
charges Q̂a generate symmetry transformations of the field operators
iθa [Q̂a , ϕi (x)] = θa δ a ϕi (x) .
(11.132)
In contrast, space-like components of the Noether currents usually can satisfy
the commutation rules analogous to (11.130) only in the theory of noninteracting fields. For general interactions Hint (ϕ) it is only possible to infer
(using the Lorentz covariance) that
(x)
ik
ja0 (t, y), jbi (t, x) = jci (t, x) ifc ab δ (3) (x − y) + Sab
(t, x) ∂k δ (3) (x − y) ,
ik
where Sab
(t, x) are the so-called Schwinger terms. Even in theories (like electrodynamics of spin 1/2 particles) in which the canonical (anti)commutation
relations formally imply the absence of Schwinger terms,32 it is possible to
prove, using arguments based on general principles of quantum mechanics,
that they cannot vanish. The contradiction with the reasoning based on the
canonical (anti)commutation relations stems from the fact that Noether currents are composite operators (products of elementary field operators taken
at the same space-time point) which are in general singular objects - their
matrix elements require some regularization; Schwinger terms can therefore
depend on the adopted regularization prescription, and a regularization can,
a priori, also induce nontrivial Schwinger terms in the commutators (11.130)
of the time components of the currents. If a given symmetry can be recovered after regularization in the renormalized theory, Schwinger terms in the
commutators like (11.130) are absent (although are generally nonvanishing in
the commutators involving spatial components of the Noether currents). In
fact, anomalies discussed in Section 23 can be understood as manifestation
of nontrivial Schwinger terms induced by the necessity of regularization in
the commutators like (11.130) of the time components of the currents.
Finally, if the Hamiltonian commutes with the Noether charges Q̂a and
the symmetry is not spontaneously broken by the vacuum state, i.e. if
Q̂a |Ωi = 0 for all a (see Section 22), the Hamiltonian eigenstates (the in
and out states) form multiplets of the symmetry algebra. Usually also the
H0 part quadratic in field operators of the full Hamiltonian H commutes
32
In theories in which symmetries are realized on scalar fields, the presence of nontrivial
(regularization independent) Schwinger terms is usually revealed already by using the
canonical commutation relations.
344
with Q̂a ’s separately, and the H0 eigenstates, i.e. particles created and annihilated by the Hermitian free-field (interaction picture) operators ϕi (x) also
form multiplets of the symmetry algebra. As all members of the symmetry
multiplets have the same mass, one then forms linear combinations of the
particle states (both in, out and the free-particle ones) which diagonalize the
Noether charges Q̂a forming the Cartan subalgebra (see Section 4) of the full
symmetry algebra. It is then convenient to form also the appropriate (complex in general) linear combinations of the free-field operators ϕi (x) creating
and destroying one-particle eigenstates of the the Cartan subalgebra generators.33 In the case of the symmetry multiplets transforming as complex
representations, such that each particle (except for essentially neutral ones)
finds its antiparticle within the same multiplet, this corresponds precisely to
forming (non-Hermitian) field operators creating a particle and annihilating
its antiparticle as described in section 8.2.
Existence of conserved charges has also important consequences for statistical properties of the system of quantum fields. If the Hamiltonian commutes with Noether charges Q̂a , the quantum numbers corresponding to the
Cartan subalgebra generators can have simultaneously definite values (because all these generators commute with each other) which are constant of
motion. This would have to be taken into account in the Gibbs Canonical
Ensemble statistical sum Zstat (T, V ), making it essentially intractable analytically (similarly as in the case of a system consisting of a fixed number
N of particles) even for noninteracting fields. One then passes to the Grand
Canonical Ensemble introducing the chemical potentials µa for each Cartan
subalgebra generator. Choosing in the Hilbert space the basis in which the
Cartan subalgebra generators are diagonal facilitates then computation of
the statistical sum Ξ(T, V, µa ).
For example, the Lagrangian
2
1X
L=
∂µ ϕl ∂ µ ϕl − M 2 ϕ2l ,
2 l=1
(11.133)
of two noninteracting scalar fields ϕ1 and ϕ2 has the O(2) ≃ U(1) symmetry
whose conserved Noether current is jµ = ϕ2 ∂µ ϕ1 −ϕ1 ∂µ ϕ2 . After quantization
it is the operators a† (k) and ac† (k) formed as appropriate linear combinations
of a†l (k), l = 1, 2 which, acting on the vacuum |Ωi, generate the states
33
The corresponding Heisenberg picture operators obtained from the free-field ones as in
(11.113) (or, more generally, as in (11.105)) have then “diagonal” matrix elements between
the one-particle states of the full Hamiltonian H and the vacuum: hΩ|φiH |p, σ, ji ∝ δ ij
with the same proportionality constant for all φiH and all states |p, σ, ji belonging to the
same symmetry multiplet.
345
|nk , . . . , nck , . . .i of the Hilbert space basis in which the Hamiltonian
X
c
~ωk a†k ak + ac†
a
+
1
,
(11.134)
H=
k k
k
where ωk =
p
c2 k2 + c4 M 2 /~2 , and the Noether charge
X †
c
Q̂ =
(ak ak − ac†
k ak ) ,
(11.135)
k
are simultaneously diagonal. Obtaining the potential Ω(T, V, µ) = −kB T ln Ξ
by computing the statistical sum
Ξ = e−β Ω = Tr e−β(H−µQ̂) ,
is then straightforward:
Z 3
dk Ω(T, V, µ) = V
~ωk + kB T ln 1 − e−(~ωk −µ)/kB T
3
(2π)
+kB T ln 1 − e−(~ωk +µ)/kB T .
(11.136)
(11.137)
We have passed already to the continuum (thermodynamical) limit as in
(11.101). The value of the chemical potential µ determines the mean (in the
sense of the ensemble) total charge Q of the field in the box V . Convergence of the summations leading to (11.137) imposes the constraint |µ| <
M =mink ωk , where M is the mass of the particles. Of course, for µ = 0 the
system is neutral (Q = 0 on average), and |µ| grows as the (mean) charge
|Q| of the system grows. Eventually, if |Q| becomes so large that |µ| → M,
the occupancy of the zero momentum (k = 0) state becomes macroscopical
(it has to be extracted before the transition to the continuous normalization)
and the usual Bose-Einstein condensation occurs.
11.3
Another view on fields quantization
In the preceding Section we have quantized field(s) not subject to any special
boundary conditions, i.e. defined in the empty space. We have used the
Schrödinger picture to determine the Hamiltonian eigenstates (in the case
of free fields) and performed the transition to the Heisenberg picture only
at the later stage. For greater generality we have allowed for Hamiltonians
explicitly dependent on time, but such cases, even if the Hamiltonian is at
most quadratic in field variables, are difficult to analyze using the formalism
presented in Section 11.2 because of technical difficulties in constructing the
evolution operator (11.106).
346
Here we present another view on fields quantization which makes clear the
central role played in this procedure by the complete set of solutions to the
classical equations of motion (11.4) derived from the underlying Lagrangian.
The advantages offered by this view are twofold. Firstly, it allows to take
into account various types of boundary conditions imposed on fields, usually
intended to model complicated interactions of fields of interest with external
agents. Secondly, it allows to quantize fields directly in the Heisenberg picture
even for systems whose Hamiltonians depend on time explicitly, without
solving34 the complicated operator equations (11.106), (11.107). Thus, in this
case, this view facilitates studies of the time evolution of quantum systems.
One should stress however, that the approach discussed here, similarly as
the one presented in the preceding Section, applies evidently only to systems
described by Lagrangians depending on first derivatives of field variables
only, or more precisely, to systems which classically admit the Hamiltonian
description.
To begin with, we quantize once again, this time directly in the Heisenberg picture, the free real field ϕ in the empty space whose classical dynamics
is governed by the Lagrangian density (11.65). As no special boundary conditions are imposed, the most general solution to the classical Klein-Gordon
equation (∂µ ∂ µ + M 2 )ϕ(x) = 0 following from (11.65) reads
Z 4
dk
ϕ(x) =
A(k) e−ik·x 2πδ(k 2 − M 2 ) ,
(11.138)
(2π)4
where (because ϕ is a real field) A(k) = A∗ (−k). Using then the formula
δ(k 2 − M 2 ) =
1
1
δ(k 0 − E(k)) +
δ(k 0 + E(k)) ,
2E(k)
2E(k)
(11.139)
we get
ϕ(x) =
Z
d3 k
A(k 0 = E, k) e−iEt+ik·x + A(k 0 = −E, k) e+iEt+ik·x .
3
(2π) 2E(k)
In the second term we change the integration variable k → −k and introduce
the new coefficients a(k) ≡ A(E, k) and a∗ (k) ≡ A(−E, −k) so that the most
general solution to the classical Klein-Gordon equation takes the form
Z
ϕ(x) = dΓk a(k) e−ik·x + a∗ (k) eik·x .
(11.140)
34
Of course, to apply this method one has to find the complete set of solutions to the
classical equations of motion, which is usually not possible in a closed form for Lagrangian
densities not quadratic in field variables. The only possible quantizaton method is then
the one presented in the preceding Secion.
347
The canonical momentum Π(x) is obtained by using the classical definition
(11.72) as ϕ̇(x). One then quantizes the system by promoting the arbitrary
coefficients a(k) and a∗ (k) (which classically are determined by some initial
data ϕ(t0 , x) = q(x), Π(t0 , x) = p(x) given at some t0 , or more generally on
some space-like hypersurface), to operators and imposing on them the commutation rules ensuring that the Heisenberg picture field operators ϕ(t, x)
and Π(t, x) satisfy the canonical equal time commutation rules (11.108)
[ϕ(t, x), Π(t, y)] = iδ (3) (x − y) ,
[ϕ(t, x), ϕ(t, y)] = [Π(t, x), Π(t, y)] = 0 .
(11.141)
It is easy to see that this requires a(k) and a∗ (k) to be promoted to annihilation and creation operators, respectively with the commutation rule
(11.118). Thus, we directly recover the Heisenberg field operator ϕH (t, x)
given by (11.116).
The same procedure, with appropriate modifications can be applied to
quantize the field ϕ with periodic boundary conditions. Strictly speaking, it
is just the approach adopted here that lends justification to the quantization
of the field ϕ in a box of volume V performed in the Schrödinger picture
in the preceding Section: as a matter of facts, we have there written down
ϕ(x) and Π(x) = ϕ̇(x) as most general functions of x consistent with the
Klein-Gordon equation and the periodic boundary conditions and only then
imposed the appropriate commutation relation on the coefficients of these
functions so that to satisfy the canonical commutation rules (11.77).
The equivalence of the quantization method in the Heisenberg picture
described here and the one based on the Schrödinger picture, the evolution
operator (11.106) and the equations (11.107), follows immediately if one notices that the Heisenberg picture operators ϕH (t, x) and ΠH (t, x) obtained
from the prescription (11.105) satisfy the canonical equal-time commutation relations (11.141) as well as the “classical” field equations (11.110) and
(11.111), respectively; in the method described above, field operators having the same properties are constructed directly, using the complete set of
classical equation of motions.35
In order demonstrate the working of this approach in nontrivial circumstances, let us consider a real scalar field ϕ between two large parallel plates
(perpendicular to the z axis) of area L2 each, separated by the distance d.
Let us impose as the boundary conditions that the field ϕ vanishes on the
plates and satisfies periodicity conditions (with the period L) in the remaining directions. Furthermore, for simplicity let us assume that the Lagrangian
35
This suggests that it should be possible to construct the evolution operator U (t, t0 )
defined in (11.106) using the complete set of solutions of the classical equations of motion.
348
density of the field ϕ is given by (11.65) with M = 0. Classically, ϕ satisfies
the Klein-Gordon equation ∂µ ∂ µ ϕ(t, x) = 0 with the boundary conditions
ϕ(t, x, y, z = 0) = ϕ(t, x, y, z = d) = 0 ,
ϕ(t, x + L, y, z) = ϕ(t, x, y + L, z) = ϕ(t, x, y, z) .
The most general form of the field ϕ therefore is
∞
∞
∞
π 2π
X
X
X
ϕnml (t) sin l z ei L (nx+my) ,
ϕ(t, x) =
d
n=−∞ m=−∞ l=1
(11.142)
(11.143)
with the coefficients ϕnml (t) satisfying the equation
d2
2
ϕnml (t) + ωnml
ϕnml (t) = 0 ,
dt2
(11.144)
where
2
2
ωnml
2 π
=
l
+ (m2 + n2 )
c2
d
2π
L
2
.
The general solution for ϕnml (t) can be written as
r
~
ϕnml (t) =
Anml e−iωnml t + Bnml e+iωnml t
2
ωnml L d
(11.145)
(11.146)
(the prefactor has been chosen for further convenience). Reality of the field
ϕ(x) imposes the relation Bnml = A∗−n,−m,l . Thus,
∞
∞
∞ r
π X
X
X
~
2π
sin l z Anml e−iωnml t ei L (nx+my)
ϕ(t, x) =
2
ωnml L d
d
n=−∞ m=−∞ l=1
2π
+A∗nml eiωnml t e−i L (nx+my) , (11.147)
after changing in the second term −n → n, −m → m. It is now easy to
check that if the coefficients Anml and A∗nml are promoted to annihilation
and creation operators, respectively, with the commutation rules
i
h
†
(11.148)
Anml , An′ m′ l′ = δnn′ δmm′ δll′ ,
the field operators ϕ(t, x) and Π(t, x) = ϕ̇(t, x) satisfy the right canonical
equal time commutation rules36
h
i
ϕ̂(t, x), Π̂(t, y) = i~δ (3) (x − y) ,
h
i
[ϕ̂(t, x), ϕ̂(t, y)] = Π̂(t, x), Π̂(t, y) = 0 .
(11.149)
36
The necessary formulae are
Z d
π Z d
π d
π π ′
dz cos l′ z cos l z = δl′ l ,
dz sin l z sin l z =
d
d
d
d
2
0
0
349
It is then straightforward to express the Hamiltonian
Z
Z L Z L
1 d
dz
dx
dy ϕ̇2 + c2 (∇ϕ)2 ,
H=
2 0
0
0
in terms of the operators Anml and A†nml :
H=
∞
X
∞
∞
X
X
n=−∞ m=−∞ l=1
1
†
~ωnml Anml Anml +
.
2
(11.150)
so that the energy of the vacuum fluctuation of the field ϕ(t, x) for L ≫ d
can be written as
r
Z
∞
X
π2
~c L2
2
2
2
k
+
l
Ebc ≡ hΩbc |H|Ωbc i =
d
k
,
(11.151)
k
k
2 (2π)2
d2
l=1
where |Ωbc i is the ground state of the field quantized with the boundary
conditions (11.142), annihilated by all Anml , and where we have replaced the
summations over n and m by the two dimensional integral (L/2π)2 dkx dky .
One can now compare the energy contained in the box L × L × d of the
field ϕ(t, x) subject to the boundary conditions (11.142) with the energy
Eper contained in the same volume L2 d of the field ϕ quantized in the box
L × L × L with the periodic boundary conditions (with period L) in all
the three directions. We are therefore interested in the difference ∆E ≡
Ebc − Eper , in which
Z
q
~c L2 d
L2 d
3
2
d
k
. (11.152)
k2k + k⊥
Eper ≡ 3 × hΩper |H|Ωper i =
L
2 (2π)3
Writing k⊥ formally as l(π/d) and integrating over dl instead of dk⊥ , we get
)
(∞ r
Z
Z ∞ r
2
2
X
π
π
~c L2
k2k + l2 2 −
∆E =
d2 kk
dl k2k + l2 2 . (11.153)
2
2 (2π)
d
d
0
l=1
This energy difference should be finite. To jsutify this claim we can appeal
to the physics of the problem. Nontrivial boundary conditions as (11.142)
∞
X
l=1
Z
0
L
2π
2π
′
π π d
sin l z sin l z ′ = δ(z − z ′ ) ,
d
d
2
dx ei L nx ei L n x = L δn′ n ,
∞
X
n=−∞
350
2π
′
ei L n(x−x ) = L δ(x − x′ ) .
considered here, arise usually when fields are quantized in a non-empty space.
The complete Hamiltonian of the system should then, strictly speaking, include terms describing interactions of the considered fields with external
agents. As these are usually complicated, they are modeled by imposing
appropriate boundary conditions on fields themselves (from the quantum
theory point of view, this means imposing some restrictions on the Hilbert
space of field states). Of course, by itself the example of the scalar field is
not very physical. One can however interpret it as a simplified model of the
electromagnetic field quantized between two conducting plates. Classically
in this case, the component of the electric field E tangent to the conductor
surface vanishes at this surface, at least for electrostatic fields, i.e. for fields
with small frequencies ω. Microscopically this is of course due to complicated interactions between the field and electrons in a metal out of which the
plates are made. But one may expect that field modes whose frequencies are
much higher than any frequencies that can be associated with electrons in
matter, almost do not feel the presence of the plates. Thus, the microscopic
physics of the interaction between the field and the metal provides an effective ultraviolet cutoff on the expression (11.153): high frequency modes of
the field ϕ (11.147) should effectively approach modes of the field quantized
without the presence of the plates. The attractive force between the two
plates, which will be foud below, is mostly due to the difference in the low
frequency modes.
Despite the argument in favour of the finiteness of ∆E given above, the
practical implementation of the cancellation in the actual calculation requires
some care. To facilitate the calculation one traditionally regularizes both
infinite sums introducing a cutoff function f (ω) which model the intuitive
picture discussed above. We assume f (ω) is such that f (0) = 1, all its
derivatives f (n) (0) = 0 for n ≥ 1, and vanishing for ω → ∞. Thus, we
rewrite the expression (11.153) as
)
(∞
Z ∞
Z
X
∆E
~c ∞
ω(k, l) f (ω(k, l)) −
dl ω(k, l) f (ω(k, l)) ,
dk k
=
L2
4π 0
0
l=1
p
with ω(k, l) = k 2 + l2 π 2 /d2 . Introducing the variable u = k 2 (d2 /π 2 ) brings
this into the form
(∞
Z
π √
X√
∆E
π 2 ~c ∞
2f
2
u
+
l
u
+
l
=
du
L2
8d3 0
d
l=1
Z ∞ √
π √
2
2
u+l
−
dl u + l f
d
0
Z ∞
π 2 ~c
≡
F (1) + F (2) + . . . −
dl F (l) ,
(11.154)
8d3
0
351
where
F (l) =
Z
0
Z l
π √
π √
2
2
2
u + l = −2
du u + l f
ξ . (11.155)
dξ ξ f
d
d
+∞
∞
One can then use the Euler-Mclaurin formula
Z ∞
∞
X
1
1
1
F (0) +
F (n) −
dn F (n) = − B2 F ′ (0) − B4 F ′′′ (0) + . . . (11.156)
2
2!
4!
0
n=1
in which Bi are the Bernoulli numbers: B2 = 1/6, B4 = −1/30, etc., to
obtain the final result: since we have assumed the f (n) (0) = 0 for n ≥ 1 it is
easy to see that F ′ (0), F ′′ (0) and all derivatives F (n) (0) with n > 3 vanish.
The only nonzero contribution to (11.154) comes from F (0) and F ′′′ (0) = −4:
∆E
~cπ 2
1
1
=
− F (0) + B4 .
(11.157)
L2
8d3
2
3!
It is also easy to see (by appropriately rescaling the integration variable),
that F (0) ∝ d3 , so that the force per area between the plates, due to the
fluctuations of the quantized field,
∂ ∆E
1 π 2 ~c
π 2 ~c
F
=
−
=
B
=
−
(11.158)
4
L2
∂d L2
2
8d4
480d4
is attractive37 and does not depend on the details of the cutoff imposed in
order to regularize the expression (11.153). This is the celebrated Casimir
effect which has been experimentally confirmed and measured in 1958. Being
initially only a curiosity, it plays now an important role in planning nanoscale
devices.
As another application of this approach to field quantization we consider
a real scalar field coupled linearly to an external source. (This example is
analogous to the one considered in Section 1.2). The Lagrangian density is
1
1
L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 + ϕJ ,
2
2
(11.159)
where J(t, x) is a given function. We assume also that J vanishes for t → ±∞
(or, better, it is just zero for |t| > T ), because then the states of the quantized
37
Of course, for M 2 = 0 proportionality of F/L2 to ~c/d3 follows from the dimensional
analysis (at least for L → ∞, in which limit the result should be independent of L). The
computation is needed only to establish the dimensionless proportionality constant and
the sign. Notice, that from the thermodynamical point of view the negative sign means
that the quantum vacuum behaves as a gas exerting negative pressure on the walls of the
recipient!
352
field can have particle interpretation in the far past and in the far future.
The classical field equation following from the Lagrangian (11.159) is
∂µ ∂ µ + M 2 ϕ(x) = J(x) .
