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Quantum field theory for matter under extreme conditions Prof. Dr. David Blaschke Institute for Theoretical Physics, University of Wroclaw Winter semester 2006 - 2007 Abstract The series of lectures gives an introduction to the modern formulation of quantum field theories using Feynman path integrals and the hierarchy of Dyson-Schwinger equations. The formalism is developed for the examples of scalar fields, fermionic fields and gauge boson fields and extended to the conditions of finite temperatures, densities and strong fields with special emphasis on phase transitions in QCD, relevant for the physics of ultrarelativistic heavy-ion collisions and the physics of compact stars. Nonequilibrium quantum field theory is considered for processes of particle creation in strong, electromagnetic and weak interactions relevant for the above fields of application and for the high-intensity optical and X-ray lasers. Contents 1 Quantum Fields at Zero Temperature 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Minkowski Space Conventions . . . . . . . . . . . . . . . . 1.2.1 Four Vectors . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Dirac Matrices . . . . . . . . . . . . . . . . . . . . 1.3 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Free Particle Solutions . . . . . . . . . . . . . . . . 1.4 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Free-Fermion Propagator . . . . . . . . . . . . . . . 1.4.2 Green Function for the Interacting Theory . . . . . 1.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Path Integral in Quantum Mechanics . . . . . . . . . . . . 1.6 Functional Integral in Quantum Field Theory . . . . . . . 1.6.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Lagrangian Formulation of Quantum Field Theory 1.6.3 Quantum Field Theory for a Free Scalar Field . . . 1.6.4 Scalar Field with Self-Interactions . . . . . . . . . . 1.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Functional Integral for Fermions . . . . . . . . . . . . . . . 1.7.1 Finitely Many Degrees of Freedom . . . . . . . . . 1.7.2 Fermionic Quantum Field . . . . . . . . . . . . . . 1.7.3 Generating Functional for Free Dirac Fields . . . . 1.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Functional Integral for Gauge Field Theories . . . . . . . . 1.8.1 Faddeev-Popov Determinant and Ghosts . . . . . . 1.8.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Dyson-Schwinger Equations . . . . . . . . . . . . . . . . . 1.9.1 Photon Vacuum Polarization . . . . . . . . . . . . . 1.9.2 Fermion Self Energy . . . . . . . . . . . . . . . . . 1.9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 1.10.1 Quark Self Energy . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 3 4 6 6 9 10 13 14 14 16 17 19 20 21 22 23 23 26 28 31 31 34 42 42 42 47 49 50 50 1.10.2 Dimensional Regularization . . 1.10.3 Regularized Quark Self Energy 1.10.4 Exercises . . . . . . . . . . . . . 1.11 Renormalized Quark Self Energy . . . 1.11.1 Renormalized Lagrangian . . . 1.11.2 Renormalization Schemes . . . 1.11.3 Renormalized Gap Equation . . 1.11.4 Exercises . . . . . . . . . . . . . 1.12 Dynamical Chiral Symmetry Breaking 1.12.1 Euclidean Metric . . . . . . . . 1.12.2 Chiral Symmetry . . . . . . . . 1.12.3 Mass Where There Was None . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 57 59 60 60 63 66 68 68 68 73 75 82 2 Quantum Fields at Finite Temperature and Density 2.1 Ensembles and Partition Function . . . . . . . . . . . . 2.2 Bosonic Fields . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Neutral Scalar Field . . . . . . . . . . . . . . . 2.3 Fermionic Fields . . . . . . . . . . . . . . . . . . . . . . 2.4 Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Quantizing the Electromagnetic Field . . . . . . 2.4.2 Blackbody radiation . . . . . . . . . . . . . . . 2.5 Interactions: Hubbard-Stratonovich Trick . . . . . . . . 2.5.1 Nambu–Jona-Lasinio (NJL) Model . . . . . . . 2.5.2 Mesonic correlations at finite temperature . . . 2.5.3 Matsubara frequency sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 84 86 86 89 92 92 94 95 96 97 98 3 Applications to Heavy-Ion Collisions and Compact Stars 103 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 Particle Production by Strong Fields 4.1 Quantum kinetic equation for particle production . 4.1.1 Introduction . . . . . . . . . . . . . . . . . . 4.1.2 Dynamics of pair creation . . . . . . . . . . 4.1.3 Discussion of the source term . . . . . . . . 4.1.4 Summary . . . . . . . . . . . . . . . . . . . 4.2 Time-dependent masses: The Inertial Mechanism . 4.2.1 Scalar field . . . . . . . . . . . . . . . . . . 4.2.2 Fermion field . . . . . . . . . . . . . . . . . 4.2.3 Massive vector bosons . . . . . . . . . . . . 4.2.4 Application to conformal cosmology models 4.2.5 The systems with unstable vacuum . . . . . 4.2.6 Summary . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 125 125 126 132 137 140 141 148 153 162 167 177 Chapter 1 Quantum Fields at Zero Temperature 1.1 Introduction This is the first part of a series of lectures whose aim is to provide the tools for the completion of a realistic calculation in quantum field theory (QFT) as it is relevant to Hadron Physics. Hadron Physics lies at the interface between nuclear and particle (high energy) physics. Its focus is an elucidation of the role played by quarks and gluons in the structure of, and interactions between, hadrons. This was once particle physics but that has since moved to higher energy in search of a plausible grand unified theory and extensions of the so-called Standard Model. The only high-energy physicists still focusing on hadron physics are those performing the numerical experiments necessary in the application of lattice gauge theory, and those pushing at the boundaries of applicability of perturbative QCD or trying to find new kinematic regimes of validity. There are two types of hadron: baryons and mesons: the proton and neutron are baryons; and the pion and kaon are mesons. Historically the names distinguished the particle classes by their mass but it is now known that there are structural differences: hadrons are bound states, and mesons and baryons are composed differently. Hadron physics is charged with the responsibility of providing a detailed understanding of the differences. To appreciate the difficulties inherent in this task it is only necessary to remember that even the study of two-electron atoms is a computational challenge. This is in spite of the fact that one can employ the Schrödinger equation for this problem and, since it is not really necessary to quantize the electromagnetic field, the underlying theory has few complications. The theory underlying hadron physics is quantum chromodynamics (QCD), and its properties are such that a simple understanding and simple calculations 1 are possible only for a very small class of problems. Even on the domain for which a perturbative application of the theory is appropriate, the final (observable) states in any experiment are always hadrons, and not quarks or gluons, so that complications arise in the comparison of theory with experiment. The premier experimental facility for exploring the physics of hadrons is the Thomas Jefferson National Accelerator Facility (TJNAF), in Newport News, Virginia. Important experiments are also performed at the Fermilab National Accelerator Facility (FermiLab), in Batavia, Illinois, and at the Deutsches Elektronensynchrotron (DESY) in Hamburg. These facilities use high-energy probes and/or large momentum-transfer processes to explore the transition from the nonperturbative to the perturbative domain in QCD. On the basis of the introduction to nonperturbative methods in QFT in the vacuum, we will develop in the second part of the lecture series the tools for a generalization to the situation many-particle systems of hadrons at finite temperatures and densities in thermodynamical equilibrium within the Matsubara formalism. The third part in the series of lectures is devoted to applications for the QCD phase transition from hadronic matter to a quark-gluon plasma (QGP) in relativistic heavy-ion collisions and in the interior of compact stars. Data are provided from a completed program of experiments at the CERN-SPS and presently running programs at RHIC Brookhaven. The future of this direction will soon open a new domain of energy densities (temperatures) at CERN-LHC (2007) and at the future GSI facility FAIR, where construction shall be completed in 2015. The CBM experiment will then allow insights into the QCD phase transition at relatively low temperatures and high baryon densities, a situation which bears already similarities with the interior of compact stars, formed in supernova explosions and observed as pulsars in isolation or in binary systems. Modern astrophysical data have an unprecendented level of accuracy allowing for new stringent constraints on the behavior of the hadronic equation of state at high densities. The final part of lectures enters the domain of nonequilibrium QFT and will focus on a particular problem which, however, plays a central role: particle production in strong time-dependent external fields. The Schwinger mechanism for pair production as a strict result of quantum electrodynamics (QED) is still not experimentally verified. We are in the fortunate situation that developments of modern laser facilities in the X-ray energy domain with intensities soon reaching the Schwinger limit for electron- positron pair creation from vacuum will allow new insights and experimental tests of approaches to nonperturbative QFT in the strong field situation. These insights will allow generalizations for the other field theories such as the Standard Model and QCD, where still the puzzles of initial conditions in heavy-ion collisions and the origin of matter in the Universe remain to be solved. 2 1.2 Minkowski Space Conventions In the first part of this lecture series I will use the Minkowski metrics. Later I will employ a Euclidean metric because that is most useful and appropriate for nonperturbative calculations. 1.2.1 Four Vectors Normal spacetime coordinates are denoted by a contravariant four-vector: xµ := (x0 , x1 , x2 , x3 ) ≡ (t, x, y, z). (1.1) Throughout: c = 1 = h̄, and the conversion between length and energy is just: 1fm = 1/(0.197327 GeV) = 5.06773 GeV −1 . (1.2) The covariant four-vector is obtained by changing the sign of the spatial components of the contravariant vector: xµ := (x0 , x1 , x2 , x3 ) ≡ (t, −x, −y, −z) = gµν xν , (1.3) where the metric tensor is gµν = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1 . (1.4) The contracted product of two four-vectors is (a, b) := gµν aµ bν = aν bν : i.e., a contracted product of a covariant and a contravariant four-vector. The Poincaréinvariant length of any vector is x2 := (x, x) = t2 − ~x2 . Momentum vectors are similarly defined: and pµ = (E, px , py , pz ) = (E, p~). (1.5) (p, k) = pν k ν = Ep Ek − p~~k . (1.6) (x, p) = tE − ~xp~ . (1.7) ∂ ∂ 1~ = (i , ∇) =: i∇µ ∂xµ ∂t i (1.8) Likewise, The momentum operator pµ := i 3 transforms as a contravariant four-vector, and I denote the four-vector analogue of the Laplacian as ∂ ∂ . (1.9) ∂ 2 := −pµ pµ = ∂xµ ∂xµ The contravariant four-vector associated with the electromagnetic field is ~ Aµ (x) = (φ(x), A(x)) (1.10) with the electric and magnetic field strengths obtained from F µν = ∂ ν Aµ − ∂ µ Aν . (1.11) For example, ~ = −∇φ ~ − ∂ A. ~ Ei = F 0i ; E (1.12) ∂t Similarly, B i = εijk F jk ; j, k = 1, 2, 3. Analogous definitions hold in QCD for the chromomagnetic field strengths. 1.2.2 Dirac Matrices The Dirac matrices are indispensable in a manifestly Poincaré covariant description of particles with spin; i.e., intrinsic angular momentum, such as fermions with spin 12 . The Dirac matrices are defined by the Clifford Algebra {γ µ , γ ν } = 2g µν (1.13) and one common 4 × 4 representation is [each entry represents a 2 × 2 matrix] γ0 = " 1 0 0 −1 # , ~γ = " 0 −i i 0 # 0 ~σ −~σ 0 # , (1.14) where ~σ are the usual Pauli Matrices: 1 σ = " 0 1 1 0 # 2 ,σ = " 3 ,σ = " 1 0 0 −1 # , (1.15) and 1 = diag[1, 1]. Clearly γ0† = γ0 ; and ~γ † = −~γ . NB. These properties are not specific to this representation; e.g., γ 1 γ 1 = −1, an analogue of the properties of a purely imaginary number. In discussing spin, two combinations of Dirac matrices frequently appear: and I note that i σ µν = [γ µ , γν] , γ 5 = iγ 0 γ 1 γ 2 γ 3 = γ5 , 2 (1.16) i γ 5 σ µν = εµνρσ σρσ , 2 (1.17) 4 with εµνρσ the completely antisymmetric Lévi-Civita tensor: ε01223 = +1, εµνρσ = −εµνρσ . In the representation introduced above, 5 γ = Furthermore, " 0 1 1 0 # . (1.18) {γ5 , γµ} = 0 , γ5† = γ5 . (1.19) ~, γ µ Aµ =: A / = γ 0 A0 − ~γ A γ µ pµ =: /p = γ 0 E − ~γ p~ , ∂ ~ = iγ µ ∂ . γ µ pµ =: i∇ / ≡ i∂/ = iγ 0 + i~γ ∇ ∂t ∂xµ (1.20) (1.21) γ5 plays a special role in the discussion of parity and chiral symmetry, two key aspects of the Standard Model. The “slash” notation is a frequently used shorthand: (1.22) (1.23) The following identities are important in evaluating the cross sections for decay and scattering processes: trγ5 tr1 tra //b tra /1 a /2 a /3 a /4 tra /1 . . . a /n trγ5 a //b trγ5 a /1 a /2 a /3 a /4 γµ a /γ µ γµ a //bγ µ γµ a //b/cγ µ = = = = = = = = = = 0, 4, 4(a, b) , 4[(a1 , a2 )(a3 , a4 ) − (a1 , a3 )(a2 , a4 ) + (a1 , a4 )(a2 , a3 )] 0 , for n odd , 0, 4iεµνρσ aµ1 aν2 aρ3 aσ4 , −2a /, 4(a, b) , −2c //ba /. (1.24) (1.25) (1.26) ,(1.27) (1.28) (1.29) (1.30) (1.31) (1.32) (1.33) (1.34) They can all be derived using the fact that Dirac matrices satisfy the Clifford algebra. For example (remember trAB = trBA): tra //b = aµ bν trγ µ γ ν 1 = aµ bν tr[γ µ γ ν + γ ν γ µ ] 2 1 = aµ bν tr[2g µν 1] 2 1 µν = aµ bν 2g 4 = 4(a, b) . 2 (1.35) (1.36) (1.37) (1.38) (1.39) 5 1.3 Dirac Equation The unification of special relativity (Poincaré covariance) and quantum mechanics took some time. Even today many questions remain as to a practical implementation of a Hamiltonian formulation of the relativistic quantum mechanics of interacting systems. The Poincaré group has ten generators: the six associated with the Lorentz transformations – rotations and boosts – and the four associated with translations. Quantum mechanics describes the time evolution of a system with interactions, and that evolution is generated by the Hamiltonian. However, if the theory is formulated with an interacting Hamiltonian then boosts will almost always fail to commute with the Hamiltonian and thus the state vector calculated in one momentum frame will not be kinematically related to the state in another frame. That makes a new calculation necessary in every frame and hence the discussion of scattering, which takes a state of momentum p to another state with momentum p0 , is problematic. (See, e.g., Ref. [2]). The Dirac equation provides the starting point for a Lagrangian formulation of the quantum field theory for fermions interacting via gauge boson exchange. For a free fermion [i∂ / − m] ψ = 0 , (1.40) where ψ(x) is the fermion’s “spinor” – a four component column vector, while in the presence of an external electromagnetic field the fermion’s wave function obeys [i∂ / − eA / − m] ψ = 0 , (1.41) which is obtained, as usual, via “minimal substitution:” pµ → pµ − eAµ in Eq. (1.40). These equations have a manifestly covariant appearance but proving their covariance requires the development of a representation of Lorentz transformations on spinors and Dirac matrices: ψ 0 (x0 ) = S(Λ) ψ(x) , Λνµ γ µ = S −1 (Λ)γ ν S(Λ) , i 2 S(Λ) = exp[− σµν ω µν ] , (1.42) (1.43) (1.44) where ω µν are the parameters characterising the particular Lorentz transformation. (Details can be found in the early chapters of Refs. [1, 3].) 1.3.1 Free Particle Solutions As usual, to obtain an explicit form for the free-particle solutions one substitutes a plane wave and finds a constraint on the wave number. In this case there are two qualitatively different types of solution, corresponding to positive and negative energy. (An appreciation of the physical reality of the negative energy 6 solutions led to the prediction of the existence of antiparticles.) One inserts ψ (+) = e−i(k,x) u(k) , ψ (−) = e+i(k,x) v(k) , (1.45) into Eq. (1.40) and obtains (k/ − m) u(k) = 0 , (k/ + m) v(k) = 0 . (1.46) Assuming that the particle’s mass in nonzero then working in the rest frame yields (γ 0 − 1) u(m, ~0) = 0 , (γ 0 + 1) v(m, ~0) = 0 . (1.47) There are clearly (remember the form of γ 0 ) two linearly-independent solutions of each equation: 1 0 0 0 0 1 0 0 0 0 1 0 0 0 u(1) (m, ~0) = , u(2) (m, ~0) = , v (1) (m, ~0) = , v (2) (m, ~0) = . 0 1 (1.48) The solution in an arbitrary frame can be obtained simply via a Lorentz boost but it is even simpler to observe that (k/ − m) (k/ + m) = k 2 − m2 = 0 , (1.49) with the last equality valid when the particles are on shell, so that the solutions for arbitrary k µ are: for positive energy (E > 0), u (α) (k) = q k/ + m u 2m(m + E) (α) (m, ~0) = E + m 1/2 2m σ·k q 2m(m + E) φα (m, ~0) , (1.50) φα (m, ~0) with the two-component spinors, obviously to be identified with the fermion’s spin in the rest frame (the only frame in which spin has its naive meaning) φ (1) = 1 0 ! , φ (2) = 0 1 ! ; (1.51) and, for negative energy, v (α) (k) = q −k/ + m 2m(m + E) v (α) (m, ~0) = σ·k q 2m(m + E) E + m 1/2 2m 7 χ (m, ~0) α χα (m, ~0) (1.52) , with χ(α) obvious analogues of φ(α) in Eq. (1.51). (This approach works because it is clear that there are two, and only two, linearly-independent solutions of the momentum space free-fermion Dirac equations, Eqs. (1.46), and, for the homogeneous equations, any two covariant solutions with the correct limit in the rest-frame must give the correct boosted form.) In quantum field theory, as in quantum mechanics, one needs a conjugate state to define an inner product. For fermions in Minkowski space that conjugate is ψ̄(x) = ψ † (x)γ 0 , which satisfies ← ψ̄(i ∂/ + m) = 0 . (1.53) This equation yields the following free particle spinors in momentum space k/ + m ū(α) (k) = ū(α) (m, ~0) q 2m(m + E) v̄ (α) (k) = v̄ (α) (m, ~0) q −k/ + m 2m(m + E) (1.54) , (1.55) where I have used γ 0 (γ µ )† γ 0 = γ µ , which relation is easily derived and is particularly important in the discussion of intrinsic parity; i.e., the transformation properties of particles and antiparticles under space reflections (an improper Lorentz transformation). The momentum space free-particle spinors are orthonormalised ū(α) (k) u(β) (k) = δαβ v̄ (α )(k) v (β) (k) = −δαβ ū(α )(k) v (β) (k) = 0 . v̄ (α )(k) u(β) (k) = 0 (1.56) It is now possible to construct positive and negative energy projection operators. Consider Λ+ (k) := X α=1,2 u(α) (k) ⊗ ū(α) (k) . (1.57) It is plain from the orthonormality relations, Eqs. (1.56), that Λ+ (k) projects onto positive energy spinors in momentum space; i.e., Λ+ (k) u(α) (k) = u(α) (k) , Λ+ (k) v (α) (k) = 0 . (1.58) Now, since X α=1,2 u (α) (m, ~0) ⊗ ū (α) (m, ~0) = 1 0 0 0 ! = 1 + γ0 , 2 (1.59) then 1 1 + γ0 Λ+ (k) = (k/ + m) (k/ + m) . 2m(m + E) 2 8 (1.60) Noting that for k 2 = m2 ; i.e., on shell, (k/ + m) γ 0 (k/ + m) = 2E (k/ + m) , (k/ + m) (k/ + m) = 2m (k/ + m) , (1.61) (1.62) one finally arrives at the simple closed form: Λ+ (k) = k/ + m . 2m (1.63) The negative energy projection operator is Λ− (k) := − X α=1,2 v (α) (k) ⊗ v̄ (α) (k) = −k/ + m . 2m (1.64) The projection operators have the following important properties: Λ2± (k) = Λ± (k) , tr Λ± (k) = 2 , Λ+ (k) + Λ− (k) = 1 . (1.65) (1.66) (1.67) Such properties are a characteristic of projection operators. 1.4 Green Functions The Dirac equation is a partial differential equation. A general method for solving such equations is to use a Green function, which is the inverse of the differential operator that appears in the equation. The analogy with matrix equations is obvious and can be exploited heuristically. Equation (1.41): [i∂ /x − eA /(x) − m] ψ(x) = 0 , (1.68) yields the wave function for a fermion in an external electromagnetic field. Consider the operator obtained as a solution of the following equation [i∂ /x0 − eA /(x0 ) − m] S(x0 , x) = 1 δ 4 (x0 − x) . (1.69) It is immediately apparent that if, at a given spacetime point x, ψ(x) is a solution of Eq. (1.68), then Z ψ(x0 ) := d4 x S(x0 , x) ψ(x) (1.70) is a solution of [i∂ /x0 − eA /(x0 ) − m] ψ(x0 ) = 0 ; i.e., S(x0 , x) has propagated the solution at x to the point x0 . 9 (1.71) This effect is equivalent to the application of Huygen’s principle in wave mechanics: if the wave function at x, ψ(x), is known, then the wave function at x0 is obtained by considering ψ(x) as a source of spherical waves that propagate outward from x. The amplitude of the wave at x0 is proportional to the amplitude of the original wave, ψ(x), and the constant of proportionality is the propagator (Green function), S(x0 , x). The total amplitude of the wave at x0 is the sum over all the points on the wavefront; i.e., Eq. (1.70). This approach is practical because all physically reasonable external fields can only be nonzero on a compact subdomain of spacetime. Therefore the solution of the complete equation is transformed into solving for the Green function, which can then be used to propagate the free-particle solution, already found, to arbitrary spacetime points. However, obtaining the exact form of S(x0 , x) is impossible for all but the simplest cases. 1.4.1 Free-Fermion Propagator In the absence of an external field Eq. (1.69) becomes [i∂ /x0 − m] S(x0 , x) = 1 δ 4 (x0 − x) . (1.72) Assume a solution of the form: S0 (x0 , x) = S0 (x0 − x) = Z d4 p −i(p,x0 −x) e S0 (p) , (2π)4 (1.73) so that substituting yields (p / − m) S0 (p) = 1 ; i.e., S0 (p) = /p + m . p2 − m2 (1.74) To obtain the result in configuration space one must adopt a prescription for handling the on-shell singularities in S(p), and that convention is tied to the boundary conditions applied to Eq. (1.72). An obvious and physically sensible definition of the Green function is that it should propagate positive-energyfermions and -antifermions forward in time but not backwards in time, and vice versa for negative energy states. As we have already seen, the wave function for a positive energy free-fermion is ψ (+) (x) = u(p) e−i(p,x) . (1.75) The wave function for a positive energy antifermion is the charge-conjugate of the negative-energy fermion solution: ψc(+) (x) = C γ 0 v(p) ei(p,x) 10 ∗ = C v̄(p)T e−i(p,x) , (1.76) where C = iγ 2 γ 0 and (·)T denotes matrix transpose. (This defines the operation of charge conjugation.) It is thus evident that our physically sensible S0 (x0 − x) can only contain positive-frequency components for x00 − x0 > 0. One can ensure this via a small modification of the denominator of Eq. (1.74): S0 (p) = /p + m /p + m → 2 , 2 2 p −m p − m2 + iη (1.77) with η → 0+ at the end of all calculations. Inserting this form in Eq. (1.73) is equivalent to evaluating the p0 integral by employing a contour in the complex-p0 that is below the real-p0 axis for p0 < 0, and above it for p0 > 0. This prescription defines the Feynman propagator. To be explicit: d3 p i~p·(~x0 −~x) 1 e (2π)3 2 ω(~ p) " # Z ∞ 0 dp /p + m /p + m −ip0 (x00 −x0 ) −ip0 (x00 −x0 ) −e × e , p0 − ω(~ p) + iη p0 + ω(~ p) − iη −∞ 2π (1.78) √ where the energy ω(~ p) = p~2 + m2 . The integrals are easily evaluated using standard techniques of complex analysis, in particular, Cauchy’s Theorem. Focusing on the first term of the sum inside the square brackets, it is apparent that the integrand has a pole in the fourth quadrant of the complex p0 -plane. For x00 − x0 > 0 we can evaluate the p0 integral by considering a contour closed by a semicircle of radius R → ∞ in the lower half of the complex p0 -plane: the integrand vanishes exponentially along that arc, where p0 = −iy, y > 0, because (−i) (−iy) (x00 − x0 ) = −y (x00 − x0 ) < 0. The closed contour is oriented clockwise so that S0 (x0 − x) = Z ∞ −∞ Z dp0 −ip0 (x00 −x0 ) /p + m −ip0 (x00 −x0 ) e = (−) i e (p / + m) 0 0 + p =ω(~ p)−iη + 2π p − ω(~ p) + iη 0 = −i e−iω(~p)(x0 −x0 ) (γ 0 ω(~ p) − γ · p~ + m) −iω(~ p)(x00 −x0 ) = −i e 2m Λ+ (p) . (1.79) For x00 − x0 < 0 the contour must be closed in the upper half plane but therein the integrand is analytic and hence the result is zero. Thus Z ∞ −∞ /p + m dp0 −ip0 (x00 −x0 ) 0 e = −i θ(x00 − x0 ) e−iω(~p)(x0 −x) 2m Λ+ (~ p) . 