(11.160)
The solutions can be written as the sum of two terms:
Z
ϕ(x) = ϕ0 (x) + i d4 y D(x − y)J(y) ,
(11.161)
where ϕ0 (x) satisfies the homogeneous equation (∂µ ∂ µ + M 2 )ϕ0 (x) = 0. The
Green’s function D(x − y) is defined by
∂µ ∂ µ + M 2 D(x − y) = −iδ (4) (x − y) .
(11.162)
and can formally be written as the Fourier integral
Z 4
i
dk
e−ik·(x−y) .
D(x − y) =
4
2
2
(2π) k − M
(11.163)
Different prescriptions
of going around the singularities at k 0 = ±E(k),
√
2
where E(k) ≡ k + M 2 , specify different Green’s functions (satisfying different boundary conditions) of the Klein-Gordon equation. In particular,
writing the denominator k 2 − M 2 as [k 0 − E(k) + i0][k 0 + E(k) + i0] gives the
retarded Green’s function Dret (x − y) vanishing for x0 − y 0 < 0, while writing
it with two −i0 factors gives the advanced function Dadv (x − y) vanishing for
x0 − y 0 > 0.
We can thus write down two different forms of the complete solution to
the classical equation of motion (11.160)
Z
ϕ(x) = ϕin (x) + i d4 y Dret (x − y)J(y) ,
Z
ϕ(x) = ϕout (x) + i d4 y Dadv (x − y)J(y) .
(11.164)
Classically, ϕin (x) is the initial field, before the source J started to act, while
ϕout (x) is the final field. Of course, both ϕin (x) and ϕout (x) can be written in
the form (11.140) with some coefficients ain (k), a∗in (k) and aout (k), a∗out (k).
Specifying ain (k) and a∗in (k) or aout (k) and a∗out (k) determines uniquely ϕ(x)
for all x.
Quantization consists of promoting ϕ(x) to the operator ϕH (x) satisfying
with the conjugated momentum ΠH (x) = ϕ̇H (x) the canonical commutation
relations (11.141). This can be done in two ways, by imposing the commutation rules on the in and out creation and annihilation operators. In this
way the same operators ϕH (x) and ΠH (x) are represented in two alternative
353
ways given in (11.164). In both representations ϕH (x) and ΠH (x) satisfy
the fundamental commutation rules (11.141) and the two representations are
unitarily equivalent.38 The operators ain (k) and a†in (k) and the operators
aout (k) and a†out (k) define two different bases in the same Hilbert space of
the field states. These bases have obvious physical meaning: the Hamiltonian
Z
1 2
1
1 2 2
2
3
H = d x ϕ̇H + (∇ϕH ) + M ϕH − ϕH J
(11.165)
2
2
2
in the first (the in) representation (11.164) of the field ϕH (t, x) takes in the
far past the form
Z
H = dΓk E(k) a†in (k)ain (k) + const ,
(11.166)
because for x0 → −∞ the retarder Green’s function Dret (x − y) vanishes
and ϕH (t, x) → ϕin (t, x). Similarly, the Hamiltonian takes the form (11.166)
with a†out (k)aout (k) replacing a†in (k)ain (k) in the far future. The states |Ω+ i,
annihilated by all ain (k), a†in (k)|Ω+ i, etc., are therefore the eigenstates of
H(−∞) having the particle interpretation in the far past39 whereas the analogous states |Ω− i, a†out (k)|Ω− i etc., have the similar interpretation in the far
future.
It is straightforward to relate the in and out operators to each other.
From (11.164) we get
Z
ϕout (x) = ϕin (x) + i d4 y [Dret (x − y) − Dadv (x − y)] J(y) . (11.167)
Explicitly the difference of the Green’s functions is
d3 k
(2π)3
∞
0
0
0
dk 0 e−ik (x −y ) ik·(x−y)
Dret (x − y) − Dadv (x − y) =
e
2E(k)
−∞ 2π
i
i
i
i
.
−
−
+
k 0 − E(k) + i0 k 0 + E(k) + i0 k 0 − E(k) − i0 k 0 + E(k) − i0
Z
Z
Upon using the formula
38
1
1
=P
∓ iπδ(x − a) ,
x − a ± i0
x−a
(11.168)
In the similar problem of the interaction of the electromagnetic field the two representations might turn out not to be unitarily equivalent, which means that the states created
by a†in (k) from |Ω+ i have all zero scalar products with the states created by a†out (k) from
|Ω− i. Physically this corresponds to creation by the external source of an infinite number
of arbitrarily soft photons.
39
Since H depends on time it cannot have global, time independent eigenstates.
354
it can be brought into the form
Dret (x − y) − Dadv (x − y) =
Thus, from (11.167) we get
Z
dΓk e−ik·(x−y) − eik·(x−y) .
aout (k) = ain (k) + iJ(E(k), k) ,
a†out (k) = a†in (k) − iJ ∗ (E(k), k) ,
(11.169)
(11.170)
where J(k) = J(k 0 , k) = J ∗ (−k 0 , −k) is defined by
Z 4
d k −ik·y
e
J(k) .
J(y) =
(2π)4
(11.171)
One can now introduce the unitary operator S, S −1 = S † , such that
(compare (1.32) and (7.22))
S|α− i = |α+ i .
(11.172)
It must therefore have the following properties:
S a†out (k) S † = a†in (k) ,
S aout (k) S † = ain (k) ,
(11.173)
S † a†in (k) S = a†out (k) ,
S † ain (k) S = aout (k) .
(11.174)
and
It is not difficult to construct the operator S explicitly
Z
Z
iδ
4
iδ′
4
S = e exp i d x ϕin (x)J(x) = e exp i d x ϕout (x)J(x) , (11.175)
where δ is an arbitrary phase factor and δ ′ can be easily related to δ with
the help of (11.167). Using the operator identity
1
eA+B = aA eB e− 2 [A,
B]
1
= eB eA e 2 [A,
B]
,
(11.176)
valid for [A, [A, B]] = [B, [A, B]] = 0, the operators S and S † can be
written in the forms which are more convenient for computation of their
matrix elements (we omit the phase factor δ):
S = ei
R
R
R
dΓk a†in (k) J(k) i dΓk ain (k) J ∗ (k) − 12 dΓk |J(k)|2
S † = e−i
e
R
e
R
dΓk ain (k) J ∗ (k) −i dΓk a†in (k) J(k)
e
355
1
e2
R
,
dΓk |J(k)|2
(11.177)
.
(11.178)
Similar formulae in terms of the out operators can also be written down.
The formulae (11.174) can be now checked with the help of the operator
expansion
eA B e−A = B + [A, B] +
1
[A, [A, B]] + . . .
2!
(11.179)
it is easy to check that indeed
S † a†in (k) S = a†in (k) − iJ ∗ (k) ,
(11.180)
etc.
Various transition probabilities can now be computed. For example, the
vacuum persistence probability is
P (0 → 0) = |hΩ− |Ω+ i|2 = |hΩ+ |S|Ω+ i|2 = e−
R
dΓk |J(k)|2
.
(11.181)
Similarly, the probability of creating n quanta of the field ϕ by the source is
given by
Z
Z
1
P (0 → n) =
dΓk1 . . . dΓkn |h(k1 , . . . kn )+ |S|Ω+ i|2
n!
1
(11.182)
= (n)n e−n
n!
(the Rfactor 1/n! is inserted because of identity of the field quanta), where
n = dΓk |J(k)|2 , i.e. it is given by the Poisson distribution, whereas the
mean total energy emitted by the source can be computed as
Z 3
d k
1
†
|J(k)|2 .
(11.183)
Erad = hΩ+ |H(aout aout )|Ω+ i =
2 (2π)3
where H(a†out aout ) stands for the Hamiltonian taken at t = +∞ which when
expressed in terms of the out operators has the free field form. The field
radiated by the source if the initial state of the field at t = −∞ was |Ψ+ i
can in turn be identified with the matrix element
lim hΨ+ |ϕH (t, x) − ϕH (−t, x)|Ψ+ i = lim hΨ+ |ϕout (t, x) − ϕin (−t, x)|Ψ+ i .
t→∞
t→∞
For |Ψ+ i = |Ω+ i, using (11.167), this gives (as hΩ+ |ϕin(x)|Ω+ i = 0)
Z
lim
hΩ+ |ϕout (x)|Ω+ i = 0lim i d4 y [Dret (x − y) − Dadv (x − y)] J(y) .
0
x →∞
x →∞
Finally we apply this approach to quantization to the case of a scalar field
in a curved space-time i.e. in a nontrivial gravitational background. We will
356
consider only the simplest case of the Friedman-Robertson-Walker (FRW)
metric in which fixed x0 sections are flat spaces. Although very simple, this
example has important application in modern cosmology.
We start with the simplest scalar field action in the general space-time40
Z
1 µν
4 √
g ∂µ ϕ∂ν ϕ − V (ϕ) ,
(11.184)
I[ϕ] = d x −g
2
where g = det(gµν ) is the determinant of the metric tensor. The equation
of motion obtained by varying the field ϕ (with the usual requirements that
the variation vanishes at large distances and at the initial and final time
hypersurfaces) reads
√
1
√
∂µ −gg µν ∂ν ϕ + V ′ (ϕ) = 0 .
−g
(11.185)
We now specify to V (ϕ) = 12 M 2 ϕ2 and to the spatially flat FRW metric with
the line element
ds2 ≡ gµν dxµ dxν = dt2 − a2 (t)(dx)2 = a2 (η) dη 2 − (dx)2 , (11.186)
where we have introduced the conformal time coordinate η defined by
Z t
dt
.
(11.187)
η(t) =
t0 a(t)
Expressed through the conformal time variable, the equation of motion (11.185)
takes the form
a′
ϕ′′ + 2 ϕ′ − ∇2 ϕ + M 2 a2 ϕ = 0 .
a
(11.188)
By introducing the rescaled field
χ(η, x) ≡ a(η)ϕ(η, x)
(11.189)
the equation (11.188) can be simplified to
χ′′ − ∇2 χ + m2eff (η) χ = 0 ,
(11.190)
i.e. to the Klein-Gordon equation but with an η dependent effective mass
term
m2eff (η) ≡ M 2 a2 (η) −
40
a′′ (η)
.
a(η)
(11.191)
The most general Raction√quadratic in the real scalar field ϕ and its first derivatives
includes also the term d4 x −g 21 ξRϕ2 , where R is the Ricci scalar. For ξ = 61 the action
I[ϕ] is conformally invariant. We set ξ = 0 for simplicity.
357
The equation of motion (11.190) follows of course from the effective action
Z Z
1
(11.192)
Ieff = dη d3 x (χ′ )2 − (∇χ)2 − m2eff (η) χ2 ,
2
which serves to define the momentum Π canonically conjugated to χ: Π = χ′ .
One can then write down the (conformal) time-dependent Hamiltonian
Z
1
H=
d3 x Π2 + (∇χ)2 + m2eff (η) χ2 .
(11.193)
2
To quantize the field χ(η, x) following the procedure adopted in this Section, we must first write down the most general solution to the classical
equation of motion (11.190). The solution can be written in the form41
Z 3
dk
χ(η, x) =
χ(η, k) eik·x ,
(11.194)
(2π)3
where for each k the function χ(η, k) satisfies independently the equation
χ′′ (η, k) + ω 2 (η, k)χ′′ (η, k) = 0 ,
(11.195)
ω 2 (η, k) ≡ k2 + m2eff (η) .
(11.196)
with
Since the equation (11.195) is a linear (in χ) second order ordinary differential equation with the real coefficient ω 2 , it has two linearly independent
solutions. The important property of the equation (11.195) is that for any
two its solutions, χ1 and χ2 , their combination
W (χ1 , χ2 ) ≡ χ′1 χ2 − χ1 χ′2 ,
(11.197)
called the Wrońskian, is independent of η:
d
W (χ1 , χ2 ) = 0 .
dη
(11.198)
It is also clear that W (χ1 , χ2 ) = 0, if χ1 and χ2 are linearly dependent.
Furthermore, from any two real linearly independent solutions χ1 and χ2 , of
(11.195) a complex solution of the equation (11.195)
v(η, k) = χ1 (η, k) + iχ2 (η, k) ,
41
(11.199)
We exploit the fact that the metric is spatially flat. For more general metrics with
nonzero spatial curvature, but still with g0i = gi0 we would have to do first the harmonic
analysis on spatial sections of the space-time.
358
can be formed. For such complex solutions
Im (v ′ v ∗ ) = −W (χ1 , χ2 ) .
(11.200)
With the help of the complex functions v(η, k) the most general solution
of the equation of motion (11.190), such that χ(η, x) is a real function can
be written in the form
Z 3
d k ik·x
e
[av (k) v ∗ (η, k) + a∗v (−k) v(η, −k)] . (11.201)
χ(η, x) =
(2π)3
Quantization of the field χ(η, x) directly in the Heisenberg picture consists
now of promoting χ(η, x) and Π(η, x) = χ′ (η, x), and hence the coefficients
av (k) and a∗v (k), to operators
Z 3
dk ∗
ik·x
†
−ik·x
a
(k)
v
(η,
k)
e
+
a
(k)
v(η,
k)
e
, (11.202)
χ(η, x) =
v
v
(2π)3
Z 3
d k ′∗
ik·x
†
′
−ik·x
Π(η, x) =
a
(k)
v
(η,
k)
e
+
a
(k)
v
(η,
k)
e
,
v
v
(2π)3
satisfying the canonical equal-(conformal)time commutation relation
[χ(η, x), Π(η, y)] = iδ (3) (x − y) .
(11.203)
It is then easy to check, that this is satisfied for av (k) and a†v (k) such that
(11.204)
av (k), a†v (k′ ) = (2π)3 δ (3) (k − k′ ) ,
and provided the complex solutions are normalized so that
v ∗ (η, k) v ′ (η, k) − v(η, −k) v ′∗ (η, −k) = i .
(11.205)
The normalization (11.205) is consistent with the (conformal)time evolution of the functions v(η, k), provided they satisfy the relation42 v(η, k) =
v(η, −k); this follows from the relation (11.200) and the property (11.198).
Complex solutions of (11.195) normalized in this way are called mode functions. It is however important to remember that the normalization condition
(11.205) by no means specifies the mode functions uniquely: different choices
of the basis of the real solutions χ1 and χ2 of (11.195) lead to different (conformal)time dependence of the functions v(η, k); by a trivial rescaling (by a
complex factor) any such function can be adjusted to satisfy (11.205). For
42
Since ω 2 defined by (11.195) depends only on the length |k| of k and not on its direction, its solutions v(η, k) can also be chosen to be isotropic, i.e. v(η, k) = v(η, |k|).
In specific situations it might prove more convenient to chose the functions v(η, k) nonisotropic but they always must satisfy the equality v(η, k) = v(η, −k).
359
example, in the standard quantization of the real scalar field ϕ in the flat
Minkowski space-time (in which η = t) one takes χ1 (t, k) = (2ω)−1/2 cos(ωt),
χ2 (t, k) = (2ω)−1/2 sin(ωt), so that v ∗ = (2ω)−1/2 exp(−iωt), but one could
also take v1 = A1 cos(ωt) + B1 sin(ωt) and v2 = A2 cos(ωt) + B2 sin(ωt) with
real coefficients A1,2 and B1,2 ; however in the flat Minkowski space-time only
the first choice makes the Hamiltonian diagonal (see below).
The operators av (k), a†v (k) formally resemble the ordinary creation and
annihilation operators. In particular, one can define the “vacuum” state |0(v) i
annihilated by all av (k) associated with the given set of complex solutions
v(η, k) and the Hilbert space of states of the quantum field χ(η, x) is spanned
by the vectors of the form a†v (k)|0(v) i, a†v (k2 )a†v (k1 )|0(v) i, etc. similar to
the vectors (11.102). However, while owing to our choice (11.186) of the
gravitational background these states transform under spatial rotations as
do the ordinary particles, they do not have all the necessary properties to
be justly called particles. Similarly, as we will see, neither the state |0(v) i
can be termed the vacuum. To see this in more detail, let’s choose another
set of linearly independent complex solutions u(η, k) and u∗ (η, k) to the
equation (11.195). It must of course be possible to express v(η, k) as a linear
combination43 of u(η, k) and u∗ (η, k):
v ∗ (η, k) = α(k) u∗ (η, k) + β(k) u(η, k) ,
(11.206)
with the coefficients α(k) and β(k) independent of η and such that α(−k) =
α(k), β(−k) = β(k) (as follows from the condition v(η, k) = v(η, −k)). The
normalization condition (11.205) satisfied by both sets of complex solutions
imposes the constraint
|α(k)|2 − |β(k)|2 = 1
(11.207)
on the Bogolyubov coefficients. By comparing the field operator χ(η, x)
written in terms of the new mode functions u(η, k) and u∗ (η, k) and the new
operators au (k) and a†u (k)
Z 3
dk ∗
ik·x
†
−ik·x
χ(η, x) =
a
(k)
u
(η,
k)
e
+
a
(k)
u(η,
k)
e
, (11.208)
u
u
(2π)3
with the same field operator given in (11.202) in terms of the operators a(k)
and a† (k) and substituting in the latter form (11.206) one finds that
au (k) = α(k) av (k) + β ∗ (k) a†v (−k) ,
a†u (k) = α∗ (k) a†v (k) + β(k) av (−k) ,
43
(11.209)
Due to the choice (11.186) of the metric v(η, k) depends only on u(η, k) and u∗ (η, k)
with the same k; in spatially non-flat gravitational backgrounds v(η) would in general be
linear combinations of all functions (with all k) of the alternative complete set of solutions.
360
where we have taken into account that α(−k) = α(k) and β(−k) = β(k).
The relations (11.209) are called the Bogolyubov transformations and the
coefficients α(k) and β(k) are the Bogolyubov coefficients.
The relations (11.209) show that the state |0(v) i annihilated by all operators av (k) is not annihilated by the operators au (k), which all annihilate the
state |0(u) i and vice versa. In particular, since
h0(v) |a†u (k)au (k)|0(v) i = |β(k)|2 (2π)3 δ (3) (0) ,
(11.210)
the quantity |β(k)|2 can be given the meaning of the density of the “uquanta” associated with the solutions u in the “vacuum” associated with
the v modes (the factor (2π)3 δ (3) (0) in (11.210) should be interpreted as the
volume factor). It is also possible to show that if the field χ(η, x) is quantized
in the finie spatial volume V = L3 , so that k = (2π/L)n, these two “vacuum”
states are related by44
Y
1
β ∗ (k) †
†
p
exp −
|0(u) i =
av (k)av (−k) |0(v) i .
(11.211)
2α(k)
|α(k)|
k
This shows, that with respect to |0(v) i the state |0(u) i is the so-called squeezed
state.
Expressing the Hamiltonian (11.193) in terms of the operators av (k) and
by inserting the expansions (11.202) one finds
Z 3
d k †
1
av (k)av (k) + av (k)a†v (k) E(η, k)
H=
3
2 (2π)
+av (k)av (−k)F ∗ (η, k) + a†v (k)a†v (−k)F (η, k) ,
a†v (k)
where
2
E(η, k) = |v ′ (η, k)| + ω 2 (η, k) |v(η, k)|2 ,
2
F (η, k) = (v ′ (η, k)) + ω 2 (η, k)v 2 (η, k) ,
(11.212)
and ω 2 (η, k) = k2 + m2eff (η). One could try to select the solutions v(η, k) so
that F (η, k) ≡ 0. However, the resulting differential equation for v(η, k) is
compatible with the basic equation (11.195) only if ω is η-independent. This
simply reflects the fact that the (conformal)time-dependent Hamiltonian does
not have stationary eigenstates. For η-independent ω the function F (η, k)
vanishes identically for
v(η, k) = p
1
2ω(k)
eiω(k)η ,
(11.213)
Q
If the product k |α(k)|−1/2 is not convergent, the two sets of creation and annihilation operators simply act in two different Hilbert spaces providing nonequivalent representations of the same commutation relations.
44
361
and the operators χ(η, x) and Π(η, x) have the same form as (11.89) and
(11.90), respectively in the ordinary Minkowski space-time.
For ω 2(η, k) depending on η in a nontrivial way one possible choice of the
functions v(η, k) is such that |0(v) i is the state in which the Hamiltonian taken
at one particular instant η0 of the conformal time has the lowest expectation
value. Minimizing h0(v) |H(η0 )|0(v) i with respect to possible choices of the
functions v(η, k) one gets the conditions
1
,
v ′ (η0 , k) = iω(η0 , k)v(η0 , k) , (11.214)
v(η0 , k) = p
2ω(η0 , k)
up to a phase factor common for v and v ′ . In this case H(η0 ) is diagonal,
i.e. F (η0 , k) = 0. The state |0(v) i is therefore called the instantaneous
vacuum. It should be clear that |0(v) i need not be neither the eigenstate
nor the state giving the lowest energy expected value for the Hamiltonian
H(η1 ) for η1 6= η0 . Moreover, it is important to remember, that the energy
of the zero point fluctuations at η0 cannot be simply subtracted, because
from the global point of view this contribution is eta-dependent and more
complicated renormalization procedure is required to remove this kind of
infinities in the curved space-time. Finally, in some cases it may happen that
ω 2 (η, k) defined in (11.196) is negative for small |k| because of the last term in
m2eff in (11.191). In such a case, the contribution to the vacuum expectation
value h0(v) |H(η0 )|0(v) i of the modes corresponding to ω 2 (η, k) < 0 cannot
be minimized and the instantaneous vacuum does not exist. In such cases
one considers other choices of the functions v(η, k) defining other types of
“vacuum” states (e.g. the Bunch-Davis vacuum, or the adiabatic vacuum)
which we will not discuss here.
A relatively simple situation occurs when the metric becomes Minkowskian
in the far past (η → −∞) and in the far future (η → +∞), so that
ω(η, k) → ωin (k) and ω(η, k) → ωout (k) in these limits. One can then define
the vacua |0ini associated with the mode functions v(η, k) such that
1
lim v(η, k) = p
eiωin (k)η ,
(11.215)
η→−∞
2ωin(k)
and |0out i associated with the functions u(η, k) having the property
1
eiωout (k)η .
(11.216)
lim u(η, k) = p
η→+∞
2ωout (k)
The states created by the operators a†v (k) related to the mode functions
v(η, k) and by the operators a†u (k) related to the mode functions u(η, k)
can be then interpreted as normal particles in the far past and in the far
future, respectively. Using the Bogolyubov coefficients one can compute then
probabilities of creating some number of particles by a variable gravitational
field.
362
11.4
Lagrangian of the electromagnetic field
Canonical quantization of fields transforming under changes of the reference frame as nontrivial representations of the Lorentz group requires special
treatment because the assumption that generalized velocities (time derivatives of canonical variables) can be expressed as in (11.9) through the conjugated momenta is usually not fulfilled. This is, in particular, the case of
the electromagnetic and Proca fields (describing after quantization massless
and massive spin 1 particles, respectively). One method of dealing with this
difficulty consists of eliminating some of the canonical variables (reducing
the number of independent ones and therefore the number of conjugated momenta). This can be done either directly (as is possible for the Proca field
- see Section 11.5) or through the use of the Dirac’s quantization formalism
adapted to system with constraints, which will be presented in Section 11.6.
We first discuss possible forms of the action I (11.1) governing classical
dynamics of vector fields. We begin with the familiar case of the electromagnetic field. The Maxwell equations (in the Gauss’ system of units, in which
the fields E and B have the same dimension - the relation of electromagnetic
quantities in the Gauss system to their counterparts in the “official” SI of
units is recalled in Appendix F) read
1 ∂B
= 0,
c ∂t
4π
1 ∂E
=
j,
∇×B −
c ∂t
c
∇×E +
∇·B = 0 ,
(11.217)
∇·E = 4πρ .
(11.218)
The first two are automatically satisfied if the scalar and vector potentials,
forming together a four-vector Aµ = (ϕ, A) are introduced, in terms of which
B = ∇×A ,
E = −∇ϕ −
1 ∂A
.
c ∂t
(11.219)
The two other equations, (11.218), follow from the Lagrangian density45
LEM = −
1
1
fµν f µν − eJµ Aµ .
16π
c
(11.220)
The electromagnetic field strength tensor fµν is defined as
fµν = ∂µ Aν − ∂ν Aµ .
(11.221)
∂
1
1 µν
∂LEM
=−
fλρ f λρ = −
f .
∂(∂µ Aν )
8π ∂(∂µ Aν )
4π
(11.222)
As it is easy to find that
45
It is more natural to factorize e - the fundamental coupling constant - out of the
four-current J ν .
363
The Euler-Lagrange equations (11.4) which here read
∂µ
∂LEM
∂LEM
,
=
∂(∂µ Aν )
∂Aν
(11.223)
therefore give
4π ν
eJ .
(11.224)
c
Since the four-divergence of the left hand side of (11.224) vanishes by antisymmetry of the field strength tensor f µν (11.221), the current J ν to which
the electromagnetic field couples must be conserved ∂µ J µ = 0 (otherwise the
equations (11.224) are inconsistent). It is instructive to examine the content
of (11.224). For ν = 0, since f 00 ≡ 0 by antisymmetry, on the left hand side
we get46
1 ∂A
i 0
0 i
0
i
∂i (∂ A − ∂ A ) = ∂i (−∂i A − ∂0 A ) = ∇· −∇ϕ −
,
(11.225)
c ∂t
∂µ f µν =
so that the second Maxwell equation (11.218) is recovered for
e J 0 = cρ .
(11.226)
For ν = k from (11.224) we get
∂0 (∂0 Ak + ∂k A0 ) + ∂i f ik =
4π k
eJ .
c
(11.227)
Furthermore, it is easy to check, that
f ij ≡ −∂i Aj + ∂j Ai = −ǫijk B k ,
so that (11.227) becomes
1∂ k
4π k
1∂
∂k ϕ +
A − ǫikl ∂i B l =
eJ ,
c ∂t
c ∂t
c
(11.228)
which, if the second relation (11.219) is taken into account, is just the first
equation (11.218), provided one identifies eJ with j.
It is sometimes useful to have explicit forms of the tensors Fµν and F µν
in terms of the Cartesian components of the ordinary three-vectors E and B:




0 −E x −E y −E z
0
Ex
Ey
Ez
 Ex
 −E x
0
−B z B y 
0
−B z B y 
µν
.
,
 y
f
=
fµν = 
z
y
z
x
E
 −E
B
0
−B x 
B
0
−B 
E z −B y B x
0
−E z −B y B x
0
46
Since we want to keep contact with the ordinary three-dimensional notation, we convert all expressions to a form in which A0 and Ai have always upper indices whereas the
derivatives ∂0 and ∂i have always lower indices, so that ∂i = ∂/∂xi is the ordinary gradient
in the contravariant coordinates xi .
364
Having the explicit formulae it is easy to see that
1
1
E2 − B2 − eJ µ Aµ .
(11.229)
LEM =
8π
c
There exist also another term bilinear in the field Aµ , which, being a scalar
with respect to proper ortochroneous Lorentz transformations,47 could be
added to the Lagrangian density (11.220):
∆L ∝ f˜µν fµν ,
(11.230)
where (recall we use ǫ0123 = +1)

1
f˜µν ≡ ǫµνλρ fλρ
2
0
 B1
=
 B2
B3
−B 1
0
−E 3
E2
−B 2
E3
0
−E 1

−B 3
−E 2 
.
E1 
0
(11.231)
∆L is, however, a total four-divergence: f˜µν fµν = ∂µ (2ǫµνλρ Aν ∂λ Aρ ) and,
as such, it does not modify the Euler-Lagrange (i.e. Maxwell) equations.
Being a total four-divergence, it has neither any impact on the dynamics
of the quantized electromagnetic fields, because no topologically nontrivial
configurations of the electromagnetic field exist classically (this statement
cannot be given a justification here). Such topologically nontrivial configurations do exist, however, for nonabelian gauge fields (see Section 20) and
the nonabelian analog of ∆L is not innocuous in the quantum theory of such
fields.
The Maxwell equations (11.217), (11.218) are written in the Gauss system
of units, in which the Coulomb potential looks simple (it does not have the
1/4π factor nor the ǫ0 factor) and the fine structure constant (defined by the
Thomson limit of the Compton scattering cross section) is αEM = e2 /~c =
1/137.035999679(94) ≈ 1/137. This system of units is not very convenient√for
the quantum theory of the electromagnetic field because of the factor 4π
which would appear in many places (e.g. in free field operators - see e.g.
W.B.Bierestecki, J.M. Lifshitz, L.P.Pitaevski Quantum Electrodynamics). It
is much more practical to go over to the Heaviside-Lorentz system of units
in which electromagnetic quantities have the same physical dimensions
as in
√
the Gauss system but are rescaled by the appropriate factors of 4π:
√
√
Aµ → 4πAµ ,
eJ µ → eJ µ / 4π ,
(11.232)
√
(i.e. AGauss
= 4πAHeaviside−Lorentz
etc.), so that in the rescaled variables the
µ
µ
Lagrangian density (11.220) takes the form
LEM = −
1
1
fµν f µν − eJµ Aµ .
4
c
47
(11.233)
As a result of the parity and time reversal transformations L′ changes sign; it is
therefore neither P - nor T - (and, hence, neither CP -) invariant.
365
√
√
This amounts to the rescalings E → 4π E and B → 4π B. The corresponding changes of ρ and j in the Maxwell equations (11.217), (11.218), are,
µ
since the elementary charge e has been
√ facorized out from the four-current
√J ,
equivalent to the rescaling e → e/ 4π (i.e. eGauss = eHeaviside−Lorentz / 4π).
As a result in the Heaviside-Lorentz units the fine structure constant αEM
reads
αEM =
e2
.
4π~c
(11.234)
In the Lagrangian density (11.233) the current J µ couples to Aµ linearly.
The task of quantizing the electromagnetic field coupled to a given external
classical current seems therefore similar to the second example discussed in
Section 11.3.
If the electromagnetic field interacts with “matter” (particles or other
fields), and both, the electromagnetic field and the matter, are treated as a
single dynamical quantum system, the current j and the charge density ρ,
that is J µ , are given in terms of the dynamical variables describing matter.
In such a case the current J µ can, if the electromagnetic field is coupled to
a system of scalar fields, even depend on the variable Aµ (but not on its
derivatives); “matter” is then coupled nonlinearly to Aµ . In all these cases
quantization of the system as a whole in the Schrödinger picture, as in (11.2),
should proceed essentially as if the electromagnetic field was free - the interactions do not modify the canonical momenta. Such a canonical quantization
of the electromagnetic field encounters, however, immediately an obstacle. If
the four components Aµ are taken for independent dynamical variables, one
finds that the momentum canonically conjugated to A0 vanishes identically:
Πν ≡
∂LEM
1
∂LEM
=
−
f0ν ,
=
∂(c ∂ 0 Aν )
c
∂ Ȧν
(11.235)
which implies that Π0 ∝ f00 ≡ 0. The electromagnetic field is therefore an
example of a system subject to constraints.
The peculiarity of the electromagnetic field is twofold: one is that Π0 = 0.
The other one is its gauge invariance - the symmetry of the Lagrangian
density (11.233) under (local) transformations
Aµ (x) → A′µ (x) = Aµ (x) +
1
∂µ θ(x) ,
e
(11.236)
in which θ is an arbitrary differentiable function of the space-time coordinates
xµ . The invariance of the f µν fµν term of (11.233) is obvious as under the
transformations (11.236) the tensor fµν does not change. Assuming that Jµ
itself does not change, the eJµ Aµ term of (11.233) transforms into eJ µ Aµ +
366
R
J µ ∂µ θ; the term J µ ∂µ θ does not contribute to the action I = d4 x LEM if the
current is conserved, which is anyway the necessary condition for consistency
of the field equations (11.224).
Invariance of the current Jµ with respect to the transformations (11.236)
and its conservation, ∂µ J µ = 0, are easy to check if it depends only, as in
the case of nonrelativistic charged particles coupled to the electromagnetic
field, discussed in Section 11.7, on dynamical variables describing “matter”.
In more complicated cases, as for example in the case of scalar fields coupled
to the electromagnetic field, the basic assuption is the gauge invariance of
the complete action
Z
1
d4 x fµν f µν + Irest [Aµ , ”matter”, ”fields”] ,
I[Aµ , ”matter”, ”fields”] ≡ −
4
by which one understands its invariance with respect to the transformation
(11.236) of the electromagnetic field potentials supplemented with appropriate, dependent on θ(x), that is local, transformations of all other fields.
For θ independent of x these transformations must form a U(1) group of
global symmetries (which were discussed in Section 11.1) of the action Irest .
For x-dependent θ this group becomes a local U(1) symmetry group of the
complete action I[Aµ , ”matter”, ”fields”]. Gauge invariance of the complete
action may be viewed as a fundamental principle, which must always be
respected. This principle, generalized to nonabelian symmetry groups (see
Section 20), underlies, in fact, the whole modern development of quantum
field theory.48 In the case of the electromagnetic field coupled to matter or
other fields, gauge invariance ensures that the generalized electromagnetic
current J µ , which enters the right hand side of the Euler-Lagrange equation
of the electromagnetic field
∂µ f µν =
e ν
J ,
c
defined by49
−eJµ (x) ≡
δ
δAµ (x)
Irest [Aµ , ”matter”, ”fields”] ,
48
(11.237)
It will be seen, however, (Section 20.3) that to properly formulate quantum theories
of gauge fields corresponding to nonabelian symmetry groups it is advantageous to take a
more general approach and to start with the extended action I = Igauge inv + Inon−gauge inv
which as a whole is not gauge invariant but posseses a continuous global symmetry of the
supersymmetric type, called the BRST symmetry (Section 20.2). Quantization with the
help of the Dirac method of Section 11.6 is then fairly easy and the BRST invariance of
the quantum theory ensures that all the implications of gauge invariance are recovered in
the properly defined physical subspace of the extended Hilbert space.
49
In most cases this current coincides with the Noether current of the globa U (1) symmetry.
367
even if it depends on Aµ , is gauge invariant and conserved. To see this let us
consider Irest of the general form
Z
Irest [Aµ , φi ] = d4 x Lrest (Aµ , φi , ∂φi ) ,
(11.238)
depending on Aµ and a set of fields φi , which under the changes (11.236) of
the gauge transform according to the rule
φi (x) → φ′i (x) = e−iθ(x)T ij φj (x) ≈ φi (x) − iθ(x)Tij φj (x) , (11.239)
in which T is a matrix representation of the U(1) group generator. From the
assumed gauge invariance of Irest for an infinitesimal gauge transformation
we get
0 = Irest [A′µ , φ′i ] − Irest [Aµ , φi ]
Z
1 δIrest
= d4 x
∂µ θ(x)
e δAµ (x)
Z
∂Lrest
∂Lrest
4
θ(x)Tij φj (x) +
∂µ [θ(x)Tij φj ] .
−i d x
∂φi (x)
∂(∂µ φi (x))
The last line is zero, as can be seen by integrating by parts and using the
Euler-Lagrange equations of motion50 (11.4) of the fields φi . In view of the
arbitrarines of θ(x) one gets (integrating by parts also in the first term)
therefore the condition of the conservation of the current (11.237). Gauge
invariance of the current (11.237) is of course guaranteed by the gauge invariance of the action (11.238). But it is precisely the gauge invariance of
the complete action I[Aµ , ”matter”, ”fields”] describing interactions of the
electromagnetic field, which is responsible for additional difficulties which we
will have to overcome quantizing the electromagnetic field in Section 11.7.
11.5
The Proca field and its quantization
Before dealing with the difficulties of the electromagnetic field, it is instructive to quantize first a simpler system with constraints, the so-called Proca
field V µ (a vector field with a nonzero mass) whose classical dynamics is
governed by the Lagrangian density51
LProca = −
1
1
Vµν V µν + M 2 Vµ V µ − g Vµ J µ ,
4
2
50
(11.240)
The reasoning remains valid also if the action Irest depends on higher derivatives of the
fields φi , provided one uses the appropriate generalization of the Euler-Lagrange equations
of motion.
51
We denote this field Vµ , instead of Aµ , to distinguish it from the electromagnetic field.
368
in which Vµν ≡ ∂µ Vν − ∂ν Vµ and g is the coupling constant (we have set
~ = c = 1). In the following we assume that the current J µ is independent
of Vµ .
The transformation (11.236) is not a symmetry of the the Lagrangian
(11.240) due to the mass term 21 M 2 Vµ V µ by which it differs from (11.233).
Still (11.240) leads to the similar (at first sight) obstacle in the canonical
quantization as (11.233): Π0 ≡ 0.
Before we overcome this problem we comment on the form of the Lagrangian density (11.240). Since it is not invariant under the gauge transformations (11.236), there is a priori no reason for which its derivative part
should take the restricted form as in (11.240). In principle, we could write
down the most general Lagrangian density, at most quadratic in Vµ and with
at most two derivatives in the form
1
1
1
LProca = − ∂µ Vν ∂ µ V ν − κ ∂µ Vν ∂ ν V µ + M 2 Vµ V µ − g Vµ J µ , (11.241)
2
2
2
with an arbitrary real κ (any other Lagrangian density satisfying our requirements differ from the above by a total derivative; an arbitrary real negative
constant, which could multiply the first term, can always be made equal to
−1/2 by an appropriate rescaling Vµ , κ, M 2 and g Jµ ). However, taking the
four-divergence of both sides of the Euler-Lagrange equation
∂µ ∂ µ V ν + κ ∂ ν (∂µ V µ ) + M 2 V ν = g J ν ,
(11.242)
resulting from (11.241), we discover that the four-divergence ∂ν V ν ≡ φ satisfies the independent equation
(1 + κ)∂µ ∂ µ φ + M 2 φ = g J˜ ,
(11.243)
in which J˜ = ∂µ J µ . Therefore, the Lagrangian density (11.241) describes
among other things also the dynamics of a scalar field52 coupled to the scalar
˜ To remove this propagation of an independent scalar field (that
current J.
is, to remove the homogeneous part of the classical solution for φ, in the case
of a conserved current J µ ) we set κ = −1; taking then the four-divergence of
both sides of the Euler-Lagrange equation (11.242) we find
M 2 ∂µ V µ = g ∂µ J µ ,
(11.244)
which shows that φ can be expressed in terms of the external current (which
depends on other fields). For κ = −1 the Lagrangian density (11.241) is just
the Proca Lagrangian density (11.240).
52
That an unconstrained vector field Vµ describes in addition a massive spin zero particle was also revealed in Section 8.4 in the course of constructing the free-field operator
transforming as a vector representation of the Lorentz group uder changes of the reference
frame.
369
We now quantize the Proca field. From its Lagrangian density (11.240)
the canonical momenta Πν conjugate to the variables V µ are obtained:
Πi =
∂LProca
= −V0i = V̇ i + ∂i V 0 ,
i
∂ V̇
Π0 = 0 .
(11.245)
As we have stressed, Π0 ≡ 0 is an obstacle in the canonical quantization.
The problem lies in the fact that the naive Hamilton’s formalism constructed
ignoring the difficulty does not lead to equations equivalent to the original
Euler-Lagrange equations
∂µ V µν = −M 2 V ν + g J ν ,
(11.246)
following from the Proca Lagrangian density (11.240), that is to the equations
∂i Vi0 = M 2 V 0 − g J 0 ,
(11.247)
∂0 (∂0 V k + ∂k V 0 ) + ∂i (−∂i V k + ∂k V i ) = −M 2 V k + g J k ,
(11.248)
obtained by setting in (11.246) ν = 0 and ν = k, respectively. The Hamiltonian corresponding to the Proca Lagrangian should be constructed following
the standard recipe:
H = Π0 V̇ 0 + Πi V̇ i − LProca
1
1
= Π0 V̇ 0 + Πi V̇ i − (V̇ i + ∂i V 0 )(V̇ i + ∂i V 0 ) + ∂i V j (∂i V j − ∂j V i )
2
2
1 2 i i 1 2 0 0
+ M V V − M V V − g V i J i + g V 0 J 0 . (11.249)
2
2
Eliminating V̇ i by using (11.245), i.e. substituting V̇ i = Πi − ∂i V 0 , one gets
1
1
H = Πi Πi + Π0 V̇ 0 (V λ , Πλ ) − Πi ∂i V 0 + (∇×V)2
2
2
1 2 i i 1 2 0 0
+ M V V − M V V − g V iJ i + g V 0J 0 .
2
2
(11.250)
Of course, since Π0 ≡ 0, there is no way to express the generalized velocity
V̇ 0 through the canonical variables V λ and Πλ . Let us therefore see, what
one gets, setting naively Π0 = 0 in the Hamiltonian
R 3 (11.250). The canonical
equations following from the Hamiltonian H = d x H obtained in this way
read
∂H
,
V̇ i = V i , H PB =
∂Πi
∂H
V̇ 0 = V 0 , H PB =
≡ 0,
∂Π0
∂H
, (11.251)
∂V i
∂H
= − 0 6= 0 ,
∂V
Π̇i = {Πi , H}PB = −
Π̇0 = {Π0 , H}PB
370
and are evidently incompatible with the Euler-Lagrange equation (11.246):
the latter imply firstly that Π0 = 0 for any time t, whereas here, even if we
set Π0 = 0 for t = 0, a nonzero Π0 will be generated for t 6= 0 due to a
nonzero derivative Π̇0 . Secondly, from (11.247) expressed in terms of Πi it
follows that the time evolution of V 0 is fully determined in terms of J 0 and
∂i Πi :
1
(11.252)
V 0 = 2 ∂i Πi + g J 0 .
M
However, the third canonical equation (11.251) implies that V 0 is constant.