0 + 2π p − ω(~ p) + iη Observe that the projection operator is only a function of p~ because p0 = ω(~ p). Consider now the second term in the brackets. The integrand has a pole in the second quadrant. For x00 − x0 > 0 the contour must be closed in the lower half 11 plane, the pole is avoided and hence the integral is zero. However, for x00 − x0 < 0 the contour should be closed in the upper half plane, where the integrand is obviously zero on the arc at infinity. The contour is oriented anticlockwise so that Z ∞ −∞ dp0 −ip0 (x00 −x0 ) /p + m 0 −ip0 (x00 −x) e = i θ(x − x ) e (p / + m) 0 0 0 p =−ω(~ p)+iη + 2π p0 + ω(~ p) − iη + 0 = i θ(x0 − x00 ) e+iω(~p)(x0 −x) (−γ 0 ω(~ p) − γ · p~ + m) 0 +iω(~ p)(x00 −x) = i θ(x0 − x0 ) e 2m Λ− (−~ p) . (1.80) Putting these results together, changing variables p~ → −~ p in Eq. (1.80), we have 0 S0 (x − x) = −i Z d3 p m h 0 0 θ(x0 − x0 ) e−i(p̃,x −x) Λ+ (~ p) 3 (2π) ω(~ p) 0 i + θ(x0 − x00 ) ei(p̃,x −x) Λ− (~ p) , (1.81) [(p̃µ ) = (ω(~ p), p~)] which, using Eqs. (1.58) and its obvious analogue for Λ− (~ p), manifestly propagates positive-energy solutions forward in time and negative energy solutions backward in time, and hence satisfies the physical requirement stipulated above. Another useful representation is obtained merely by observing that Z ∞ −∞ dp0 −ip0 (x00 −x) /p + m e = 2π p0 − ω(~ p) + iη + Z = Z /p + m dp0 −ip0 (x00 −x) e = 2π p0 + ω(~ p) − iη + Z ∞ −∞ ∞ −∞ p) − ~γ · p~ + m dp0 −ip0 (x00 −x) γ 0 ω(~ e 2π p0 − ω(~ p) + iη + 0 dp −ip0 (x00 −x) 1 e 2mΛ+ (~ p) 0 , 2π p − ω(~ p) + iη + (1.82) and similarly Z ∞ −∞ = Z ∞ −∞ ∞ −∞ p) − ~γ · p~ + m dp0 −ip0 (x00 −x) −γ 0 ω(~ e 2π p0 + ω(~ p) − iη + 0 1 dp −ip0 (x00 −x) e 2mΛ− (−~ p) 0 , 2π p + ω(~ p) − iη + (1.83) so that Eq. (1.78) can be rewritten 0 S0 (x − x) = Z d4 p −i(p,x0 −x) m 1 1 Λ+ (~ p) 0 . − Λ− (−~ p) 0 e 4 (2π) ω(~ p) p − ω(~ p) + iη p + ω(~ p) − iη (1.84) " 12 # The utility of this representation is that it provides a single Fourier amplitude for the free-fermion Green function; i.e., an alternative to Eq. (1.77): " # m 1 1 S0 (p) = Λ+ (~ p) 0 , − Λ− (−~ p) 0 ω(~ p) p − ω(~ p) + iη p + ω(~ p) − iη (1.85) which is indispensable in making a connection between covariant perturbation theory and time-ordered perturbation theory: the second term generates the Zdiagrams in loop integrals. 1.4.2 Green Function for the Interacting Theory We now return to Eq. (1.69): /(x0 ) − m] S(x0 , x) = 1 δ 4 (x0 − x) , [i∂ /x0 − eA (1.86) which defines the Green function for a fermion in an external electromagnetic field. As mentioned, a closed form solution of this equation is impossible in all but the simplest field configurations. Is there, nevertheless, a way to construct a systematically improvable approximate solution? To achieve that one rewrites the equation: [i∂ /x0 − m] S(x0 , x) = 1 δ 4 (x0 − x) + eA /(x0 ) S(x0 , x) , (1.87) which, it is easily seen by substitution, is solved by 0 0 S(x , x) = S0 (x − x) + e = S0 (x0 − x) + e +e 2 Z 4 d y1 Z Z Z d4 y S0 (x0 − y)A /(y) S(y, x) (1.88) d4 y S0 (x0 − y)A /(y) S0(y − x) d4 y2 S0 (x0 − y1 )A /(y1 ) S0 (y1 − y2 )A /(y2 ) S0 (y2 − x) + . . . (1.89) This perturbative expansion of the full propagator in terms of the free propagator provides an archetype for perturbation theory in quantum field theory. One obvious application is the scattering of an electron/positron by a Coulomb field, which is a worked example in Sec. 2.5.3 of Ref. [3]. Equation (1.89) is a first example of a Dyson-Schwinger equation [4]. This Green function has the following interpretation: 1. It creates a positive energy fermion (antifermion) at spacetime point x; 2. Propagates the fermion to spacetime point x0 ; i.e., forward in time; 3. Annihilates this fermion at x0 . 13 The process can equally well be viewed as 1. The creation of a negative energy antifermion (fermion) at spacetime point x0 ; 2. Propagation of the antifermion to the spacetime point x; i.e., backward in time; 3. Annihilation of this antifermion at x. Other propagators have similar interpretations. 1.4.3 Exercises 1. Prove these relations for on-shell fermions: (k/ + m) γ 0 (k/ + m) = 2E (k/ + m) , (k/ + m) (k/ + m) = 2m (k/ + m) . 2. Obtain the Feynman propagator for the free-field Klein Gordon equation: (∂x2 + m2 )φ(x) = 0 , (1.90) in forms analogous to Eqs. (1.81), (1.84). 1.5 Path Integral in Quantum Mechanics Local gauge theories are the keystone of contemporary hadron and high-energy physics. Such theories are difficult to quantise because the gauge dependence is an extra non-dynamical degree of freedom that must be dealt with. The modern approach is to quantise the theories using the method of functional integrals, attributed to Feynman and Kac. References [3, 5] provide useful overviews of the technique, which replaces canonical second-quantisation. It is customary to motivate the functional integral formulation by reviewing the path integral representation of the transition amplitude in quantum mechanics. Beginning with a state (q1 , q2 , . . . , qN ) at time t, the probability that one 0 will obtain the state (q10 , q20 , . . . , qN ) at time t0 is given by (remember, the time evolution operator in quantum mechanics is exp[−iHt], where H is the system’s Hamiltonian): 0 hq10 , q20 , . . . , qN ; t0 |q1 , q2 , . . . , qN ; ti = lim n→∞ × exp i n+1 X j=1 N Z Y n Y α=1 i=1 dqα (ti ) Z n+1 Y i=1 pα (tj )[qα (tj ) − qα (tj−1 )] − H(p(tj ), 14 dpα (ti ) 2π qtj + qtj −1 , )(1.91) 2 where tj = t + j, = (t0 − t)/(n + 1), t0 = t, tn+1 = t0 . A compact notation is commonly introduced to represent this expression: hq 0 t0 |qtiJ = Z [dq] Z ( Z [dp] exp i t0 t ) dτ [p(τ )q̇(τ ) − H(τ ) + J(τ )q(τ )] , (1.92) where J(t) is a classical external “source.” NB. The J = 0 exponent is nothing but the Lagrangian. Recall that in Heisenberg’s formulation of quantum mechanics it is the operators that evolve in time and not the state vectors, whose values are fixed at a given initial time. Using the previous formulae it is a simple matter to prove that the time ordered product of n Heisenberg position operators can be expressed as hq 0 t0 |T {Q(t1 ) . . . Q(tn )}|qti = Z ( Z Z [dq] [dp] q(t1 ) q(t2 ) . . . q(tn ) exp i t0 t ) dτ [p(τ )q̇(τ ) − H(τ )] . (1.93) NB. The time ordered product ensures that the operators appear in chronological order, right to left. Consider a source that “switches on” at ti and “switches off” at tf , t < ti < tf < t0 , then 0 0 J hq t |qti = Z dqi dqf hq 0 t0 |qf tf i hqf tf |qi ti iJ hqi ti |qti . (1.94) Alternatively one can introduce a complete set of energy eigenstates to resolve the Hamiltonian and write hq 0 t0 |qti = X n t0 →−i∞ t → +i∞ = 0 hq 0 |φn i e−iEn (t −t) hφn |qi 0 hq 0 |φ0 i e−iE0 (t −t) hφ0 |qi ; (1.95) (1.96) i.e., in either of these limits the transition amplitude is dominated by the ground state. It now follows from Eqs. (1.94), (1.96) that hq 0 t0 |qti lim = 0 −iE0 (t −t) hq 0 |φ i hφ |qi 0 0 t0 →−i∞ e t → +i∞ Z dqi dqf hφ0 |qf tf i hqf tf |qi ti iJ hqi ti |φ0 i =: W [J] ; (1.97) i.e., the ground-state to ground-state transition amplitude (survival probability) in the presence of the external source J. From this it will readily be apparent that, with tf > t1 > t2 > . . . > tm > ti , δ m W [J] = im δJ(t1 ) . . . δJ(t1 ) J=0 Z dqi dqf hφ0 |qf tf i hqf tf |T {Q(t1 ) . . . Q(tn )}|qi ti i hqi ti |φ0 i, (1.98) 15 which is the ground state expectation value of a time ordered product of Heisenberg position operators. The analogues of these expectation values in quantum field theory are the Green functions. δ , is defined analogously to the The functional derivative introduce here: δJ(t) derivative of a function. It means to write J(t) → J(t) + (t); expand the functional in (t); and identify the leading order coefficient in the expansion. Thus for Z Hn [J] = dt0 J(t0 )n (1.99) δHn [J] = δ = Z Z dt0 J(t0 )n = Z dt0 [J(t0 ) + (t0 )]n − dt0 n J(t0 )n−1 (t0 ) + [. . .]. Z dt0 J(t0 )n (1.100) (1.101) So that taking the limit δJ(t0 ) = (t0 ) → δ(t − t0 ) one obtains δHn [J] = n J(t)n−1 . δJ(t) (1.102) The limit procedure just described corresponds to defining δJ(t) = δ(t − t0 ) . δJ(t0 ) (1.103) This example, which is in fact the “product rule,” makes plain the very close analogy between functional differentiation and the differentiation of functions so that, with a little care, the functional differentiation of complicated functions is straightforward. Another note is in order. The fact that the limiting values in Eqs. (1.96), (1.97) are imaginary numbers is a signal that mathematical rigour may be found more easily in Euclidean space where t → −itE . Alternatively, at least in principle, the argument can be repeated and made rigorous by making the replacement En → En − iη, with η → 0+ . However, that does not help in practical calculations, which proceed via Monte-Carlo methods; i.e., probability sampling, which is only effective when the integrand is positive definite. 1.6 Functional Integral in Quantum Field Theory The Euclidean functional integral is particularly well suited to a direct numerical evaluation via Monte-Carlo methods because it defines a probability measure. However, to make direct contact with the perturbation theory of canonical secondquantised quantum field theory, I begin with a discussion of the Minkowski space formulation. 16 1.6.1 Scalar Field I will consider a scalar field φ(t, x), which is customary because it reduces the number of indices that must be carried through the calculation. Suppose that a large but compact domain of space is divided into N cubes of volume 3 and label each cube by an integer α. Define the coordinate and momentum via qα (t) := φα (t) = 1 3 Z Vα d3 x φ(t, x) , q̇α (t) := φ̇α (t) = 1 3 Z Vα d3 x ∂φ(t, x) ; (1.104) ∂t i.e., as the spatial averages over the cube denoted by α. The classical dynamics of the field φ is described by a Lagrangian: L(t) = Z d3 x L(t, x) → N X 3 Lα (φ̇α (t), φα (t), φα±s (t)) , (1.105) α=1 where the dependence on φα±s (t); i.e., the coordinates in the neighbouring cells, is necessary to express spatial derivatives in the Lagrangian density, L(x). With the canonical conjugate momentum defined as in a classical field theory ∂Lα ∂L = 3 =: 3 πα (t) , ∂ φ̇α (t) ∂ φ̇α (t) pα (t) := (1.106) the Hamiltonian is obtained as H= X α pα (t) q̇α (t) − L(t) =: X 3 Hα , (1.107) α Hα (πα (t), φα (t), φα±s (t)) = πα (t) φ̇α (t) − Lα . (1.108) The field theoretical equivalent of the quantum mechanical transition amplitude, Eq. (1.91), can now be written N Z Y n Y lim n→∞,→0+ α=1 i=1 dφα (ti ) n+1 X X dπα (ti ) φα (tj ) − φα (tj−1 ) 3 exp i 3 πα (tj ) 2π α i=1 j=1 =: Z ( Z [Dφ] [Dπ] exp i ( ! φα (tj ) + φα (tj−1 ) φα±s (tj ) + φα±s (tj−1 ) πα (tj ), , 2 2 − Hα Z Z Y n t0 t dτ Z " ∂φ(τ, ~x) d x π(τ, ~x) − H(τ, ~x) ∂τ 3 #) , (1.109) where, as classically, π(t, ~x) = ∂L(t, ~x)/∂ φ̇(t, ~x) and its average over a spacetime cube is just πα (t). Equation (1.109) is the transition amplitude from an initial field configuration φα (t0 ) := φα (t) to a final configuration φα (tn+1 ) := φα (t0 ). 17 Generating Functional In quantum field theory all physical quantities can be obtained from Green functions, which themselves are determined by vacuum-to-vacuum transition amplitudes calculated in the presence of classical external sources. The physical or interacting vacuum is the analogue of the true ground state in quantum mechanics. And, as in quantum mechanics, the fundamental quantity is the generating functional: 1 W [J] := N Z Z Z [Dφ] [Dπ] exp i 4 d x [π(x)φ̇(x) − H(x) + 1 iηφ2 (x) + J(x)φ(x)] 2 (1.110) where N is chosen so that W [0] = 1, and a real time-limit is implemented and made meaningful through the addition of the η → 0+ term. (NB. This subtracts a small imaginary piece from the mass.) It is immediately apparent that , 1 δ n W [J] h0̃|T {φ̂(x1 )φ̂(x2 ) . . . φ̂(xn )}|0̃i =: G(x1 , x2 , . . . , xn ) , = in δJ(x1 )δJ(x2 ) . . . δJ(xn ) J=0 h0̃|0̃i (1.111) where |0̃i is the physical vacuum. G(x1 , x2 , . . . , xn ) is the complete n-point Green function for the scalar quantum field theory; i.e., the vacuum expectation value of a time-ordered product of n field operators. The appearance of the word “complete” means that this Green function includes contributions from products of lower-order Green functions; i.e., disconnected diagrams. The fact that the Green functions in a quantum field theory may be defined via Eq. (1.111) was first observed by Schwinger [6] and does not rely on the functional formula for W [J], Eq. (1.110), for its proof. However, the functional formulism provides the simplest proof and, in addition, a concrete means of calculating the generating functional: numerical simulations. Connected Green Functions It is useful to have a systematic procedure for the a priori elimination of disconnected parts from a n-point Green function because performing unnecessary work, as in the recalculation of m < n-point Green functions, is inefficient. A connected n-point Green function is given by Gc (x1 , x2 , . . . , xn ) = (−i) where Z[J], defined via n−1 δ n Z[J] , δJ(x1 )δJ(x2 ) . . . δJ(xn ) J=0 W [J] =: exp{iZ[J]} , is the generating functional for connected Green functions. 18 (1.112) (1.113) It is instructive to illustrate this for a simple case (recall that I have already proven the product rule for functional differentiation): δ 2 Z[J] δ 2 ln W [J] Gc (x1 , x2 ) = (−i) = − δJ(x1 ) δJ(x2 ) J=0 δJ(x1 ) δJ(x2 ) J=0 " # J=0 δW [J] − δJ(x2 ) δ 1 δW [J] = − δJ(x1 ) W [J] δJ(x2 ) 1 δW [J] = + 2 W [J] δJ(x1 ) J=0 h0̃|φ̂(x1 )|0̃i h0̃|φ̂(x2 )|0̃i h0̃|T {φ̂(x1 )φ̂(x2 )}|0̃i i − i2 h0̃|0̃i h0̃|0̃i h0̃|0̃i = −G(x1 ) G(x2 ) + G(x1 , x2 ) . = i 1.6.2 1 δ 2 W [J] W [J] δJ(x1 ) δJ(x2 ) J=0 (1.114) Lagrangian Formulation of Quantum Field Theory The double functional integral employed above is cumbersome, especially since it involves the field variable’s canonical conjugate momentum. Consider then a Hamiltonian density of the form 1 ~ H(x) = π 2 (x) + f [φ(x), ∇φ(x)] . 2 (1.115) In this case Eq. (1.110) involves Z 1 2 [Dπ] exp i d x [− π (x) + π(x)φ̇(x)] 2 Z Z R 4 2 i {i d x [φ̇(x)] } 4 2 = e [Dπ] exp − d x [π(x) − φ̇(x)] 2 R 4 2 Z = e{i 4 d x [φ̇(x)] } × N, (1.116) where “N” is simply a constant. (This is an example of the only functional integral than can be evaluated exactly: the Gaussian functional integral.) Hence, with a Hamiltonian of the form in Eq. (1.115), the generating functional can be written Z Z N 1 W [J] = [Dφ] exp i d4 x [L(x) + iηφ2 (x) + J(x)φ(x)] . (1.117) 2 N Recall now that the classical Lagrangian density for a scalar field is L(x) = L0 (x) + LI (x) , L0 (x) = 1 [∂µ φ(x) ∂ µ φ(x) − m2 φ2 (x)] , 2 (1.118) (1.119) with LI (x) some functional of φ(x) that usually does not depend on any derivatives of the field. Hence the Hamiltonian for such a theory has the form in Eq. (1.115) and so Eq. (1.117) can be used to define the quantum field theory. 19 1.6.3 Quantum Field Theory for a Free Scalar Field The interaction Lagrangian vanishes for a free scalar field so that the generating functional is, formally, 1 W0 [J] = N̄ Z Z [Dφ] exp i 4 dx 1 [∂µ φ(x) ∂ µ φ(x) − m2 φ2 (x) + iηφ2 (x)] + J(x)φ(x) 2 (1.120) The explicit meaning of Eq. (1.120) is 1 W0 [J] = lim+ →0 N̄ Z Y α X dφα exp i 4 X α β 4 1 2 φα Kαβ φβ + X α 4 where α, β label spacetime hypercubes of volume and Kαβ satisfies lim+ Kαβ = [−∂ 2 − m2 + iη]δ 4 (x − y) , 4 Jα φ α , (1.121) is any matrix that (1.122) →0 →0+ →0+ →0+ d4 x; i.e., Kαβ is any matrix whose where α → x, β → y and α 4 → continuum limit is the inverse of the Feynman propagator for a free scalar field. (NB. The fact that there are infinitely many such matrices provides the scope for “improving” lattice actions since one may choose Kαβ wisely rather than for simplicity.) Recall now that for matrices whose real part is positive definite Z n Y Rn i=1 dxi exp{− P n 1 X 2 xi Aij xj + i,j=1 R n X i=1 b i xi } n (2π)n/2 (2π)n/2 1 1 X = √ exp bt A−1 b .(1.123) bi (A−1 )ij bj } = √ exp{ 2 2 detA detA i,j=1 Hence Eq. (1.121) yields W0 [J] = lim+ →0 1 1 1 1X 4 X 4 √ Jα 8 (K −1 )αβ Jβ } , exp{ 0 2 α N̄ i detA β (1.124) where, obviously, the matrix inverse is defined via X Kαγ (K −1 )γβ = δαβ . (1.125) γ Almost as obviously, consistency of limits requires Z X 1 4 4 δαβ = δ (x − y) , lim+ lim = d4 x , →0 →0+ 4 α so that, with O(x, y) := lim+ →0 20 1 (K −1 )αβ , 8 (1.126) (1.127) . the continuum limit of Eq. (1.125) can be understood as follows: lim+ →0 Z ⇒ . X 4 Kαγ γ 1 1 (K −1 )γβ = lim+ 4 δαβ 8 →0 d4 w [−∂x2 − m2 + iη]δ 4 (x − w) O(w, y) = δ 4 (x − y) [−∂x2 − m2 + iη] O(x, y) = δ 4 (x − y) . ·· (1.128) Hence O(x, y) = ∆0 (x − y); i.e., the Feynman propagator for a free scalar field: ∆0 (x − y) = Z d4 p −i(q,x−y) 1 e . 4 2 (2π) q − m2 + iη (1.129) (NB. This makes plain the fundamental role of the “iη + ” prescription in Eq. (1.77): it ensures convergence of the expression defining the functional integral.) Putting this all together, the continuum limit of Eq. (1.124) is 1 i W [J] = exp − 2 N̂ 1.6.4 Z 4 dx Z 4 d y J(x) ∆0 (x − y) J(y) . (1.130) Scalar Field with Self-Interactions A nonzero interaction Lagrangian, LI [φ(x)], provides for a self-interacting scalar field theory (from here on I will usually omit the constant, nondynamical normalisation factor in writing the generating functional): W [J] = Z Z [Dφ] exp i " Z d x LI " Z d 4 x LI = exp i = exp i 4 d4 x [L0 (x) + LI (x) + J(x)φ(x)] !# Z 1 δ [Dφ] exp i d4 x [L0 (x) + J(x)φ(x)] (1.131) i δJ(x) !# Z Z 1 δ i 4 4 exp − d x d y J(x) ∆0 (x − y) J(y) , 2 i δJ(x) (1.132) Z where " Z exp i 4 d x LI 1 δ i δJ(x) !# := ∞ X n=0 in LI n! " 1 δ i δJ(x) !#n . (1.133) Equation (1.132) is the basis for a perturbative evaluation of all possible Green functions for the theory. As an example I will work through a first-order calculation of the complete 2-point Green function in the theory defined by LI (x) = − λ 4 φ (x) . 4! 21 (1.134) The generating functional yields W [0] = λ 1−i 4! Z δ d4 x δJ(x) !4 λZ 4 = 1−i d x 3[i∆0 (0)]2 . 4! exp − i 2 Z d4 u Z d4 v J(u) ∆0 (u − v) J(v) J=0 (1.135) The 2-point function is λ Z 4 δ 2 W [J] d x [i∆0 (0)]2 [i∆0 (x1 − x2 )] = −i∆0 (x1 − x2 ) + i δJ(x1 ) δJ(x2 ) 8 Z λ +i d4 x [i∆0 (0)][i∆0 (x1 − x)][i∆0 (x − x2 )] . (1.136) 2 Using the definition, Eq. (1.111), and restoring the normalisation, 1 1 δ 2 W [J] (1.137) i2 W [0] δJ(x1 ) δJ(x2 ) Z λ d4 x [i∆0 (0)][i∆0 (x1 − x)][i∆0 (x − x(1.138) = i∆0 (x1 − x2 ) − i 2 )] , 2 G(x1 , x2 ) = where I have used a + λb = a + λ(b − ac) + O(λ2 ) . (1.139) 1 + λc This complete Green function does not contain any disconnected parts because the vacuum is trivial in perturbation theory; i.e., 1 δW [J] h0̃|φ̂(x)|0̃i =0 := G(x)|J=0 = i δJ(x) J=0 h0̃|0̃i (1.140) so that the field does not have a nonzero vacuum expectation value. This is the simplest demonstration of the fact that dynamical symmetry breaking is a phenomenon inaccessible in perturbation theory. 1.6.5 Exercises 1. Repeat the derivation of Eq. (1.114) for Gc (x1 , x2 , x3 ). 2. Prove Eq. (1.131). 3. Derive Eq. (1.136). 4. Prove Eq. (1.140). 22 1.7 1.7.1 Functional Integral for Fermions Finitely Many Degrees of Freedom Fermionic fields do not have a classical analogue: classical physics does not contain anticommuting fields. In order to treat fermions using functional integrals one must employ Grassmann variables. Reference [7] is the standard source for a rigorous discussion of Grassmann algebras. Here I will only review some necessary ideas. The Grassmann algebra GN is generated by the set of N elements, θ1 , . . . , θN which satisfy the anticommutation relations {θi , θj } = 0 , i, j = 1, 2, . . . , N . (1.141) It is clear from Eq. (1.141) that θi2 = 0 for i = 1, . . . , N . In addition, the elements {θi } provide the source for the basis vectors of a 2n -dimensional space, spanned by the monomials: 1, θ1 , . . . , θN , θ1 θ2 , . . . θN −1 θN , . . . , θ1 θ2 . . . θN ; (1.142) i.e., GN is a 2N -dimensional vector space. (NB. One can always choose the pdegree monomial in Eq. (1.142): θi1 θi2 . . . θIN , such that i1 < i2 < . . . < iN .) Obviously, any element f (θ) ∈ GN can be written f (θ) = f0 + X i1 f1 (i1 ) θi1 + X f2 (i1 , i2 ) θi1 θi2 +. . .+ i1 ,i2 X fN (i1 , i2 , . . . , iN ) θ1 θ2 . . . θN , i1 ,i2 ,...,iN (1.143) where the coefficients fp (i1 , i2 , . . . , ip ) are unique if they are chosen to be fully antisymmetric for p ≥ 2. Both “left” and “right” derivatives can be defined on GN . As usual, they are linear operators and hence it suffices to specify their operation on the basis elements: ∂ θi θi θi = δsi1 θi2 . . . θip − δsi2 θi1 . . . θip + . . . + (−)p−1 δsip θi1 θi2 . . . θ(1.144) ip−1 , ∂θs 1 2 p ← θ i1 θ i2 θ ip ∂ = δsip θi1 . . . θip−1 − δsip−1 θi1 . . . θip−2 θip + . . . + (−)p−1 δsi1 θi(1.145) 2 θ ip . ∂θs The operation on a general element, f (θ) ∈ GN , is easily obtained. It is also obvious that ∂ ∂ ∂ ∂ f (θ) = − f (θ) . (1.146) ∂θ1 ∂θ2 ∂θ2 ∂θ1 A definition of integration requires the introduction of Grassmannian line elements: dθi , i = 1, . . . , N . These elements also satisfy Grassmann algebras: {dθi , dθj } = 0 = {θi , dθj } , i, j = 1, 2, . . . N . 23 (1.147) The integral calculus is completely defined by the following two identities: Z dθi = 0 , Z dθi θi = 1 , i = 1, 2, . . . , N . (1.148) For example, it is straightforward to prove, using Eq. (1.143), Z dθN . . . dθ1 f (θ) = N ! fN (1, 2, . . . , N ) . (1.149) In standard integral calculus a change of integration variables is often used to simplify an integral. That operation can also be defined in the present context. Consider a nonsingular matrix (Kij ), i, j = 1, . . . , N , and define new Grassmann variables ξ1 , . . . , ξN via θi = N X Kij ξj . (1.150) (K −1 )ji dξj (1.151) Z (1.152) i=j With the definition dθi = N X j=1 one guarantees It follows immediately that Z dθi θj = δij = dξi ξj . θ1 θ2 . . . θN = (det K) ξ1 ξ2 . . . ξN dθN dθN −1 . . . dθ1 = (det K −1 ) dξN dξN −1 . . . dξ1 , (1.153) (1.154) and hence Z dθN . . . dθ1 f (θ) = (det K −1 ) Z dξN . . . dξ1 f (θ(ξ)) . (1.155) In analogy with scalar field theory, for fermions one expects to encounter integrals of the type I := Z dθN . . . dθ1 exp N X θi Aij θj , (1.156) i,j=1 where (Aij ) is an antisymmetric matrix. NB. Any symmetric part of the matrix,A, cannot contribute since: X θi Aij θj relable = X θj Aji θi (1.157) θj Aij θi (1.158) j,i i,j use sym. = X i,j anticom. = ..