Thus, the dynamics of V µ following from the Hamiltonian (11.250) is not
the same as the one generated by the Euler-Lagrange equations (11.247) and
(11.248).
The equivalence is restored if in the Hamiltonian (11.250), in addition to
setting Π0 = 0, one substitutes for V 0 the expression (11.252), that is, if one
“algebraically” eliminates one of the canonical variables (in other words, if
one expresses its value at any t by a combination of other canonical variables
taken at the same instant t). One then obtains (integrating by parts):
1
1
1
(∂i Πi )2 + (∇×V)2
H = Πi Πi +
2
2
2M
2
2
1
g
g
+ M 2 V iV i − g V iJ i +
J 0 J 0 + 2 J 0 ∂i Πi .
2
2
2M
M
The canonical equations now become
(11.253)
1
g
∂H
= Πi − 2 ∂i (∂j Πj ) − 2 ∂i J 0 ,
(11.254)
∂Πi
M
M
∂H
(11.255)
Π̇i = − i = −ǫijk ǫklm ∂j ∂l V m − M 2 V i + g J i .
∂V
Their equivalence with the Euler-Lagrange (11.246) equations can be seen
as follows: If we define an auxiliary variable V 0 as in (11.252); the equation
(11.247) is then automatically satisfied. Next, the equation (11.254) can be
written in the form Πi = V̇ i + ∂i V 0 . Differentiating this relation with respect
to time and inserting Π̇i obtained in this way into (11.255) transforms the
latter equation into (11.248). Thus, by eliminating “algebraically”one of the
variables, we have constructed the Hamilton’s formalism which is equivalent
to the field equations following from the original Lagrangian (11.240).
V̇ i =
Quantization is now simple: one promotes V i (x, t) and Πi (x, t) at t = 0
to operators in the Schrödinger picture and imposes the standard canonical
commutation relations
h
i
i
V̂ (x), Π̂j (y) = i V i (x), Πj (y) PB = iδji δ(x − y) ,
h
i h
i
V̂ i (x), V̂ j (y) = Π̂i (x), Π̂j (y) = 0 .
(11.256)
371
on the operators representing the independent canonical variables. In this
way quantized get only physical degrees of freedom of the system of fields.
If the current J µ depends on field variables φa and Πa other than V i and Πi
themselves, the states |Ψi of the system can be represented as wave functionals Ψ[V i (x), φa (x), t] = h [V i ], [φa ]|Ψ(t)i on which the operators V̂ i (x) and
Π̂i (x) act as
V̂ i (x) = V i (x) ,
Π̂i (x) = −i
δ
.
δV i (x)
(11.257)
The next task is to find such a representation of the operators53 V i (x) and
Πi (x) in terms of the creation and annihilation operators (having the standard commutation rules) that the free part H0 of the Hamiltonian H (that
is, its part independent of the current J µ ) becomes diagonal. To simplify
this task we can apply the method outlined in Section 11.3 allowing to easily
quantize the free field (i.e. for J µ = 0) in the Heisenberg picture; the freefield Heisenberg picture operators VHi (t, x), ΠH
i (t, x) taken at t = 0 will just
provide the necessary representations of the Schrödinger picture operators of
the interacting (i.e. for J µ 6= 0) Proca field.
To this end first write down the the most general solution to the classical
field equations (11.242) and (11.244) following from the Proca Lagrangian,
which for J µ = 0 read
(∂ν ∂ ν + M 2 )V µ (t, x) = 0 ,
∂µ V µ (t, x) = 0 .
(11.258)
The most general classical solution of the first equation can be written in the
form (11.116). In order to satisfy the second of these equations we make in
(11.116) the replacement
X
a(k) →
a(k, λ) ǫµ (k, λ) ,
(11.259)
λ=0,±1
where ǫµ (k, λ) are three vectors satisfying the conditions
kµ ǫµ (k, λ) = 0 .
(11.260)
(since k 2 = M 2 6= 0 there can be only three linearly independent such vectors). Two of them, corresponding to λ = ±1, can be chosen to have only
space-like components (in the plane perpendicular to k) and will be normalized so that
ǫµ (k, λ) ǫ∗µ (k, λ) = −1 .
53
We suppress hats on operators from now on.
372
(11.261)
(i.e. ǫ(k, λ) · ǫ∗ (k, λ) = 1). For the third one, corresponding to λ = 0, we
take the four-vector
|k| k E(k)
µ
ǫ (k, λ = 0) =
,
(11.262)
,
M |k| M
which also satisfies the conditions (11.260) and (11.261). The vectors ǫµ (k, λ)
satisfy then the sum rule
X
λ=0,±1
ǫµ (k, λ) ǫ∗ν (k, λ) = −gµν +
kµ kν
.
M2
(11.263)
Thus, for J µ = 0 for the free Heisenberg picture field operators we take
Z
X µ
VH (t, x) = dΓk
a(k, λ) ǫµ (k, λ) e−iEt+ik·x
λ=0,±1
+a† (k, λ) ǫµ∗ (k, λ) eiEt−ik·x , (11.264)
√
where, as usually, dΓk = d3 k/(2π)3 2E and E = k2 + M 2 (of the four
components of VHµ only VHi ’s are independent). We then recall that in the
classical Proca theory Πi (t, x) = V̇ i (t, x) + ∂i V 0 (t, x). In agreement with
this relation, for the free operator ΠH
i (in the Heisenberg picture) we take
the expression
Z
X 1
H
a(k, λ) ǫ̃i (k, λ) e−iEt+ik·x
dΓk E(k)
Πi (t, x) =
i
λ=0,±1
−a† (k, λ) ǫ̃i∗ (k, λ) eiEt−ik·x , (11.265)
where
ǫ̃i (k, λ) ≡ ǫi (k, λ) −
ki 0
ǫ (k, λ) .
E
(11.266)
Obviously, ǫ̃i (k, λ) = ǫi (k, λ) for λ = ±1 and for λ = 0 one finds ǫ̃i (k, λ =
0) = (k i /|k|)(M/E). It is then easy to verify that VH0 (t, x) is correctly
reproduced by (11.252) for J µ = 0. Furthermore, one can check that the
canonical equal-time commutation rules (11.256) are satisfied if
a(k, λ), a† (k′ , λ′ ) = (2π)3 2E(k)δλλ′ δ (3) (k − k′ ) ,
[a(k, λ), a(k′ , λ′ )] = a† (k, λ), a† (k′ , λ′) = 0 .
(11.267)
It follows that for the Schrödinger picture operators V i (x) and Πi (x), which
at t = 0 are equal to the Heisenberg picture operators of the interacting
Proca field (which for t 6= 0 do not have the forms (11.264) and (11.265)!)
one can take VHi (t = 0, x) and ΠH (t = 0, x), respectively. In terms of the
373
creation and annihilation operators introduced in this way, the free part H0
of the Hamiltonian (11.253) takes the form
Z
X
H0 = dΓk E(k)
a† (k, λ) a(k, λ) ,
(11.268)
λ=0,±1
plus an infinite constant which we discard. It is also easy to see that the
operators (11.264) and (11.265), which in the theory of the interacting Proca
field become the interaction picture (free) operators VIi (x) and ΠIi (x) (see
Section 11.9), are obtained from the Schrödinger picture ones as
VIi (t, x) = eiH0 t/~ V i (x) e−iH0 t/~ ,
ΠIi (t, x) = eiH0 t/~ Πi (x) e−iH0 t/~ ,
as they should.
Finally, one can compute the commutators of V 0 (x) which for nonzero J 0
is given by (11.252) with J 0 depending on the canonical variables of systems
coupled to the Proca field but not on the canonical variables V i or Πi of the
vector field itself. We find
1 i
V i (x), V 0 (y) = 2 V i (x), ∂jy Πj (y) = − 2 ∂ix δ (3) (x − y) ,
M
M
Πi (x), V 0 (y) = 0 .
(11.269)
Thus, the operator corresponding to the time component of V µ does not
commute with the operators corresponding to the remaining components.
11.6
The Dirac method
In Section 11.5 the Proca vector field has been quantized by expressing “algebraically”one of its components (V 0 ) in terms of the remaining independent
canonical variables (V i and Πi ). The form of the canonical commutation
relations imposed on the independent canonical variables (that is the right
quantization prescription) was then obvious and they determined also the
commutation relations satisfied by the operator V 0 . Explicit solutions for
dependent variables (like V 0 ) in terms of the other variables, chosen as independent, even if possible in principle, may not always be easy (in field theories solving “algebraically” may mean solving differential equations in the
space variable x - the term “algebraic” refers only to the fact that no time
derivatives are involved). Therefore, in this section we describe a systematic
method proposed by Dirac, which allows to find the right commutation relations which must be imposed on the original set of canonical variables along
with a set of identities which must be satisfied by operators representing these
374
variables, in order to properly quantize a system subject to constraints.54 In
the case of systems subject to constraints (called second class) similar to the
ones encountered in the Proca theory of the vector field the Dirac’s method
allows to impose the right commutation relations directly on the original set
of canonical variables without solving explicitly for dependent ones. The
same method proves useful also for systems subject to another class of constraints (first class ones), which, as in the case of the electromagnetic field,
reflect invariance of the (classical) physical state of the system with respect to
changes of “gauge” - i.e. with respect to appropriate transformations of the
canonical variables. Finally, the Dirac’s method allows for an easy canonical
quantization of theories of non-Abelian Yang-Mills fields which in the BRST
formulation become systems subject to second class constraints (see Section
20.3).
Canonical quantization consists of the identification of the canonical variables q i , i = 1, . . . , n and the conjugated momenta pi and their subsequent
promotion to the Schrödinger picture operators q̂ i and p̂i satisfying the commutation rule
i
q̂ , p̂j = i~ q i , pj PB = i~ δ i j .
The method in this simple form is applicable if the canonical variables and
their momenta are all independent. Classical systems may, however, be subject to constraints as a result of which not all their canonical variables are
independent. The primary constraints ΦM (q, p) = 0, M = 1, . . . , n−r, follow
from the structure of the Lagrangian and reflect the impossibility to solve
for q̇ j , j = r + 1, . . . , n the equations
pj =
∂
L(q, q̇) ,
∂ q̇ j
(11.270)
for all q̇ j . In such a case the original Euler-Lagrange equations of motion are
equivalent (see Appendix I) to the set of canonical equations which follow
from the extended Hamiltonian HE of the form
!
n
n−r
X
X
˙
∂L
i
HE (q, p) =
q − L(q, q̇)
+
ΦM (q, p) uM
i
∂
q̇
i
i=1
M =1
q̇=q̇ (q,p,q̇)
≡ H(q, p) +
n−r
X
ΦM (q, p) uM ,
(11.271)
M =1
combined with the set of (primary) constraints ΦM = 0. The subscript
q̇ = q̇ i (q, p, q̇) on the bracket in the first term of (11.271) means that r of the
54
The method itself, however, does not give any clues, how to satisfy these commutation
relations and the identities by expanding field operators taken at t = 0 into creation and
annihilation operators.
375
generalized velocities q̇ i (those for which the relations (11.270) can be solved
for) are expressed in terms of the canonical variables and the remaining n − r
generalized velocities.55 The sum in the second term runs over the set of n−r
primary constraints ΦM (q, p) = 0. The factors uM , which can be viewed as
Lagrange multipliers, are just those n − r generalized velocities q̇ i , for which
the equations (11.270) cannot be solved. By construction (see Appendix I)
primary constraints ΦM (q, p) = 0 have the structure pM − fM (q, pk ) = 0,
M = r + 1, . . . , n, and k = 1, . . . , r, where pM are the canonical momenta
corresponding to the velocities uM . The mentioned equivalence is to be
understood in the sense that the constraints ΦM (q, p) = 0 differentiated with
respect to time, combined with the evolution equations q̇ i = {q i , HE }PB and
ṗi = {pi , HE }PB obtained from the extended Hamiltonian HE (11.271) yield
second order differential equations for q i which are equivalent to the original
Euler-Lagrange equations. The dynamics can be cast into a fully Hamiltonian
form by giving the factors uM an appropriate dependence on pi and q i . This
is done as follows.
The constraints ΦM (q, p) = 0 should be preserved by the dynamics generated by the Hamiltonian (11.271) - in this case it will suffice to impose
them on the initial data only. This means that Φ̇N (q, p), that is the Poisson
brackets {ΦN , HE }PB , must be made vanishing. Investigating these Poisson
brackets one can encounter different situations. Barring the case of obvious
contradictions, like 1 = 0, the first possibility is that {ΦN , HE }PB vanishes
if the already identified constraints are imposed after computing the Poisson
bracket (the Poisson brackets are computed treating all variables q i and pi
as independent). We denote this
{ΦN , HE }PB ≃ 0 ,
where the symbol ≃ means “vanishes weakly” (Dirac’s terminology). The
second possibility is that the investigated Poisson bracket of the constraint
ΦN with the Hamiltonian (11.271) does not vanish, even weakly, and the
result does not depend on the (yet) unknown coefficients uN . This means
that the new constraint ΦK ≡ {ΦN , HE }PB = 0 has to be added to the list
of constraints. Constraints identified in this way are called secondary. The
obvious next step is to investigate the Poisson bracket of the newly identified
constraint ΦK with the Hamiltonian (11.271) which can lead to a yet new
constraint and so on. Proceeding in this way, one eventualy identifies all the
constraints a given system is subject to. Finally, there are Poisson brackets
of ΦN with the Hamiltonian HE which do not vanish weakly but depend on
the coefficients uM . Requiring that all such Poisson brackets, obtained for
55
In Appendix I it is shown that the content of the bracket, that is H(q, p) in the second
line of (11.271), is independent of the n − r velocities q̇ j for which the relations (11.270)
cannot be solved.
376
primary and all secondary constraint (weakly) vanish one gets the system of
linear equations
{ΦN , H}PB +
n−r
X
{ΦN , ΦM }PB uM ≃ 0 ,
(11.272)
M =1
in which the sum runs over the n − r primary constraints. The solution of
this system of linear inhomogeneous equations takes the general form
u
M
M
= c (q, p) +
p
X
M
v(a)
(q, p) sa ,
(11.273)
a=1
M
in which v(a)
(q, p), a = 1, . . . , p are all linearly independent solutions of the
homogeneous system
n−r
X
M
{ΦN , ΦM }PB v(a)
(q, p) ≃ 0 .
(11.274)
M =1
In this way the extended Hamiltonian (11.271) gets fixed and all the consistency conditions {ΦN , HE }PB ≃ 0 are satisfied.
An important feature of the extended Hamiltonian (11.271) is its possible dependence on the coefficients sa , which may be arbitrary functions of
time. They are present whenever there exist linear combinations Φa (q, p) ≡
M
ΦM (q, p) v(a)
(q, p) of the primary constraints which, as follows form (11.274),
have (weakly) vanishing Poisson brackets with all other constraints. Linear
combinations Φa (q, p) such that
{ΦN , Φa }PB ≃ 0
for all
N
(11.275)
(the Poisson brackets should be computed as if the variables were unconstrained; the constraints ΦM = 0 are imposed afterwards) are called first
class constraints. The remaining ones, which have at least one nonzero Poisson bracket with the remaining constraints, are called second class. It should
be noted at this point that the consistency conditions ensured by fixing the
coefficient functions uM (q, p) as in (11.273) would not be spoiled if other
second class constraints, formed as linear combinations which include also
secondary constraints, are included into the Hamiltonian HE (11.271). Thus
for the complete Hamiltonian HT of the system one may take the expression
HT (q, p) = H(q, p) +
2k
X
ΦM (q, p) cM (q, p) +
X
Φa (q, p) sa (t) ,
(11.276)
a
M =1
in which H is defined by (11.271), the sum over M runs over all 2k ≤ n − r
second class constraints (formed as linear combinations of the promary and
377
secondary constraints) and the sum over a runs over all primary first class
constraints. Consistency of the constraints with the dynamics generated by
HT is ensured by giving (solving the equations like (11.272)) the coefficients
cM (q, p) the appropriate dependence on q and p (it is then sufficient to impose the constraints on the initial data only), while the factors sa (t) can be
arbitrary functions of time without affecting this consistency.
Due to the presence of the arbitrary functions sa (t), which is always the
case if there are primary first class constraints, the time evolution of the
canonical variables q i (t) and pi (t) generated by the Hamiltonian (11.276) is
not unique - the initial data q i (t0 ) and pi (t0 ) set at some t = t0 do not specify uniquely the values of q i (t) and pi (t) at other times t. Another possible
situation arises if there are secondary first class constraints (and no primary
first class ones): the time evolution is then unique, once the initial data are
given, but their choice is not. Thus, first class constraints are always associated with some kind of “gauge invariance” by which one means a situation
in which the (classical) physical state of a system subject to such constraints
does not change if the values of the variables used to describe it are changed
appropriately. In other words, infinitely many different values of the same
canonical variables correspond to the same (classical) physical state. It is
easy to see, that the infinitesimal transformations of the canonical variables
which do not change the physical state are generated by the primary (?) first
class constraints Φa :
δq i = θa {q i , Φa }PB ,
δpi = θa {pi , Φa }PB ,
with arbitrary time dependent parameters θa .
There are two ways of quantizing systems subject to first class constraints.
One consists of working in the extended Hilbert space of states |Ψi. The
physical states |Ψphys i, forming a subspace of the full Hilbert space, are then
selected by the conditions
Φa (q̂, p̂)|Ψphys i = 0 ,
for all
a.
(11.277)
This approach which in principle requires some care in ordering the operators
q̂, p̂ so that [Φ̂a , Φ̂b ] = cd (q̂, p̂)Φ̂d (as is necessary for consistency of the
conditions (11.277)), is used e.g. to quantize the electromagnetic field in
the covariant Lorentz gauge ∂µ Aµ = 0 (this is very similar to the GuptaBleuler method of the electromagnetic field quantization - see Section 11.11)
and in the quantization of the relativistic string (see the old review by C.
Rebbi). The alternative way, which will be demonstrated on the example of
the electromagnetic field in the next section, is to fix the gauge completely.
378
This is done by imposing additional constraints Φ(q, p) = 0 from outside,56
so that the entire system of constraints becomes second class and the time
evolution of the variables becomes uniquely determined by the equations of
motion and the initial data; the resulting classical systems subject to second
class constraints can be quantized with the help of the Dirac prescription
which we describe below and apply next to the Proca theory of the vector
field. Finally, it may happen that the additional constraints do not fix the
gauge completely and there remain some secondary first class constraints (but
all primary constraints are already second class). In this case the classical
time evolution is uniquely determined but there is a freedom in fixing the
initial data. The quantum theory can be then constructed by treating the
second class constraints á la Dirac (see below) and the remaining first class
constraints should be imposed as subsidiary conditions selecting those vectors
of the Hilbert space which represent quantum states of the physical system.
To quantize a theory in which first class constraints are absent from the
beginning or have been eliminated by imposing additional constraints from
outside, one forms the matrix CN M :
CN M ≡ {ΦN , ΦM }PB .
(11.278)
Since the matrix CN M is antisymmetric (the basic property of the Poisson
brackets) and det(CN M ) 6= 0 (vanishing of this determinant would mean that
one can form at least one more linear combination of the constraints which
has zero Poisson brackets with all others and, hence, belongs to to the first
class constraints which we have assumed to be already eliminated), it must be
of even dimension. The Dirac quantization prescription for systems subject
to second class constraints then reads
[q̂, p̂] = i~ {q, p}D ,
ΦN (q̂, p̂) ≡ 0 ,
for all N ,
(11.279)
that is, the commutators (or anticommutators in the case of fermionic fields
- see section 11.8) are determined by the Dirac brackets (instead of Poisson
brackets) and all the constraints ΦM = 0 are realized as strong operator
identities. The Dirac bracket {A(p, q), B(p, q)}D of two functions of the
canonical variables is defined as follows:
X
N M
{A, B}D ≡ {A, B}PB −
{A, ΦN }PB C −1
{ΦM , B}PB , (11.280)
N,M
56
This is possible because, as explained above, the physical state of the system does not
fix the values of the canonical variables uniquely; the added constraints can be then made
consistent with the dynamics generated by HE (or HT ) from which arbitrary functions sa
are now absent.
379
(again the Poisson brackets have to be computed as if there were no constraints). The Dirac bracket shares all the algebraic properties of the Poisson
bracket: it is bilinear in canonical variables, antisymmetric and satisfies the
Jacobi identity. It is also a matter of a straightforward algebra to check that
the definition (11.280) is invariant with respect to linear changes
N
ΦM (q, p) = OM
(q, p) Φ̃N (q, p) ,
N
of the basis of constraints, provided the matrix OM
(q, p) is nonsingular (invertible) on the surface of constraints. Finally, the Dirac bracket has the
(easy to verify) property
{A, ΦN }D = 0 ,
(11.281)
which ensures compatibility of the commutation relation (11.279) with the
constraints realized as strong operator identities. This also means that the
Hamiltonian operator is obtained from the H part of the extended classical
Hamiltonian HE (11.271): the sum of the primary constraints (which in
the classical theory must be kept in order to ensure the compatibility of
constraints with the dynamics) is in the quantum theory just a zero operator.