· X θi Aij θj = i,j 24 − X θi Aij θj . (1.159) 0 for A = At . (1.160) i,j Assume for the moment that A is a real matrix. Then there is an orthogonal matrix S (SS t = I) for which S t AS = 0 λ1 0 0 −λ1 0 0 0 0 0 0 λ2 0 0 −λ2 0 ... ... ... ... ... ... ... ... ... =: à . (1.161) PN Sij ξj and using Consequently, applying the linear transformation θi = Eq. (1.155), we obtain I= Z dξN . . . dξ1 exp Hence I = R R N X ξi Ãij ξj i,j=1 i=1 . (1.162) o n dξN . . . dξ1 exp 2 [λ1 ξ1 ξ2 + λ2 ξ3 ξ4 + . . . + λN/2 ξN −1 ξN ] = 2N/2 λ1 λ2 . . . λN , N even n o dξN . . . dξ1 exp 2 [λ1 ξ1 ξ2 + λ2 ξ3 ξ4 + . . . + λ(N −1)/2 ξN −2 ξN −1 ] = 0 , N odd i.e., since det A = det Ã, I= √ det 2A , . (1.163) (1.164) Equation (1.164) is valid for any real matrix, A. Hence, by the analytic function theorem, it is also valid for any complex matrix A. The Lagrangian density associated with the Dirac equation involves a field ψ̄, which plays the role of a conjugate to ψ. If we are express ψ as a vector in GN then we will need a conjugate space in which ψ̄ is defined. Hence it is necessary to define θ̄1 , θ̄2 , . . . , θ̄N such that the operation θi ↔ θ̄i is an involution of the algebra onto itself with the following properties: i) (θ̄i ) = θi ii) (θi θj ) = θ̄j θ̄i iii) λ θi = λ∗ θ̄i , λ ∈ C . (1.165) The elements of the Grassmann algebra with involution are θ1 , θ2 , . . . , θN , θ̄1 , θ̄2 . . . , θ̄N , each anticommuting with every other. Defining integration via obvious analogy with Eq. (1.148) it follows that Z dθ̄N dθN . . . dθ1 dθ1 exp − 25 N X i,j=1 θ̄i Bij θj = det B , (1.166) for any matrix B. (NB. This is the origin of the fermion determinant in the quantum field theory of fermions.) This may be compared with the analogous result for commuting real numbers, Eq. (1.123): Z 1.7.2 RN N Y dxi exp{−π i=1 N X i,j=1 xi Aij xj } = √ 1 . det A (1.167) Fermionic Quantum Field To describe a fermionic quantum field the preceding analysis must be generalised to the case of infinitely many generators. A rigorous discussion can be found in Ref. [7] but here I will simply motivate the extension via plausible but formal manipulations. Suppose the functions {un (x) , n = 0, . . . , ∞} are a complete, orthonormal set that span a given Hilbert space and consider the Grassmann function θ(x) := ∞ X un (x) θn , (1.168) n=0 where θn are Grassmann variables. Clearly {θ(x), θ(y)} = 0 . (1.169) The elements θ(x) are considered to be the generators of the “Grassman algebra” G and, in complete analogy with Eq. (1.143), any element of G can be uniquely written as f= ∞ Z X dx1 dx2 . . . dxN θ(x1 )θ(x2 ) . . . θ(xN ) fn (x1 , x2 , . . . , xN ) , (1.170) n=0 where, for N ≥ 2, the fn (x1 , x2 , . . . , xN ) are fully antisymmetric functions of their arguments. In another analogy, the left- and right-functional-derivatives are defined via their action on the basis vectors: δ θ(x1 )θ(x2 ) . . . θ(xn ) = δθ(x) δ(x − x1 ) θ(x2 ) . . . θ(xn ) − . . . + (−)n−1 δ(x − xn ) θ(x1 ) . . . θ(xn−1 )(1.171) , ← δ = δθ(x) δ(x − xn ) θ(x1 ) . . . θ(xn−1 ) − . . . + (−)n−1 δ(x − x1 ) θ(x2 ) . . . θ(xn )(1.172) , θ(x1 )θ(x2 ) . . . θ(xn ) cf. Eqs. (1.144), (1.145). 26 Finally, we can extend the definition of integration. Denoting [Dθ(x)] := lim dθN . . . dθ2 dθ1 , (1.173) N →∞ consider the “standard” Gaussian integral I := Z [Dθ(x)] exp Z dxdy θ(x)A(x, y)θ(y) (1.174) where, clearly, only the antisymmetric part of A(x, y) can contribute to the result. Define Z Aij := dxdy ui (x)A(x, y)uj (y) , (1.175) then I = lim N →∞ Z dθN . . . dθ2 dθ1 e so that, using Eq. (1.164), I = lim N →∞ q PN i=1 θi Aij θj det 2AN , (1.176) (1.177) where, obviously, AN is the N × N matrix in Eq. (1.175). This provides a definition for the formal result: Z Z √ (1.178) dxdy θ(x)A(x, y)θ(y) = Det2A , I = [Dθ(x)] exp where I will subsequently identify functional equivalents of matrix operations as proper nouns. The result is independent of the the basis vectors since all such vectors are unitarily equivalent and the determinant is cylic. (This means that a new basis is always related to another basis via u0 = U u, with U U † = I. Transforming to a new basis therefore introduces a modified exponent, now involving the matrix U AU † , but the result is unchanged because det U AU † = det A.) In quantum field theory one employs a Grassmann algebra with an involution. In this case, defining the functional integral via [D θ̄(x)][Dθ(x)] := lim dθ̄N dθN . . . dθ̄2 dθ2 dθ̄1 dθ1 , N →∞ (1.179) one arrives immediately at a generalisation of Eq. (1.166) Z [D θ̄(x)][Dθ(x)] exp − Z dxdy θ̄(x) B(x, y) θ(x) = DetB , (1.180) which is also a definition. The relation ln det B = tr ln B , 27 (1.181) valid for any nonsingular, finite dimensional matrix, has a generalisation that is often used in analysing quantum field theories with fermions. Its utility is to make possible a representation of the fermionic determinant as part of the quantum field theory’s action via DetB = exp {Tr LnB} . (1.182) I note that for an integral operator O(x, y) Tr O(x, y) := Z d4 x tr O(x, x) , (1.183) which is an obvious analogy to the definition for finite-dimensional matrices. Furthermore a functional of an operator, whenever it is well-defined, is obtained via the function’s power series; i.e., if f (x) = f0 + f1 x + f2 x2 + [. . .] , (1.184) then 4 f [O(x, y)] = f0 δ (x − y) + f1 O(x, y) + f2 1.7.3 Z d4 w O(x, w)O(w, y) + [. . .] . (1.185) Generating Functional for Free Dirac Fields The Lagragian density for the free Dirac field is Lψ0 (x) = Z d4 x ψ̄(x) (i∂/ − m) ψ(x) . (1.186) Consider therefore the functional integral ¯ ξ] = W [ξ, Z Z [D ψ̄(x)][Dψ(x)] exp i ¯ d4 x ψ̄(x) i∂/ − m + iη + ψ(x) + ψ̄(x)ξ(x) + ξ(x)ψ(x) h 28 . (1.187) Here ψ̄(x), ψ(x) are identified with the generators of G, with the minor additional complication that the spinor degree-of-freedom is implicit; i.e., to be explicit, Q Q one should write 4r=1 [D ψ̄r (x)] 4s=1 [Dψs (x)]. This only adds a finite matrix degree-of-freedom to the problem, so that “Det A” will mean both a functional and a matrix determinant. This effect will be encountered again; e.g., with the appearance of fermion colour and flavour. In Eq. (1.187) I have also introduced ¯ anticommuting sources: ξ(x), ξ(x), which are also elements in the Grassmann algebra, G. The free-field generating functional involves a Gaussian integral. To evaluate that integral I write O(x, y) = (i∂/ − m + iη + )δ 4 (x − y) i (1.188) and observe that the solution of Z d4 w O(x, w) P (w, y) = I δ 4 (x − y) (1.189) i.e., the inverse of the operator O(x, y) is (see Eq. 1.72) precisely the free-fermion propagator: P (x, y) = S0 (x − y) . (1.190) Hence I can rewrite Eq. (1.187) in the form ¯ ξ] = W [ξ, Z Z [D ψ̄(x)][Dψ(x)] exp i ¯ d xd y ψ̄ (x)O(x, y)ψ (y) − ξ(x)S 0 (x − y)ξ(y) 4 h 4 0 0 (1.191) where 0 ψ̄ (x) := ψ̄(x) + Z ¯ d w ξ(w) S0 (w − x) , ψ(x) := ψ(x) + 4 Z d4 w S0 (x − w) ξ(w) . (1.192) Clearly, ψ̄ (x) and ψ (x) are still in G and hence related to the original variables by a unitary transformation. Thus changing to the “primed” variables introduces a unit Jacobian and so 0 0 ¯ ξ] = exp −i W [ξ, × Z Z 0 ¯ S0 (x − y) ξ(y) d xd y ξ(x) 4 4 0 Z [D ψ̄ (x)][Dψ (x)] exp i Z = det[−iS0−1 (x − y)] exp −i = 4 1 exp −i N0ψ Z 4 0 0 d xd y ψ̄ (x)O(x, y)ψ (y) ¯ S0 (x − y) ξ(y) d4 xd4 y ξ(x) ¯ S0 (x − y) ξ(y) , d4 xd4 y ξ(x) (1.193) (1.194) (1.195) where N0ψ := det[iS0 (x − y)]. Clearly. ¯ ξ] N0ψ W [ξ, ξ̄=0=ξ = 1. (1.196) The 2 point Green function for the free-fermion quantum field theory is now easily obtained: N0ψ = ˆ ¯ ξ] δ 2 W [ξ, h0|T {ψ̂(x)ψ̄(y)}|0i = ¯ (−i)δξ(y) h0|0i iδ ξ(x) ξ̄=0=ξ Z Z [D ψ̄(x)][Dψ(x)] ψ(x)ψ̄(y) exp i = i S0 (x − y) ; 4 d x ψ̄(x) i∂/ − m + iη + ψ(x) (1.197) i.e., the inverse of the Dirac operator, with exactly the Feynman boundary conditions. 29 i As in the example of a scalar quantum field theory, the generating functional ¯ ξ], defined via: for connected n-point Green functions is Z[ξ, ¯ ξ] =: exp iZ[ξ, ¯ ξ] . W [ξ, n o (1.198) Hitherto I have not illustrated what is meant by “DetO,” where O is an integral operator. I will now provide a formal example. (Rigour requires a careful consideration of regularisation and limits.) Consider a translationally invariant operator Z d4 p O(x, y) = O(x − y) = O(p) e−i(p,x−y) . (1.199) (2π)4 Then, for f as in Eq. (1.185), o d4 p n 2 f + f O(p) + f O (p) + [. . .] e−i(p,x−y) 0 1 2 (2π)4 Z d4 p f (O(p)) e−i(p,x−y) . (1.200) = (2π)4 f [O(x − y)] = Z I now apply this to N0ψ := Det[iS0 (x − y)] and observe that Eq. (1.182) means one can begin by considering TrLn iS0 (x − y). Writing " # /p 1 S0 (p) = m ∆0 (p ) 1 + , ∆0 (p2 ) = 2 , m p − m2 + iη + 2 (1.201) the free fermion propagator can be re-expressed as a product of integral operators: S0 (x − y) = Z d4 w m ∆0 (x − w) F (w − y) , (1.202) with ∆0 (x − y) given in Eq. (1.129) and F (x − y) = Z d4 p /p e−i(p,x−y) . 1+ 4 (2π) m # " (1.203) It follows that n h TrLn iS0 (x − y) = Tr Ln i m∆0 (x − y) + Ln δ 4 (x − y) + F (x − y) io . (1.204) Using Eqs. (1.183), (1.200), one can express the second term as h TrLn δ 4 (x − y) + F (x − y) i d4 p tr ln [1 + F (p)] (2π)4 " # Z Z d4 p p2 4 = dx 2 ln 1 − 2 (1.205) (2π)4 m = 30 Z d4 x Z and, applying the same equations, the first term is TrLn im∆0 (x − y) = Z 4 dx Z h i d4 p 2 2 2 ln i m∆ (p ) , 0 (2π)4 (1.206) where in both cases d4 x measures the (infinite) spacetime volume. Combining these results one obtains R Ln N0ψ = TrLn iS0 (x − y) = Z 4 dx Z d4 p 2 ln ∆0 (p2 ) , 4 (2π) (1.207) where the factor of 2 reflects the spin-degeneracy of the free-fermion’s eigenvalues. (Including a “colour” degree-of-freedom, this would become “2Nc ,” where Nc is the number of colours.) 1.7.4 Exercises 1. Verify Eq. (1.149). 2. Verify Eqs. (1.153), (1.154). 3. Verify Eqs. (1.163), (1.164). 4. Verify Eqs. (1.166). 5. Verify Eq. (1.181). 6. Verify Eq. (1.197). 7. Verify Eq. (1.205). 1.8 Functional Integral for Gauge Field Theories To begin I will consider a non-Abelian gauge field theory in the absence of couplings to matter field, which is described by a Lagrangian density: 1 a (x)Faµν (x), L(x) = − Fµν 4 a Fµν (x) = ∂µ Bνa − ∂ν Bµa + gfabc Bµb Bνc , (1.208) (1.209) where g is a coupling constant and fabc are the structure constants of SU (Nc ): i.e., with {T a : a = 1, . . . , Nc2 − 1} denoting the generators of the group [Ta , Tb ] = ifabc Tc . 31 (1.210) a In the fundamental representation {Ta } ≡ { λ2 }, where {λa } are the generalization of the eight Gell-Mann matrices, while in the adjoint representation, relevant to the realization of transformations on the gauge fields, {T a }bc = −ifabc . (1.211) An obvious guess for the form of the generating functional is W [J] = Z [DBµa ] exp Z i 4 d x[L(x) + Jaµ (x)Bµa ] , (1.212) where, as usual, Jaµ (x) is a (classical) external source for the gauge field. It will immediately be observed that this is a Lagrangian-based expression for the generating functional, even though the Hamiltonian derived from Eq. (1.208) is not of the form in Eq. (1.115). It is nevertheless (almost) correct (I will motivate the modifications that need to be made to make it completely correct) and provides the foundation for a manifestly Poincaré covariant quantisation of the field theory. Alternatively, one could work with Coulomb gauge, build the Hamiltonian and construct W [J] in the canonical fashion, as described in Sec. ??, but then covariance is lost. The Coulomb gauge procedure gives the same Smatrix elements (Green functions) as the (corrected-) Lagrangian formalism and hence they are completely equivalent. However, manifest covariance is extremely useful as it often simplifies the allowed form of Green functions and certainly simplifies the calculation of cross sections. Thus the Lagrangian formulation is most often used. The primary fault with Eq. (1.212) is that the free-field part of the Lagrangian density is singular: i.e., the determinant encountered in evaluating the free-gaugeboson generating functional vanishes, and hence the operator cannot be inverted. This is easily demonstrated. Observe that Z 1 d xL0 (x) = − d4 x[∂µ Bνa (x) − ∂ν Bµa (x)][∂ µ Baν (x) − ∂ ν Baµ (x)](1.213) 4Z n o 1 = − d4 xd4 yBµa (x) [−g µν ∂ 2 + ∂ µ ∂ ν ]δ 4 (x − y) Bνa (y) (1.214) 2 Z 1 d4 xd4 yBµa (x)K µν (x, y)Bνa (y) . (1.215) =: 2 Z 4 The operator K µν (x, y) thus determined can be expressed µν K (x − y) = Z d4 q µν 2 µ ν −g q + q q e−i(p,x−y) 4 (2π) (1.216) from which it is apparent that the Fourier amplitude is a projection operator; i.e., a key feature of K µν (x, y) is that it projects onto the space of transverse gauge field configurations: qµ (g µν q 2 − q µ q ν ) = 0 = (g µν q 2 − q µ q ν )qν , 32 (1.217) where q µ is the four-momentum associated with the gauge field. It follows that the W0 [J] obtained from Eq. (1.212) has no damping associated with longitudinal gauge fields and is therefore meaningless. A simple analogy is Z ∞ −∞ dx Z ∞ 2 −∞ dye−(x−y) , (1.218) in which the integrand does not damp along trajectories in the (x, y)-plane related via a spatial translation: (x, y) → (x + a, y + a). Hence there is an overall divergence associated with the translation of the centre of momentum: X = (x + y)/2. Y = (x − y): X → X + (a, a), Y → Y , Z ∞ −∞ dx Z ∞ −∞ dye−(x−y) 2 = = Z ∞ dX −∞ Z ∞ −∞ Z dX ∞ −∞ dY e−Y √ 2 π=∞. (1.219) (1.220) The underlying problem, which is signalled by the behavior just identified, is the R gauge invariance of the action: L(x); i.e., the action is invariant under local field transformation Bµ (x) := igBµa (x)T a → G(x)Bµ (x)G−1 (x) + [∂µ G(x)]G−1 (x) , (1.221) G(x) = exp{−igT a Θa (x)} . (1.222) This means that, given a reference field configuration Bˆµ (x), the integrand in the generating functional, Eq. (1.212), is constant along the path through the gauge field manifold traversed by the applying gauge transformations to Bˆµ (x). Since the parameters characterizing the gauge transformations, Θa , are continuous functions, each such gauge orbit contains an uncountable infinity of gauge field configurations. It is therefore immediately apparent that the generating functional, as written, is is undefined: it contains a multiplicative factor proportional to the length (or volume) of the gauge orbit. (NB. While there is, in addition, an uncountable infinity of distinct reference configurations, the action changes upon any shift orthogonal to a gauge orbit.) Returning to the example in Eq. (1.219), the analogy is that the “Lagrangian density” l(x, y) = (x − y)2 is invariant under translations; i.e., the integrand is invariant under the operation ( ) ∂ ∂ gs (x, y) = exp s +s ∂x ∂y 0 0 (x, y) → (x , y ) = gs (x, y)(x, y) = (x + s, y + s) . (1.223) (1.224) Hence, given a reference point P = (x0 , y0 ) = (1, 0), the integrand is constant along the path (x, y) = P0 + (s, s) through the (x, y)-plane. (This is a translation of the centre-of-mass: X0 = (x0 + y0 )/2 = 1/2 → X0 + s.) Since s∈(−∞, ∞), this 33 path contains an uncountable infinity of points, and at each one the integrand has precisely the same value. The integral thus contains a multiplicative factor proportional to the length of the translation path, which clearly produces an infinite (meaningless) result for the integral. (NB. The value of the integrand changes upon a translation orthogonal to that just identified.) 1.8.1 Faddeev-Popov Determinant and Ghosts This problem with the functional integral over gauge fields was identified by Faddeev and Popov. They proposed to solve the problem by identifying and extracting the gauge orbit volume factor. A Simple Model Before describing the procedure in quantum field theory it can be illustrated using the simple integral model. One begins by defining a functional of our “field variable”, (x, y), which intersects the centre-of-mass translation path once, and only once: f (x, y) = (x + y)/2 − a = 0 . (1.225) Now define a functional ∆f such that ∆f [x, y] Z ∞ −∞ daδ((x + y)/2 − a) = 1 . (1.226) It is clear that ∆f [x, y] is independent of a: 0 0 −1 ∆f [x + a , y + a ] = ã=a−a0 = ∞ Z −∞ Z ∞ −∞ daδ((x + a0 + y + a0 )/2 − a) (1.227) dãδ((x + y)/2 − ã) (1.228) ∆f [x, y]−1 . = (1.229) Using ∆f the model generating functional can be rewritten Z ∞ −∞ dx Z ∞ −∞ dye −l(x,y) = Z ∞ −∞ da Z ∞ −∞ dx Z ∞ −∞ dye−l(x,y) ∆f [x, y]δ((x + y)/2 − a) (1.230) and now one performs a centre-of-mass translation: x → x0 = x + a, y → y 0 = y + a, under which the action is invariant so that the integral becomes Z = ∞ −∞ Z ∞ −∞ da da Z ∞ −∞ Z ∞ −∞ dx dx Z ∞ −∞ Z ∞ −∞ dye−l(x,y) ∆f [x, y]δ((x + y)/2 − a) (1.231) dye−l(x,y) ∆f [x + a, y + a]δ((x + y)/2) . (1.232) 34 Now making use of the a−independence of ∆f [x, y], Eq. (1.227), = Z ∞ −∞ da Z Z ∞ −∞ dx Z ∞ −∞ dye−l(x,y) ∆f [x, y]δ((x + y)/2) . (1.233) In this last line the “volume” or “path length” has been explicitely factored out at the cost of introducing a δ-function, which fixes the centre-of-mass; i.e., a single point on the path of translationally equivalent configurations, and a functional ∆f , which, we will see, is the analogue in this simple model of the Fadeev-Popov determinant. Hence the “correct” definition of the generating functional for this model is w(~j) = Z ∞ −∞ dx Z ∞ −∞ dye−l(x,y)+jx x+jy y ∆f [x, y]δ((x + y)/2) . 35 (1.234) Gauge Fixing Conditions To implement this idea for the real case of non-Abelian gauge fields one envisages an hypersurface, lying in the manifold of all gauge fields, which intersects each gauge orbit once, and only once. This means that if fa [Bµa (x); x] = 0 , a = 1, 2, . . . , Nc2 − 1 , (1.235) is the equation describing the hypersurface, then there is a unique element in each gauge orbit that satisfies one Eq. (1.235), and the set of these unique elements, none of which cannot be obtained from another by a gauge transformation, forms a representative class that alone truly characterises the physical configuration of gauge fields. The gauge-equivalent, and therefore redundant, elements are absent. Equations (1.235) can also be viewed as defining a set of non-linear equations for G(x), Eq (??). This in the sense that for a given field configuration, Bµ (x), it is always possible to find a unique gauge transformation, G1 (x), that yields a gauge transformed field BµG1 (x), from Bµ (x) via Eq. (??), which is the one and only solution of fa [Bµa,G1 (x); x] = 0. Equation (1.235) therefore defines a gauge fixing condition. In order for Eqs. (1.235) to be useful it must be possible that, when given a configuration Bµ (x), for which fa [Bµ (x); x] 6= 0, the equation fa [BµG (x); x] = 0 (1.236) can be solved for the gauge transformation G(x). To see a consequence of this requirement consider a gauge configuration Bµb (x) that almost, but not quite, satisfies Eqs. (1.235). Applying an infinitesimal gauge transformation to this Bµb (x) the requirement entails that it must be possible to solve fa [Bµb (x) → Bµb (x) − g fbdc Bµd (x) δΘc (x) − ∂µ δΘb (x); x] = 0 (1.237) for the infinitesimal gauge transformation parameters δΘa (x). Equation (1.237) can be written (using the chain rule) fa [Bµb (x)] − Z i δfa [Bµb (x); x] h cd e δ ∂ + gf B (y) δΘd (y) = 0 . dy ν ced ν c δBν (y) 4 (1.238) ~ which has a solution for ~θ if, and This looks like the matrix equation f~ = O θ, only if, det O 6= 0, and a similar constraint follows from Eq. (1.238): the gauge fixing conditions can be solved if, and only if, Det Mf i δfa [Bµb (x); x] h cd e := Det − δ ∂ − gf B (y) ν cde ν δBνc (y) ( ) δfa [Bµb (x); x] =: Det − [Dν (y)]cd 6= 0 . δBνc (y) ( ) 36 (1.239) Equation (1.239) is the so-called admissibility condition for gauge fixing conditions. A simple illustration is provided by the lightlike (Hamilton) gauges, which are specified by nµ Bµa (x) = 0 , n2 > 1 , a = 1, 2, . . . , (N 2 − 1) . (1.240) Choosing (nµ ) = (1, 0, 0, 0), the equation for G(x) is, using Eq. (??) ∂ G(t, ~x) = − G(t, ~x) B 0 (t, ~x) , ∂t (1.241) and this nonperturbative equation has the unique solution G(t, ~x) = T exp − Z t −∞ 0 ds B (s, ~x) , (1.242) where T is the time ordering operator. One may compare this with Eq. (1.238), which only provides a perturbative [in g] solution. While that may be an advantage, Eq. (1.242) is not a Poincaré covariant constraint and that makes it difficult to employ in explicit calculations. A number of other commonly used gauge fixing conditions are ∂ µ Bµa (x) = 0 , ∂ µ Bµa (x) = Aa (x) , nµ Bµa (x) = 0 , n2 < 0 , nµ Bµa (x) = 0 , n2 = 0 , ~ · B~a (x) = 0 , ∇ Lorentz gauge Generalised Lorentz gauge Axial gauge Light-like gauge Coulomb gauge (1.243) and the generalised axial, light-like and Coulomb gauges, where an arbitrary function, Aa (x), features on the r.h.s. All of these choices satisfy the admissability condition, Eq. (1.239), for small gauge field variations but in some cases, such as Lorentz gauge, the uniqueness condition fails for large variations; i.e., those that are outside the domain of perturbation theory. This means that there are at least two solutions: G1 , G2 , of Eq. (1.236), and perhaps uncountably many more. Since no nonperturbative solution of any gauge field theory in four spacetime dimensions exists, the actual number of solutions is unknown. If it is infinite then the Fadde’ev-Popov definition of the generating functional fails in that gauge. These additional solutions are called Gribov Copies and their existence raises questions about the correct way to furnish a nonperturbative definition of the generating functional [9], which are currently unanswered. Isolating and Eliminating the Gauge Orbit Volume To proceed one needs a little information about the representation of non-Abelian groups. Suppose u is an element of the group SU (N ). Every such element can be 37 characterised by (N 2 − 1) real parameters: {Θa , a = 1, . . . , N 2 − 1}. Let G(u) be the representation of u under which the gauge fields transform; i.e., the adjoint representation, Eqs. (??), (??). For infinitesimal tranformations G(u) = I − igT a Θa (x) + O(Θ2 ) (1.244) where {T a } are the adjoint representation of the Lie algebra of SU (N ), Eq. (??). Clearly, if u, u0 ∈ SU (N ) then uu0 ∈ SU (n) and G(u)G(u0 ) = G(uu0 ). (These are basic properties of groups.) To define the integral over gauge fields we must properly define the gaugefield “line element”. This is the Hurwitz measure on the group space, which is invariant in the sense that du0 = d(u0 u). In the neighbourhood of the identity one may always choose Y du = dΘa (1.245) a and the invariance means that since the integration represents a sum over all possible values of the parameters Θa , relabelling them as Θ̃a cannot matter. It is now possible to quantise the gauge field; i.e., properly define Eq. (??). Consider ∆f [Bµa ] defined via, cf. Eq. (??), ∆f [Bµb ] Z Y du(x) x δ[fa {Bµb,u (x); x}] = 1 , Y x,a (1.246) where Bµb,u (x) is given by Eq. (??) with G(x) → u(x). ∆f [Bµa ] is gauge invariant: Z Y b,u ∆−1 f [Bµ ] = du0 (x) x,a x Z Y = 0 δ[fa {Bµb,u u (x); x}] Y d(u0 (x)u(x)) x,a x Z Y = Y du00 (x) x 0 δ[fa {Bµb,u u (x); x}] 00 b δ[fa {Bµb,u (x); x}] = ∆−1 f [Bµ ] . Y x,a (1.247) Returning to Eq. (??), one can write W [0] = Z Y Z du(x) [DBµa ] ∆f [Bµa ] x Y x,b δ[fb {Bµa,u (x); x}] exp{i Bµa (x) Now execute a gauge transformation: → invariance of the measure and the action, one has W [0] = [ Z Y x du(x)] Z [DBµa ] ∆f [Bµa ] Y x,b 38 −1 Bµa,u (x), δ[fb {Bµa (x); x}] Z d4 x L(x)} . (1.248) so that, using the Z exp{i d4 x L(x)} , (1.249) where now the integrand of the group measure no longer depends on the group element, u(x) (cf. Eq. (??)). The gauge orbit volume has thus been identified and can be eliminated so that one may define Z W [Jµa ] = [DBµa ] ∆f [Bµa ] Y x,b δ[fb {Bµa (x); x}] exp{i Z d4 x [L(x) + Jaµ (x)Bµa (x)]} . (1.250) Neglecting for now the possible existence of Gribov copies, Eq. (1.250) is the foundation we sought for a manifestly Poincaré covariant quantisation of the gauge field. However, a little more work is needed to mould a practical tool. Ghost Fields A first step is an explicit calculation of ∆f [Bµa ]. Since it always appears multiplied by a δ-function it is sufficient to evaluate it for those field configurations that satisfy Eq. (1.235). Recalling Eqs. (1.238), (1.239), for infinitesimal gauge transformations fa [Bµb,u (x); x] = fa [Bµb (x); x] + = 0+ Z Z d4 y (Mf )ac Θc (y) d4 y (Mf )ac Θc (y) . (1.251) Hence, using the definition, Eq. (1.246), b ∆−1 f [Bµ ] = Z Y a dΘ (x) δ x,a Z 4 d y (Mf )ac Θc (y) (1.252) , (1.253) and changing variables: Θa → Θ̃a = (Mf )ac Θc (y), this gives b ∆−1 f [Bµ ] = Det Mf Z Yn h dΘ̃a (x) δ Θ̃a (y) x,a io where the determinant is the Jacobian of the transformation, so that ∆f [Bµb ] = Det Mf . (1.254) Now recall Eq. (1.180). This means Eq. (1.254) can be expressed as a functional integral over Grassmann fields: φ̄a , φb , a, b = 1, . . . , (N 2 − 1), ∆f [Bµb ] Z = [D φ̄b ][Dφa ] exp − Z 4 4 d xd y φ̄b (x) (Mf )bc (x, y) φa (y) , . (1.255) Consequently, absorbing non-dynamical constants into the normalisation, W [Jµa ] = Z [DBµa ] [D φ̄b ][Dφa ] Y x,b Z 4 × exp i d x [L(x) + δ[fb {Bµa (x); x}] Jaµ (x)Bµa (x)] 39 +i Z 4 4 d xd y φ̄b (x) (Mf )bc (x, y) φa (y) . (1.256) The Grassmann fields {φ̄a , φb } are the Fadde’ev-Popov Ghosts. They are an essential consequence of gauge fixing. As one concrete example, consider the Lorentz gauge, Eq. (1.243), for which h (ML )ab = δ 4 (x − y) δ ab ∂ 2 − gfabc ∂ µ Bµc (x) and therefore W [Jµa ] Z = [DBµa ] [D φ̄b ][Dφa ] Y δ[∂ x,a µ Bµa (x)] Z exp i 4 i (1.257) 1 4 a d x − Faµν (x) Fµν (x) − ∂µ φ̄a (x) ∂ ν φa (x) + gfabc [∂ µ φ̄a (x)]φb (x)Bµc (x) + Jaµ (x)Bµa (x)(1.258) . This expression makes clear that a general consequence of the Fadde’ev-Popov procedure is to introduce a coupling between the gauge field and the ghosts. Thus the ghosts, and hence gauge fixing, can have a direct impact on the behaviour of gauge field Green functions. As another, consider the axial gauge, for which i h (MA )ab = δ 4 (x − y) δ ab nµ ∂ µ , . (1.259) ∂ ν Bµa (x) = Aa (x) , (1.260) Expressing the related determinant via ghost fields it is immediately apparent that with this choice there is no coupling between the ghosts and the gauge field quanta. Hence the ghosts decouple from the theory and may be discarded as they play no dynamical role. However, there is a cost, as always: in this gauge Q the effect of the delta-function, x,a δ[nµ Bµa (x)], in the functional integral is to complicate the Green functions by making them depend explicitly on (nµ ). Even the free-field 2-point function exhibits such a dependence. It is important to note now that this decoupling of the ghost fields is tied to an “accidental” elimination of the fabc term in Dµ (y), Eq. (1.239). That term is always absent in Abelian gauge theories, for which quantum electrodynamics is the archetype, because all generators commute and analogues of fabc must vanish. Hence in Abelian gauge theories ghosts decouple in every gauge. To see how δ[fb {Bµa (x); x}] influences the form of Green functions, consider the generalised Lorentz gauge: where {Aa (x)} are arbitrary functions. The Fadde’ev-Popov determinant is the same in generalised Lorentz gauges as it is in Lorentz gauge and hence ∆GL [Bµa ] = ∆L [Bµa ] , (1.261) where the r.h.s. is given in Eq. (1.257). The generating functional in this gauge is therefore W [Jµa ] = Z [DBµa ] [D φ̄b ][Dφa ] Y x,a δ[∂ µ Bµa (x) a Z − A (x)] exp i 4 1 4 a d x − Faµν (x) Fµν (x) − ∂µ φ̄a (x) ∂ ν φa (x) + gfabc [∂ µ φ̄a (x)]φb (x)Bµc (x) + Jaµ (x)Bµa (x) 40 . Gauge invariance of the generating functional, Eq. (1.246), means that one can integrate over the {Aa (x)}, with a weight function to ensure convergence: Z [DAa ] exp − i Z 4 a d x A (x) Aa (x) , 2λ (1.262) to arrive finally at the generating functional in a covariant Lorentz gauge, specified by the parameter λ: Z W [Jµa , ξ¯ga , ξga ] = [DBµa ] [D φ̄b ][Dφa ] 1 µ a [∂ Bµ (x)] [∂ ν Bνa (x)] 2λ − ∂µ φ̄a (x) ∂ ν φa (x) + gfabc [∂ µ φ̄a (x)]φb (x)Bµc (x) Z exp i 1 4 a d4 x − Faµν (x) Fµν (x) − +Jaµ (x)Bµa (x) + ξ¯ga (x)φa (x) + φ̄a (x)ξga (x) , (1.263) where {ξ¯ga , ξga } are anticommuting external sources for the ghost fields. (NB. To complete the definition one should add convergence terms, iη + , for every field or, preferably, work in Euclidean space.) Observe now that the free gauge boson piece of the action in Eq. (1.263) is := Z 1 2 Z 1 2 d4 x d4 y Bµa (x) K µν (x − y; λ) Bνa (y) 1 d4 x d4 y Bµa (x) [g µν ∂ 2 − ∂ µ ∂ ν (1 − ) − ig µν η + ]δ 4 (x − y) Bνa(1.264) (y) λ The operator K µν (x − y; λ) thus defined can be expressed µν K (x − y) = Z d4 q (2π)4 1 −g (q + iη ) + q q [1 − ] e−i(q,x−y) , λ µν 2 + µ ν (1.265) cf. Eq. (??), and now 1 ν q (1.266) λ so that in this case the action does damp variations in the longitudinal components of the gauge field. K µν (x − y; λ) is the inverse of the free gauge boson propagator; i.e., the free gauge boson 2-point Green function, D µν (x − y), is obtained via Z d4 w Kρµ (x − w) D ρν (w − y) = g µν δ 4 (x − y) , (1.267) qµ K µν = − and hence µν D (x − y) = Z d4 q (2π)4 −g µν qµqν + (1 − λ) 2 q + iη + ! q2 1 e−i(q,x−y) . + + iη (1.268) The obvious λ dependence is a result of the presence of δ[fb {Bµa (x); x}] in the generating functional, and this is one example of the δ-function’s direct effect on the form of Green functions: they are, in general, gauge parameter dependent. 41 1.8.2 Exercises 1. Verify Eq. (1.257). 2. Verify Eq. (1.258). 3. Verify Eq. (1.259). 4. Verify Eq. (1.268). 1.9 Dyson-Schwinger Equations It has long been known that, from the field equations of quantum field theory, one can derive a system of coupled integral equations interrelating all of a theory’s Green functions [6, 10]. This tower of a countable infinity of equations is called the complex of Dyson-Schwinger equations (DSEs). This intrinsically nonperturbative complex is vitally important in proving the renormalisability of quantum field theories and at its simplest level provides a generating tool for perturbation theory. In the context of quantum electrodynamics (QED) I will illustrate a nonperturbative derivation of two equations in this complex. The derivation of others follows the same pattern. 1.9.1 Photon Vacuum Polarization Generating Functional for QED The action for QED with Nf flavours of electomagnetically active fermion, is Nf X 1 1 µ / ψ f − Fµν F µν − S[Aµ , ψ, ψ̄] = d4 x ∂ Aµ (x) ∂ ν Aν (x) , ψ̄ f i∂/ − mf0 + ef0 A 4 2λ 0 f =1 (1.269) f f where: ψ̄ (x), ψ (x) are the elements of the Grassmann algebra that describe the fermion degrees of freedom, mf0 are the fermions’ bare masses and ef0 their charges; and Aµ (x) describes the gauge boson [photon] field, with Z Fµν = ∂µ Aν − ∂ν Aµ , (1.270) and λ0 the bare covariant-Lorentz-gauge fixing parameter. (NB. To describe an electron the physical charge ef < 0.) The derivation of the generating functional in Eq. (1.263) can be employed with little change here. In fact, in this context it is actually simpler because ghost fields decouple. Combining the procedure for fermions and gauge fields, 42 described in Secs. 1.7, ?? respectively, one arrives at Z W [Jµ , ξ, ξ̄] = [DAµ ] [Dψ][D ψ̄] Z exp i + Nf X f =1 4 1 4 d x − F µν (x) Fµν (x) − 1 µ ∂ Aµ (x) ∂ ν Aν (x) 2λ0 / ψ f +J µ (x)Aµ (x) + ξ¯f (x)ψ f (x) + ψ̄ f (x)ξ f (x) ψ̄ f i∂/ − mf0 + ef0 A , where Jµ is an external source for the electromagnetic field, and ξ f , ξ¯f are external sources for the fermion field that, of course, are elements in the Grassmann algebra. (NB. In Abelian gauge theories there are no Gribov copies in the covariant-Lorentz-gauges.) Functional Field Equations As described in Sec. 1.6.1, it is advantageous to work with the generating func¯ ξ] defined via tional of connected Green functions; i.e., Z[Jµ , ξ, n o W [Jµ , ξ, ξ̄] =: exp iZ[Jµ , ξ, ξ̄] . (1.271) The derivation of a DSE now follows simply from the observation that the integral of a total derivative vanishes, given appropriate boundary conditions. Hence, for example, Z Z i h f δ 0 = [DAµ ] [Dψ][D ψ̄] exp i S[Aµ , ψ, ψ̄] + d4 x ψ ξ f + ξ¯f ψ f + Aµ J µ δAµ (x) = Z [DAµ ] [Dψ][D ψ̄] ( ) Z i h f δS + Jµ (x) exp i S[Aµ , ψ, ψ̄] + d4 x ψ ξ f + ξ¯f ψ f + Aµ J µ δAµ (x) ( " # ) δ δ δS δ , = ,− + Jµ (x) W [Jµ , ξ, ξ̄] , (1.272) δAµ (x) iδJ iδ ξ¯ iδξ where the last line has meaning as a functional differential operator on the generating functional. Differentiating Eq. (1.269) gives X f f 1 δS e0 ψ (x)γµ ψ f (x) = ∂ρ ∂ ρ gµν − 1 − ∂µ ∂ν Aν (x) + δAµ (x) λ0 f (1.273) so that the explicit meaning of Eq. (1.272) is −Jµ (x) = ρ ∂ρ ∂ gµν X f δZ 1 − 1− ∂µ ∂ν + e0 λ0 δJν (x) f 43 " δZ δ δ iZ δZ γµ ¯f + f γµ ¯f − f δξ (x) δ ξ (x) δξ (x) δ ξ (x) #! , (1.274) where I have divided through by W [Jµ , ξ, ξ̄]. Equation (1.274) represents a compact form of the nonperturbative equivalent of Maxwell’s equations. One-Particle-Irreducible Green Functions The next step is to introduce the generating functional for one-particle-irreducible (1PI) Green functions: Γ[Aµ , ψ, ψ̄], which is obtained from Z[Jµ , ξ, ξ̄] via a Legendre transformation Z[Jµ , ξ, ξ̄] = Γ[Aµ , ψ, ψ̄] + Z f d4 x ψ ξ f + ξ¯f ψ f + Aµ J µ . h i (1.275) A one-particle-irreducible n-point function or “proper vertex” contains no contributions that become disconnected when a single connected m-point Green function is removed; e.g., via functional differentiation. This is equivalent to the statement that no diagram representing or contributing to a given proper vertex separates into two disconnected diagrams if only one connected propagator is cut. (A detailed explanation is provided in Ref. [3], pp. 289-294.) A simple generalisation of the analysis in Sec. 1.6.1 yields δZ δZ δZ = ψ(x) , = Aµ (x) , ¯ = −ψ̄(x) , µ δJ (x) δξ(x) δ ξ(x) (1.276) where here the external sources are nonzero. Hence Γ in Eq. (1.275) must satisfy δΓ δΓ δΓ = −Jµ (x) , = ξ¯f (x) . = −ξ f (x) , µ f f δA (x) δψ (x) δ ψ̄ (x) (1.277) (NB. Since the sources are not zero then, e.g., δA 6= 0 , δJ µ (x) Aµ (x) = Aµ (x; [Jµ , ξ, ξ̄]) ⇒ (1.278) with analogous statements for the Grassmannian functional derivatives.) It is easy to see that setting ψ̄ = 0 = ψ after differentiating Γ gives zero unless there are equal numbers of ψ̄ and ψ derivatives. (This is analogous to the result for scalar fields in Eq. (1.140).) Now consider the product (with spinor labels r, s, t) − Z δ2Z δ2Γ d4 z f . g ¯ h h ¯ δξr (x)ξt (z) δψt (z)ψ s (y) ξ = ξ = 0 ψ=ψ=0 (1.279) Using Eqs. (1.276), (1.277), this simplifies as follows: = Z δψth (z) δξsg (y) δξsg (y) dz f = f = δrs δ f g δ 4 (x − y) . ¯ h ξ = ξ = 0 δξr (x) δψt (z) δξr (x) ψ = ψ = 0 ψ=ψ=0 (1.280) 4 44 Now returning to Eq. (1.274) and setting ξ¯ = 0 = ξ one obtains i X f h 1 δΓ ν f ρ A (x) − i e tr γ S (x, x; [A ]) , = ∂ ∂ g − 1 − ∂ ∂ µ µ ρ µν µ ν 0 δAµ (x) ψ=ψ=0 λ0 f (1.281) after making the identification S f (x, y; [Aµ ]) = − δ2Z δ2Z = (no summation on f ) , δξ f (y)ξ¯f (x) δ ξ¯f (x)ξ f (y) (1.282) which is the connected Green function that describes the propagation of a fermion with flavour f in an external electromagnetic field Aµ (cf. the free fermion Green function in Eq. (1.197).) I observe that it is a direct consequence of Eq. (1.279) that δ2Γ f −1 , (1.283) S (x, y; [A]) = δψ f (x)δ ψ̄ f (y) ψ=ψ=0 and it is a general property that such functional derivatives of the generating functional for 1PI Green functions are related to the associated propagator’s inverse. Clearly the vacuum fermion propagator or connected fermion 2-point function is S f (x, y) := S f (x, y; [Aµ = 0]) . (1.284) Such vacuum Green functions are of primary interest in quantum field theory. To continue, one differentiates Eq. (1.281) with respect to Aν (y) and sets Jµ (x) = 0, which yields δ2Γ δAµ (x)δAν (y) Aµ = 0 ψ=ψ=0 = ∂ρ ∂ ρ gµν − 1 − −i X f 1 ∂µ ∂ν δ 4 (x − y) λ0 −1 δ2Γ δ ef0 tr γµ . δAν (y) δψ f (x)δ ψ̄ f (x) ψ=ψ=0 (1.285) The l.h.s. is easily understood. Just as Eqs. (1.283), (1.284) define the inverse of the fermion propagator, so here is δ2Γ (D ) (x, y) := µ ν δA (x)δA (y) −1 µν Aµ = 0 ψ=ψ=0 . (1.286) The r.h.s., however, must be simplified and interpreted. First observe that −1 δ δ2Γ f f δAν (y) δψ (x)δ ψ̄ (x) ψ=ψ=0 = 45 − Z −1 δ2Γ d4 ud4 w f f δψ (x)δ ψ̄ (w) ψ=ψ=0 (1.287) which is an analogue of the result for finite dimensional matrices: i dA−1 (x) d h dA(x) −1 A (x) + A(x) A(x)A−1 (x) = I = 0 = dx dx dx dA(x) dA−1 (x) = −A−1 (x) A−1 (x) .(1.288) ⇒ dx dx Equation (1.287) involves the 1PI 3-point function ef0 Γfµ (x, y; z) := δ2Γ . δAν (z) δψ f (x)δ ψ̄ f (y) δ (1.289) This is the proper fermion-gauge-boson vertex. At leading order in perturbation theory Γfν (x, y; z) = γν δ 4 (x − z) δ 4 (y − z) , (1.290) a result which can be obtained via the explicit calculation of the functional derivatives in Eq. (1.289). Now, defining the gauge-boson vacuum polarisation: Πµν (x, y) = i X (ef0 )2 f Z h i d4 z1 d4 z2 tr γµ S f (x, z1 )Γfν (z1 , z2 ; y)S f (z2 , x) , (1.291) it is immediately apparent that Eq. (1.285) may be expressed as −1 µν ρ (D ) (x, y) = ∂ρ ∂ gµν 1 − 1− ∂µ ∂ν δ 4 (x − y) + Πµν (x, y) . λ0 (1.292) In general, the gauge-boson vacuum polarisation, or “self-energy,” describes the modification of the gauge-boson’s propagation characteristics due to the presence of virtual fermion-antifermion pairs in quantum field theory. In particular, the photon vacuum polarisation is an important element in the description of process such as ρ0 → e+ e− . The propagator for a free gauge boson was given in Eq. (1.268). In the presence of interactions; i.e., for Πµν 6= 0 in Eq. (1.292), this becomes D µν (q) = −1 δ2Γ δ2Γ δ , f f f f δAν (y) δψ (u)δ ψ̄ (w) δψ (w)δ ψ̄ (x) ψ=ψ=0 −g µν + (q µ q ν /[q 2 + iη]) 1 qµqν − λ , 0 q 2 + iη 1 + Π(q 2 ) (q 2 + iη)2 (1.293) where I have used the “Ward-Takahashi identity:” qµ Πµν (q) = 0 = Πµν (q) qν , 46 (1.294) which means that one can write Πµν (q) = −g µν q 2 + q µ q ν Π(q 2 ) . (1.295) Π(q 2 ) is the polarisation scalar and, in QED, it is independent of the gauge parameter, λ0 . (NB. λ0 = 1 is called “Feynman gauge” and it is useful in perturbative calculations because it obviously simplifies the Π(q 2 ) = 0 gauge boson propagator enormously. In nonperturbative applications, however, λ0 = 0, “Landau gauge,” is most useful because it ensures the gauge boson propagator itself is transverse.) Ward-Takahashi identities (WTIs) are relations satisfied by n-point Green functions, relations which are an essential consequence of a theory’s local gauge invariance; i.e., local current conservation. They can be proved directly from the generating functional and have physical implications. For example, Eq. (1.295) ensures that the photon remains massless in the presence of charged fermions. (A discussion of WTIs can be found in Ref. [1], pp. 299-303, and Ref. [3], pp. 407-411; and their generalisation to non-Abelian theories as “Slavnov-Taylor” identities is described in Ref. [5], Chap. 2.) In the absence of external sources for fermions and gauge bosons, Eq. (1.291) can easily be represented in momentum space, for then the 2- and 3-point functions that appear explicitly must be translationally invariant and hence can be simply expressed in terms of Fourier amplitudes. This yields iΠµν (q) = − X f (ef0 )2 Z d4 ` tr[(iγµ )(iS f (`))(iΓf (`, ` + q))(iS(` + q))] . (1.296) (2π)d It is the reduction to a single integral that makes momentum space representations most widely used in continuum calculations. In QED the vacuum polarisation is directly related to the running coupling constant. This connection makes its importance obvious. In QCD the connection is not so direct but, nevertheless, the polarisation scalar is a key component in the evaluation of the strong running coupling. In the above analysis we saw that second derivatives of the generating functional, Γ[Aµ , ψ, ψ̄], give the inverse-fermion and -photon propagators and that the third derivative gave the proper photon-fermion vertex. In general, all derivatives of this generating functional, higher than two, produce the corresponding proper Green’s functions, where the number and type of derivatives give the number and type of proper Green functions that it can serve to connect. 1.9.2 Fermion Self Energy Equation (1.274) is a nonperturbative generalisation of Maxwell’s equation in quantum field theory. Its derivation provides the model by which one can obtain 47 an equivalent generalisation of Dirac’s equation. To this end consider that Z Z h g i δ 4 g g g µ ¯ 0 = [DAµ ] [Dψ][D ψ̄] f exp i S[Aµ , ψ, ψ̄] + d x ψ ξ + ξ ψ + Aµ J δ ψ̄ (x) Z = [DAµ ] [Dψ][D ψ̄] ) ( h g i δS f 4 g ¯g ψ g + Aµ J µ + ξ (x) exp i S[A , ψ, ψ̄] + d x ψ ξ + ξ µ δ ψ̄ f (x) " # ) ( δ δS δ δ f + η (x) W [Jµ , ξ, ξ̄] ,− = , δ ψ̄ f (x) iδJ iδ ξ¯ iδξ " ! # δ δ f f = ξ f (x) + i∂/ − m0 + e0 γ µ W [Jµ , ξ, ξ̄] . iδJ µ (x) iδ ξ¯f (x) Z (1.297) (1.298) This is a nonperturbative, funcational equivalent of Dirac’s equation. One can proceed further. A functional derivative with respect to ξ f : δ/δξ f (y), yields 4 δ (x − y)W [Jµ ] − i∂/ − mf0 + ef0 γ µ δ iδJ µ (x) ! W [Jµ ] S f (x, y; [Aµ ]) = (1.299) 0, after setting ξ f = 0 = ξ¯f , where W [Jµ ] := W [Jµ , 0, 0] and S(x, y; [Aµ ]) is defined in Eq. (1.282). Now, using Eqs. (1.271), (1.277), this can be rewritten 4 δ (x − y) − i∂/ − mf0 + ef0 A /(x; [J]) + ef0 γ µ δ iδJ µ (x) ! S f (x, y; [Aµ ]) = (1.300) 0, which defines the nonperturbative connected 2-point fermion Green function (This is clearly the functional equivalent of Eq. (1.86).) The electromagentic four-potential vanishes in the absence of an external source; i.e., Aµ (x; [J = 0]) = 0, so it remains only to exhibit the content of the remaining functional differentiation in Eq. (1.300), which can be accomplished using Eq. (1.287): δ iδJ µ (x) S f (x, y; [Aµ ]) = = Z −1 δAν (z) δ δ2Γ d4 z µ f f iδJ (x) δAν (z) δψ (x)δ ψ̄ (y) ψ=ψ=0 −ef0 = −ef0 Z Z d4 z d 4 u d4 w δAν (z) f S (x, u) Γν (u, w; z) S(w, y) iδJµ (x) d4 z d4 u d4 w iDµν (x − z) S f (x, u) Γν (u, w; z) S(w, y) , (1.301) where, in the last line, I have set J = 0 and used Eq. (1.286). Hence, in the absence of external sources, Eq. (1.300) is equivalent to δ 4 (x − y) = i∂/ − mf0 S f (x, y) − i (ef0 )2 Z d4 z d4 u d4 w D µν (x, z) γµ S(x, u) Γν (u, w; z) S(w, y) = δ 4 (x (1.302) − y) . 48 Just as the photon vacuum polarisation was introduced to simplify, or reexpress, the DSE for the gauge boson propagator, Eq. (1.291), one can define a fermion self-energy: Z Σf (x, z) = i(ef0 )2 d4 u d4 w D µν (x, z) γµ S(x, u) Γν (u, w; z) , (1.303) so that Eq. (1.302) assumes the form Z d4 z h i i∂/x − mf0 δ 4 (x − z) − Σf (x, z) S(z, y) = δ 4 (x − y) . (1.304) Again using the property that Green functions are translationally invariant in the absence of external sources, the equation for the self-energy can be written in momentum space: Σf (p) = i (ef0 )2 Z d4 ` D µν (p − `) [iγµ ] [iS f (`)] [iΓfν (`, p)] . (2π)4 (1.305) In terms of the self-energy, it follows from Eq. (1.304) that the connected fermion 2-point function can be written in momentum space as S f (p) = /p − mf0 1 . − Σf (p) + iη + (1.306) Equation (1.305) is the exact Gap Equation. It describes the manner in which the propagation characteristics of a fermion moving through the ground state of QED (the QED vacuum) is altered by the repeated emission and reabsorption of virtual photons. The equation can also describe the real process of Bremsstrahlung. Furthermore, the solution of the analogous equation in QCD its solution provides information about dynamical chiral symmetry breaking and also quark confinement. 1.9.3 Exercises 1. Verify Eq. (1.277). 2. Verify Eq. (1.287). 3. Verify Eq. (1.296). 4. Verify Eq. (1.300). 49 1.10 Perturbation Theory 1.10.1 Quark Self Energy A key feature of strong interaction physics is dynamical chiral symmetry breaking (DCSB). In order to understand it one must first come to terms with explicit chiral symmetry breaking. Consider then the DSE for the quark self-energy in QCD: −i Σ(p) = −g02 Z d4 ` i D µν (p − `) λa γµ S(`) Γaν (`, p) , 4 2 (2π) (1.307) where I have suppressed the flavour label. The form is precisely the same as that in QED, Eq. (1.305), with the only difference being the introduction of the colour (Gell-Mann) matrices: {λa ; a = 1, . . . , 8} at the fermion-gauge-boson vertex. The interpretation of the symbols is also analogous: D µν (`) is the connected gluon 2-point function and Γaν (`, `0 ) is the proper quark-gluon vertex. The one-loop contribution to the quark’s self-energy is obtained by evaluating the r.h.s. of Eq. (1.307) using the free quark and gluon propagators, and the quark-gluon vertex: i (1.308) Γaν (0) (`, `0 ) = λa γν , 2 which appears to be a straightforward task. To be explicit, the goal is to calculate (2) −g02 −i Σ (p) = Z i 2 d4 k (2π)4 × λa γ µ −g µν kµ kν + (1 − λ0 ) 2 k + iη + ! 1 i a λ γµ k6 + /p − m0 + iη + 2 k2 1 + iη + (1.309) and one may proceed as follows. First observe that Eq. (1.309) can be re-expressed as (2) −i Σ (p) = −g02 × C2 (R) ( Z d4 k 1 1 2 4 2 2 + (2π) (k + p) − m0 + iη k + iη + ) (k, p)6k γ (6k + /p + m0 ) γµ − (1 − λ0 ) (6k − /p + m0 ) − 2 (1 − λ0 ) 2 k + iη + (1.310) µ where I have used 1 a1 a λ λ 2 2 = C2 (R) I c ; C2 (R) = Nc2 − 1 , 2Nc (1.311) with Nc the number of colours (Nc = 3 in QCD) and I c , the identity matrix in colour space. Now note that 2 (k, p) = [(k + p)2 − m20 ] − [k 2 ] − [p2 − m20 ] 50 (1.312) and hence (2) −i Σ (p) = −g02 ( C2 (R) Z d4 k 1 1 2 4 2 + 2 (2π) (k + p) − m0 + iη k + iη + γ µ (6k + /p + m0 ) γµ + (1 − λ0 ) (p / − m0 ) 2 + (1 − λ0 ) (p − m20 ) ) k6 k6 − (1 − λ0 ) [(k + p)2 − m20 ] 2 . 2 + k + iη k + iη + (1.313) Focusing on the last term: d4 k 1 1 k6 [(k + p)2 − m20 ] 2 2 4 2 + 2 + (2π) (k + p) − m0 + iη k + iη k + iη + Z 1 k6 d4 k =0 (1.314) = 4 2 + 2 (2π) k + iη k + iη + Z because the integrand is odd under k → −k, and so this term in Eq. (1.313) vanishes, leaving Z −i Σ(2) (p) = −g02 C2 (R) ( 1 1 d4 k 2 4 2 + 2 (2π) (k + p) − m0 + iη k + iη + γ µ (6k + /p + m0 ) γµ + (1 − λ0 ) (p / − m0 ) ) k6 + (1 − λ0 ) (p2 − m20 ) 2 . k + iη + (1.315) Consider now the second term: (1 − λ0 ) (p / − m0 ) Z d4 k 1 1 . 2 4 2 2 + (2π) (k + p) − m0 + iη k + iη + In particular, focus on the behaviour of the integrand at large k 2 : 1 1 1 k 2 →±∞ ∼ . 