The Dirac prescription can be justified as follows. First of all, one can
notice that if the first class constraints are absent, that is the last sum is
absent in the Hamiltonian HT (11.276), all the factors cM (q, p) are determined
from the consistency conditions {ΦN , HT }PB ≃ 0 and are given by
cM (q, p) = −(C −1 )M N {ΦN , HT }PB .
The canonical equtions of motion q̇ i = {q i , HT }PB and ṗi = {pi , HT }PB take
then just the form
i
i
i
q̇ = {q , HT }PB = {q , H}PB +
2k
X
{q i , ΦM }PB cM (q, p) ≡ {q i , H}D ,
M =1
and similarly for ṗi . Since in the quantum theory the equations of motion
satisfied by the Heisenberg picture operators are given by their commutators
with the Hamiltonian, the above form of the classical equations of motion
strongly supports the Dirac quantization rule. More formally, it can be shown
that for the even number of second class constraints ΦM , M = 1, 2, . . . , 2k
there always exists a canonical transformation (qi , pi ) → (Qi (q, p), Pi (q, p)),
i = 1, 2, . . . , n, such that in the new canonical variables the constraints ΦM
take the simple form
Qi = 0 ,
Pi = 0
for
i = 1, . . . , k.
(11.282)
One can therefore quantize the system in the remaining new variables (Qi , Pi )
with i = k + 1, . . . , n forgetting about the first k + k = 2k variables which
380
are zero. The commutator of the quantum operators (q̂i , p̂j ) when expressed
in terms of the commutators [Q̂i , P̂j ] = i~ δij for i, j > k are then given by
the Dirac bracket (times i~). The advantage of the Dirac method for finding the right commutation relations which must be satisfied by the canonical variables is that one does not have to find the canonical transformation
(qi , pi ) → (Qi (q, p), Pi (q, p)) explicitly nor does one have to solve the constraints for some variables in terms of others.
In the case of the Proca field the primary constraint57
Φ1 ≡ Π0 (t, x) = 0 ,
(11.283)
follow from the structure of the Lagrangian (11.240). The extended Hamiltonian takes therefore the form
1
1
1
HE = Πi Πi − Πi ∂i V 0 + (∇×V)2 + M 2 V i V i
2
2
2
1 2 0 0
− M V V + Π0 u1 − g V i J i + g V 0 J 0 , (11.284)
2
with an initially unknown function u1 . Since Φ̇1 = {Φ1 , HE }PB does not
vanish (even weakly), the secondary constraint
Φ2 ≡ M 2 V 0 (t, x) − ∂i Πi (t, x) − g J 0 (t, x) = 0 ,
(11.285)
must be imposed as the consistency condition. There no more constraints vanishing of Φ̇2 = {Φ2 , HE }PB can be ensured by adjusting the function u1 .
This completes the Hamiltonization of the classical dynamics of the Proca
field.
In the quantum theory Φ1 and Φ2 become operator identities; therefore,
Π0 must be represented by the zero operator and through (11.285) the operator V 0 becomes completely determined by Πi and J 0 . The canonical
commutation relations of the V i , Πi , V 0 and Π0 operators are given by the
Dirac prescription (11.279) which ensures their compatibility with the operator relations Φ1 = 0 and Φ2 = 0. The matrix CN M (11.278) has in this case
the form
C1x,2y = {Φ1x , Φ2y }PB = {Φ1 (x), Φ2 (y)}PB = −M 2 δ (3) (x − y) , (11.286)
and its inverse reads
C
−1
=
− M12
0
(3)
δ (x − y)
57
1
M2
δ (3) (x − y)
0
.
(11.287)
In field theory the index i of q i in the formulae (11.270)-(11.282) includes also the
space variable x. Hence, each ΦM stands for an infinite set of constraints ΦMx ≡ ΦM (x).
381
The Dirac bracket of any two functions A(x) and B(y) of the canonical
variables V µ and Πµ therefore reads
{A(x), B(y)}D = {A(x), B(y)}PB
Z
1
− 2 d3 z {A(x), Π0 (z)}PB M 2 V 0 (z) − ∂i Πi (z) − gJ 0 (z), B(y) PB
M
Z
1
+ 2 d3 z A(x), M 2 V 0 (z) − ∂i Πi (z) − gJ 0 (z) PB {Π0 (z), B(y)}PB .
M
Computing using this formula the commutator
i
V (x), V 0 (y) = i~ V i (x), V 0 (y) D ,
(11.288)
one finds the same result as was obtained in section 11.5 by treating the V 0
operator as constructed out of Πi and J 0 in agreement with the constraint
Φ2 = 0. It is also easy to check that the Dirac prescription gives all the
remaining commutators in their standard forms. The rest of the quantization
procedure is then unmodified.
11.7
Quantization of the electromagnetic field
We now deviate a little bit from the main logic of this Section (devoted mainly
to quantization of systems of relativistic fields) and consider quantization of
the system consisting of the classical electromagnetic field coupled to N classical nonrelativistic charged particles. The reason for this deviation is that
in this way we will obtain a working theory, called nonrelativistic quantum
electrodynamics (NRQED), which is applicable to a wide range of physical
phenomena related to interaction of light with nonrelativistic matter, so long
as spin effects can be neglected. It also constitutes a modern starting point
for calculations of properties of bound states in fully relativistic quantum
electrodynamics.
The Lagrangian describing classical dynamics of such a system is given
by the sum of three terms: L = LEM + Lmatt + Lint , where
Z
1
d3 x fµν f µν ,
(11.289)
LEM = −
4
N
Lmatt
N
N
X
1X
1X
mi vi2 −
V (ri ) −
V (ri − rj ) ,
=
2 i=1
2
i=1
(11.290)
i6=j
Lint
Z
N h
i
X
qi
e
= −e
qi φ(ri ) − A(ri ) ·vi ≡ −
d3 x Jµ (x)Aµ (x) , (11.291)
c
c
i=1
382
where the current in the last formula has the form (qi are particle charges in
units of e > 0)
0
J (t, x) =
J(t, x) =
N
X
i=1
N
X
i=1
qi c δ (3) (x − ri (t)) ,
qi vi δ (3) (x − ri (t)) .
(11.292)
For greater generality we have allowed in Lmatt also for 1-particle interactions
V (ri ) with external potentials (which can be also of electromagnetic origin e.g. the electrostatic interactions binding electrons in atoms, which we may
treat separately from the dynamical electromagnetic field) and two-particle
interactions V (ri − rj ) of non-electromagnetic origin.
To quantize this system we first try to take for its canonical variables and
conjugated momenta ri (t), Pi (t), Aµ (t, x) and Πµ (t, x). In the matter part
everything goes in the standard way:
Pi (t) =
qi
∂L
= mi vi (t) + eA(t, ri (t)) .
∂vi (t)
c
(11.293)
Expressing vi (t) in the matter part of the Hamiltonian through Pi (t) we get
H = HE +
N
2
X
1 qi
Pi − eA(ri )
2mi
c
i=1
+e
N
X
qi φ(ri ) +
i=1
N
X
V (ri ) +
i=1
N
1X
V (ri − rj ) ,
2 i6=j
(11.294)
where HE is the current-independent (here “current-independent” before expressing φ in terms of J 0 !) part of the (extended) electromagnetic field
Hamiltonian (11.319) which we derive below. Quantization of the matter
part is standard: we promote ri (t) and Pi (t) taken at t = 0 to Schrödinger
picture operators satisfying the commutation rules [r̂i , P̂j ] = i~ δij . We can
then work either in the position representation in which P̂i = −i~ ∂/∂ri , or
we can go over to the second quantization formalism (presented in Section 5),
which is especially convenient if the matter particles are identical.
The electromagnetic part of the system is more troublesome. Written in
terms of the potentials the Lagrangian density L (11.233) of the electromagnetic field coupled to the current J µ reads (c is kept for decoration, φ ≡ A0 ):
LEM
1
=
2
1∂
A + ∇φ
c ∂t
2
−
1
1
1
(∇ × A)2 − eJ 0 φ + eJ·A . (11.295)
2
c
c
383
R
Of course, d3 x LEM = LEM + Lint in the notation of (11.289-11.291). As
already checked in Subsection 11.4, the Euler-Lagrange equations following
from this Lagrangian density are (ρ ≡ eJ 0 )
∇×B −
1
1 ∂E
= eJ ,
c ∂t
c
∇·E = ρ .
(11.296)
As the canonical momenta Πi we get
Πi =
1 ∂Ai 1
1
+ ∂i φ = − E i ,
2
c ∂t
c
c
(11.297)
and Π0 = 0, similarly as in the case of the Proca vector field. We have
therefore the primary constraint Φ0 ≡ Π0 and we have to check whether it
is compatible with the Hamilton’s canonical equations. To this end we form
the Hamiltonian density HE ≡ HEM + Hint + Φ0 u0 according to the Dirac
method expressing Ȧi through Πi by using the relation (11.297) and adding
the primary constraint with an unknown coefficient u0 . This gives:58
HE = Πi Ȧi + u0 Φ0 − L
1
1
1
c2
= Πi Πi − c Πi ∂i φ + (∇×A)2 + Π0 u0 + eJ 0 φ − eJ·A
2
2
c
c
1
1 2
2
0
E + B + E·∇φ + Π0 u + ρ φ − eJ·A ,
(11.298)
=
2
c
Using this Hamiltonian density we find59
Z
∂HE
∂u0
1
3
Φ̇0 ≡ Π0 ,
d x HE
=−
= −Π0
− c ∂i Πi − eJ 0
0
∂φ
∂A
c
PB
1
(11.299)
≃ −c ∂i Πi − eJ 0 ,
c
As this is not zero (even weakly), the secondary constraint Φ2 ≡ c2 ∂i Πi + eJ 0
has to be imposed. This is obviously the Gauss law, which is the second of the
Euler-Lagrange equations (11.296), but is lost in the Hamilton’s formalism
because of vanishing of Π0 . The constraint Φ2 is already compatible with
the dynamics: {Φ2 , HE }PB = c ∂µ (eJ µ ) vanishes provided the current is
conserved.60
58
Note, that J depends now on r̂i and P̂i .
In computing Poisson brackets all variables and momenta have to be treated as independent; therefore, for any two functionals G and H we have
Z
δH
δG
δH
δG
µ
µ
3
.
−
G[A , Πµ ], H[A , Πµ ]
≡ d x
δAλ (x) δΠλ (x) δAλ (x) δΠλ (x)
PB
59
60
To derive the relation {Φ2 , HE }PB = c ∂µ (eJ µ ) one must take into account the full
Hamiltonian of the theory which in the case considered here includes, in addition to the
terms displayed in (11.298), also the term Hmatt governing the dynamics of the nonrelativistic particles; one then finds that {J 0 , HE }PB = c ∂0 J 0 .
384
The system is therefore subject to two constraints, Φ0 and Φ2 , but they
turn out to be the first class:
{Φ0 , Φ2 }PB = 0 ,
and the Dirac prescription for quantization cannot be applied because the
(classical) state of the electromagnetic field, fully characterized by E and B,
does not determine uniquely the potentials Aµ . The mathematical reason for
this is the gauge invariance of the Lagrangian (11.295). The Euler-Lagrange
equations do not determine Aµ (t, x) uniquely for t 6= 0 from initial conditions
Aµ (x) and Ȧµ (x) specified at t = 0: one can always imagine a function θ(t, x)
such that ∂ µ θ(t, x) = 0 and ∂ µ θ̇(t, x) = 0 for t = 0, but nonvanishing for
t 6= 0. If Aµ (t, x) solves the Euler-Lagrange equations with the given initial
conditions, then so does Aµ (t, x) + ∂ µ θ(t, x) with the same initial conditions
at t = 0. Since the constraints Φ0 = 0 and Φ2 = 0 essentially make the
canonical Hamilton’s equations equivalent to the Euler-Lagrange equations,
it is clear that they cannot help to determine Aµ (t, x) uniquely for all t as it
was possible for the Proca vector field.
Unique determination of the time evolution of Aµ (t, x) becomes possible
if one chooses a gauge. In a fixed gauge the Euler-Lagrange equations can
determine Aµ (t, x) for all t unambiguously and it should be possible to find
an equivalent Hamilton’s canonical formulation too. One possibility is the
Coulomb gauge
∇·A = 0 .
(11.300)
Another, convenient in some applications, is the temporal gauge A0 = 0.
However, while the latter makes the time evolution unique, it does not fix
uniquely the initial data: transformations with time independent function
θ(x) are still possible. As a result, the constraint Φ2 = 0 (the Gauss law)
remains in this approach first class and has to be imposed as a subsidiary
condition selecting physical state-vectors of the Hilbert space. For this reason
we here choose to work in the Coulomb gauge (11.300) which classically has
a clear physical interpretation: it disentangles electrostatic effects from those
due to radiation.
We restart, therefore, the whole procedure with the same Lagrangian
density (11.295) and one primary constraint
Φ1 ≡ ∇·A ≡ ∂i Ai ,
(11.301)
imposed from outside. As previously one finds Φ0 ≡ Π0 as the primary
constraint and has to consider the extended Hamiltonian which now involves
two initially unspecified functions u0 and u1 :
HE =
c2
1
1
1
Πi Πi − c Πi ∂i φ + (∇×A)2 + Φ0 u0 + Φ1 u1 + eJ 0 φ − eJ·A .
2
2
c
c
385
The consistency condition {Φ0 , HE }PB ≃ 0, leads again to the secondary
constraint (the Gauss law)
Φ2 ≡ c2 ∂i Πi + eJ 0 .
(11.302)
In turn the requirement that {Φ1 , HE }PB ≃ 0 leads to yet another constraint
Φ3 ≡ c2 ∂i Πi − c ∂i ∂i φ .
(11.303)
There are no more constraints: conservation of the current J µ implies vanishing of Φ̇2 , provided the function u1 is set to zero, while u0 can be adjusted
to ensure Φ̇3 = 0.
It is convenient to replace the constraint Φ3 (11.303) by the linear combination Φ3′ of (11.303) and (11.302):
Φ3′ ≡ c ∂i ∂i φ + eJ 0 = 0 .
(11.304)
This shows that in the quantum theory the operator A0 (t, x) ≡ φ(t, x) is
related to the operator J 0 by the identity:61
Z
J 0 (t, y)
e
d3 y
.
(11.305)
φ(t, x) =
4πc
|x − y|
Thus, the operator A0 (t, x) ≡ φ(t, x) becomes completely determined in
terms of the matter canonical variables only (note, that J 0 given by (11.292)
does not depend on vi and, hence, J 0 , unlike J, when expressed through r̂i
and P̂i does not depend on Ai ). Since Π0 must be represented by the zero
operator, the constraints Φ0 and Φ2 effectively eliminate these variables as
independent operators from the quantum theory.
The complete system of constraints Φ0 , Φ1 , Φ2 and Φ3′ is of second class.
The corresponding 4×4 matrix CM N = −CN M is nonsingular - it has nonzero
elements on the antidiagonal:
C1x,2y = {Φ1 (x), Φ2 (y)}PB
(x) (y) i
(x) (x)
= c2 ∂i ∂j
A (x), Πj (y) PB = −c2 ∂i ∂i δ (3) (x − y) , (11.306)
and
C0x,3′ y = {Φ0 (x), Φ3′ (y)}PB =
1
(x) (x)
C1x,2y = −c ∂k ∂k δ (3) (x − y) . (11.307)
c
The canonical commutation relation which must be imposed on the operators Ai , Πi , A0 and Π0 are determined by the Dirac prescription (11.279).
61
1
Recall that ∇2 |x|
= −4π δ (3) (x).
386
The inverse (C −1 )N x,M y of the CM N matrix has also nonzero elements only
on its antidiagonal. These are
C −1
1x,2y
=−
1
4πc2 |x
− y|
=
0x,3′ y
1
C −1
,
c
(11.308)
Using this matrix one determines the basic commutator:
1 i
A (x), Πj (y) = Ai (x), Πj (y) D ≡ Ai (x), Πj (y) PB
(11.309)
i~ Z
Z
2z,1w
− d3 z d3 w Ai (x), Φ2 (z) PB C −1
{Φ1 (w), Πj (y)}PB
(terms with the remaining pairs of constraints give all zero). Using the
Poisson brackets
i
(x)
A (x), Φ2 (z) PB = −c2 ∂i δ (3) (x − z) ,
(w)
{Φ1 (w), Πj (y)}PB = ∂j δ (3) (w − y) ,
it is easy to find that
i
(x) (x)
A (x), Πj (y) D = δji δ (3) (x − y) + ∂i ∂j
1
.
4π|x − y|
(11.310)
Proceeding in the similar way, one can check that all the remaining Dirac
brackets among the system of variables Ai , Πj , A0 and Π0 vanish (vanishing
of the Dirac brackets of Π0 ≡ Φ0 with all functions of the canonical variables
of the entire theory is ensured by the property (11.281)). In particular,
i
A (x), Aj (y) D = {Πi (x), Πj (y)}D = 0 .
(11.311)
Therefore, the basic quantization conditions read
1
(x) (x)
,
Ai (x), Πj (y) = i~ δji δ (3) (x − y) + i~ ∂i ∂j
4π|x − y|
i
A (x), Aj (y) = [Πi (x), Πj (y)] = 0 .
(11.312)
The function on the right hand side of the first commutation rule (11.312)
is called the transverse delta function. It has the following momentum space
representation
1
4π|x − y|
Z 3
d k ik·(x−y) ij k i k j
ij
δ −
,
e
≡ δtr (x − y) =
(2π)3
|k|2
j
i
δ ij δ (3) (x − y) + ∂(x)
∂(x)
387
(11.313)
which makes it explicit that
(x) (x)
x
∂i δji δ (3) (x − y) + ∂i ∂j
1
4π|x − y|
= 0,
(11.314)
and, therefore, that the commutation rules (11.312) are compatible with the
constraint (11.301).
The Hamiltonian density operator
c2
1
1
1
Πi Πi + (∇×A)2 − c Πi ∂i φ + eJ 0 φ − eJ·A , (11.315)
2
2
c
c
in which φ is now a shorthand for the operator (11.305) and Π0 has been set
to zero, together with the quantization rules (11.312) specify in principle the
dynamics of the quantized electromagnetic field. There is, however, one technical problem that the operator Πi (x) does not commute with matter sector
canonical variables. This is because the Dirac brackets of Πi (x) with functions F depending on the matter sector canonical variables do not necessarily
vanish.62 Indeed, according to the Dirac prescription such commutators are
given by i~ times the Dirac brackets
H=
{F (mat), Πi (x)}D = {F (mat), Πi (x)}PB
Z
Z
M z,N w
3
− d z d3 w {F (mat), ΦM (z)}PB C −1
{ΦN (w), Πi (x)}PB .
The Poisson bracket in the first line is of course zero. To the second line
contributes only the term with MN = 21. This line therefore gives
Z
Z
1
(w)
3
∂i δ (3) (w − x)
− d z d3 w {F (mat), Φ2 (z)}PB
2
4πc |z − w|
Z
1
(x)
= d3 z F (mat), eJ 0 (z) PB ∂i
.
2
4πc |z − x|
This is evidently nonzero, if F is a function of Pj because J 0 (z) given by
(11.292) depends on ri , and {ri , Pj }PB = δij . Dealing with such an operator
would be difficult. Therefore, one defines another operator
1
∂i φ ,
(11.316)
ΠT
i = Πi −
c
which in fact is just (1/c2 )Ȧi (cf. (11.297)), i.e. the transverse (the divergenceless) part of the electric field, which has better properties. Firstly,
1 (x)
− ∂i {F (mat), φ(x)}D
c
Z
Z
M z,N w
1 (x)
3
= ∂i
d z d3 w {F (mat), ΦM (z)}PB C −1
{ΦN (w), φ(x)}PB
c
Z
1
1 (x)
d3 z F (mat), eJ 0 (z) PB
= − ∂i
c
4πc|z − x|
62
It is easy to check that {F (mat), Ai (x)}D = 0.