2 2 2 2 + 2 + (k + p) − m0 + iη k + iη (k − m0 + iη + ) (k 2 + iη + ) (1.316) The integrand has poles in the second and fourth quadrant of the k0 -plane but vanishes on any circle of radius R → ∞. That means one may rotate the contour anticlockwise to find Z ∞ 1 dk 0 2 2 (k − m0 + iη + ) (k 2 + iη + ) 0 Z i∞ 1 = dk 0 2 0 ([k 0 ]2 − ~k 2 − m0 + iη + )([k 0 ]2 − ~k 2 + iη + ) Z ∞ 1 k 0 →ik4 dk4 = i . (1.317) 0 (−k42 − ~k 2 − m20 ) (−k42 − ~k 2 ) 51 Performing a similar analysis of the Z R0 −∞ part one obtains the complete result: d4 k 1 = i 2 4 2 (2π) (k − m0 + iη + ) (k 2 + iη + ) Z d3 k (2π)3 Z 1 dk4 . 2 2 −∞ 2π (−~ k − k4 − m20 ) (−~k 2 − k42 ) (1.318) ∞ These two steps constitute what is called a “Wick rotation.” The integral on the r.h.s. is defined in a four-dimensional Euclidean space; i.e., k 2 := k12 + k22 + k32 + k42 ≥ 0 . . . k 2 is nonnegative. A general vector in this space can be written in the form: (k) = |k| (cos φ sin θ sin β, sin φ sin θ sin β, cos θ sin β, cos β) , (1.319) and clearly k 2 = |k|2 . In this space the four-vector measure factor is Z d4E k f (k1 , . . . , k4 ) = 1 2 Z ∞ 0 2 2 dk k Z π 0 2 dβ sin β Z π 0 dθ sin θ Z 2π 0 dφ f (k, β, θ, φ) (1.320) Returning now to Eq. (1.316) the large k behaviour of the integral can be determined via Z Z ∞ d4 k i 1 1 1 ≈ dk 2 2 2 4 2 + 2 + 2 (2π) (k + p) − m0 + iη k + iη 16π 0 (k + m20 ) Z Λ2 i 1 = lim dx 2 16π Λ→∞ 0 x + m20 i 2 2 lim ln 1 + Λ /m → ∞; = 0 16π 2 Λ→∞ (1.321) 2 i.e., after all this work, the result is meaningless: the one-loop contribution to the quark’s self-energy is divergent! Such “ultraviolet” divergences, and others which are more complicated, appear whenever loops appear in perturbation theory. (The others include “infrared” divergences associated with the gluons’ masslessness; e.g., consider what would happen in Eq. (1.321) with m0 → 0.) In a renormalisable quantum field theory there exists a well-defined set of rules that can be used to render perturbation theory sensible. First, however, one must regularise the theory; i.e., introduce a cutoff, or some other means, to make finite every integral that appears. Then every step in the calculation of an observable is rigorously sensible. Renormalisation follows; i.e, the absorption of divergences, and the redefinition of couplings and masses, so that finally one arrives at S-matrix amplitudes that are finite and physically meaningful. The regularisation procedure must preserve the Ward-Takahashi identities (the Slavnov-Taylor identities in QCD) because they are crucial in proving that a theory can be sensibly renormalised. A theory is called renormalisable if, and only if, there are a finite number of different types of divergent integral so that only a finite number of masses and coupling constants need to be renormalised. 52 1.10.2 Dimensional Regularization The Pauli-Villars prescription is favoured in QED and that is described, for example, in Ref. [3]. In perturbative QCD, however, “dimensional regularisation” is the most commonly used procedure and I will introduce that herein. The key to the method is to give meaning to the divergent integrals by changing the dimension of spacetime. Returning to the exemplar, Eq. (1.316), this means we consider T = Z dD k 1 1 2 D 2 + 2 (2π) (k + p) − m0 + iη k + iη + (1.322) where D is the dimension of spacetime and is not necessarily four. Observe now that Γ(α + β) Z 1 1 xα−1 (1 − x)β−1 = , dx a α bβ Γ(α) Γ(β) 0 [a x + b (1 − x)]α+β (1.323) where Γ(x) is the gamma-function: Γ(n + 1) = n!. This is an example of what is commonly called “Feynman’s parametrisation,” and one can now write T = Z 1 0 dx Z dD k 1 .(1.324) 2 (2π)D [(k − xp)2 − m0 (1 − x) + p2 x(1 − x) + iη + ]2 (1.325) The momentum integral is well-defined for D = 1, 2, 3 but, as we have seen, not for D = 4. One proceeds under the assumption that D is such that the integral is convergent then a shift of variables is permitted: T Z k→k−xp = 1 0 dx Z dD k 1 , D 2 2 (2π) [k − a + iη + ]2 (1.326) where a2 = m20 (1 − x) − p2 x(1 − x). I will consider a generalisation of the momentum integral: In = Z 1 dD k , D 2 2 (2π) [k − a + iη + ]n (1.327) and perform a Wick rotation to obtain In = i (−1)n (2π)D Z dD k [k 2 1 . + a 2 ]n (1.328) The integrand has an O(D) spherical symmetry and therefore the angular integrals can be performed: SD := Z 2π D/2 dΩD = . Γ(D/2) 53 (1.329) Clearly, S4 = 2π 2 , as we saw in Eq. (1.321). Hence In (−1)n 1 = i D−1 D/2 2 π Γ(D/2) Z ∞ 0 dk k D−1 . (k 2 + a2 )n (1.330) Writing D = 4 + 2 one arrives at i In = (−a2 )2−n 2 (4π) a2 4π ! Γ(n − 2 − ) . Γ(n) (1.331) (NB. Every step is rigorously justified as long as 2n > D.) The important point for continuing with this procedure is that the analytic continuation of Γ(x) is unique and that means one may use Eq. (1.331) as the definition of In whenever the integral is ill-defined. I observe that D = 4 is recovered via the limit → 0− and the divergence of the integral for n = 2 in this case is encoded in Γ(n − 2 − ) = Γ(−) = 1 − γE + O() ; − (1.332) i.e., in the pole in the gamma-function. (γE is the Euler constant.) Substituting Eq. (1.331) in Eq. (1.326) and setting n = 2 yields i Γ(−) T = (gν ) (4π)2 (4π) 2 Z 1 0 p2 m20 dx (1 − x) − 2 x(1 − x) ν2 ν " # , (1.333) wherein I have employed a nugatory transformation to introduce the mass-scale ν. It is the limit → 0− that is of interest, in which case it follows that (x = exp ln x ≈ 1 + ln x) 1 m20 1 p2 i dx ln − − γ + ln 4π − (1 − x) − x(1 − x) T = (gν )2 E (4π)2 ν2 ν2 0 ( ! " #) i 1 p2 m20 m20 2 = (gν ) − − γE + ln 4π + 2 − ln 2 − 1 − 2 ln 1 − 2 . (4π)2 ν p m0 (1.334) ( Z " #) It is important to understand the physical content of Eq. (1.334). While it is only one part of the gluon’s contribution to the quark’s self-energy, many of its properties hold generally. 1. Observe that T (p2 ) is well-defined for p2 < m20 ; i.e., for all spacelike momenta and for a small domain of timelike momenta. However, at p2 = m20 , T (p2 ) exhibits a ln-branch-point and hence T (p2 ) acquires an imaginary part for p2 > m20 . This imaginary part describes the real, physical process by which a quark emits a massless gluon; i.e., gluon Bremsstrahlung. In QCD this is one element in the collection of processes referred to as “quark fragmentation.” 54 2. The mass-scale, ν, introduced in Eq. (1.333), which is a theoretical artifice, does not affect the position of the branch-point, which is very good because that branch-point is associated with observable phenomena. While it may appear that ν affects the magnitude of physical cross sections because it modifies the coupling, that is not really so: in going to D = 2n − dimensions the coupling constant, which was dimensionless for D = 4, has acquired a mass dimension and so the physical, dimensionless coupling constant is α := (gν )2 /(4π). It is this dependence on ν that opens the door to the generation of a “running coupling constant” and “running masses” that are a hallmark of quantum field theory. 3. A number of constants have appeared in T (p2 ). These are irrelevant because they are eliminated in the renormalisation procedure. (NB. So far we have only regularised the expression. Renormalisation is another step.) 4. It is apparent that dimensional regularisation gives meaning to divergent integrals without introducing new couplings or new fields. That is a benefit. The cost is that while γ5 = iγ 0 γ 1 γ 2 γ 3 is well-defined with particular properties for D = 4, a generalisation to D 6= 4 is difficult and hence so is the study of chiral symmetry. D-dimensional Dirac Algebra When one employs dimensional regularisation all the algebraic manipulations must be performed before the integrals are evaluated, and that includes the Dirac algebra. The Clifford algebra is unchanged in D-dimensions {γ µ , γ ν } = 2 g µν ; µ, ν = 1, . . . , D − 1 , (1.335) gµν g µν = D (1.336) D 1D , (2 − D) γ ν , 4 g νλ 1D + (D − 4) γ ν γ λ , −2 γ ρ γ λ γ ν + (4 − D) γ ν γ λ γ ρ , (1.337) (1.338) (1.339) (1.340) but now and hence γµ γ ν γµ γ ν γ µ γµ γ ν γ λ γ µ γµ γ ν γ λ γ ρ γ µ = = = = where 1D is the D × D-dimensional unit matrix. It is also necessary to evaluate traces of products of Dirac matrices. For a D-dimensional space, with D even, the only irreducible representation of the Clifford algebra, Eq. (1.335), has dimension f (D) = 2D/2 . In any calculation it is the (anti-)commutation of Dirac matrices that leads to physically important 55 factors associated with the dimension of spacetime while f (D) always appears as a common multiplicative factor. Hence one can just set f (D) ≡ 4 (1.341) in all calculations. Any other prescription merely leads to constant terms of the type encountered above; e.g., γE , which are eliminated in renormalisation. The D-dimensional generalisation of γ5 is a more intricate problem. However, I will not use it herein and hence will omit that discussion. Observations on the Appearance of Divergences Consider a general Lagrangian density: L(x) = L0 (x) + X gi Li (x) , (1.342) i where L0 is the sum of the free-particle Lagrangian densities and Li (x) represents the interaction terms with the coupling constants, gi , written explicitly. Assume that Li (x) has fi fermion fields (fi must be even since fermion fields always appear in the pairs ψ̄, ψ), bi boson fields and n∂i derivatives. The action S= Z d4 x L(x) (1.343) must be a dimensionless scalar and therefore L(x) must have mass-dimension M 4 . Clearly a derivative operator has dimension M . Hence, looking at the freeparticle Lagrangian densities, it is plain that each fermion field has dimension M 3/2 and each boson field, dimension M 1 . It follows that a coupling constant multiplying the interaction Lagrangian density Li (x) must have mass-dimension 3 2 [gi ] = M 4−di , di = fi + bi + n∂i . (1.344) It is a fundamentally important fact in quantum field theory that if there is even one coupling constant for which [gi ] < 0 (1.345) then the theory possesses infinitely many different types of divergences and hence cannot be rendered finite through a finite number of renormalisations. This defines the term nonrenormalisable. In the past nonrenormalisable theories were rejected as having no predictive power: if one has to remove infinitely many infinite constants before one can define a result then that result cannot be meaningful. However, the modern view is different. Chiral perturbation theory is a nonrenormalisable “effective theory.” However, at any finite order in perturbation theory there is only a finite number 56 of undetermined constants: 2 at leading order; 6 more, making 8 in total, when one-loop effects are considered; and more than 140 new terms when two-loop effects are admitted. Nevertheless, as long as there is a domain in some physical external-parameter space on which the one-loop corrected Lagrangian density can be assumed to be a good approximation, and there is more data that can be described than there are undetermined constants, then the “effective theory” can be useful as a tool for correlating observables and elucidating the symmetries that underly the general pattern of hadronic behaviour. 1.10.3 Regularized Quark Self Energy I can now return and re-express Eq. (1.315): (2) 2 −i Σ (p) = −(g0 ν ) C2 (R) ( Z dD k 1 1 1 2 4 2 2 + 2 (2π) ν (k + p) − m0 + iη k + iη + γ µ (6k + /p + m0 ) γµ + (1 − λ0 ) (p / − m0 ) 2 + (1 − λ0 ) (p − m20 ) ) k6 . k 2 + iη + (1.346) It can be separated into a sum of two terms, each proportional to a different Dirac structure: Σ(p /) = ΣV (p2 ) /p + ΣS (p2 ) 1D , (1.347) that can be obtained via trace projection: ΣV (p2 ) = 1 1 1 2 tr [p / Σ(p / )] , Σ (p ) = trD [Σ(p /)] . D S f (D) p2 f (D) (1.348) These are the vector and scalar parts of the dressed-quark self-energy, and they are easily found to be given by 2 dD k 1 (2π)4 ν 2 ( 1 [(k + p)2 − m20 + iη + ] [k 2 + iη + ] #) " pµ k µ 2 µ 2 2 2 . × (2 − D)(p + pµ k ) + (1 − λ0 )p + (1 − λ0 )(p − m0 ) 2 k + iη + (1.349) Z D d k 1 m0 (D − 1 + λ0 ) ΣS (p2 ) = −i (g0 ν )2 C2 (R) . (1.350) (2π)4 ν 2 [(k + p)2 − m20 + iη + ] [k 2 + iη + ] 2 2 p ΣV (p ) = −i (g0 ν ) C2 (R) Z (NB. As promised, the factor of f (D) has cancelled.) 57 These equations involve integrals of the general form dD k (2π)4 Z dD k J µ (α, β; p2 , m2 ) = (2π)4 2 Z 2 I(α, β; p , m ) = 1 1 ,(1.351) 2 2 2 ν [(k + p) − m + iη + ]α [k 2 + iη + ]β 1 kµ ,(1.352) ν 2 [(k + p)2 − m2 + iη + ]α [k 2 + iη + ]β the first of which we have already encountered in Sec. 1.10.2. The general results are (D = 4 + 2) i I(α, β; p2 , m2 ) = (4π)2 p2 × − 4πν 2 ! 1 p2 m2 1− 2 p !α+β−2 !2+−α−β i J (α, β; p , m ) = p (4π)2 µ 2 2 Γ(α + β − 2 − ) Γ(2 + − β) Γ(α) Γ(2 + ) 1 p2 µ ! m2 p2 1 − × − 4πν 2 p2 =: pµ J(α, β; p2 , m2 ) , 2 F1 (α !α+β−2 !2+−α−β + β − 2 − , 2 + − β, 2 + ; 1 ), 1 − (m2 /p2 ) (1.353) Γ(α + β − 2 − ) Γ(3 + − β) Γ(α) Γ(3 + ) 2 F1 (α + β − 2 − , 3 + − β, 3 + ; 1 ) 1 − (m2 /p2 ) (1.354) where 2 F1 (a, b, c; z) is the hypergeometric function. Returning again to Eqs. (1.349), (1.350) it is plain that ΣV (p2 ) = −i (g0 ν )2 C2 (R) h nh 2(1 + ) J(1, 1) − (1 + 2) I(1, 1) − (p2 − m20 ) J(1, 2) +λ0 (p2 − m20 ) J(1, 2) − I(1, 1) io , i (1.355) ΣS (p2 ) = −i (g0 ν )2 C2 (R) m0 (3 + λ0 + 2) I(1, 1; p2, m2 ) , (1.356) where I have omitted the (p2 , m20 ) component in the arguments of I, J. The integrals explicitly required are i p2 1 I(1, 1) = − + ln 4π − γE − ln − 2 + 2 (4π)2 ν ! ! !) 2 2 2 m m m2 m , − 2 ln − 2 − 1 − 2 ln 1 − 2 p p p p ( ! i 1 1 p2 J(1, 1) = − + ln 4π − γE − ln − 2 + 2 (4π)2 2 ν ( 2 − m p2 ! 2 2− m p2 ! 2 ln − m p2 ! − 1 − 2 58 2 " (1.357) # 2 2 m m + 2 2 p p ln 2 1− m p2 ! − 2 m , p2 (1.358) ( 2 2 i 1 m m − 2 ln − 2 2 2 (4π) p p p J(1, 2) = ! 2 + 2 m m ln 1 − 2 2 p p ! +1 ) . (1.359) Using these expressions it is straightforward to show that the λ0 -independent term in Eq. (1.355) is identically zero in D = 4 dimensions; i.e., for − > 0− , and hence (gν )2 C2 (R) ΣV (p2 ) = λ0 (4π)2 m2 −1 − 20 + 1 − p ( 1 m2 − ln 4π + γE + ln 20 ν ! !) 4 2 m0 p ln 1 − 2 . 4 p m (1.360) It is obvious that in Landau gauge: λ0 = 0, in four spacetime dimensions: ΣV (p2 ) ≡ 0, at this order. The scalar piece of the quark’s self-energy is also easily found: (gν )2 ΣS (p ) = m0 C2 (R) (4π)2 2 m20 1 −(3 + λ0 ) − ln 4π + γE + ln 2 ν ! !) 2 2 p m . (1.361) +2(2 + λ0 ) − (3 + λ0 ) 1 − 20 ln 1 − 2 p m0 ( " # Note that in Yennie gauge: λ0 = −3, in four dimensions, the scalar piece of the self-energy is momentum-independent, at this order. We now have the complete regularised dressed-quark self-energy at one-loop order in perturbation theory and its structure is precisely as I described in Sec. 1.10.2. Renormalisation must follow. One final observation: the scalar piece of the self-energy is proportional to the bare current-quark mass, m0 . That is true at every order in perturbation theory. Clearly then lim ΣS (p2 , m20 ) = 0 (1.362) m0 →0 and hence dynamical chiral symmetry breaking is impossible in perturbation theory. 1.10.4 Exercises 1. Verify Eq. (1.321). 2. Verify Eq. (1.328). 3. Verify Eq. (1.334). 4. Verify Eqs. (1.349) and (1.350). 5. Verify Eqs. (1.360) and (1.361). 59 1.11 Renormalized Quark Self Energy Hitherto I have illustrated the manner in which dimensional regularisation is employed to give sense to the divergent integrals that appear in the perturbative calculation of matrix elements in quantum field theory. It is now necessary to renormalise the theory; i.e., to provide a well-defined prescription for the elimination of all those parts in the calculated matric element that express the divergences and thereby obtain finite results for Green functions in the limit → 0− (or with the removal of whatever other parameter has been used to regularise the divergences). 1.11.1 Renormalized Lagrangian The bare QCD Lagrangian density is 1 2 1 − 2 1 − 4 L(x) = − ∂µ Bνa (x)[∂ µ B νa (x) − ∂ ν B µa (x)] − 1 ∂µ Bνa (x)∂µ Bνa (x) 2λ g fabc [∂ µ B νa (x) − ∂ ν B µa (x)]Bµb (x)Bνc (x) g 2 fabc fade Bµb (x)Bνc (x)B µd (x)B νe (x) −∂µ φ̄a (x)∂ µ φa (x) + g fabc ∂µ φ̄a (x) φb (x)B µc (x) +q̄ f (x)i∂/q f (x) − mf q̄ f (x)q f (x) + 1 2 g q̄ f (x)λaB / a (x)q f (x) ,(1.363) where: Bµa (x) are the gluon fields, with the colour label a = 1, . . . , 8; φ̄a (x), φa (x) are the (Grassmannian) ghost fields; q̄ f (x), q f (x) are the (Grassmannian) quark fields, with the flavour label f = u, d, s, c, b, t; and g, mf , λ are, respectively, the coupling, mass and gauge fixing parameter. (NB. The QED Lagrangian density is immediately obtained by setting fabc ≡ 0. It is clearly the non-Abelian nature of the gauge group, SU (Nc ), that generates the gluon self-couplings, the triple-gluon and four-gluon vertices, and the ghost-gluon interaction.) The elimination of the divergent parts in the expression for a Green function can be achieved by adding “counterterms” to the bare QCD Lagrangian density, one for each different type of divergence in the theory; i.e., one considers the renormalised Lagrangian density LR (x) := L(x) + Lc (x) , (1.364) with Lc (x) = C3Y M 1 2 +C1Y M 1 4 ∂µ Bνa (x)[∂ µ B νa (x) − ∂ ν B µa (x)] + C6 1 2 1 ∂µ Bνa (x)∂µ Bνa (x) 2λ g fabc [∂ µ B νa (x) − ∂ ν B µa (x)]Bµb (x)Bνc (x) +C5 g 2 fabc fade Bµb (x)Bνc (x)B µd (x)B νe (x) 60 +C̃3 ∂µ φ̄a (x)∂ µ φa (x) − C̃1 g fabc ∂µ φ̄a (x) φb (x)B µc (x) −C2F q̄ f (x)i∂/q f (x) + C4 mf q̄ f (x)q f (x) − C1F 1 2 g q̄ f (x)λaB / a (x)q f (x) . (1.365) To prove the renormalisability of QCD one must establish that the coefficients, Ci , each understood as a power series in g 2 , are the only additional terms necessary to remove all the ultraviolet divergences in the theory at every order in the perturbative expansion. In the example of Sec. 1.10.2 I illustrated that the divergent terms in the regularised self-energy are proportional to g 2 . This is a general property and hence all of the Ci begin with a g 2 term. The Ci -dependent terms can be treated just as the terms in the original Lagrangian density and yield corrections to the expressions we have already derived that begin with an order-g 2 term. Returning to the example of the dressed-quark self-energy this means that we have an additional contribution: ∆Σ(2) (p /) = C2F /p − C4 m , (1.366) and one can choose C2F , C4 such that the total self-energy is finite. The renormalisation constants are introduced as follows: Zi := 1 − Ci (1.367) so that Eq. (1.365) becomes Z3Y M Z ∂µ Bνa (x)[∂ µ B νa (x) − ∂ ν B µa (x)] − 6 ∂µ Bνa (x)∂µ Bνa (x) 2 2λ Z1Y M µ νa ν µa b g fabc [∂ B (x) − ∂ B (x)]Bµ (x)Bνc (x) − 2 Z − 5 g 2 fabc fade Bµb (x)Bνc (x)B µd (x)B νe (x) 4 LR (x) = − −Z̃3 ∂µ φ̄a (x)∂ µ φa (x) + Z̃1 g fabc ∂µ φ̄a (x) φb (x)B µc (x) +Z2F q̄ f (x)i∂/q f (x) − Z4 mf q̄ f (x)q f (x) + Z1Y M 2 g q̄ f (x)λaB / a (x)q f (x) , (1.368) NB. I have implicitly assumed that the renormalisation counterterms, and hence the renormalisation constants, are flavour independent. It is always possible to choose prescriptions such that this is so. I will now introduce the bare fields, coupling constants, masses and gauge 61 fixing parameter: 1/2 1/2 B0µa (x) := Z3Y M B µa (x) , q0f (x) := Z2F q f (x) , 1/2 φ̄a0 (x) := Z̃3 φ̄a (x) , 1/2 φa0 (x) := Z̃3 φa (x) , −3/2 g0Y M := Z1Y M Z3Y M g , −1/2 −1/2 g̃0 := Z̃1 Z̃3−1 Z3Y M , (1.369) 1/2 −1 g0F := Z1F Z3Y M Z2F g, −1 g05 := Z5 Z3Y M g −1 mf0 := Z4 Z2F mf , λ0 := Z6−1 Z3Y M λ . The fields and couplings on the r.h.s. of these definitions are called renormalised, and the couplings are finite and the fields produce Green functions that are finite even in D = 4 dimensions. ( NB. All these quantities are defined in D = 4 + 2dimensional space. Hence one has for the Lagrangian density: [L(x)] = M D , and the field and coupling dimensions are [q(x)] = [q̄(x)] = M 3/2+ , [B µ (x)] = M 1+ , [φ(x)] = [φ̄(x)] = M 1+ , [g] = M − , [λ] = M 0 , [m] = M 1 .) (1.370) The renormalised Lagrangian density can be rewritten in terms of the bare quantities: 1 2 1 − 2 1 − 4 a LR (x) = − ∂µ B0ν (x)[∂ µ B0νa (x) − ∂ ν B0µa (x)] − 1 a a ∂µ B0ν (x)∂µ B0ν (x) 2λ0 a a g0Y M fabc [∂ µ B0νa (x) − ∂ ν B0µa (x)]B0µ (x)B0ν (x) 2 b c g05 fabc fade B0µ (x)B0ν (x)B0µd (x)B0νe (x) −∂µ φ̄a0 (x)∂ µ φa0 (x) + g̃0 fabc ∂µ φ̄a0 (x) φb0 (x)B0µc (x) +q̄0f (x)i∂/q0f (x) − mf0 q̄0f (x)q0f (x) + 1 2 g0F q̄0f (x)λaB / a0 (x)q0f (x) . (1.371) It is apparent that now the couplings are different and hence LR (x) is not invariant under local gauge transformations (more properly BRST transformations) unless g0Y M = g̃0 = g0F = g05 = g0 . (1.372) Therefore, if the renormalisation procedure is to preserve the character of the gauge theory, the renormalisation constants cannot be completely arbitrary but must satisfy the following “Slavnov-Taylor” identities: g0Y M = g̃0 g0Y M = g0F g0Y M = g05 Z̃3 Z3Y M = , Z1Y M Z̃1 Z2F Z3Y M = , ⇒ Z1Y M Z1F Z2 ⇒ Z5 = 1Y M . Z3Y M ⇒ 62 (1.373) In QED the second of these equations becomes the Ward-Takahashi identity: Z1F = Z2F . 1.11.2 Renormalization Schemes At this point we can immediately write an expression for the renormalised dressedquark self-energy: (2) ΣR (p /) = ΣV (p2 ) + C2F /p + ΣS (p2 ) − C4 m . (1.374) The subtraction constants are not yet determined and there are many ways one may choose them in order to eliminate the divergent parts of bare Green functions. Minimal Subtraction In the minimal subtraction (MS) scheme one defines a dimensionless coupling α := (gν )2 4π (1.375) and considers each counterterm as a power series in α with the form Ci = j ∞ X X (2j) Ci,k j=1 k=1 1 k α π j , (1.376) where the coefficients in the expansion may, at most, depend on the gauge parameter, λ. Using Eqs. (1.360), (1.361) and (1.374) we have (2) ΣR (p /) m2 α 1 1 λ C2 (R) − ln 4π + γE + ln 2 (1.377) = /p π 4 ν ! !# ) m2 m4 p2 −1 − 2 + 1 − 4 ln 1 − 2 + C2F (1.378) p p m ( " # α1 1 m2 +m C2 (R) −(3 + λ) − ln 4π + γE + ln 2 (1.379) π 4 ν ! ! ) p2 m2 +2(2 + λ) − (3 + λ) 1 − 2 ln 1 − 2 − C4 . (1.380) p m ( " Now one chooses C2F , C4 such that they cancel the 1/ terms in this equation and therefore, at one-loop level, α 1 1 λ C2 (R) , π 4 α 1 1 = 1 + (3 + λ) C2 (R) , 4 π Z2F = 1 − C2F = 1 + Z4 = 1 − C 4 63 (1.381) (1.382) and hence Eq. (1.380) becomes (2) ΣR (p /) m2 α 1 λ C2 (R) − ln 4π + γE + ln 2 = /p π 4 ν ! !# ) 2 4 2 m m p −1 − 2 + 1 − 4 ln 1 − 2 + C2F p p m ( ! m2 α1 C2 (R) −(3 + λ) − ln 4π + γE + ln 2 +m π 4 ν ! ! ) 2 2 m p +2(2 + λ) − (3 + λ) 1 − 2 ln 1 − 2 − C4 p m " ( ! (1.383) (1.384) (1.385) , (1.386) which is the desired, finite result for the dressed-quark self-energy. It is not common to work explicitly with the counterterms. More often one uses Eq. (1.371) and the definition of the connected 2-point quark Green function (an obvious analogue of Eq. (1.282)): iSRf (x, y; m, λ, α) = − δ 2 ZR [Jµa , ξ, ξ̄] −1 = h0|q f (x) q̄ f (y)|0i = Z2F h0|q0f (x) q̄0f (y)|0i f f ¯ δξ (y)δ ξ (x) (1.387) to write n o −1 SR (p /; m, λ, α) = lim Z2F S0 (p /; m0 , λ0 , α0 ; ) , →0 (1.388) where in the r.h.s. m0 , λ0 , α0 have to be substituted by their expressions in terms of the renormalised quantities and the limit taken order by order in α. To illustrate this using our concrete example, Eq. (1.388) yields n o n 1 − ΣV R (p2 ; m, λ, α) /p − m 1 + ΣSR (p2 ; m, λ, α)/m = lim− Z2F →0 nh i h o 1 − ΣV (p2 ; m0 , λ0 , α0 ) /p − m0 1 + ΣS (p2 ; m0 , λ0 , α0 )/m0 io (1.389) and taking into account that m = Z4−1 Z2F m0 then 1 − ΣV R (p2 ; m, λ, α) = 1 + ΣSR (p2 ; m, λ, α)/m = lim− Z2F →0 n n o 1 − ΣV (p2 ; m0 , λ0 , α0 ) , (1.390) o lim− Z4 1 + ΣS (p2 ; m0 , λ0 , α0 )/m0 .(1.391) → Now the renormalisation constants, Z2F , Z4 , are chosen so as to exactly cancel the 1/ poles in the r.h.s. of Eqs. (1.390), (1.391). Using Eqs. (1.360), (1.361), Eqs. (1.381), (1.382) are immediately reproduced. Modified Minimal Subtraction The modified minimal substraction scheme MS is also often used in QCD. It takes advantage of the fact that the 1/ pole obtained using dimensional regularisation 64 always appears in the combination 1 − ln 4π + γE (1.392) so that the renormalisation constants are defined so as to eliminate this combination, in its entirety, from the Green functions. At one-loop order the renormalisation constants in the MS scheme are trivially related to those in the MS scheme. At higher orders there are different ways of defining the scheme and the relation between the renormalisation constants is not so simple. Momentum Subtraction In the momentum subtraction scheme (µ-scheme) a given renormalised Green function, GR , is obtained from its regularised counterpart, G, by subtracting from G its value at some arbitrarily chosen momentum scale. In QCD that scale is always chosen to be a Euclidean momentum: p2 = −µ2 . Returning to our example of the dressed-quark self-energy, in this scheme ΣAR (p2 ; µ2 ) := ΣA (p2 ; ) − ΣA (p2 ; ) ; A = V, S , (1.393) and so ! ( α(µ) 1 1 1 = λ(µ) C2 (R) −m2 (µ) 2 + 2 4 π p µ !) ! ! ! 4 2 m (µ) p m4 (µ) µ2 + 1− , ln 1 − − 1− ln 1 + 2 p4 m(µ)2 µ4 m (µ) (1.394) α(µ) 1 (2) C2 (R) {−[3 + λ(µ)] ΣSR (p2 ; µ2 ) = m(µ) 4 π " ! ! ! !#) m2 (µ) p2 m2 (µ) µ2 × 1− ln 1 − 2 − 1+ ln 1 + 2 p2 m (µ) µ2 m (µ) (1.395) (2) ΣV R (p2 ; µ2 ) where the renormalised quantities depend on the point at which the renormalisation has been conducted. Clearly, from Eqs. (1.390), (1.391), the renormalisation constants in this scheme are (2) Z2F (2) Z4 (2) = 1 + ΣV (p2 = −µ2 ; ) = 1− (2) ΣS (p2 2 = −µ ; )/m(µ) . (1.396) (1.397) It is apparent that in this scheme there is at least one point, the renormalisation mass-scale, µ, at which there are no higher order corrections to any of the Green functions: the corrections are all absorbed into the redefinitions of the coupling constant, masses and gauge parameter. This is valuable if the coefficients of 65 the higher order corrections, calculated with the parameters defined through the momentum space subtraction, are small so that the procedure converges rapidly on a large momentum domain. In this sense the µ-scheme is easier to understand and more intuitive than the MS or MS schemes. Another advantage is the manifest applicability of the “decoupling theorem,” Refs. [11], which states that quark flavours whose masses are larger than the scale chosen for µ are irrelevant. This last feature, however, also emphasises that the renormalisation constants are flavour dependent and that can be a nuisance. Nevertheless, the µ-scheme is extremely useful in nonperturbative analyses of DSEs, especially since the flavour dependence of the renormalisation constants is minimal for light quarks when the Euclidean substraction point, µ, is chosen to be very large; i.e., much larger than their current-masses. 1.11.3 Renormalized Gap Equation Equation (1.307) is the unrenormalised QCD gap equation, which can be rewritten as −iS0−1 (p) = −i(p / − m0 ) + g02 Z d4 ` i D0µν (p − `) λa γµ S0 (`) Γa0ν (`, p) . (1.398) 4 2 (2π) The renormalised equation can be derived directly from the generating functional defined using the renormalised Lagrangian density, Eq. (1.368), simply by repeating the steps described in Sec. 1.9.2. Alternatively, one can use Eqs. (1.371) to derive an array of relations similar to Eq. (1.387): µν −1 a ΓRν (k, p) , D0µν (k) = Z3Y M DR (k) , Γa0ν (k, p) = Z1F (1.399) and others that I will not use here. (NB. It is a general feature that propagators are multiplied by the renormalisation constant and proper vertices by the inverse of renormalisation constants.) Now one can replace the unrenormalised couplings, masses and Green functions by their renormalised forms: −1 −iZ2−1 SR−1 (p) = −ip / + iZ2F Z 4 mR Z 4 d` i µν 2 −1 −2 2 −1 a +Z1F Z3Y Z3Y M DR (p − `) λa γµ Z2F SR (`) Z1F ΓRν (`, p) , M Z2F gR 4 2 (2π) (1.400) which simplifies to −iΣR (p) = i(Z2F − 1) /p − i(Z4 − 1) mR Z d4 ` i −Z1F gR2 D µν (p − `) λa γµ SR (`) ΓaRν (`, p) 2 (2π)4 R =: i(Z2F − 1) /p − i(Z4 − 1) mR − iΣ0 (p) , (1.401) 66 where Σ0 (p) is the regularised self-energy. In the simplest application of the µ-scheme one would choose a large Euclidean mass-scale, µ2 , and define the renormalisation constants such that ΣR (/ p + µ = 0) = 0 (1.402) which entails Z2F = 1 + Σ0V (p / + µ = 0) , Z4 = 1 − Σ0S (p / + µ = 0)/mR (µ) , (1.403) where I have used Σ0 (p) = Σ0 (p) /p + Σ0S (p). (cf. Eqs. (1.396), (1.397).) This is simple to implement, even nonperturbatively, and is always appropriate in QCD because confinement ensures that dressed-quarks do not have a mass-shell. On-shell Renormalization If one is treating fermions that do have a mass-shell; e.g., electrons, then an on-shell renormalisation scheme may be more appropriate. One fixes the renormalisation constants such that SR−1 (p /) p /=mR = /p − mR , (1.404) which is interpreted as a constraint on the pole position and the residue at the pole; i.e., since S −1 (p /) = [/ p − mR − ΣR (p /)]|p/=mR " d − [/ p − mR ] ΣR (p /) d/ p then Eq. (1.404) entails ΣR (p /)|p/=mR # +... (1.405) p /=mR d = 0, ΣR (p /) = 0. d/ p p /=mR (1.406) The second of these equations requires Z2F d d 0 2 (1.407) = 1 + Σ0V (m2R ) + 2 m2R 2 Σ0V (p2 ) 2 2 + 2 mR Σ (p ) 2 2 p =mR dp dp2 S p =m R and the first: Z4 = Z2F − Σ0V (m2R ) − Σ0S (m2R ) . 67 (1.408) 1.11.4 Exercises 1. Using Eqs. (1.360), (1.361) derive Eqs. (1.381), (1.382). 2. Verify Eqs. (1.396), (1.397). 3. Verify Eq. (1.401). 4. Verify Eq. (1.403). 5. Beginning with Eq. (1.404), derive Eqs. (1.407), (1.408). NB. /p/p = p2 and d d hence dp f (p2 ) = dp f (p //p) = 2 /p dpd2 f (p2 ). / / 1.12 Dynamical Chiral Symmetry Breaking This phenomenon profoundly affects the character of the hadron spectrum. However, it is intrinsically nonperturbative and therefore its understanding is best sought via a Euclidean formulation of quantum field theory, which I will now summarise. 1.12.1 Euclidean Metric I will formally describe the construction of the Euclidean counterpart to a given Minkowski space field theory using the free fermion as an example. The Lagrangian density for free Dirac fields is given in Eq. (1.186): Lψ0 (x) = Z ∞ −∞ dt Z (1.409) dtE f (−itE ) (1.410) d3 x ψ̄(x) i∂/ − m + iη + ψ(x) . Recall now the observations at the end of Sec. 1.5 regarding the role of the iη + in this equation: it was introduced to provide a damping factor in the generating functional. The alternative proposed was to change variables and introduce a Euclidean time: t → −itE . As usual Z ∞ −∞ dt f (t) = Z ∞ −∞ d(−itE ) f (−itE ) = −i Z ∞ −∞ if f (t) vanishes on the curve at infinity in the second and fourth quadrants of the complex-t plane and is analytic therein. To complete this Wick rotation, however, we must also determine its affect on i∂/, since that is part of the “f (t);” i.e., the integrand: i∂/ = iγ 0 ∂ ∂ + iγ i i ∂t ∂x M →E → = =: 68 ∂ ∂ + iγ i i E ∂(−it ) ∂x ∂ ∂ −γ 0 E − (−iγ i ) i ∂t ∂x ∂ −γµE E , ∂xµ iγ 0 (1.411) (1.412) where (xE x, x4 := −it) and the Euclidean Dirac matrices are µ ) = (~ γ4E = γ 0 γiE = −iγ i . (1.413) These matrices are Hermitian and satisfy the algebra {γµE , γνE } = 2δµν ; µ, ν = 1, . . . , 4, (1.414) where δµν is the four-dimensional Kroncker delta. Henceforth I will adopt the notation a E · bE = 4 X E aE µ bν . (1.415) µ=1 Now, assuming that the integrand is analytic where necessary, one arrives formally at Z ∞ −∞ = −i Z dt ∞ Z −∞ d3 x ψ̄(x) i∂/ − m + iη + ψ(x) dtE Z d3 x ψ̄(xE ) −γ E · ∂ E − m + iη + ψ(xE ) . (1.416) Intepreted naively, however, the action on the r.h.s. poses problems: it is not real if one inteprets ψ̄(xE ) = ψ † (xE ) γ4 . Let us study its involution AE = Z d4 xE ψ̄(xE ) (−γ E · ∂ E − m + iη + ) ψ(xE ) (1.417) where ψ̄r (xE ) and ψs (xE ) are intepreted as members of a Grassmann algebra with involution, as described in Sec. 1.7.1: AE = Z = Z E ] − m δ + iη + δ ) ψ (xE ) d4 xE ψ̄r (xE ) (−[γ E · ∂→ rs rs rs s 4 E E d x ψ̄s (x ) −[γ E · E] ∂← rs ∗ + ! − m δrs − iη δrs ψr (xE ) . (1.418) We know that the mass is real: m∗ = m, and that as an operator the gradient is anti-Hermitian; i.e., E = −∂ E ∂← → (1.419) [γµE ]rs = [(γµE )T ]sr . (1.420) (γµE )T := −CE γµE CE† , (1.421) but that leaves us with If we define 69 where CE = γ2 γ4 is the Euclidean charge conjugation matrix then AE = A E (1.422) when ψ̄ and ψ are members of a Grassmann algebra with involution and we take the limit η + = 0, which we will soon see is permissable. Return now to the generating functional of Eq. (1.187): ¯ ξ] = W [ξ, Z Z [D ψ̄(x)][Dψ(x)] exp i ¯ d4 x ψ̄(x) i∂/ − m + iη + ψ(x) + ψ̄(x)ξ(x) + ξ(x)ψ(x) h Z h d4 xE ψ̄(xE ) γ · ∂ + m − iη + ψ(xE ) i (1.424) so that the Euclidean generating functional ¯ ξ] := W E [ξ, Z × exp − [D ψ̄(xE )][Dψ(xE )] Z ¯ E )ψ(xE ) d x ψ̄(x ) (γ · ∂ + m) ψ(x ) + ψ̄(x )ξ(x ) + ξ(x 4 h E E E E i (1.425) involves a positive-definite measure because the free-fermion action is real, as I have just shown. Hence the “+iη + ” convergence factor is unnecessary here. Euclidean Formulation as Definitive Working in Euclidean space is more than simply a pragmatic artifice: it is possible to view the Euclidean formulation of a field theory as definitive (see, for example, Refs. [12, 13, 14, 15]). In addition, the discrete lattice formulation in Euclidean space has allowed some progress to be made in attempting to answer existence questions for interacting gauge field theories [15]. The moments of the Euclidean measure defined by an interacting quantum field theory are the Schwinger functions: S n (xE,1 , . . . , xE,n ) (1.426) which can be obtained as usual via functional differentiation of the analogue of Eq. (1.425) and subsequently setting the sources to zero. The Schwinger functions are sometimes called Euclidean space Green functions. 70 . (1.423) A Wick rotation of the action transforms the source-independent part of the integrand, called the measure, into exp − i Given a measure and given that it satisfies certain conditions [i.e., the Wightman and Haag-Kastler axioms], then it can be shown that the Wightman functions, W n (x1 , . . . , xn ), can be obtained from the Schwinger functions by analytic continuation in each of the time coordinates: W n (x1 , . . . , xn ) = lim S n ([~x1 , x14 + ix01 ], . . . , [~xn , xn4 + ix0n ]) i x4 →0 (1.427) with x01 < x02 < . . . < x0n . These Wightman functions are simply the vacuum expectation values of products of field operators from which the Green functions [i.e., the Minkowski space propagators] are obtained through the inclusion of step[θ-] functions in order to obtain the appropriate time ordering. (This is described in some detail in Refs. [13, 14, 15].) Thus the Schwinger functions contain all of the information necessary to calculate physical observables. This notion is used directly in obtaining masses and charge radii in lattice simulations of QCD, and it can also be employed in the DSE approach since the Euclidean space DSEs can all be derived from the appropriate Euclidean generating functional using the methods of Sec. 1.9 and the solutions of these equations are the Schwinger functions. All this provides a good reason to employ a Euclidean formulation. Another is a desire to maintain contact with perturbation theory where the renormalisation group equations for QCD and their solutions are best understood [16]. Collected Formulae for Minkowski ↔ Euclidean Transcription To make clear my conventions (henceforth I will omit the supescript E used to denote Euclidean four-vectors): for 4-vectors a, b: a · b := aµ bν δµν := 4 X a i bi , (1.428) i=1 so that a spacelike vector, Qµ , has Q2 > 0; the Dirac matrices are Hermitian and defined by the algebra {γµ , γν } = 2 δµν ; (1.429) and I use γ5 := − γ1 γ2 γ3 γ4 (1.430) tr [γ5 γµ γν γρ γσ ] = −4 εµνρσ , ε1234 = 1 . (1.431) so that The Dirac-like representation of these matrices is: ~γ = 0 −i~τ i~τ 0 ! , γ4 = 71 τ0 0 0 −τ 0 ! , (1.432) where the 2 × 2 Pauli matrices are: 1 0 0 1 0 τ = ! 0 1 1 0 1 , τ = ! 0 −i i 0 2 , τ = ! 3 , τ = 1 0 0 −1 ! . (1.433) Using these conventions the [unrenormalised] Euclidean QCD action is S[B, q, q̄] = Z 1 1 a a Fµν + ∂ · B a ∂ · B a + d4 x Fµν 4 2λ Nf X f =1 1 2 q̄f γ · ∂ + mf + ig λa γ · B a qf , (1.434) a where Fµν = ∂µ Bνa − ∂ν Bµa − gf abc Bµb Bνc . The generating functional follows: W [J, ξ, ξ̄] = Z dµ(q̄, q, B, ω̄, ω) exp Z i h d4 x q̄ ξ + ξ¯ q + Jµa Aaµ , (1.435) with sources: η̄, η, J, and a functional integral measure dµ(q̄, q, B, ω̄, ω) := (1.436) Y Y YY a g a a Dq̄φ (x)Dqφ (x) Dω̄ (x)Dω (x) DBµ (x) exp(−S[B, q, q̄] − S [B, ω, ω̄]) , x µ a φ where φ represents both the flavour and colour index of the quark field, and ω̄ and ω are the scalar, Grassmann [ghost] fields. The normalisation W [η̄ = 0, η = 0, J = 0] = 1 (1.437) is implicit in the measure. As we saw in Sec. 1.8.1, the ghosts only couple directly to the gauge field: Sg [B, ω, ω̄] = Z h i d4 x −∂µ ω̄ a ∂µ ω a − gf abc ∂µ ω̄ a ω b Bµc , (1.438) and restore unitarity in the subspace of transverse [physical] gauge fields. I note that, practically, the normalisation means that ghost fields are unnecessary in the calculation of gauge invariant observables using lattice-regularised QCD because the gauge-orbit volume-divergence in the generating functional, associated with the uncountable infinity of gauge-equivalent gluon field configurations in the continuum, is rendered finite by the simple expedient of only summing over a finite number of configurations. It is possible to derive every equation introduced above, assuming certain analytic properties of the integrands. However, the derivations can be sidestepped using the following transcription rules: 72 Configuration Space 1. Z M d4 xM → −i Momentum Space Z E d 4 xE 1. Z M d4 k M → i Z E 2. ∂ / → iγ E · ∂ E 2. k / → −iγ E · k E 3. A / → −iγ E · AE 3. A / → −iγ E · AE 4. Aµ B µ → −AE · B E 4. kµ q µ → −k E · q E 5. xµ ∂µ → xE · ∂ E 5. kµ xµ → −k E · xE d4 k E These rules are valid in perturbation theory; i.e., the correct Minkowski space integral for a given diagram will be obtained by applying these rules to the Euclidean integral: they take account of the change of variables and rotation of the contour. However, for the diagrams that represent DSEs, which involve dressed n-point functions whose analytic structure is not known as priori, the Minkowski space equation obtained using this prescription will have the right appearance but it’s solutions may bear no relation to the analytic continuation of the solution of the Euclidean equation. The differences will be nonperturbative in origin. 1.12.2 Chiral Symmetry Gauge theories with massless fermions have a chiral symmetry. Its effect can be visualised by considering the helicity: λ ∝ J · p, the projection of the fermion’s spin onto its direction of motion. λ is a Poincaré invariant spin observable that takes a value of ±1. The chirality operator can be realised as a 4 × 4-matrix, γ 5 , and a chiral transformation is then represented as a rotation of the 4 × 1-matrix quark spinor field q(x) → eiγ5 θ q(x) . (1.439) A chiral rotation through θ = π/2 has no effect on a λ = +1 quark, qλ= + → qλ= + , but changes the sign of a λ = −1 quark field, qλ= − → − qλ= − . In composite hadrons this is manifest as a flip in their parity: J P = + ↔ J P = − ; i.e., a θ = π/2 chiral rotation is equivalent to a parity transformation. Exact chiral symmetry therefore entails that degenerate parity multiplets must be present in the spectrum of the theory. For many reasons, the masses of the u- and d-quarks are expected to be very small; i.e., mu ∼ md ΛQCD . Therefore chiral symmetry should only be weakly broken, with the strong interaction spectrum exhibiting nearly degenerate parity partners. The experimental comparison is presented in Eq. (1.440): + N ( 12 , 938) − N ( 21 , 1535) π(0− , 140) a0 (0+ , 980) 73 ρ(1− , 770) . a1 (1+ , 1260) (1.440) Clearly the expectation is very badly violated, with the splitting much too large to be described by the small current-quark masses. What is wrong? Chiral symmetry can be related to properties of the quark propagator, S(p). For a free quark (remember, I am now using Euclidean conventions) m − iγ · p (1.441) S0 (p) = m2 + p 2 and as a matrix m −iγ · p + e2iγ5 θ 2 (1.442) S0 (p) → eiγ5 θ S0 (p)eiγ5 θ = 2 2 p +m p +m2 under a chiral transformation. As anticipated, for m = 0, S0 (p) → S0 (p); i.e., the symmetry breaking term is proportional to the current-quark mass and it can be measured by the “quark condensate” Z Z d4 p m d4 p tr [S(p)] ∝ , (1.443) −hq̄qi := 4 4 2 (2π) (2π) p + m2 which is the “Cooper-pair” density in QCD. For a free quark the condensate vanishes if m = 0 but what is the effect of interactions? As we have seen, interactions dress the quark propagator so that it takes the form −iγ · pA(p2 ) + B(p2 ) 1 = 2 2 2 , (1.444) S(p) := iγ · p + Σ(p) p A (p ) + B 2 (p2 ) where Σ(p) is the self energy, expressed in terms of the scalar functions: A and B, which are p2 -dependent because the interaction is momentum-dependent. On the valid domain; i.e., for weak-coupling, they can be calculated in perturbation theory and at one-loop order, Eq. (1.395) with p2 → −(pE )2 µ2 , m2 (µ), αS p2 B(p ) = m 1 − ln π m2 " 2 #! , (1.445) which is ∝ m. This result persists: at every order in perturbation theory every mass-like correction to S(p) is ∝ m so that m is apparently the only source of chiral symmetry breaking and hq̄qi ∝ m → 0 as m → 0. The current-quark masses are the only explicit chiral symmetry breaking terms in QCD. However, symmetries can be “dynamically” broken. Consider a point-particle in a rotationally invariant potential V (σ, π) = (σ 2 + π 2 − 1)2 , where (σ, π) are the particle’s coordinates. In the state depicted in Fig. 1.1, the particle is stationary at an extremum of the action that is rotationally invariant but unstable. In the ground state of the system, the particle is stationary at any point (σ, π) in the trough of the potential, for which σ 2 + π 2 = 1. There are an uncountable infinity of such vacua, |θi, which are related one to another by rotations in the (σ, π)plane. The vacua are degenerate but not rotationally invariant and hence, in general, hθ|σ|θi 6= 0. In this case the rotational invariance of the Hamiltonian is not exhibited in any single ground state: the symmetry is dynamically broken with interactions being responsible for hθ|σ|θi 6= 0. 74 • 1.5 1 1 0.5 0.5 0 0 -1 -0.5 -0.5 0 0.5 -1 1 Figure 1.1: A rotationally invariant but unstable extremum of the Hamiltonian obtained with the potential V (σ, π) = (σ 2 + π 2 − 1)2 . 1.12.3 Mass Where There Was None The analogue in QCD is hq̄qi 6= 0 when m = 0. At any finite order in perturbation theory that is impossible. However, using the Dyson-Schwinger equation [DSE] for the quark self energy [the QCD “gap equation”]: iγ · p A(p2 ) + B(p2 ) = Z2 iγ · p + Z4 m Z Λ d4 ` 2 λa 1 +Z1 g D (p − `) γ Γa (`, p) , (1.446) µν µ 4 2 (2π) 2 iγ · `A(` ) + B(`2 ) ν depicted in Fig. 1.2, it is possible to sum infinitely many contributions. That allows one to expose effects in QCD which are inaccessible in perturbation theR ory. [NB. In Eq. (1.446), m is the Λ-dependent current-quark bare mass and Λ represents mnemonically a translationally-invariant regularisation of the integral, with Λ the regularisation mass-scale. The final stage of any calculation is to remove the regularisation by taking the limit Λ → ∞. The quark-gluon-vertex and quark wave function renormalisation constants, Z1 (ζ 2 , Λ2 ) and Z2 (ζ 2 , Λ2 ), depend on the renormalisation point, ζ, and the regularisation mass-scale, as does the mass renormalisation constant Zm (ζ 2 , Λ2 ) := Z2 (ζ 2 , Λ2 )−1 Z4 (ζ 2 , Λ2 ).] The quark DSE is a nonlinear integral equation for A and B, and it is the nonlinearity that makes possible a generation of nonperturbative effects. The 75 Σ D = γ S Γ Figure 1.2: DSE for the dressed-quark self-energy. The kernel of this equation is constructed from the dressed-gluon propagator (D - spring) and the dressedquark-gluon vertex (Γ - open circle). One of the vertices is bare (labelled by γ) as required to avoid over-counting. kernel of the equation is composed of the dressed-gluon propagator: g 2 Dµν (k) = δµν kµ kν − 2 k ! g2 G(k 2 ) 2 , G(k ) := , k2 [1 + Π(k 2 )] (1.447) where Π(k 2 ) is the vacuum polarisation, which contains all the dynamical information about gluon propagation, and the dressed-quark-gluon vertex: Γaµ (k, p). The bare (undressed) vertex is Γaµ (k, p)bare = γµ λa . 2 (1.448) Once Dµν and Γaµ are known, Eq. (1.446) is straightforward to solve by iteration. One chooses an initial seed for the solution functions: 0A and 0B, and evaluates the integral on the right-hand-side (r.h.s.). The bare propagator values: 0A = 1 and 0B = m are often adequate. This first iteration yields new functions: 1A and 1B, which are reintroduced on the r.h.s. to yield 2A and 2B, etc. The procedure is repeated until nA = n+1A and nB = n+1B to the desired accuracy. It is now easy to illustrate DCSB, and I will use three simple examples. Nambu–Jona-Lasinio Model The Nambu–Jona-Lasinio model [17] has been popularised as a model of lowenergy QCD. The commonly used gap equation is obtained from Eq. (1.446) via the substitution 1 g 2 Dµν (p − `) → δµν 2 θ(Λ2 − `2 ) (1.449) mG in combination with Eq. (1.448). The step-function in Eq. (1.449) provides a momentum-space cutoff. That is necessary to define the model, which is not 76 renormalisable and hence can be regularised but not renormalised. Λ therefore persists as a model-parameter, an external mass-scale that cannot be eliminated. The gap equation is iγ · p A(p2 ) + B(p2 ) = iγ · p + m + 4 3 1 m2G Z d4 ` −iγ · `A(`2 ) + B(`2 ) 2 2 θ(Λ − ` ) γ γ(1.450) µ µ, (2π)4 `2 A2 (`2 ) + B 2 (`2 ) where I have set the renormalisation constants equal to one in order to complete the defintion of the model. Multiplying Eq. (1.450) by (−iγ · p) and tracing over Dirac indices one obtains 2 2 2 p A(p ) = p + 8 3 1 m2G Z d4 ` A(`2 ) 2 2 θ(Λ − ` ) p · ` 2 2 2 , (2π)4 ` A (` ) + B 2 (`2 ) (1.451) from which it is immediately apparent that A(p2 ) ≡ 1 . (1.452) This property owes itself to the the fact that the NJL model is defined by a fourfermion contact interaction in configuration space, which entails the momentumindependence of the interaction in momentum space. Simply tracing over Dirac indices and using Eq. (1.452) one obtains B(p2 ) = m + 16 3 1 Z d4 ` B(`2 ) 2 2 , θ(Λ − ` ) m2G (2π)4 `2 + B 2 (`2 ) (1.453) from which it is plain that B(p2 ) = constant = M is the only solution. This, too, is a result of the momentum-independence of the model’s interaction. Evaluating the angular integrals, Eq. (1.453) becomes 1 1 Z Λ2 M 1 1 M =m+ 2 2 , dx x = m + M 2 2 C(M 2 , Λ2 )(1.454) 2 3π mG 0 x+M 3π mG h C(M 2 , Λ2 ) = Λ2 − M 2 ln 1 + Λ2 /M 2 i (1.455) Λ defines the model’s mass-scale and I will henceforth set it equal to one so that all other dimensioned quantities are given in units of this scale, in which case the gap equation can be written M =m+M 1 1 C(M 2 , 1) . 3π 2 m2G (1.456) Irrespective of the value of mG , this equation always admits a solution M 6= 0 when the current-quark mass m 6= 0. Consider now the chiral-limit, m = 0, wherein the gap equation is M =M 1 1 C(M 2 , 1) . 