388
′
(only the element (C −1 )3 z,0w contributes to the sum over constraints; the
Poisson bracket {F (mat), φ(x)}PB obviously vanishes), so this precisely cancels out the unwanted term in the Dirac bracket of Πi with F (mat). Hence,
the operator ΠT
i does commute with the matter variables. Secondly, since
1 (x) i
A (x), φ(y) D = 0
− ∂i
c
(the Poisson bracket of Ai (x) with φ(y) vanishes, Ai has nonzero Poisson
bracket only with Φ2 , while φ only with Φ0 and (C −1 )2z,0w = 0, so the
second term in the Dirac bracket vanishes too), it follows that
i
(x
(x)
i (3)
A (x), ΠT
j (y) = i~ δj δ (x − y) + i~ ∂i )∂j
1
,
4π|x − y|
(11.317)
just as for Πi (y). Finally, one has to check the Dirac bracket
T
1 (x) (y)
Πi (x), ΠT
j (y) D = {Πi (x), Πj (y)}D + 2 ∂i ∂j {φ(x), φ(y)}D
c
1 (y)
1 (x)
− ∂i {φ(x), Πj (y)}D − ∂j {Πi (x), φ(y)}D .
c
c
As already has been established in (11.311), the first term on the right hand
side is zero. To show that the remaining three terms also vanish, we note
that
{φ(x), φ(y)}D = {φ(x), φ(y)}PB
Z
Z
M z,N w
3
{ΦN (w), φ(y)}PB = 0 :
− d z d3 w {φ(x), ΦM (z)}PB C −1
the Poisson bracket of two φ’s vanishes and the second term vanishes because
φ’s have nonzero Poisson bracket
Π0 and (C −1 )0z,0w = 0. To com only with
y
plete the proof, we consider Πi (x), ∂j φ(y) D : the corresponding Poisson
bracket is zero and
Z
Z
M z,N w 3
ΦN (w), ∂jy φ(y) PB
− d z d3 w {Πi (x), ΦM (z)}PB C −1
also vanishes because Πi has a nonzero Poisson bracket only with Φ1 , φ only
with Φ0 and (C −1 )1z,0w = 0. We have thus shown, that the new operator ΠT
i
has all the properties required from the canonical momentum operator but
in contrast to Πi , it commutes with operators representing the matter sector
variables. In addition, ΠT
i satisfies a simpler constraint than Φ2 = 0:
Φ2′ ≡ ∂i ΠT
i = 0.
389
(11.318)
Expressing now the Hamiltonian density operator (11.315) through ΠT
i
we obtain
H=
1
1
1
c2 T T 1
Πi Πi + (∇×A)2 − ∂i φ∂i φ + e J 0 φ − eJ·A
2
2
2
c
c
c2 T T 1
1
1
= Πi Πi + (∇×A)2 + eJ 0 φ − e J·A .
2
2
2c
c
(11.319)
The noncovariantly looking term eJ 0 φ/2c (obtained after integrating by parts
and using the constraint Φ3′ = 0 (11.304)) produces in the Hamiltonian the
term representing energy of the ordinary electrostatic Coulomb interaction
Z
Z
e2 J 0 (x)J 0 (y)
1
3
d x d3 y
.
(11.320)
H⊃
2
4πc2 |x − y|
It remains to find the representation of the Schrödinger picture operators
A and ΠT
simi in terms of the creation and annihilation operators satisfying
R
ple commutation rules and diagonalizing the Hamiltonian H0EM = d3 x H0EM
obtained from (11.319) for J µ = 0. In addition, these representations should
automatically ensure that the constraints Φ1 = Φ2′ = 0 hold as operator
identities. It is easy to guess that
Z 3 s 2 Xh
i
~c
dk
†
i
ik·x
i∗
−ik·x
, (11.321)
ǫ
(k,
λ)a
(k)e
+
ǫ
(k,
λ)a
(k)e
Ai (x) =
λ
λ
(2π)3 2ωk λ=±1
i
where ωk = c|k| and ǫi (k, λ) are two polarization vectors such that
k·ǫ(k, λ) = 0
(11.322)
(so that the constraint Φ1 ≡ ∇·A = 0 is satisfied) and
ǫ(k, λ)·ǫ∗ (k, λ′ ) = δλλ′ .
(11.323)
Because together with k/|k| the two vectors ǫ(k, λ) form an orthonormal
basis of the three-dimensional space, the following summation rule holds
X
λ=±1
ǫi (k, λ) ǫj∗ (k, λ) = δ ij −
kikj
.
|k|2
For the operator ΠT
i we then postulate the expansion
r
Z
~ωk X i
1 d3 k
T
ǫ (k, λ) aλ (k) eik·x
Πi (x) =
3
2
i (2π)
2c λ=±1
(11.324)
i
− ǫi∗ (k, λ) a†λ (k) e−ik·x . (11.325)
390
It is then straightforward to check that with the commutation rules
h
i
†
′
aλ (k), aλ′ (k ) = (2π)3 δλλ′ δ(k − k′ ) ,
i
h
(11.326)
[aλ (k), aλ′ (k′ )] = a†λ (k), aλ′ (k′ ) = 0 ,
all the fundamental commutation relations between Ai (x) and ΠT
i (x) are
satisfied and the free part of the Hamiltonian (11.319) takes (after discarding
an infinite constant) the form63
Z 3
X †
dk
~ω
aλ (k)aλ (k) .
(11.327)
HEM =
k
(2π)3
λ=±1
As the full Hamiltonian of the electromagnetic field coupled to a system
of N charged particles we therefore get
H=
N
2
X
X †
1 qi
d3 k
a
(k)a
(k)
+
P̂
−
~ω
eA(r̂
)
(11.328)
λ
i
k
i
λ
(2π)3
2m
c
i
i=1
λ=±1
Z
Z
N
N
X
1 X
1
e2 J 0 (x)J 0 (y)
3
+
V (r̂i ) +
V (r̂i − r̂j ) +
d x d3 y
.
2 i6=j=1
2
4πc2 |x − y|
i=1
Z
where A(r̂i ) is the operator (11.321) depending now on the charged particle
position operators r̂i . This dependence is introduced into A by the integral
in Lint (11.291) and the delta function in the current (11.292). For J 0 of the
form (11.292) the last term in the Hamiltonian reduces to
N
1 X e2 qi qj
,
2 i,j=1 4π|r̂i − r̂j |
which is just the Coulomb electrostatic energy of a system of charged pointlike particles (in the Heaviside system of units). Note however, that the
√
By the rescaling 2~ωk aλ (k) → aλ (k) the first of the rules (11.326) can be brought
into the standard relativistic form
63
[aλ (k), a†λ′ (k′ )] = δλλ′ δΓ (k − k′ ) .
The expansions of the operators Ai and ΠT
i then take the simple forms
Z
X Ai (x) = ~c dΓk
ǫi (k, λ) aλ (k) eik·x + H.c. ,
λ=±1
i
ΠT
i (x) = −
c
Z
dΓk ~ωk ǫi (k, λ) aλ (k) eik·x − H.c. ,
and d3 k/(2π)3 in (11.327) gets replaced by dΓk ≡ d3 k/(2π)3 2~ωk .
391
summation includes infinite terms arising for i = j which have to be subtracted by hand.
There can be no question about the relativistic covariance of the theory just constructed, because matter particles are treated nonrelativistically.
Therefore we cosider here covariance of the quantum theory of the free (not
coupled to matter particles) electromagnetic field. In this case from the Lagrangian density (11.233) one obtains the canonical energy-momentum tensor
(we set ~ = c = 1)
µν
Tcan
= −f µλ ∂ νAλ +
1 µν λκ
g f fλκ ,
4
(11.329)
which is not symmetric. The symmetric Belinfante energy-momentum tensor
takes the form
µν
Tsymm
= −f µλ f νλ +
1 µν λκ
g f fλκ ,
4
(11.330)
and gives64
0
P =
Z
d
3
00
x Tsymm
1
=
2
Z
i
h
d3 x Ȧi Ȧi + (∇×A)2 ,
which is just (11.319) for zero external current J µ and
Z
i
P = d3 x Πk ∂ iAk .
(11.331)
(11.332)
µνλ
The tensor Mcan
(11.61) of the electromagnetic field takes the form
µνλ
µλ
µν
νλ κ
Mcan
= xν Tcan
− xλ Tcan
+ if µκ Jvec
Aρ .
(11.333)
ρ
The generators J ij and J 0i of the Lorentz transformations take then the form
Z
o
n
ij
J = d3 x Ȧk (xi ∂ j − xj ∂ i )Ak − Ȧi Aj − Ȧj Ai ,
Z
h
i
2
0i
i
3
i 1
k k
J = tP − d x x
Ȧ Ȧ + (∇×A) .
(11.334)
2
The operators H ≡ P 0 , J i ≡ 12 ǫkl J kl , P i and K i ≡ J 0i can be shown
to satisfy the Poincaré algebra commutation rules (6.21) by virtue of the
canonical commutation relations (11.317). However, while
µ
µ
eiaµ P Ak (x) e−iaµ P = Ak (x + a) ,
k
i
ij
i
ωij J ij k
− 2i ωij J ij
− 2 ωij Jvec
2
Al (Λ−1 (ω)·x) ,
A (x) e
= e
e
l
64
i
For the free electromagnetic field A0 = 0 and ΠT
i = Πi = Ȧ .
392
as usually (here Λ is a pure rotation), performing the transformation of Ak (x)
corresponding to a Lorentz boost one finds that
k
0i
0i
0i
eiω0i J Ak (x) e−iω0i J = e−iω0i Jvec
Al (Λ−1 (ω)·x) + ∆Ak (x) ≡ A′k (x) ,
l
where the extra term ∆Ak (x) ensures that the ∂k A′k (x) = 0 as it must be
for consistency: the divergence of the left hand side of the above formula
obviously vanishes. This means that a transformations of the photon field
operator Ai (x) generated by K i consist of the corresponding Lorentz boost
supplemented with a suitable gauge transformation. Thus, transformations
generated by the conserved Noether charges (11.329) and (11.333) in the
Coulomb gauge automatically preserve this gauge.
Finally we consider the thermodynamical properties of the free electromagnetic field in equilibrium with the walls of a box of volume V = L3 . If
the field is quantized in the box, the Hamiltonian takes the form
XX
HEM =
~ωk a†kλ akλ ,
(11.335)
k
λ=±1
where now the wave vectors k are discrete, k = (2π/L)n, ωk = c|k| and
[ak′ λ′ , a†kλ ] = δk′ k δλ′ λ .
(11.336)
The Hilbert space is spanned by the HEM eigenstates |nk1 λ+ , nk1 λ− , nk2 λ+ . . .i,
where each occupation number nki λi can run from 0 to infinity. This allows
to easily compute the Gibbs Canonical Ensemble partition function


∞
∞

Y X
X
−~ ωk /kB T nkλ+
−~ ωk /kB T nkλ−
Zstat =
e
e


nkλ+ =0
nkλ− =0
k
2
Y
1
.
(11.337)
=
−~ ωk /kB T
1
−
e
k
The Helmholtz free energy is then65
X
2 ln 1 − e−~ ωk /kB T ,
F (T, V ) = kB T
k
or, in the thermodynamical limit,
Z
2V
3
−~ ωk /kB T
d
k
ln
1
−
e
F (T, V ) = kB T
(2π)3
Z ∞
V
= kB T 2 3
dω ω 2 ln 1 − e−~ ω/kB T .
π c 0
65
(11.338)
Had we not subtracted the
P (infinite) energy of the zero point oscillations, we would
get in F (T, V ) an extra term k ~ωk .
393
All well known properties of the electromagnetic radiation in equilibrium,
including the celebrated equation of state p = 31 u(T ), where u(T ) = U(T )/V
is the radiation energy density, can be derived from the free energy F (T, V ).
11.8
Canonical construction of half-integer spin quantum fields
As demonstrated in sections 11.2, 11.3, 11.5 and 11.7, the formal procedure
of canonical quantization (supplemented in the case of systems subject to
constraints by the Dirac method described in section 11.6) applied to classical (c-number) fields which under spatial rotations of the reference frame
transform as integer spin representations of the rotation group allows to
build theories of quantum fields whose elementary excitations (“quanta”)
are bosons. In the interaction picture (see section 11.9), i.e. when the basic
field operators of such quantum theories are represented in Fock spaces of
eigenstates of the free Hamiltonians of such theories, one recovers the same
Feynman rules for computing S matrix elemens, which in sections 8 and 9
were obtained in the approach based on quantum mechanics of relativistic
particles, without any reference to classical fields.
Apart from making (owing to the Noether theorem presented in section 11.1) extraction of consequences of various possible symmetries more
straightforward, the most important virtue of the approach based on field
quantization is that being essentially nonperturbative, it allows for a deeper
insight into the structure of the quantum theory (see Section 13).
We here want to lay similar foundations for relativistic interactions of halfinteger spin particles. Superficially, it could seem it should suffice to apply
the same formalism as previously to classical c-number fields but transforming under changes of the Lorentz frame as half-integer spin representations
of the Spin(1, 3) ≃ SL(2, C) group (the universal covering of the Lorentz
SO(1, 3) group). This is indeed so for the best known kind of half-integer
spin particles - massive charged fermions accompanied by their antifermions:
as in many textbooks one can in this case start with a relativistically invariant Lagrangian density of a c-number complex Dirac field ψ transforming as
the ( 12 , 21 ) representation of the SL(2, C) group arguing that to obtain a sensible quantum theory it is necessary to replace the Poisson brackets with the
anticommutators instead of commutators (one usually passes freely over the
presence of constraints in this case). Yet, not for all types of half-integer spin
particles relativistic interactions can be obtained in this way. The simplest
example is the massive neutral spin 1/2 particle (called Majorana fermion)
whose field operator was constructed at the end of section 8.3. This is be394
cause already the corresponding free Lagrangian density cannot simply be
written down: in the expression
1
L = λ̄iσ̄ µ ∂µ λ − m(λλ + λ̄λ̄)
2
(11.339)
(see section 8.6 for the Lorentz transformation properties of λα and λ̄α̇ ) the
term proportional to m is simply zero, if λα and λ̄α̇ are treated as c-number
spinors. This shows that the consistent field-based approach to constructing
quantum theories of fermions must start with Lagrangian densities which are
(bosonic) functions of fields taking values in the Grassmann algebra generated
by the infinite set of anticommuting generators.
To get acquinted with these notions it is convenient to consider first a classical bosonic system having a finite number of degrees of freedom. Dynamics
of such a system described by n generalized (real) variables q i can be formulated as a mapping from R (time) into the n-fold direct product F × . . . × F
of an abstract algebra F of functions f (a1 , . . . , a2n ) of 2n variables a1 , . . . , a2n
which are identified with initial values q01 , . . . , q0n , q̇01 , . . . , q̇0n of the variables q i
and the corresponding generalized velocities q̇ i (t). Indeed, for a fixed instant t
the values q i (t) can be treated as a set of n functions q i (t) = f i (t, a1 , . . . , a2n )
of the variables a1 , . . . , a2n . With an appropriate topology (to define convergence of sequences) such functions f i (t, a1 , . . . , a2n ) can be viewed as elements
of an abstract commutative and associative algebra over R generated by the
set of 2n commuting generators a1 , . . . , a2n (ak al = al ak ):
f (a1 , . . . , a2n ) = f0 +
X
fk ak +
k
1X
fk k ak1 ak2
2 k ,k 1 2
1
2
1 X
+
fk1 k2 k3 ak1 ak2 ak3 + . . .
3! k ,k .k
1
2
(11.340)
3
Differentiation can in the algebra F be defined as a linear algebraic operation
by the basic rules
∂ i
a = δji ,
j
∂a
and
∂ 1
∂ k 1
k
l
a
.
.
.
a
.
.
.
a
=
a a . . . al = a1 . . . al .
k
k
∂a
∂a
and their extension to all elements of F using linearity. Thus each q i (t) is
for fixed t an element of F ; a classical trajectory (q 1 (t), . . . , q n (t)) can be,
therefore, viewed as a mapping t → F × . . . × F .
A Lagrangian which determines the dynamics of such a system is in this
picture a mapping of the 2n-fold direct product of the algebra F into F itself:
F × . . . × F → L(f 1 , . . . , f 2n ) ∈ F and the action functional I[q(t)] is given
395
by
Z
t2
dt L(a1 , . . . , a2n )|a1 =q1 (t),...,a2n =q̇2n (t)
t
Z t2
Z 1t2
1
2n dt L(q 1 (t), . . . , q̇ 2n (t)) . (11.341)
dt L(a , . . . , a ) ≡
≡
I[q(t)] =
t1
t1
Its variation with respect to the trajectory is to be understood as
Z t2 X
n ∂L d i
∂L i
δq (t) + n+i δq (t) ,
(11.342)
δI[q(t)] =
dt
i
∂a
∂a
dt
t1
i=1
where each δq i (t) is again a mapping from R into F reducing to the zero
element of F for t = t1 and t = t2 .
The fermionic counterpart of the algebra F of functions described above
is the Grassmann algebra G over C generated by 2m elements (generators)
ξ 1 , . . . , ξ 2m which anticommute:
ξ α ξ β = −ξ β ξ α .
Each element of G is of the form
X
X
gα1 α2 ξ α1 ξ α2 + . . .
g = g0 +
gα ξ α +
α
+
(11.343)
α1 <α2
X
gα1 ...α2m ξ α1 . . . ξ α2m .
(11.344)
α1 <...<α2m
In contrast to (11.340), the number of terms in the above sum is finite owing
to the property (11.343) which implies that ξ αξ α = 0 (the zero element of the
algebra G). In the natural way the Grassmann algebra splits into the direct
sum of Geven and Godd (the only common element of these two subspaces being
the zero element of G) whose elements have the general forms
X
X
gα1 ...α2m ξ α1 . . . ξ α2m ,
gα1 α2 ξ α1 ξ α2 + . . . +
geven = g0 +
α1 <...<α2m
α1 <α2
godd =
X
α
α
gα ξ + . . . +
X
gα1 ...α2m−1 ξ α1 . . . ξ α2m−1 ,
α1 <...<α2m
respectively. Elements of Geven and Godd can be distinguished by assigning
them the G-algebra parities Pg which assume values 0 and 1 respectively (a
general element of G has no definite parity).
In the algebra G one defines two linear operations called left- and rightderivatives with respect to the generator ξ α by the rules
∂
∂
β
ξ
=
ξ β = δαβ ,
α
α
∂L ξ
∂R ξ
396
∂
∂
ξ γ . . . ξ α . . . ξ σ = (−1)L
ξ αξ γ . . . ξ σ = (−1)L ξ γ . . . ξ σ ,
α
∂L ξ
∂L ξ α
∂
∂
ξ γ . . . ξ α . . . ξ σ = (−1)R
ξ γ . . . ξ σ ξ α = (−1)R ξ γ . . . ξ σ ,
α
α
∂R ξ
∂R ξ
where L and R are the numbers of interchanges of the generators needed to
place ξ α to the left or right of the string of the generators.
Analogously to the bosonic case, “classical” dynamics of a fermionic system described by m variables ψ α is a mapping from R into an m-fold direct
product Godd × . . . × Godd with the generators ξ 1 , . . . , ξ m and ξ m+1 , . . . , ξ 2m
having the interpretation of the initial “values” of ψ α (t) and ψ̇ α (t). The Lagrangian is in this case a mapping of the 2m-fold product Godd × . . . × Godd
into Geven :
L(ψ 1 (t), . . . , ψ̇ 1 (t), . . .) ≡ L(ξ 1 , . . . , ξ 2m )|ξ1=ψ1 (t),..., ξm+1 =ψ̇1 (t),... . (11.345)
Dynamics (the equations of motion) follows from the condition δI[ψ(t)] = 0
where the variation is written in terms of the right derivatives66
Z t2 X
m ∂L α
∂L d
α
dt
δI[ψ(t)] =
δψ (t) +
δψ (t) , (11.346)
α
m+α dt
∂
ξ
∂
ξ
R
R
t1
α=1
where each δψ α (t) is an arbitrary mapping from R into Godd reducing to the
zero element of Godd for t = t1 and t = t2 . The resulting “classical” dynamics is then rather abstract and, as has been discussed in the introduction
to this chapter, no picture of fluctuating fields can be associated with the
corresponding quantum theory.
Finally, the mathematical structure allowing for a uniform treatment of
mixed bosonic and fermionic systems is the Bieriezin algebra B obtained from
the Grassmann algebra by treating the coefficient functions gα1 ... in (11.344)
as elements of the algebra F of functions.67 Thus, the Bieriezin algebra is
generated by 2n + 2m elements (generators) z a , z i = ai , i = 1, . . . , 2n and
z 2n+α = ξ α, α = 1, . . . , 2m, which have the property
z a z b = (−1)Pa Pb z b z a ,
(11.347)
with the natural assignments of the B-algebra parities Pa (which as in the
Grassmann algebra can assume values 0 and 1). Obviously, the parity of a
product b1 b2 of two elements of B having well-defined parities Pb1 and Pb2
66
Alternatively it can be written in terms of the left derivatives with the variations
placed to the left - see the firmula (11.349).
67
Of course real and imaginary parts of each gα1 ... are treated as two independent
elements of F .
397
is Pb1 + Pb2 (mod 2) and b1 b2 = (−1)Pb1 Pb2 b2 b1 . Similarly as the Grassmann
algebra, the Bieriezin algebra naturally splits into the direct sum of the even
and odd subspaces Beven and Bodd whose elements have well defined parities (0
and 1, respectively). Left- and right derivatives with respect to the generators
z a are linear operations defined by the rules similar to the ones in the Galgebra
k
X
∂
b1
bk
z
.