2 2 3π mG 77 (1.457) This equation admits a solution M ≡ 0, which corresponds to the perturbative case considered above: when the bare mass of the fermion is zero in the beginning then no mass is generated via interactions. It follows from Eq. (1.443) that the condensate is also zero. This situation can be described as that of a theory without a mass gap: the negative energy Dirac sea is populated all the way up to E = 0. Suppose however that M 6= 0 in Eq. (1.457), then the equation becomes 1= 1 1 C(M 2 , 1) . 3π 2 m2G (1.458) It is easy to see that C(M 2 , 1) is a monotonically decreasing function of M with a maximum value at M = 0: C(0, 1) = 1. Consequently Eq. (1.458) has a M 6= 0 solution if, and only if, 1 1 > 1; (1.459) 3π 2 m2G i.e., if and only if Λ2 ' (0.2 GeV )2 (1.460) 3π 2 for a typical value of Λ ∼ 1 GeV. Thus, even when the bare mass is zero, the NJL model admits a dynamically generated mass for the fermion when the coupling exceeds a given minimum value, which is called the critical coupling: the chiral symmetry is dynamically broken! (In this presentation the critical coupling is expressed via a maximum value of the dynamical gluon mass, mG .) At these strong couplings the theory exhibits a nonperturbatively generated gap: the initially massless fermions and antifermions become massive via interaction with their own “gluon” field. Now the negative energy Dirac sea is only filled up to E = −M , with the positive energy states beginning at E = +M ; i.e., the theory has a dynamically-generated, nonperturbative mass-gap ∆ = 2M . In addition the quark condensate, which was zero when evaluated perturbatively because m = 0, is now nonzero. Importantly, the nature of the solution of Eq. (1.456) also changes qualitatively when mG is allowed to fall below it’s critical value. It is in this way that dynamical chiral symmetry breaking (DCSB) continues to affect the hadronic spectrum even when the quarks have a small but nonzero current-mass. m2G < Munczek-Nemirovsky Model The gap equation for a model proposed more recently [18], which is able to represent a greater variety of the features of QCD while retaining much of the simplicity of the NJL model, is obtained from Eq. (1.446) by using G(k 2 ) = (2π)4 G δ 4 (k) k2 78 (1.461) in Eq. (1.447) with the bare vertex, Eq. (1.448). Here G defines the model’s mass-scale. The gap equation is iγ · p A(p2 ) + B(p2 ) = iγ · p + m + G γµ −iγ · p A(p2 ) + B(p2 ) γµ (,1.462) p2 A2 (p2 ) + B 2 (p2 ) where again the renormalisation constants have been set equal to one but in this case because the model is ultraviolet finite; i.e., there are no infinities that must be regularised and subtracted. The gap equation yields the following two coupled equations: A(p2 ) p2 A2 (p2 ) + B 2 (p2 ) B(p2 ) B(p2 ) = m + 4 2 2 2 , p A (p ) + B 2 (p2 ) A(p2 ) = 1 + 2 (1.463) (1.464) where I have set the mass-scale G = 1. Consider the chiral limit equation for B(p2 ): B(p2 ) = 4 B(p2 ) . p2 A2 (p2 ) + B 2 (p2 ) (1.465) The existence of a B 6≡ 0 solution; i.e., a solution that dynamically breaks chiral symmetry, requires p2 A2 (p2 ) + B 2 (p2 ) = 4 , (1.466) measured in units of G. Substituting this identity into equation Eq. (1.463) one finds 1 A(p2 ) − 1 = A(p2 ) ⇒ A(p2 ) ≡ 2 , (1.467) 2 which in turn entails 2 q B(p ) = 2 1 − p2 . (1.468) The physical requirement that the quark self energy be real in the spacelike region means that this solution is only acceptable for p2 ≤ 1. For p2 > 1 one must choose the B ≡ 0 solution of Eq. (1.465), and on this domain we then find from Eq. (1.463) that q 2 1 2 ⇒ A(p ) = A(p ) = 1 + 2 1 + 1 + 8/p2 . 2 p A(p2 ) 2 (1.469) Putting this all together, the Munczek-Nemirovsky model exhibits the dynamical chiral symmetry breaking solution: A(p2 ) = B(p2 ) = 2; q 1 1 + 1 + 8/p2 2 ( √ 1 − p2 ; 0; 79 p2 ≤ 1 p2 > 1 . ; p2 ≤ 1 p2 > 1 (1.470) (1.471) which yields a nonzero quark condensate. Note that the dressed-quark self-energy is momentum dependent, as is the case in QCD. It is important to observe that this solution is continuous and defined for all 2 p , even p2 < 0 which corresponds to timelike momenta, and furthermore p2 A2 (p2 ) + B 2 (p2 ) > 0 , ∀ p2 . (1.472) This last fact means that the quark described by this model is confined: the propagator does not exhibit a mass pole! I also note that there is no critical coupling in this model; i.e., the nontrivial solution for B always exists. This exemplifies a contemporary conjecture that theories with confinement always exhibit DCSB. In the chirally asymmetric case the gap equation yields 2 B(p2 ) , m + B(p2 ) 4 [m + B(p2 )]2 B(p2 ) = m + . B(p2 )([m + B(p2 )]2 + 4p2 ) A(p2 ) = (1.473) (1.474) The second is a quartic equation for B(p2 ) that can be solved algebraically with four solutions, available in a closed form, of which only one has the correct p2 → ∞ limit: B(p2 ) → m. Note that the equations and their solutions always have a smooth m → 0 limit, a result owing to the persistence of the DCSB solution. Renormalization-Group-Improved Model Finally I have used the bare vertex, Eq. (1.448) and G(Q) depicted in Fig. 1.3, in solving the quark DSE in the chiral limit. If G(Q = 0) < 1 then B(p2 ) ≡ 0 is the only solution. However, when G(Q = 0) ≥ 1 the equation admits an energetically favoured B(p2 ) 6≡ 0 solution; i.e., if the coupling is large enough then even in the absence of a current-quark mass, contrary to Eq. (1.445), the quark acquires a mass dynamically and hence hq̄qi ∝ Z d4 p B(p2 ) 6= 0 for m = 0 . (2π)4 p2 A(p2 )2 + B(p2 )2 (1.475) These examples identify a mechanism for DCSB in quantum field theory. The nonzero condensate provides a new, dynamically generated mass-scale and if its magnitude is large enough [−hq̄qi1/3 need only be one order-of-magnitude larger than mu ∼ md ] it can explain the mass splitting between parity partners, and many other surprising phenomena in QCD. The models illustrate that DCSB is linked to the long-range behaviour of the fermion-fermion interaction. The same is true of confinement. The question is then: How does the quark-quark interaction behave at large distances in QCD? It remains unanswered. 80 1.0 0.8 G(Q) 0.6 0.4 0.2 0.0 0 1 10 100 Q (GeV) Figure 1.3: Illustrative forms of G(Q): the behaviour of each agrees with perturbation theory for Q > 1 GeV. Three possibilities are canvassed in Sec.1.12.3: G(Q = 0) < 1; G(Q = 0) = 1; and G(Q = 0) > 1. 81 Bibliography [1] J.D. Bjorken and S.D. 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Dyson, “The S Matrix In Quantum Electrodynamics,” Phys. Rev. 75 (1949) 1736. [11] K. Symanzik, “Infrared Singularities And Small Distance Behavior Analysis,” Commun. Math. Phys. 34 (1973) 7; T. Appelquist and J. Carazzone, “Infrared Singularities And Massive Fields,” Phys. Rev. D 11 (1975) 2856. [12] K. Symanzik in Local Quantum Theory (Academic, New York, 1969) edited by R. Jost. 82 [13] R.F. Streater and A.S. Wightman, A.S., PCT, Spin and Statistics, and All That, 3rd edition (Addison-Wesley, Reading, Mass, 1980). [14] J. Glimm and A. Jaffee, Quantum Physics. A Functional Point of View (Springer-Verlag, New York, 1981). [15] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Theory and Statistical Mechanics (Springer-Verlag, New York, 1982). [16] D.J. Gross, “Applications Of The Renormalization Group To High-Energy Physics,” in Proc. of Les Houches 1975, Methods In Field Theory (North Holland, Amsterdam, 1976) pp. 141-250. [17] Y. Nambu and G. Jona-Lasinio, “Dynamical Model Of Elementary Particles Based On An Analogy With Superconductivity. I,II” Phys. Rev. 122 (1961) 345, 246 [18] H.J. Munczek and A.M. Nemirovsky, Phys. Rev. D 28 (1983) 181. 83 Chapter 2 Quantum Fields at Finite Temperature and Density 2.1 Ensembles and Partition Function In equilibrium statistical mechanice, one normally encounters three types of ensemble. The microcanonical ensemble is used to describe an isolated system which has a fixed energy E, a fixed particle number N , and a fixed volume V . The canonical ensemble is used to describe a system in contact with a heat reservoir at temperature T . The system can freely exchange energy with the reservoir. Thus T , N , and V are fixed. In the grand canonical ensemble, the system can exchange particles as well as energy with a reservoir. In this ensemble, T , V , and the chemical potential µ are fixed variables. In the latter two ensembles T −1 = β may be thought of as a Lagrange multiplier which determines the mean energy of the system. Similarly, µ may be thought of as a Lagrange multiplier which determines the mean number of particles in the system. In a relativistic quantum system where particles can be created and destroyed, it is most straightforward to compute observables in the grand canonical ensemble. So we therefore use this ensemble. This is without loss of generality since one can pass over to either of the other ensembles by performing a Laplace transform on the variable µ and/or the variable β. Consider a system described by a Hamiltonian Ĥ and a set of conserved number operators N̂i . In relativistic QED, for example, the number of electrons minus the number of positrons is a conserved quantity, not the electron number or the positron number separately. These number operators must be Hermitian and must commute with H as well as with each other. Also, the number operators must be extensive in order that the usual macroscopic thermodynamic limit can be taken. The statistical density matrix is h i ρ̂ = exp −β(Ĥ − µi N̂i ) , 84 (2.1) where a summation over i is implied. The ensemble average of an operator  is Trρ̂ . Trρ̂ A= (2.2) The grand canonical partition function is Z = Trρ̂ . (2.3) The function Z = Z(T, V, µ1 , µ2 , . . .) is the single most important function in thermodynamics. From it all other standard thermodynamic properties may be determined. For example the equations of state for, e.g., the pressure P , the particle number N , the entropy S, and the energy E are, in the infinite volume limit, ∂ ln Z , ∂V ∂ ln Z , Ni = T ∂µi ∂(T ln Z) , P = ∂T E = −P V + T S + µi Ni . P = T (2.4) (2.5) (2.6) (2.7) Note that the notion of ensembles as introduced here for the situation in thermodynamical equilibrium can be extended to the nonequilibrium situation, where a generalized Gibbs ensemble can be introduced for systems which are in a nonequilibrium situation that can be characterized by further observables such as currents or reaction variables. These additional observables shall be accounted for by an enlarged set of Lagrange multipliers thus arriving at a statistical operator of the nonequilibrium state, also called relevant statistical operator within the Zubarev formalism. In order to calculate the partition using methods of quantum field theory, we recall that in quantum statistics Z = Tr e−β(Ĥ−µi N̂i ) = Z dφa hφa |e−β(Ĥ−µi N̂i ) |φa i , (2.8) where the sum runs over all (eigen-)states. This has an appearance very similar to the transition amplitude discussed in the previous chapter. In fact, we can express the partition function Z as a functional integral over fields and their conjugate momenta. Before doing so, we must switch to an imaginary time variable τ = i t and limit the integration over τ to the region between 0 and β. The trace operation means that we have to integrate over all fields φa . Finally, if the system admits some conserved charge, then we must make the replacement H(π, φ) → K(pi, φ) = H(π, φ) − µN (π, φ) , 85 (2.9) where N (π, φ) is the conserved charge density. Thus, we finally arrive at the fundamental formula Z= Z Dπ Z periodic Dφ exp (Z β 0 Z ∂φ dx iπ − H(π, φ) + µN (π, φ) ∂τ 3 !) . (2.10) The term “periodic” means that the integration over the field is constrained so that φ(~x, 0) = φ(~x, β). This is a consequence of the trace operation, setting φa (~x) = φ(~x, 0) = φ(~x, β). There is no restriction on the π integration. The generalization of (2.10) to an arbitrary number of fields and conserved charges is obvious. 2.2 2.2.1 Bosonic Fields Neutral Scalar Field The most general renormalizable Lagrangian for a neutral scalar field is 1 1 L = ∂µ φ∂ µ φ − m2 φ2 − U (φ) , 2 2 (2.11) U (φ) = gφ3 + λφ4 , (2.12) where the potential is and λ ≥ 0 for stability of the vacuum. The momentum conjugate to the field is π= ∂L ∂φ = , ∂(∂0 φ) ∂t (2.13) and the Hamiltonian is H=π ∂φ 1 1 1 − L = π 2 + (∇φ)2 + m2 φ2 + U (φ) . ∂t 2 2 2 (2.14) There is no conserved charge. The first step in evaluating the partition function is to return to the discretized version Z = lim N →∞ exp ΠN i=1 N Z X j=1 Z ∞ −∞ ! dπi Z dφi 2π periodic d3 x iπj (φj+1 − φj ) − ∆τ (2.15) 1 2 1 1 πj + (∇φj )2 + m2 φ2j + U (φj ) 2 2 2 The momentum integrations can be done immediately since they are just products of Gaussian integrals. Divide position space into M 3 little cubes with V = L3 , L = aM , a → 0, M → ∞, M an integer. 86 . For convenience, and to ensure that Z remains explicitely dimensionless at each step in the calculation, we write πj = Aj /(a3 ∆τ )1/2 and integrate Aj from −∞ to +∞. For each cube we obtain Z ∞ −∞ 1 a3 exp − A2j + i 2 ∆τ dAj 2π !1/2 (φj+1 − φj )Aj −a3 (φj+1 − φj )2 = (2π)−1/2 exp 2∆τ " # . (2.16) Thus far we have Z = lim (2π) M,N →∞ exp ∆τ −M 3 N/2 N Z X j=1 Z h ΠN i=1 dφi 1 d 3 x − 2 i (φj+1 − φj ) ∆τ !2 Returning to the continuum limit, we obtain Z = N0 Z periodic Z Dφ exp β 0 1 1 ) . − (∇φj )2 − m2 φ2j − U (φj(2.17) 2 2 dτ Z d3 xL ! . (2.18) The Lagrangian is expressed as a functional of φ and its first derivatives. The formula (2.17) expresses Z as a functional integral over φ of the exponential of the action in imaginary time. The normalization constant is irrelevant, since multiplication of Z by any constant does not change the thermodynamics. Next, we turn to the case of noninteracting fields with U (φ) = 0. Interactions will be discussed separately. Define S= Z β 0 dτ Z 1Z β Z 3 d xL = − dτ d x 2 0 3 " ∂φ + (∇φ)2 + m2 φ2 ∂τ !# . (2.19) Integrating by parts and taking note of the periodicity of φ, we obtain 1 S=− 2 Z β 0 dτ Z ∂2 d xφ − 2 − ∇2 + m2 φ . ∂τ ! 3 (2.20) The field can be decomposed into a Fourier series according to φ(~x, τ ) = β V !1/2 ∞ X X ei(~p~x+ωn τ ) φn (~ p) , (2.21) n=−∞ p ~ where ωn = 2πnT , due to the constraint of periodicity that φ(~x, β) = φ(~x, 0) for all ~x. The normalization of (2.21) is chosen conveniently so that each Fourier 87 amplitude is dimensionless. Substituting (2.21) into (2.20), and noting that φ−n (−~ p) = φ∗n (~ p) as required by the reality of φn (~ p), we find 1 XX 2 S = − β2 (ωn + ω 2 )φn (~ p)φ∗n (~ p) 2 n p ~ (2.22) √ with ω = p~2 + m2 . The integrand depends only on the amplitude of φ and not its phase. The phases can be integrated out to get Z = N 0 Πn Πp~ Z h 1 dAn (~ p) exp − β 2 (ωn2 + ω 2 )A2n (~ p) 2 −∞ ∞ = N 0 Πn Πp~ 2π/(β 2 (ωn2 + ω 2 )) i1/2 . (2.23) Ignoring an overall multiplicative factor independent of β and V , which does not affect the thermodynamics, we arrive at h Z = Πn Πp~ β 2 (ωn2 + ω 2 ) i−1/2 . (2.24) More formally one can arrive at this result by using the general rules for Gaussian functional integrals over commuting (bosonic) variables, derived before in the QFT chapter, since (2.18) and (2.20) can be expressed as Z = N0 Z 1 Dφ exp − (φ, Dφ) = N 0 constant(det D)−1/2 , 2 (2.25) where D = β 2 (ωn2 + ω 2 ) in (~ p, ωn ) space and (φ, Dφ) denotes the inner product on the function space. Thus far we have ln Z = − i 1 XX h 2 2 ln β (ωn + ω 2 ) . 2 n p~ (2.26) Note that the sum over n is divergent. This unfortunate feature stems from our careless handling of the integration measure Dφ. A more rigorous treatment using the proper definition of Dφ gives a finite result. In order to handle (2.26), we make use of h ln (2πn)2 + β 2 ω 2 ) i Z = β 2 ω2 1 h i dΘ2 2 + ln 1 + (2πn) , Θ2 + (2πn)2 (2.27) where the last term is β− independent and thus can be ignored. Furthermore, ∞ X 1 2π 2 2 = 1 + 2 2 Θ eΘ − 1 −∞ n + (Θ/2π) hence ln Z = − XZ p ~ βω 1 1 1 dΘ + Θ 2 e −1 88 . , (2.28) (2.29) Carrying out the Θ integral, and throwing away a β− independent piece, we finally arrive at ln Z = V Z d3 p 1 − βω − ln(1 − e−βω ) , 3 (2π) 2 (2.30) from which we obtain immediately the well-known expression for the ideal Bose gas (µ = 0), once we subtract the divergent expressions for the zero-point energy E0 = − ∂ ln Z0 = V ∂β Z d3 p ω , (2π)3 2 (2.31) and for the zero-point pressure P0 = T ∂ E0 ln Z0 = − , ∂V V (2.32) which are typical for the quantum field-theoretical treatment. With this subtraction the vacuum is defined as the state with zero energy and pressure. 2.3 Fermionic Fields Dirac fermions are descibed by a four-spinor field ψ with a Lagrangian density L = ψ̄(i∂/ − m)ψ ! 0 ∂ † 0 ~ −m ψ . = ψ γ iγ + i~γ · ∇ ∂t (2.33) The momentum conjugate to this field is Π= ∂L = iψ † , ∂(∂ψ/∂t) (2.34) because γ 0 γ 0 = 1. Thus, somewhat paradoxically, ψ and ψ † must be treated as independent entities in the Hamiltonian formulation. The Hamiltonian density is found by standard procedures, ! ∂ψ ∂ ~ + m)ψ , H=Π − L = ψ† i ψ − L = ψ̄(−i~γ · ∇ ∂t ∂t (2.35) and the partition function is Z = Tr e−β(Ĥ−µQ̂) , with the conserved charge Q = reads Z= Z † Dψ Dψ exp "Z β 0 dτ R Z (2.36) d3 xψ † ψ. The functional integral representation 3 d xψ † ! ∂ ~ − m + µγ 0 ψ −γ + i~γ · ∇ ∂τ 0 89 # (2.37) As with bosons, it is most convenient to work in (~ p, ωn ) space instead of (~x, τ ) space, i.e., !1/2 ∞ X X β ψα (~x, τ ) = ei(~p~x+ωn τ ) ψ̃α;n (~ p) , (2.38) V n=−∞ p ~ where now ωn = (2n + 1)πT due to the antiperiodicity of the (Grassmannian) Fermion field at the borders of the fundamental strip 0 ≤ τ ≤ β in the imaginary time, ψ(~x, 0) = ψ(~x, β). Now we are ready to evaluate the fermionic partition function (2.37), Z = Πn Πp~ Πα XX S = n Z † idψ̃α;n (~ p)dψ̃α;n (~ p) † iψα;n (~ p)Dαρ ψρ;n (~ p) , eS , p ~ h i D = −iβ (−iωn + µ) − γ 0~γ · p~ − mγ 0 , (2.39) using our knowledge about Grassmannian integration of Gaussian functional integrals, resulting in Z = det D . (2.40) Employing the identity ln det D = Tr ln D , (2.41) and evaluating the determinant in Dirac space explicitly (Exercise !), one finds ln Z = 2 XX n p ~ h n ln β 2 (ωn + iµ)2 + ω 2 io . (2.42) Since both positive and negative frequencies have to be summed over, the latter expression can be put in a form analogous to the above expression in the bosonic case, ln Z = XXn n p ~ h ln β 2 ωn2 + (ω − µ)2 i h + ln β 2 ωn2 + (ω + µ)2 io . (2.43) In the further evaluation we can go similar steps as in the bosonic case, with two exceptions: (1) the presence of a chemical potential, splitting the contributions of particles and antiparticles; (2) the Matsubara frequencies are now odd multiples of πT , so that the infinite sum to be exploited reads ∞ X 1 1 1 1 = − 2 2 2 Θ 2 eΘ + 1 n=−∞ (2n + 1) π + Θ . (2.44) Integrating over the auxiliary variable Θ, and dropping terms independent of β and µ, we finally obtain ln Z = 2V Z i d3 p h −β(ω−µ) −β(ω+µ) βω + ln(1 + e ) + ln(1 + e ) . (2π)3 90 (2.45) Notice that the factor 2 crresponding to the spin- 21 nature of the fermions comes out automatically. Separate contributions from particles (µ) and antiparticles (-µ) are evident. Finally, the zero-point energy of the vacuum also appears in this formula. 91 2.4 2.4.1 Gauge Fields Quantizing the Electromagnetic Field In this chapter we would like to understand how the Faddeev-Popov ghosts as introduced to eliminate divergences of the gauge freedom will act in the pure gauge theory to eliminate unphysical degrees of freedom and allow to derive the blackbody radiation law with the correct number of physical degrees of freedom. We start with recalling the path integral formulation of QED for the photon field Aµ (x) with the field strength tensor Fµν = ∂µ Aν − ∂ν Aµ (2.46) and the free action functional S=− 1 4 Z d4 xFµν F µν . (2.47) This action is invariant under gauge transformations Aµ (x) → Aµ (x) = A0µ (x) + ∂µ ω(x) , (2.48) where ω(x) is a scalar function which parametrizes the gauge transformations. The momenta conjugate to the space components of Ai (x) are, up to a sign, the components Ei (x) = Ei (x) of the electric field πi = −Ei = −F0i , (2.49) while the magnetic field B(x) is Bi = εijk ∂j Ak . (2.50) We work in an axial gauge, A3 = 0 to be specific. The momenta π1 and π2 are independent variables; E3 is not an independnet variable, but it is a function of π1 and π2 , which may be computed from Gauss’s law ∇ · E = 0. There are thus two dynamical variables A1 and A2 with conjugate momenta π1 and π2 . We define π3 = −E3 (π1 , π2 , but π3 is not to be interpreted as a conjugate momentum. The partition function is written as a Hamiltonian path integral Z= Z D(π1 , π2 ) Z Ai (0)=Ai (β) 1 2 D(A , A ) exp "Z β 0 4 1 2 # d x(iπ1 ∂τ A + iπ2 ∂τ A − H) (2.51) , where we have used the notation Z β 0 d4 x = Z β 0 92 dτ Z d3 x , (2.52) while the Hamiltonian density H is 1 2 1 π1 + π22 + E23 (π1 , π2 ) + B2 H = (E2 + B2 ) = 2 2 (2.53) Equation (2.51) is then transformed by using Z 1= Dπ3 δ(π3 + E3 (π1 , π2 )) , (2.54) and ∂(∇ · π) δ(π3 + E3 (π1 , π2 )) = δ(∇ · π) det ∂π3 h ! i = det ∂3 δ 3 (x − y) δ(∇ · π) . (2.55) In the following step, one inserts an integral representation of δ(∇ · π) δ(∇ · π) = Z " Z DA4 exp i β 0 # 4 d xA4 (∇ · π) , (2.56) where A4 = iA0 , and we work in Euclidean space now: xµ = (x, x4 ) = (x, τ ), A4 = (A, A4 ). Performing the π− integration, we are left with "Z # 1 1 d x (i∂τ A − i∇A4 )2 − B2 Z = D(A1 , A2 , A4 ) det ∂3 δ (x − y) exp , 2 2 0 (2.57) where A = (A1 , A2 , 0). Note that the argument of the exponential is Z h i 3 β 4 1 2 1 2 E − B =L. 2 2 (2.58) The A− integration is rendered more aesthetic by inserting 1= Z DA3 δ(A3 ) , (2.59) and the partition function assumes the form Z= Z µ h 3 i DA δ(A3 ) det ∂3 δ (x − y) exp Z β 0 4 d xL ! . (2.60) The axial gauge A3 = 0 is not a particularly convenient gauge to use for practical computations. Furthermore, it is not immediately apparent that (2.60) is a gauge invariant expression for Z. Take an arbitrary gauge specified by F = 0, where F is some function of Aµ and its derivatives. For the gauge above, F = A3 . For this gauge, (2.60) is given by ! ! Z Z β ∂F µ 4 Z = DA δ(F ) det exp d xL . (2.61) ∂α 0 93 Equation (2.61) is manifestly gauge invariant: L is invariant, the gauge fixing factor times the Jacobian of the transformation δ(F ) det(∂F/∂α) is invariant, and the integration is over all four components of the vector potential. Equation (2.61) reduces to (2.60) in the case of the axial gauge A3 = 0. We know this is correct since it was derived from first principles in the Hamiltonian formulation of the gauge theory, Z = Tr e−β Ĥ 2.4.2 Blackbody radiation It is important to verify that (2.61) describes blackbody radiation with two polarization degrees of freedom. We will do this here in the axial gauge A3 = 0, the Feynman gauge is left as an exercise. In the axial gauge, we rewrite (2.57) as Z = S0 Z 1 = 2 D(A0 , A1 , A2 ) det(∂3 )eS0 Z dτ Z d3 x(A0 , A1 , A2 ) ∂ −∂1 ∂τ ∂22 + ∂32 + −∂1 ∂2 ∇2 ∂ × −∂1 ∂τ ∂ −∂2 ∂τ ∂ −∂2 ∂τ ∂1 ∂2 ∂12 + ∂32 + ∂2 ∂τ 2 ∂2 ∂τ 2 A0 A1 . A2 (2.62) We can express the determinant of ∂3 as a functional integral over a complex ghost field C: that is, a Grassmann field with spin-0, det(∂3 ) = Z D C̄DC exp Z β 0 dτ Z 3 d xC̄∂3 C ! . (2.63) These ghost fields C̄ and C are not physical fields since they do not appear in the Hamiltonian. Furthermore, since they are anticommuting scalar fields they violate the connection between spin and statistics. It is simply a convenient functional integral representation of the determinant of an operator. The greatest applicability of these ficticious ghost fields will be to non-Abelian gauge theories, see also Sect. 5.2.1. of Le Bellac [2]. In frequency-momentum space the partition function is expressed as ln Z = ln det(βp3 ) − D = β2 p2 −ωn p1 −ωn p1 1 ln det(D) , 2 −ωn p1 −ωn p2 ωn2 + p22 + p23 −p1 p2 . −p1 p2 ωn2 + p21 + p23 Carrying out the determinantal operation, ln Z = i h 1 1 Tr ln(β 2 p23 ) − Tr ln β 6 p23 (ωn2 + p2 )2 2 2 94 h = ln Πn Πp β 2 (ωn2 + p2 ) = 2V Z i−1 1 d3 p − βω − ln(1 − e−βω ) . 