.
.
z
=
(−1)Li δabi z b1 . . . (no z bi ) . . . z bk ,
∂L z a
i=1
k
X
∂
b1
bk
z
.
.
.
z
=
(−1)Ri δabi z b1 . . . (no z bi ) . . . z bk ,
∂R z a
i=1
(11.348)
where Li (Ri ) are the numbers of odd generator interchanges needed to place
z bi on the extreme left (right) of the string of the generators. Left- and rightderivatives with respect to bosonic generators ai are, of course, identical. A
useful property of derivatives with respect to fermionic generators ξ α are
∂
∂
(bodd ) =
(bodd ) ,
α
∂L ξ
∂R ξ α
∂
∂
(beven ) = −
(beven ) .
α
∂L ξ
∂R ξ α
(11.349)
Dynamics of a mixed bosonic-fermionic system is determined by the action principle
Z t2 ∂L
∂L d a
a
dt
δI[q, q̇] =
δq +
δq = 0 ,
∂R q a
∂R q̇ a dt
t1
in which L(z 1 , . . . , z 2n+2m ) is a function taking values in Beven and its derivatives with espect to q a and q̇ a have to be understood as in (11.342) and
(11.346). The equations of motion take the standard form
∂L
d ∂L
=
a
dt ∂R q̇
∂R q a
a = 1, . . . , n + m .
(11.350)
Hamiltonization of the dynamics of a bosonic-fermionic system is analogous to the one of the ordinary bosonic systems except that one has to fix a
convention (which is arbitrary) for the definition of the canonical momenta.
We chose to define the momenta pa (t) as right derivatives
pa (t) =
∂L
.
∂R q̇ a (t)
398
(11.351)
Consequently, the Hamiltonian must take the form
X
H=
pa q̇ a − L ,
(11.352)
a
in order for its variation to be independent of the variations δ q̇ a of the velocities. Indeed, with this definition, using
∂L
∂
∂L
∂
a
δq +
δ q̇ a,
δpb =
∂R q a ∂R q̇ b
∂R q̇ a ∂R q̇ b
one can write the variation of H
∂H
a
Pa Pb ∂
δH =
δq
+
(−1)
∂R q a
∂R q̇ a
∂L
∂R q̇ b
δ q̇ a q̇ b,
in the form
∂H
∂
δH =
−
a
∂R q
∂R q a
∂L
∂R q̇ b
q b δq a + q̇ b δpb .
The Poisson bracket of the two functions F and G of definite B-algebra
parities must be defined in agreement with the adopted convention. The
right definition is
X ∂F ∂G
PF PG ∂F ∂G
− (−1)
,
(11.353)
{F, G}PB =
a∂ p
a∂ p
∂
q
∂
q
R
L
a
R
L
a
a
with the left derivatives with respect to the momenta. It has the easy to
verify properties
{F, G}PB = −(−1)PF PG {G, F }PB ,
{F, GH}PB = {F, G}PB H + (−1)PF PG G {F, H}PB .
With the definition (11.353) of the Poisson bracket in the B-algebra the
canonical equations of motion q̇ a = {q a , H}PB , ṗa = {pa , H}PB , are, for systems not subject to constraints, equavalent to the Euler-lagrange equations
derived from the action principle δI[q(t)] = 0.
Canonical quantization of such systems consists in promoting their canonical variables q i , pi and ψ α , πα to Schrödinger picture operators q̂ i , p̂i , ψ̂ α
and π̂α satisfying the (anti)commutation relations:
[q̂ i , p̂j ] = i~{q i , pj }PB ,
{ψ̂ α , π̂β }+ = i~{ψ α , πβ }PB ,
(11.354)
in agreements with the symmetry/antisymmetry properties of the corresponding Poisson brackets. The resulting equations satisfied by the Heisenberg picture operators q̂ i (t) and ψ̂ α (t) are then formally identical with the
classical Euler-Lagrange equations (11.350).
399
The Dirac procedure allowing to handle systems subject to (second class)
constraints extends to the case of mixed bosonic-fermionic systems essentially
without modifications. The only difference is that the C matrix is in this case
a supermatrix and has more complicated symmetry/antisymmetry properties
(the Poisson bracket in the B-algebra is not simply antisymmetric).
Transition from classical mechanics to classical field theory is achieved, as
usually, by passing with the number of variables q a (and therefore also with
the number of generators of the Bieriezin algebra) to continuum, that is by
assscibing a certain number of independent variables to each space point x.
As an example of the application of this formalism we consider here quantization of the relativistic Grassmann field ψ transforming as the spinor representation of the Lorentz group, or more precisely, as the representation of
the group Spin(1, 3) - the universal covering of the Lorentz SO(1, 3) group.
The extension to Spin(1, 3) is possible because (what is clearly reflected in
the formalism of the Grassmann and Bieriezin algebras) such fields are not
classically measurable; all observables must be bosonic (i.e. belong to Geven
or Beven ) that is bilinear or quadrilinear etc. in the fields ψ and transform
as representations (scalar, vector, tensors) of the true Lorentz group.68 We
assume therefore that under the change of the reference frame x → x′ = Λ·x
the Grassmann algebra valued field ψ transforms as
µν
i
ψ ′ (x′ ) = e− 2 ωµν Jspin ψ(x) ,
(11.355)
µν
where the matrices Jspin
of the spinor representation satisfy the commutation
rule
h
i
µν
λρ
µρ νλ
µλ νρ
νρ µλ
νλ µρ
Jspin, Jspin = i g Jspin − g Jspin − g Jspin + g Jspin . (11.356)
µν
As discussed in Section 8.3 the matrices Jspin
can be constructed either by
means of the Clifford algebras or by exploiting the isomorphism of Spin(1, 3)
and SL(2, C). Here, as in (8.53) we take
µν
Jspin
=
1 µν
i µ ν
,
[γ , γ ] ≡ σ4×4
4
(11.357)
with the 4 × 4 matrices γ µ satisfying the basic Clifford algebra relation (8.52)
in the representations (8.64) or (8.66).
68
At the “classical” level this is the only argument why to consider fields transforming
under a larger symmetry group than SO(1, 3) despite the fact that only the Lorentz
group of symmetries can be inferred from physical experience. In the quantum case, as
was discussed in Section 4, going over to the universal covering of the SO(1, 3) group
is a natural consequence of the possible occurrence of the projective representations of
SO(1, 3).
400
The most frequently encountered Lagrangian density (leading to the theory of noninteracting fermions) has form69
L = iψ † γ 0 γ µ ∂µ ψ − m ψ † γ 0 ψ ≡ ψ̄(iγ µ ∂µ − m)ψ
(11.358)
(Lorentz invariance of this Lagrangian density can be shown using the formulae given in Section 8). The Grassmann algebra-valued fields ψα and ψα† are
four-component spinors. They are all treated as an independent Grassmann
variables. The Euler-Lagrange equations derived from (11.358)
γ 0 (iγ µ ∂µ − m) ψ = 0 ,
←
ψ̄ −iγ µ ∂ µ − m = 0 ,
(11.359)
are just the Dirac equations.
To set up the canonical formalism for this system we first find the canonical momenta (recall that the derivatives are the right ones)
Πψ =
∂L
= iψ † ,
∂R ψ̇
Πψ† =
∂L
= 0.
∂R ψ̇ †
(11.360)
It is clear that it is impossible to express ψ̇ and ψ̇ † through Πψ and Πψ† . The
system is therefore subject to constraints (we suppress spinor indices)
Φ1 (x) ≡ Πψ (x) − iψ † (x) = 0 ,
Φ2 (x) ≡ Πψ† (x) = 0 ,
(11.361)
which are of second class. Although in this case the Dirac procedure described in section 11.6 is not really indispensable, because the canonical
equations when restricted to ψ and Πψ are fully equivalent to the original Euler-Lagrange equation (11.359), it is instructive to go through this
procedure to see how it leads to the well established results.
Constructing the extended Hamiltonian density according to the rules of
section 11.6 one gets
HE = Πψ ψ̇ + Πψ† ψ̇ † − ψ † iψ̇ − ψ̄iγ i ∂i ψ + m ψ̄ψ
= ψ † γ 0 −iγ i ∂i + m ψ + Φ1 u1 + Φ2 u2 .
(11.362)
The functions u1 and u2 are just the velocities ψ̇ and ψ̇ † . Equating to zero
the Poisson brackets
←
{Φ1 , HE }PB = −ψ̄(iγ k ∂ k + m) − iu2 ,
{Φ2 , HE }PB = γ 0 (−iγ k ∂k + m)ψ − iu1 ,
69
Another, less frequently encountered, is the symmetric form:
i † 0 µ
i
ψ γ γ ∂µ ψ − ∂µ ψ † γ 0 γ µ ψ − m ψ † γ 0 ψ .
2
2
Using the Dirac formalism it is easy to check that it leads to the same quantum theory as
(11.358).
L=
401
of the constraints with the Hamiltonian determines these functions. No new
(secondary) constraints are needed. Combining u1 and u2 obtained in this
way with the canonical equations ψ̇ = {ψ, HE }PB = u1 , ψ̇ † = {ψ † , HE }PB =
u2 one obtains equations which are fully equivalent to the Euler-Lagrange
ones (11.359).
In order to quantize the theory one computes the Dirac brackets. The
nonzero elements of the C matrix are C1x,2y = C2x,1y ≡ {Φ1 (x), Φ2 (y)}PB =
−iδ (3) (x − y) (it is in this case symmetric). Its inverse reads
(C −1 )1x,2y = (C −1 )2x,1y = iδ (3) (x − y) .
(11.363)
The Dirac brackets computed as in (11.280) read
{ψ(x), Πψ (y)}D = δ (3) (x − y) ,
{ψ(x), ψ † (y)}D = −iδ (3) (x − y) ,
{ψ(x), Πψ† (y)}D = 0 ,
{ψ † (x), Πψ (y)}D = 0 ,
{Πψ† (x), Πψ (y)}D = 0 ,
{ψ † (x), Πψ† (y)}D = 0 .
The canonical quantization rule is now {A, B}+ = i~ {A, B}D which leads
to the canonical anticommutator
{ψα (x), ψβ† (y)}+ = ~ δαβ δ (3) (x − y) ,
(11.364)
{ψα (x), ψβ (y)}+ = {ψα† (x), ψβ† (y)}+ = 0 .
Since in the quantum theory the constraints are realized as operator identities, Πψ† becomes the zero operator and Πψ can everywhere be eliminated in
favour of iψ † . The Hamiltonian density operator reads
H = ψ † γ 0 (−iγ k ∂k + m)ψ .
(11.365)
The next step is to expand the basic operators ψα (x) and ψα† (x) into
creation and annihilation operators, satisfyng simple anticommutation rules.
Since ψα has four independent components, there must be four annihilation
and four creation operators. As the Grassmann variables ψα and ψα† are
not real c-numbers, there corresponding operators ψα and ψα† need not be
Hermitian. We write therefore their expansions in the forms
Z
X
ψα (x) = dΓp eip·x
[uα (p, σ) au (p, σ) + vα (−p, σ) av (−p, σ)] , (11.366)
σ
ψα† (x) =
Z
X
dΓp e−ip·x
u∗α (p, σ) a†u (p, σ) + vα∗ (−p, σ) a†v (−p, σ) ,
σ
402
using the functions uα (p, σ) p
and vα (p, σ) constructed in section 8.3 and
with dΓp = (2π)3 2Ep , Ep = p2 + m2 . Computing the anticommutators
(11.364) and anticipating that {au , av }+ = {a†u , a†v }+ = 0 we get
Z
Z
XX
′
†
{ψα (x), ψβ (y)}+ = dΓp dΓp′ ei(p·x−p ·y)
σ
σ′
∗
′
′
uα (p, σ)uβ (p , σ ) {au (p, σ), a†u (p′ , σ ′ )}+
+vα (−p, σ)vβ∗ (−p′ , σ ′ ) {av (−p, σ), a†v (−p′ , σ ′ )}+ .
Recalling the summation rules (8.105):
X
X
uα (p, σ)u∗β (p, σ) =
uα (p, σ)ūβ (p, σ)γ 0 = (Ep γ 0 − p · γ + m)γ 0 αβ ,
s
σ
X
σ
vα (−p, σ)vβ∗ (−p, σ) = (Ep γ 0 + p · γ − m)γ 0 αβ ,
it is easy to check that the commutation rules (11.364) are satisfied if
{au (p, σ), a†u (p′ , σ ′ )}+ = δΓ (p − p′ ) δσσ′ ,
{av (p, σ), a†v (p′ , σ ′ )}+ = δΓ (p − p′ ) δσσ′ ,
(11.367)
and the other anticommutators are zero as anticipated.
To express the Hamiltonian through the operators au etc. we use the
equality70
γ 0 (−iγ i ∂i + m)ψ(x)
Z
X
= dΓp Ep eip·x
[au (p, σ) u(p, σ) − av (−p, σ) v(−p, σ)] ,
σ
and the normalization conditions
u† (p, σ) · u(p, σ ′ ) = 2Ep δσσ′ ,
v † (p, σ) · v(p, σ ′ ) = 2Ep δσσ′ ,
u† (p, σ) · v(−p, σ ′ ) = v † (−p, σ) · u(p, σ ′) = 0 ,
70
(11.368)
It follows from the equalities
γ 0 (−iγ k ∂k + m) u(p, σ) eip·x = γ 0 (γ ·p + m) u(p, σ) eip·x = Ep u(p, σ) eip·x
γ 0 (−iγ k ∂k + m) v(−p, σ) eip·x = γ 0 (γ ·p + m) v(−p, σ) eip·x = −Ep v(−p, σ) eip·x ,
which in turm follow from the formulae (8.101) written in the form (γ 0 Ep −γ·p−m)u(p) =
0 and (γ 0 Ep + γ ·p + m)v(−p) = 0.
403
which can be derived from the rules of constructing the functions u(p, σ) and
v(p, σ) given in section 8.3. In this way one gets
Z
H = d3 x ψ † (x)γ 0 (−iγ i ∂i + m)ψ(x)
Z
X
= dΓp Ep
a†u (p, σ) au (p, σ) − a†v (p, σ) av (p, σ) . (11.369)
σ
If the basic rules (11.364) and consequently (11.367) involved commutators
(instead of anticommutators) this form of the Hamiltonian would be a disaster: from the commutation rules it would unambiguously follow (at least if
the system is quantized in a finite box) that there is a state |Ω0 i annihilated
by all au and av and that all other states are created by acting on |Ω0 i with
a†u and a†v . But the states created by a†v would then have negative energy! In
the Fock space there would exist states with arbitrarily large negative energy
and, therefore, the spectrum of the Hamiltonian would not be bounded from
below. Fortunately, with the anticommutators such a conclusion does not
follow: since the anticommutation relations are symmetric with respect to
interchanges a ↔ a† we can call av the creation operator. More formally, we
can substitute
au (p, σ) = b(p, σ) ,
av (p, σ) = d† (p, σ) ,
a†u (p, σ) = b† (p, σ) ,
a†v (p, σ) = d(p, σ) ,
(11.370)
which allows to rewrite the Hamiltonian (discarding the infinite constant) as
Z
X
H = dΓp Ep
b† (p, σ) b(p, σ) + d† (p, σ) d(p, σ) .
(11.371)
σ
The Hamiltonian (11.371) is positive semidefinite and the operators b† (p, σ),
b(p, σ) and d† (p, σ), d(p, σ) satisfying the rules
b(p, σ), b† (p′ , σ ′ ) + = δΓ (p − p′ ) δσσ′ ,
d(p, σ), d† (p′ , σ ′ ) + = δΓ (p − p′ ) δσσ′ ,
(11.372)
(with all other anticommutators zero) are the creation and annihilation operators of particles and antiparticles, respectively. This interpretation follows
from the fact that states created by b† (p, σ) and d† (p, σ) have the same
energies (and, hence same masses). Moreover, the Lagrangian (11.358) is
invariant under the global transformations
ψ → ψ ′ = e−iQθ ψ
ψ † → ψ †′ = eiQθ ψ † ,
(11.373)
forming a U(1) symmetry group with Q being the charge and θ the transformation parameter. The corresponding conserved Noether current is
j µ = ψ†γ 0 γ µψ ,
404
(11.374)
and the states created by b† and d† are eigenstates of
Z
Z
X
3
0
Q̂ = d x j (x) = Q dΓp
b† (p, σ) b(p, σ) − d† (p, σ) d(p, σ) , (11.375)
σ
with opposite eigenvalues
Q̂ b† (p, σ)|Ω0 i = Q b† (p, σ)|Ω0 i ,
Q̂ d† (p, σ)|Ω0 i = −Q d† (p, σ)|Ω0 i .
(11.376)
Quantization of the theory defined by the Lagrangian (11.339) can be
performed along the same lines. Quantization of theories of fermionic and
mixed fermionic-bosonic systems of fields whose complete Lagrangians consist of (11.358) (or (11.339)) plus terms involving higher powers of fields than
the second (examples of realistic such theories will be discussed in section
11.12), also proceed along the same lines.
As in the case of the scalar field ϕ discussed in section 11.2, one can now
introduce time dependent field operators ψ(t, x) ≡ ψ(x), ψ † (t, x) ≡ ψ † ((x)
by the formulae
ψ(t, x) = eiHt ψ(x) e−iHt ,
ψ † (t, x) = eiHt ψ † (x) e−iHt ,
(11.377)
which satisfy the canonical equations of motion
d
ψ(t, x) = i[H, ψ(t, x)] ,
dt
d †
ψ (t, x) = i[H, ψ † (t, x)] ,
dt
(11.378)
which take the form formally identical with the classical equations of motion derived from the underlying Lagrangian (that is with the equations
(11.359) in the case of the Lagrangian (11.358)). The time-dependent operators ψ(t, x) and ψ † (t, x), which in the theory of free fields are the Heisenberg
operators and in the theory of interacting fields in which the Hamiltonian
(11.365) or (11.371) play the role of the free part H0 of the full Hamiltonian,
are the interaction picture operators, take the forms
Z
X
ψα (x) = dΓp
e−ipx uα (p, σ) b(p, σ) + eipx vα (p, σ) d† (p, σ) ,
σ
ψα† (x) =
Z
dΓp
X
σ
e−ipx vα∗ (p, σ) d(p, σ) + eipx u∗α (p, σ) b† (p, σ) .
One can also find the canonical energy-momentum tensor
µν
Tcan
=
∂L
∂ ν ψ − g µν L .
∂R (∂µ ψ)
405
As long as the Lagrangian is linear in ψ and ψ † (as in (11.358)), the time
dependent (Heisenberg picture) operators satisfy the equations of motion
µν
which make L vanishing (as the operator) and Tcan
simplifies to the fierst
term only. Thus, in the theory (11.358)
µν
Tcan
= iψ † γ 0 γ µ ∂ ν ψ .
(11.379)
R
00
When expressed through the field operators d3 xTcan
is just the Hamiltonian
(11.371) and
Z
Z
X
i
3
0i
P = d x Tcan = dΓp pi
b† (p, σ)b(p, σ) + d† (p, σ)d(p, σ) , (11.380)
σ
is the momentum operator commuting with H. Finally, the tensor
µκ
µν
M µνκ = xν Tcan
− xκ Tcan
+
∂L
νκ
−iJspin
ψ,
∂R (∂µ ψ)
gives the operators generating rotations and boosts
Z
i νκ
ν κ
κ ν
νκ
3
†
J = i d x ψ x ∂ − x ∂ − σ4×4 ψ .
2
(11.381)
(11.382)
To see that the particles which are “quanta” of the field ψ have spin s = 1/2
it is enough to act with J 12 = J z on the one particle states representing
particles at rest
1
J z b† (0, ± 12 )|Ω0 i = ± b† (0, ± 21 )|Ω0 i ,
2
1
J z d† (0, ± 12 )|Ω0 i = ± d† (0, ± 21 )|Ω0 i .
2
(11.383)
More generally, one can check the commutation rules of the operators H, P, J
and K and verify that the one particle states b† (p, ± 12 )|Ω0 i and d† (p, ± 21 )|Ω0 i
transform under the action of U(Λ) ≈ 1 − 2i ωµν J µν + . . . operators in the way
appropriate for spin 1/2 particles.
The statistical properties of the quantized fermionic field can also be
readily obtained. If the it is quantized in the box of volume V = L3 the
Hamiltonian (before subtracting the infinite constant) is
XX
H=
Ep b†pσ bpσ + d†pσ dpσ − 1 ,
(11.384)
p
σ
and the Hilbert space is spanned by the states |nbpσ , . . . , ndpσ , . . .i with nbpσ , ndpσ =
0 or 1. Since there is a conserved charge Q̂, one computes the Grand Statistical Ensemble sum
Ξ(T, V, µ) = Tr e−β(H−µQ̂)
406
=
Y
=
Y
p
p


1
X
nbp =0
e2βEp
1
b
2
e 2 βEp e−β(Ep −µ)np  ×
Y
p


1
X
ndp =0
1
d
2
e 2 βEp e−β(Ep +µ)np 
2
2
1 + e−β(Ep −µ)
1 + e−β(Ep +µ) ,
(11.385)
(the squares come from the two spin states). Thus, for Ω(T, V, µ) = −kB T ln Ξ,
after going over to the continuous normalization, one gets
Z
d3 p −β(Ep −µ)
Ω(T, V, µ) = −2V
E
+
k
T
ln
1
+
e
p
B
(2π~)3
+kB T ln 1 + e−β(Ep +µ) , (11.386)
p
where Ep = c2 p2 + m2 c4 and the factor of 2 accounts for two spin states
per each p.
11.9
Transition to the interaction picture
As in the approach to quantum field theory based on quantum mechanics of
relativistic interacting particles, also in the approach relying on quantizing
fields one can assume that the full Hamiltonians of interacting fields possesses
in and out particle-like eigenstates |α± i. One is then primarily interested in
obtaining the S matrix elements
Sβα =
outhβ|αiin ,
allowing to compute various transition rates using the recipies formulated in
section 10. We have therefore to discuss how to recover in the approach based
on quantization of interacting classical fields the perturbative expansion and
Feynman rules which in section 9 was formulated within the approach based
on constructing interactions of relativistic particles. As we will see, the procedure by which this is achieved, called transition to the interaction picture,
when applied to classical relativistic fields automatically produces all the additional interactions which in the previous approach had to be included by
hands in order to obtain a Lorentz covariant S matrix.
We begin with the simplest example of the theory described by the Lagrangian density
1
1
L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − Hint (ϕ) .
2
2
(11.387)
where Hint (ϕ) is some interaction density. After quantization described in
section 11.2 the Hamiltonian of the system, expressed through the Schrödinger
407
picture (time independent) operators ϕ(x), Π(x) satisfying the rules (11.77)
is the sum of the free Hamiltonian
Z
1
d3 x Π2 (x) + (∇ϕ(x))2 + M 2 ϕ2 (x) ,
(11.388)
H0 =
2
which is quadratic in field operators and the interaction71
Z
Vint = d3 x Hint (ϕ(x), Π(x)) .
The formula for S-matrix elements
Z
Sβα = hβ0 |T exp −i
+∞
−∞
I
dt Vint
(t)
|α0 i ,
(11.389)
(11.390)
established in section 7.2 using rather general setting of quantum mechanics
should apply also to the field theory case. In (11.390), as in (7.41),
Z
I
iH0 t
d3 x Hint (ϕ(x), Π(x)) e−iH0 t
Vint (t) = e
Z
Z
3
I
= d x Hint (ϕI (t, x), ΠI (t, x)) ≡
d3 x Hint
.
(11.391)
and for the states |α0 i one should take the eigenstates of H0 (11.388) constructed in section 11.2. The time-dependent operators ϕI (x) ≡ ϕI (t, x) and
ΠI (x) ≡ ΠI (t, x) given by
ϕI (t, x) = eiH0 t ϕ(x) e−iH0 t ,
ΠI (t, x) = eiH0 t Π(x) e−iH0 t ,
(11.392)
are called in this context the interaction picture operators and have the
structure of free-field operators (discussed in Section 8). In the theory defined
by the lagrangian (11.387) they take the form
Z
ϕI (x) = dΓk a(k) e−ikx + a† (k) eikx ,
Z
1
dΓk E(k) a(k) e−ikx − a† (k) eikx .
(11.393)
ΠI (x) =
i
with kx ≡ Ek t − k·x. They satisfy (by construction) the equal-time commutation rules
[ϕI (t, x), ΠI (t, y)] = iδ (3) (x − y) ,
71
(11.394)
To be more general we write the formulae as if Hint depended also on Π(x).
408
etc. and the equation
(∂µ ∂ µ + M 2 )ϕI (t, x) = 0 .
(11.395)
It should also be clear that the free Hamiltonian H0 (11.388) can be also
written in terms of the time dependent interaction picture operators
Z
1
iH0 t
−iH0 t
d3 x Π2I (t, x) + (∇ϕI (t, x))2 + M 2 ϕ2I (t, x) .
H0 = e H0 e
=
2
In the case of the theory (11.387) it is obvious that ϕ̇I (t, x) = ΠI (t, x).
In cases like the one of electromagnetic field interacting with other fields (to
be discussed below) it is useful to remember that ϕI (t, x) and ΠI (t, y) satisfy
the equations
ϕ̇I (t, x) = i[H0 , ϕI (t, x)] ,
Π̇I (t, x) = i[H0 , ΠI (t, x)] ,
(11.396)
of which the first one unambigously fixes the relation of ϕ̇I (t, x) with ΠI (t, x).
In this way the perturbation expansion for the S matrix (which leads to
the Feynman rules discussed in sections 9.3-9.6) derived from principles of
quantum mechanics in the framework of Sections 8 and 9 is recovered in the
approach based on quantization of relativistic fields. It should be obvious,
that this perturbative expansion is based on the same assumptions as those
formulated in section 7.1, namely that the particle-like in and out states of
the full Hamiltonian are in the strict one-to-one correspondence (7.10) with
the eigenstates |α0 i of the free Hamiltonian (11.388).
Transition to the interacting picture is more subtle in the case of vector
(massive and massless) fields. Let us first consider the Lagrangian density
1
1
L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − Hint (ϕ) − J µ ∂µ ϕ + . . . ,
(11.397)
2
2
in which J µ is some four-vector constructed out of fields other than ϕ itself (the ellipses stands for terms depending on these other fields). In the
Hamilton’s formalism
∂L
Π(t, x) =
= ∂0 ϕ − J0 ,
(11.398)
∂ ϕ̇
and
Z
Z
d x Π(x)ϕ̇(x) − d3 x L(ϕ(x), ϕ̇(x))
Z
1
1
3
= d x Π (Π + J0 ) − (Π + J0 )2 + (∇ϕ)2
2
2
1 2
0
+ M ϕ(t, x) + Hint + J·∇ϕ + J (Π + J0 ) .
2
H=
3
409
(11.399)
Upon canonical quantization ϕ(x) and Π(x) become operators satisfying the
standard canonical commutation rules. Splitting the full Hamiltonian into
the free part H0 and the interaction Vint we get H0 (Π(x), ϕ(x)) as in (11.388)
and
Z
Vint = d3 x [Π(x)J 0 (x) + J(x) ·∇ϕ(x)
+
1 0 2
J (x) + Hint (ϕ(x))].
2
(11.400)
As in the preceding case H0 gets diagonalized upon introducing the creation
and annihilation operators and the Fock basis of H0 eigenstates is constructed
to play the role of the |α0 i states in the formula (11.390). The interation
picture operators ϕI (t, x) and ΠI (t, x) are obtained as in (11.392). The
important point is that the first of the equations (11.396) tells that
ΠI (t, x) = ∂0 ϕI (t, x) ,
(11.401)
and not ∂0 ϕI (t, x) − JI0 (t, x) (as for the corresponding Heisenberg picture
operator). For this reason the interaction Vint (t) used in the perturbation
expansion for the S matrix takes the form
Z
2
1 0
µ
I
3
Vint (t) = d x JI (t, x) ∂µ ϕI (t, x) +
J (t, x) + Hint (ϕI (t, x)) , (11.402)
2 I
i.e. it acquires a noncovariant term (JI0 )2 , so that the expression under the
integral over d3 x in (11.402) is not a Lorentz invariant interaction density (as
was assumed in (7.70)). However, as we have seen in Section 9.5, precisely
such a noncovariant term had to be added (by hands in the approaches of
Sections 7, 8) to compensate for the noncovariant term in the propagator of
the vector field ∂µ ϕI (x) (arising due to the singular nature of products of field
operators taken at the same space-time point) and to restore the Poincaré
covariance of the S matrix. Similarly, in the case of the Proca theory the
same procedure gives
V̇Ii (t, x) = ΠIi (t, x) −
1
∂i ∂j ΠIj (t, x) ,
M2
as in the free theory, which again leads to the presence of the noncovariant
term (9.67) in Hint (VIi (t, x), ΠIi (t, x)).
In electrodynamics in which the electromagnetic field couples to the conserved current J µ one considers the free field part
Z
n
o
2
1
d3 x ΠT (x) + [∇×A(x)]2 + H0rest = H0EM + H0mat , (11.403)
H0 =
2
410
of the full Hamiltonian consisting of (11.319) plus H mat which is built out
ou the operators forming the current J µ , and expands the operators A(x),
ΠT (x) into the creation and annihilation operators a† (k, λ), a(k, λ) associated with eigenstates of H0EM :
Z
i
Xh
†
ik·x
∗
−ik·x
A(x) = dΓk
aλ (k) ǫ(k, λ) e
+ aλ (k) ǫ (k, λ) e
, (11.404)
λ=±1
1
Π (x) =
i
T
Z
i
Xh
†
ik·x
∗
−ik·x
aλ (k) ǫ(k, λ) e
− aλ (k) ǫ (k, λ) e
,
dΓk |k|
λ=±1
so that (up to a c-number constant)
Z
X
EM
H0 = dΓk
|k| a† (k, λ) a(k, λ) .
(11.405)
λ=±1
The interaction picture operators
AI (t, x) = eiH0 t A(x) e−iH0 t ,
ΠI (t, x) = eiH0 t ΠT (x) e−iH0 t ,
(11.406)
which enter the formula (11.391), are related by ΠI (t, x) = ȦI (t, x). This
again follows from computing the commutator
Z
ij
i
i
iH0 t
d3 y δtr
(y − x) ΠTj (y) e−iH0 t = ΠIi (t, x) ,
ȦI (t, x) = i[H0 , A (x)] = e
because ∂j ΠTj = 0. Notice that because A0 (x) is given by (11.305), the
interaction picture operator A0I (t, x) is identically equal to
Z
e JI0 (t, y)
0
AI (t, x) = d3 y
,
(11.407)
4π|x − y|
and can everywhere be replaced by this combination. This allows to introduce
a dummy operator A0I (t, x) = 0, with the help of which the time dependent
interaction (7.41)
Z
Z
Z
0
0
e2
3
3 J (x)J (y)
3
I
iH0 t
dx dy
e−iH0 t
−e d x J(x)·A(x) +
Vint (t) = e
2
4π|x − y|
Z
Z
2Z
0
e
J (t, x)JI0 (t, y)
= −e d3 x JI (t, x)·AI (t, x) +
d3 x d3 y I
, (11.408)
2
4π|x − y|
411
can be written in a formally covariant form72
Z
Z
Z
J 0 (t, x)JI0 (t, y)
e2 3
µ
I
3
I
d x d3 y I
, (11.409)
Vint (t) = e d x Jµ (t, x)·AI (t, x) +
2
4π|x − y|
(H0 = H0EM + H0mat in this formula is the free part of the Hamiltonian of
the entire system of fields, including the fields forming the currents JI0 ).
As explained in (9.6), the Coulomb part of the interaction (11.409) can be
dropped provided one drops at the same time noncovariant pieces in the
photon propagator.
11.10
Corrections to external lines
In the preceding section we have shown that quantizing classical fields one
arrives essentially at the same scheme for calculating S matrix elements (assuming that particle-like in and out eigenstates of the full Hamiltonian H
exist) as in the approach of sections 7 and 9 based on constructing interactions Vint of relativistic particles. Therefore, also in the approach discussed
here one has to solve the problem what to do with the corrections to external
lines like the ones discussed in subsection 9.7. There the problem was solved
using the freedom in constructing interaction terms. Here it seems superficially we are not given such a freedom: quantized is a well defined classical
theory with a fixed Lagrangian density and no new terms should be added
to it.
However, in quantizing fields there is another freedom which allows to
find the solution of the problem73 (from the practical point of view this
solution is equivalent to the one given in subsection 9.7). As usually, we will
illustrate this on the example of the theory of a real scalar field defined by
72
Of course, if one is interested in computing amplitudes with insertions of the A0H (x)
operator (see section 13)
R
0
0
−i dt Vint (t)
|α0 i
outhβ|T (AH (x) . . .)|αiin = hβ0 |T AI (x) . . . e
one has to use as A0I (x) in the pre-exponential factor the combination (11.407), and not
the dummy operator A0I (x) = 0.
73
Recall, that the problem of external lines, we are now concerned with, has nothing to
do with the question of finitness of integrals corresponding to loops in Feynman diagrams.
One can therefore think that an ultraviolet cutoff is imposed on all loop integrals. Nevertheless, as it will become clear in section 14, only a special class of classical theories gives
rise to quantum theories (called renormalizable) in which avoiding ill defined expressions
of the sort discussed here and all other divergences (arising from integrations over loop
four-momenta) indeed does not require adding new terms to the original Lagrangian.
412
the Lagrangian density
1
λ
1
L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − ϕ4 .
2
2
4!
(11.410)
To quantize this theory we have first to choose the canonical field variable.
Instead of ϕ (to which corresponds the canonical momentum Π equal, by
classical Hamilton’s equations of motion, ∂0 ϕ), we can chose another variable
ϕ̃ related to ϕ by ϕ = Z 1/2 ϕ̃. We have then the canonical momentum Π̃,
by classical equations of motion equal Z∂0 ϕ̃ = Z 1/2 Π, which implies that
the canonical equal time commutation rules satisfied by the corresponding
operators ϕ̃H and Π̃H are the same as the ones satisfied by ϕH and ΠH :
h
i
ϕ̃H (t, x), Π̃H (t, y) = iδ (3) (x − y) .
(11.411)
We will call the operator ϕH (singled out by the canonical form of the Lagrangian kinetic term in (11.410)) the canonical or, in the context of renormalization, the bare field operator and ϕ̃H = Z −1/2 ϕH with arbitrary Z 6= 1
the renormalized (although from the point of view of the Lagrangian dynamics ϕ̃ is also a “canonical” field variable; in this context renormalized simply
means rescaled) field operator74 and the factor Z will be called the field renormalization constant (the name “wave function renormalization constant” is
also in use).
Working with ϕ̃H , Π̃H and performing the standard steps we arrive at
the Hamiltonian density
1
1
1
λ
H = Z −1 Π̃2H + Z (∇ϕ̃H )2 + ZM 2 ϕ̃2H − Z 2 ϕ̃4H ,
2
2
2
4!
which we rewrite in the form of H0
Z
h
i
1
2
H0 =
d3 x Π̃2H (0, x) + (∇ϕ̃H (0, x))2 + Mph
ϕ̃2H (0, x) ,
2
(11.412)
(11.413)
and the interaction Hamiltonian density
Hint =
1
1 −1
Z − 1 Π̃2H + (Z − 1) (∇ϕ̃H )2
2
2
1
λ
2
+ ZM 2 − Mph
ϕ̃2H + Z 2 ϕ̃4H .
2
4!
(11.414)
In this approach we have to admit that due to the self-interaction the physical
mass Mph of the particles described by H is not equal to the bare mass
parameter M in the Lagrangian (11.410).
74
For this reason in section 14 the operators like ϕ̃H will be denoted ϕR .
413
Performing now the transition to the interaction picture as in section
11.9, i.e. solving the commutation relation (11.411) at t = 0 with the help
of the standard creation and annihilation operators diagonalizing H0 (as in
subsection 11.2), we introduce the interaction picture field operators75
Z
iH0 t
−iH0 t
ϕI (t, x) ≡ e
ϕ̃H (0, x) e
= dΓk a(k) e−ik·x + a† (k) eik·x ,
ΠI (t, x) ≡ eiH0 t Π̃H (0, x) e−iH0 t = ∂0 ϕI (t, x) .
(11.415)
Expressing the interaction Hamiltonian density similarly as in (11.391) in
terms of ϕI (x) we get
2
1
1
I
2
Hint
= − (Z − 1)∂µ ϕI ∂ µ ϕI +
ZM 2 − Mph
ϕI
2
2
1 −1
λ
Z + Z − 2 ∂0 ϕI ∂0 ϕI .
+ Z 2 ϕ4I +
4!
2
(11.416)
Thus, we get the additional interaction terms of the same structure76 as in
(9.100). They allow, by properly adjusting, order by order in the perturbative
2
expansion, the factors δZ ≡ Z − 1 and δ 2 ≡ M 2 − Mphys
, to cancel the
dangerous corrections to external lines in the way explained in section 9.7.
Notice, that quantizing the theory defined by the Lorentz scalar Lagrangian
density (11.410) using the variable ϕ̃ we automatically obtain in the resulting
interaction Hamiltonian density the noncovariant term ∝ ∂0 ϕI ∂0 ϕI whose
necessity in the approach of subsection 9.7 could only be guessed at.77
As will become clear in section 13, the choice of δZ and of δM 2 such
that the corrections to the external lines are absent altogether is necessary
to satisfy the conditions underlying the scheme, developed in section 7.2 for
direct calculation of S matrix elements.78 The choice of δM 2 is obvious: the
eigenstates of H0 must correspond to particles whose mass is the same as
the mass of the 1-particle eigenstates of the full H. The choice of Z is more
subtle. In principle, any choice of the canonical field variable is good for
quantization. However, the postulated correspondence (7.10) between the
H0 eigenstates and the in and out eigenstates of H singles out, through the
75
Notice that ΠI = ∂0 ϕI does not conflict with the fact that by the operator equations
of motion Π̃H = Z∂0 ϕ̃H : the limit limt→0 (∂0 ϕ̃H (t, x)) may differ from limt→0 (∂0 ϕI (t, x)).
76
We will make the notation for coefficients of these terms more systematic in sections
13 and 14.
77
Its coefficient is also automatically fixed in terms of the coefficient of the first, covariant
term. Moreover, if Z = 1+O(λ), i.e. deviates from 1 starting with one-loop approximation
only (in fact, in the considered example it starts only with two loops), the noncovariant
term whose coefficient is Z −1 (Z − 1)2 contributes starting from yet higher order.
78
Strictly speaking Hint should be also supplemented with a constant term allowing to
renormalize to zero energy of the vacuum state. This is equivalent to the usual practice
of discarding all vacuum graphs.
414
argument presented in (9.99), one particular variable ϕ̃ which as the quantum
operator ϕ̃H creates, for t → ∓∞, from the physical vacuum |Ω± i (the ground
state of H) the physical one particle states |(p)± i with the amplitude equal
to h(p)0 |ϕI (t, x)|Ω0 i. As will be shown in section 13, this is equivalent to
working with the particular renormalized operator ϕ̃H , which can be called
physical (or renormalized in the physical way) ϕph , for which the splitting of
2
H into H0 and Vint is such that the residue of the simple pole at p2 = Mph
of the (connected) Green’s function G̃(2) (p2 ) defined by the formula
Z
Z
′
4 (4)
′
(2) 2
4
(2π) δ (p + p )G̃ (p ) ≡ d x d4 y e−ip·x e−ip ·y hΩ− |T [ϕ̃H (x)ϕ̃H (y)]|Ω+ i ,
is exactly equal to i.79 In the perturbative expansion this is ensured by adjusting Z and δM 2 , order by order, in such a way that they cancel the two first
2
terms in the Taylor expansion of the self-energy Σ(p) around p2 = Mph
. From
the perspective of the general renormalization procedure (to be discussed in
detail in section 14) the direct method of calculating S matrix elements is
intimately related to the so-called On-Shell renormalization scheme.
In section 13 it will be shown that in the approach to quantum field theory
based on Green’s functions, which is more general, it is possible to work directly with the canonical (bare) field operator ϕH (or any other renormalized
operator ϕ̃H that differs from the physical one). This approach does not rely
on the assumptions of section 7.1; S matrix elements are then not computed
directly, but must be obtained with the help of the Lehmann-SymanzikZimmermann (LSZ) prescription (see section 13.4) applied to appropriate
Green’s functions of the theory. However, within the perturbative expansion
the LSZ prescription reduces to a small modification of the rules for external
lines of Feynman diagrams.
11.11
Indefinite metric quantization of the electromagnetic field
Quantization of the electromagnetic field in the Coulomb gauge described
in section 11.7 may appear inconvenient, as it is not manifestly covariant.
It is also technically rather complicated. It is possible to quantize the electromagnetic field in a fully covariant way but this requires working in the
79
Superficially it may seem that the free particle Hilbert spaces spanned by the action
of the creation and annihilation operators contained in ϕI (x) obtained from ϕH (0, x) and
from ϕ̃H (0, x) are “identical”, so there should be no difference in the choice of the canonical
variable. The mathematical subtlety here is that for systems with infinite number of
degrees of freedom the canonical commutation relations can have unitarily inequivalent
representations. Thus, the free particle Hilbert spaces constructed from ϕH (0, x) and
ϕ̃H (0, x) are in fact “different”.
415