3 (2π) 2 (2.64) Here, ω = |p|. Comparison with the result for the scalar field case shows that (2.64) describes massless bosons with two spin degrees of freedom in thermal equilibrium; in other words, blackbody radiation. 2.5 Interactions: Hubbard-Stratonovich Trick So far we have dealt with free quantum fields in the absence of interactions and have obtained nice closed expressions for the thermodynamic potential, i.e., therefore also for the generating functionals of the thermodynamic Green functions. However, once we switch on the interactions in our model field theories, there is only a very limited class of soluble models, in general we have to apply approximations. The most common technique is based on perturbation theory, which requires a small parameter. For strong interactions at low momentum transfer (the infrared region), the coupling is nonperturbatively strong and alternative, nonperturbative methods have to be invoked. One of the strategies, which is especially suitable for the treatment of quantum field theories within the path integral formulation is based on the introduction of collective variables (auxialiary fields) by an exact integral transformation due to Stratonovich and Hubbard which allows to eliminate (integrate out) the elementary degrees of freedom. Generally, the (dual) coupling of the auxialiary fields is weak so that perturbative expansions of the nonlinear effective action make sense and provide useful results already at low orders of this expansion. A general class of interactions for which the Hubbard-Stratonovich (HS) transformation is immediately applicable, are four-fermion couplings of the currentcurrent type Lint = G(ψ̄ψ)2 . (2.65) A Fermi gas with this typ of interaction serves as a model for electronic superconductivity (Bardeen-Cooper-Schrieffer (BCS) model, 1957) or for chiral symmetry breaking in quark matter (Nambu–Jona-Lasinio (NJL) model, 1961). The HS-transformation for (2.65) reads h exp G(ψ̄ψ) 2 i =N Z σ2 + ψ̄ψσ Dσ exp 4G " # (2.66) and allows to bring the functional integral over fermionic fields into a quadratic (Gaussian) form so that fermions can be integrated out. This is also called Bosonization procedure. 95 2.5.1 Nambu–Jona-Lasinio (NJL) Model Here we will present an application of the HS technique to the NJL model for quark matter at finite densities and temperatures. This is possible since the interaction of this model is of the current-current form and therefore the HS trick for the bosonization of 4-fermion interactions applies. The Lagrangian density is given by L = q̄iα (i∂/δij δαβ − Mij0 δαβ + µij,αβ γ 0 )qjβ + GS 8 h X (q̄τfa q)2 + (q̄iγ5 τfa q)2 a=0 + GD Xh i C (q̄iα ijk αβγ qjβ )(q̄iC0 α0 i0 j 0 k α0 β 0 γ qj 0 β 0 ) k,γ i C + (q̄iα iγ5 ijk αβγ qjβ )(q̄iC0 α0 iγ5 i0 j 0 k α0 β 0 γ qj 0 β 0 ) , (2.67) where from here on the quark spinor is qiα , with the flavor index i = u, d, s and α = r, g, b stands for the color degree of freedom. Mij0 = diag(m0u , m0d , m0s ) is the current quark mass matrix in flavor space and µij,αβ is the chemical potential matrix in color and flavor space. The grand cononical thermodynamical potential Ω = T ln Z is known once we manage to evaluate the partition function in a reasonable approximation. A closed solution, even for the simple model Lagrangion (2.67), is not possible. The bosonization of the partition function proceeds as follows Z Z Z = Dq̄Dq exp d4 xL (2.68) 96 2.5.2 Mesonic correlations at finite temperature In the previous section we have seen how the concept of order parameters can be introduced in quantum field theory in the mean-field approximation. By analysis of the gap equations describing the minima of the thermodynamical potential in the space of order parameters we have we could investigate the phenomenon of spontaneous symmetry breaking, indicated by a nonvanishing value of the order parameter (gap). Important examples being: chiral symmetry breaking (mass gap) and superconductivity (energy gap). At finite temperature and density the values of these gaps change and their vanishing indicates the restoration of a symmetry. As the symmetries prevailing under given thermodynamical conditions of temperature and chemical potential (density) characterize a phase of the system, we have thus acquainted ourselves with a powerful quantum field theoretical method of analysing phase transitions. The results of such an analysis are summarized in phase diagrams. A prominent example is the phase diagram of QCD in the temperature- density plane, shown schematically in Fig. ??. It exhibits two major domains: Hadronic matter (confined quarks and gluons) and the Quark-gluon plasma (QGP), separated by the phenomenon of quark (and gluon) deconfinement under investigation in heavy-ion collisions, but also in the physics of compact stars and in simulations of Lattice-gauge QCD on modern Teraflop computers. The hadronic phase is subdivided into a nuclear matter phase at low temperatures with a gas-liquid transition (similar to the van-der-Waals treatment of real gases) and a ’plasma phase’ of a hot hadron gas with a multitude of hadronic resonances. The QGP phase is subdivided into a quark matter phase at low temperatures where most likely gluons are still condensed (confined) and the strongly correlated Fermi-liquid of quarks shoild exhibit the phenomenon of superconductivity/ superfludity with a Bose condensate of Cooper pairs of . diquarks. At asymptotic temperatures and densities one expects a system of free quarks and gluons (due to asymptotic freedom of QCD). In any real situation in terrestrial experiments or in astrophysics, one expects to be quite far from this ideal gas state. As the nature of the confinement-deconfinement transition is not yet clarified, we must speculate what to expect in the vicinity of the conjectured deconfinement phase transition. In the lattice-QCD simulations one has found evidence for strong correlations even above the critical temperature Tc ∼ 170 MeV, which is defined by an increase of the effective number of degrees of freedom ε(T )/T 4 by about one order of magnitude in a close vicinity of Tc . Indications for strong correlations in the QGP one has also found in recent RHIC experiments, where from particle production and flow one has found fast thermalization and an extremely low viscosity (’perfect fluid’). One speaks about an < 2 Tc . ’sQGP phase’ at temperatures Tc < ∼T ∼ In order to investigate the phase diagram experimentally, one has to correlate the observables with the regions. As the detectors are situated in the ’vacuum’ at zero temperature and chemical potential where confinement prevails, no direct 97 Big Bang Quark-Gluon-Plasma 1.5 0.1 DECONFINEMENT CERN-SPS FAIR (Project) AGS Brookhaven Hadron gas CONFINEMENT QCD - Lattice Gauge Theory Temperature [TH =140 MeV] RHIC, LHC (construction) s SIS Darmstadt on yI SuperNovae Nuclear matter 1 ion is oll C Quark Matter av He COLOR SUPERCONDUCTIVITY Neutron / Quark Stars Baryon Density 3 -3 [nο =0.16 fm ] Figure 2.1: The conjectured phase diagram of QCD and places where to investigate it. detection of quarks and gluons from the plasma phase is possible and one has to conjecture about the properties of the sQGP phase(s) from the traces which they might leave in the hadronic and electromagnetic spectra emitted by the hadronizing fireball formed in a heavy-ion collision or by the neutrino-emitting cooling processes of compact stars. As a prerequisite for discussing hadronic spectra and their modifications, we will study the mesonic spectral function(s), and for their analysis within effective models like the NJL or MN models, we need to evaluate loop diagrams at finite temperature. A basic ingredient are Matsubara frequency sums. 2.5.3 Matsubara frequency sums In computing Feynman diagrams with internal fermion lines we shall encounter frequency sums, and we have to learn how to evaluate them. Let us denote the quantity (ωn2 + Ep2 )−1 , which appears in the free Fermion propagator SF (iωn , p) = − Z ∞ −∞ dp0 ρF (p0 , p) m−p / = 2 2π iωn − p0 ωn − E2p 98 (2.69) ˜ n , Ep ), and its mixed representation by ∆(τ, ˜ Ep ), suppressing the subby ∆(iω script F for simplicity ˜ Ep ) = T ∆(τ, X ˜ n , Ep ) . e−iωn τ ∆(iω (2.70) n It can easily be checked (Exercise !) that i 1 h (1 − ñ(Ep ))e−Ep τ − ñ(Ep )eEp τ 2Ep X s = (1 − f˜(sEp ))e−sEp τ , s=±1 2Ep ˜ Ep ) = ∆(τ, (2.71) where the Fermi-Dirac distribution ñ(p0 ) is ñ(p0 ) = 1 eβ|p0 | + 1 . (2.72) One should note the absolute value of p0 in (2.72), in contrast with the definition of f˜(p0 ) = 1/[exp(βp0 ) + 1]. Note also that f˜(E) = ñ(E) , f˜(−E) = 1 − ñ(E) , 1 − f˜(E) − f˜(−E) = 0 . (2.73) In frequency space, the formula corresponding to (2.71) is X 1 ˜ s (iωn , Ep = ∆ 2 − Ep s=±1 X s 1 − . = 2Ep iωn − sEp s=±1 ˜ n , Ep ) = ∆(iω ωn2 (2.74) The frequency sums are performed by following the methods of analytic continuation iωn → k0 and contour integration with a function having simple poles at the discrete frequencies k0 = iωn with unit residuum and convergence for |k0 | → ∞. Let us give two general examples for Matsubara sums of one-loop diagrams with two external (amputated) legs. • Fermion-boson case, see Fig. 2.5.3: s1 s2 1 + f (s1 E1 ) − f˜(s2 E2 ) 4E1 E2 iω − s1 E1 − s2 E2 n (2.75) ˜ X ˜ s (i(ω − ωn ), E2 ) = − is2 1 + f (s1 E1 ) − f (s2 E2 ) T ωn ∆s1 (iωn , E1 )∆ 2 4E2 iω − s1 E1 − s2 E2 n (2.76) T X ˜ s2 (i(ω − ωn ), E2 ) = − ∆s1 (iωn , E1 )∆ 99 φ 1 k q φ φ 2 q−k Figure 2.2: One-loop diagram for the fermion-antifermion polarization function. φ q k φ 1 φ 2 q−k Figure 2.3: One-loop diagram for the fermion self-energy. 100 • Fermion-antifermion case, see Fig. 2.5.3: ˜ ˜ ˜ s1 (iωn , E1 )∆ ˜ s2 (i(ω − ωn ), E2 ) = − s1 s2 1 − f (s1 E1 ) − f (s2 E2 ) ∆ 4E1 E2 iω − s1 E1 − s2 E2 n (2.77) ˜ ˜ X ˜ s1 (iωn , E1 )∆ ˜ s2 (i(ω − ωn ), E2 ) = − is2 1 − f (s1 E1 ) − f (s2 E2 ) ωn ∆ T 4E2 iω − s1 E1 − s2 E2 n (2.78) T X Note that you can obtain unify these formula using the rule: f (sE) → −f˜(sE) when replacing a bosonic by a fermionic line. Exercise: Verify these expressions! 101 Bibliography [1] J.I. Kapusta, Finite-temperature Field Theory, Cambridge University Press, 1989 [2] M. Le Bellac, Thermal Field Theory, Cambridge University Press, 1996 [3] D. Blaschke, et al., Phys. Rev. D 72 (2005) XXXXXXXXXXXXXXXXXXXXXXXXXX 102 Chapter 3 Applications to Heavy-Ion Collisions and Compact Stars 3.1 Introduction 103 A typical frequency sum, which we shall encounter in the calculation of loop diagrams is X S(iωn ) = T ∆(i(ωm − ωn ))∆0 (iωn ) , (3.1) n where the oscillator frequencies of ∆ and ∆0 are ω and ω 0 , respectively. Using the Fourier representation of ∆(iωn ) ∆(iωn ) = Z ∞ 0 dτ eiωn τ ∆(τ ) (3.2) and the relation T X n where p = 0, ±1, ±2, . . ., yields eiωn τ = X p δ(τ − pβ) , S(iωn ) = 124 (3.3) (3.4) Chapter 4 Particle Production by Strong Fields 4.1 4.1.1 Quantum kinetic equation for particle production Introduction Particle production in ultrarelativistic heavy-ion collisions raises a number of challenging problems. One of these interesting questions is how to incorporate the mechanism of particle creation into a kinetic theory[1, 2, 3, 4, 5, 6, 7, 8, 9]. Within the framework of a flux tube model[4, 10, 11, 12], a lot of promising research has been carried out during the last years. In the scenario where a chromo-electric field is generated by a nucleus-nucleus collision, the production of parton pairs can be described by the Schwinger mechanism[13, 15, 15]. The produced charged particles generate a field which, in turn, modifies the initial electric field and may cause plasma oscillations. The interesting question of the back reaction of this field has been analyzed within a field theoretical approach[16, 17, 18]. The results of a simple phenomenological consideration based on kinetic equations and the field-theoretical treatment[5, 19, 20, 21] agree with each other. The source term which occurs in such a modified Boltzmann equation was derived phenomenologically in Ref.[22]. However, the systematic derivation of this source term in relativistic transport theory is not yet fully carried out. For example, recently it was pointed out that the source term may have non-Markovian character[6, 23] even for the case of a constant electric field. In the present lecture, a kinetic equation is derived in a consistent field theoretical treatment for the time evolution of the pair creation in a time-dependent and spatially homogeneous electric field. This derivation is based on the Bogoliubov transformation for field operators between the asymptotic in-state and the instantaneous state. In contrast with phenomenological approaches, but in agree125 ment with[6, 23], the source term of the particle production is of non-Markovian character. The kinetic equation derived reproduces the Schwinger result in the low density approximation in the weak field limit. 4.1.2 Dynamics of pair creation Creation of fermion pairs In this section we demonstrate the derivation of a kinetic equation with a source term for fermion-antifermion production. As an illustrative example we consider electron-positron creation, however the generalization to quark-antiquark pair creation in a chromoelectric field in Abelian approximation is straightforward. For the description of e+ e− production in an electric field we start from the QED lagrangian 1 L = ψ̄iγ µ (∂µ + ieAµ )ψ − mψ̄ψ − Fµν F µν , 4 (4.1) where F µν is the field strength, the metric is taken as g µν = diag(1, −1, −1, −1) and for the γ– matrices we use the conventional definition [24]. In the following we consider the electromagnetic field as classical and quantize only the matter field. Then the Dirac equation reads (iγ µ ∂µ − eγ µ Aµ − m)ψ(x) = 0 . (4.2) We use a simple field-theoretical model to treat charged fermions in an external electric field charactarized by the vector potential Aµ = (0, 0, 0, A(t)) with A(t) = A3 (t). The electric field E(t) = E3 (t) = −Ȧ(t) = −dA(t)/dt (4.3) is assumed to be time-dependent but homogeneous in space (E1 = E2 = 0). This quasi-classical electric field interacts with a spinor field ψ of fermions. We look for solutions of the Dirac equation where eigenstates are represented in the form (±) ψp̄r (x) = iγ 0 ∂0 + γ k pk − eγ 3 A(t) + m χ(±) (p̄, t) Rr eip̄x̄ , (4.4) where k = 1, 2, 3 and the superscript (±) denotes eigenstates with the positive and negative frequencies. Herein Rr (r = 1, 2) is an eigenvector of the matrix γ0γ3 1 0 0 1 , (4.5) , R2 = R1 = −1 0 0 −1 126 so that Rr+ Rs = 2δrs . The functions χ(±) (p̄, t) are related to the oscillator-type equation χ̈ (±) 2 (p̄, t) = − ω (p̄, t) + ieȦ(t) χ(±) (p̄, t) , (4.6) where we define the total energy ω 2 (p̄, t) = ε2⊥ + Pk2 (t), the transverse energy ε2⊥ = m2 + p̄2⊥ and Pk (t) = pk − eA(t). The solutions χ(±) (p̄, t) of Eq. (4.6) for positive and negative frequencies are defined by their asymptotic behavior at t0 = t → −∞, where A(t0 ) = 0. We obtain χ(±) (p̄, t) ∼ exp ( ± iω0 (p̄) t) , (4.7) where the total energy in asymptotic limit is given as ω0 (p̄) = ω0 (p̄, t0 ) = lim ω(p̄, t). Note that the system of the spinor functions (4.106) is complete t→−∞ and orthonormalized. The field operators ψ(x) and ψ̄(x) can be decomposed in the spinor functions (4.106) as follows: ψ(x) = X r,p̄ (−) (+) ψp̄r (x) bp̄r (t0 ) + ψp̄r (x) d+ −p̄r (t0 ) . (4.8) + + The operators bp̄r (t0 ), bp̄r (t0 ) and dp̄r (t0 ), dp̄r (t0 ) describe the creation and annihilation of electrons and positrons in the in-state |0in > at t = t0 , and satisfy the anti-commutation relations [24] {bp̄r (t0 ), bp̄+0 r0 (t0 )} = {dp̄r (t0 ), dp̄+0 r0 (t0 )} = δrr0 δp̄p̄0 . (4.9) The evolution affects the vacuum state and mixes states with positive and negative energies resulting in non-diagonal terms that are responsible for pair creation. The diagonalization of the hamiltonian corresponding to a Dirac-particle (Eq. (4.2)) in the homogeneous electric field (4.3) is achieved by a time-dependent Bogoliubov transformation bp̄r (t) = αp̄ (t) bp̄r (t0 ) + βp̄ (t) d+ −p̄r (t0 ) , dp̄r (t) = α−p̄ (t) dp̄r (t0 ) − β−p̄ (t) (4.10) b+ −p̄r (t0 ) with the condition |αp̄ (t)|2 + |βp̄ (t)|2 = 1 . (4.11) Here, the operators bp̄r (t) and dp̄r (t) describe the creation and annihilation of quasiparticles at the time t with the instantaneous vacuum |0t >. Clearly, the operator system b(t0 ), b+ (t0 ); d(t0 ), d+ (t0 ) is unitary non-equivalent to the system b(t), b+ (t); d(t), d+ (t). The substitution of Eqs. (4.10) into Eq. (4.8) leads to the new representation of the field operators ψ(x) = X (−) Ψp̄r (x) bp̄r (t) + r,p̄ 127 (+) Ψp̄r (x) d+ −p̄r (t) . (4.12) (±) The link between the new Ψp̄r (x) and the former (4.106) basis functions is given by a canonical transformation (−) (−) (+) (+) (−) Ψp̄r (x) ψp̄r (x) = αp̄ (t) Ψp̄r (x) − βp̄∗ (t) Ψp̄r (x) , (+) ψp̄r (x) αp̄∗ (t) = Ψ (x)p̄r + βp̄ (t) (4.13) . (±) Therefore it is justified to assume that the functions Ψp̄r have a spin structure (+) similar to that of ψp̄r in Eq. (4.106), (±) (±) Ψp̄r (x) = iγ 0 ∂0 + γ k pk − eγ 3 A(t) + m φp̄ (x) Rr e±iΘ(t) eip̄x̄ , (4.14) where the dynamical phase is defined as Θ(p̄, t) = Z t t0 dt0 ω(p̄, t0 ) . (4.15) (±) The functions φp̄ are yet unknown. The substitution of Eq. (4.14) into Eqs. (4.13) leads to the relations (−) (+) χ(−) (p̄, t) = αp̄ (t) φp̄ (t) e−iΘ(p̄,t) − βp̄∗ (t) φp̄ (t) eiΘ(p̄,t) , χ (+) (p̄, t) = αp̄∗ (t) (+) φp̄ (t) e iΘ(p̄,t) + βp̄ (t) (−) φp̄ (t) e −iΘ(p̄,t) (4.16) . Now we are able to find explicit expressions for the coefficients αp̄ (t) and βp̄ (t). Taking into account that the functions χ(±) (p̄, t) are defined by Eq. (4.6), we introduce additional conditions to Eqs. (4.16) according to the Lagrange method χ̇ (−) (p̄, t) = −iω(p̄, t) αp̄ (t) χ̇(+) (p̄, t) = (−) φp̄ (t) e −iΘ(p̄,t) (+) iω(p̄, t) αp̄∗ (t) φp̄ (t) eiΘ(p̄,t) + βp̄∗ (t) (+) φp̄ (t) (−) e iΘ(p̄,t) , (4.17) − βp̄ (t) φp̄ (t) e−iΘ(p̄,t) . Differentiating these equations with respect to time, using Eqs. (4.6) and (4.16) and then choosing as an Ansatz (±) φp̄ (t) = v u u ω(p̄, t) ± Pk (t) t ω(p̄, t) , (4.18) we obtain the following differential equations for the coefficients α̇p̄ (t) = eE(t)ε⊥ ∗ β (t) e2iΘ(p̄,t) , 2ω 2 (p̄, t) p̄ (4.19) β̇p̄∗ (t) = − eE(t)ε⊥ αp̄ (t) e−2iΘ(p̄,t) . 2ω 2 (p̄, t) 128 As the result of the Bogoliubov transformation we obtained the new coefficients of the instantaneous state at the time t. The relations between them read 1 (−) (−) ω(p̄, t) χ (p̄, t) + i χ̇ (p̄, t) eiΘ(p̄,t) , αp̄ (t) = q 2 ω(p̄, t) (ω(p̄, t) − Pk (t)) βp̄∗ (t) 1 = − q ω(p̄, t) χ 2 ω(p̄, t) (ω(p̄, t) − Pk (t)) (−) (p̄, t) − i χ̇ (−) (p̄, t) e −iΘ(p̄,t) It is convenient to introduce new operators which absorb the dynamical phase Bp̄r (t) = bp̄r (t) e−iΘ(p̄,t) , Dp̄r (t) = dp̄r (t) e−iΘ(p̄,t) (4.21) (4.22) satisfying the anti-commutation relations: {Bp̄r (t), Bp̄+0 r0 (t)} = {Dp̄r (t), Dp̄+0 r0 (t)} = δrr0 δp̄p̄0 . (4.23) It is easy to show that these operators satisfy the Heisenberg-like equations of motion dBp̄r (t) eE(t)ε⊥ + =− 2 D (t) + i [H(t), Bp̄r (t)] , dt 2ω (p̄, t) −p̄r (4.24) eE(t)ε⊥ + B (t) + i [H(t), Dp̄r (t)] , 2ω 2 (p̄, t) −p̄r dDp̄r (t) = dt where H(t) is the hamiltonian of the quasiparticle system H(t) = X ω(p̄, t) r,p̄ + Bp̄r (t) Bp̄r (t) − D−p̄r (t) + D−p̄r (t) . (4.25) The first term on the r.h.s. of Eqs. (4.24) is caused by the unitary non-equivalence of the in-representation and the quasiparticle one. Now we explore the evolution of the distribution function of electrons with the momentum p̄ and spin r defined as + + fr (p̄, t) =< 0in |bp̄r (t) bp̄r (t)|0in >=< 0in |Bp̄r (t) Bp̄r (t)|0in > . (4.26) According to the charge conservation the distribution functions for electrons and positrons are equal fr (p̄, t) = f¯r (p̄, t), where + f¯r (p̄, t) = < 0in |d+ −p̄r (t) d−p̄r (t)|0in > = < 0in |D−p̄r (t) D−p̄r (t)|0in > . (4.27) The distribution functions (4.26) and (4.27) are normalized to the total number of pairs N (t) of the system at a given time t X r,p̄ fr (p̄, t) = X f¯r (p̄, t) = N (t) . r,p̄ 129 (4.28) (4.20) . Time differentiation of Eq. (4.26) leads to the following equation eE(t)ε⊥ dfr (p̄, t) =− 2 Re{Φr (p̄, t)} . dt ω (p̄, t) (4.29) Herein, we have used the equation of motion (4.24) and evaluated the occurring commutator. The function Φr (p̄, t) in Eq. (4.29) describes the creation and annihilation of an electron-positron pair in the external electric field E(t) and is given as Φr (p̄, t) =< 0in |D−p̄r (t) Bp̄r (t)|0in > . (4.30) It is straightforward to evaluate the derivative of this function. Applying the equations of motion (4.24), we obtain dΦr (p̄, t) eE(t)ε⊥ 2fr (p̄, t) − 1 − 2iω(p̄, t) Φr (p̄, t) , = dt 2ω 2 (p̄, t) (4.31) where because of charge neutrality of the system, the relation fr (p̄, t) = f¯r (p̄, t) is used. The solution of Eq. (4.31) may be written in the following integral form Φr (p̄, t) = ε⊥ 2 Z t t0 dt0 eE(t0 ) 0 2fr (p̄, t0 ) − 1 e2i[Θ(p̄,t )−Θ(p̄,t)] . 2 0 ω (p̄, t ) (4.32) The functions Θ(p̄, t) and Θ(p̄, t0 ) in Eq. (4.32) can be taken at t0 (see the definition (4.15)). Hence with A(t0 ) = 0, the function Φr (p̄, t) vanishes at t0 . Inserting Eq. (4.32) into the r.h.s of Eq. (4.29) we obtain eE(t)ε⊥ dfr (p̄, t) = dt 2ω 2 (p̄, t) Z 0 0 eE(t )ε⊥ dt 2 1 ω (p̄, t0 ) −∞ t 0 0 − 2fr (p̄, t ) cos 2[Θ(p̄, t) − Θ(p̄, t )] . (4.33) Since the distribution function obviously does not depend on spin (4.33), we can define: fr = f . With the substitution f (p̄, t) → F (P̄ , t), where the 3-momentum is now defined as P̄ (p⊥ , Pk (t)), the kinetic equation (4.33) is reduced to its final form: dF (P̄ , t) ∂F (P̄ , t) ∂F (P̄ , t) = + eE(t) = S(−) (P̄ , t) , (4.34) dt ∂t ∂Pk (t) with the Schwinger source term 0 eE(t)ε⊥ Z t 0 eE(t )ε⊥ dt 1 − 2F (P̄ , t0 ) cos 2[Θ(p̄, t) − Θ(p̄, t0 )] . 2ω 2 (p̄, t) −∞ ω 2 (p̄, t0 ) (4.35) Recently in Ref.[6], a kinetic equation similar to (4.34) has been derived within a projection operator formalism for the case of a time-independent electric field where it was first noted that this source term has non-Markovian character. As well as in this method the multiple pair creation is not considered. The S(−) (P̄ , t) = 130 presence of the Pauli blocking factor [1 − 2F (P̄ , t)] in the source term has been obtained earlier in Ref.[18]. We would like to emphasize the closed form of the kinetic equation in the present work where the source term does not include the anomalous distribution functions (4.30) for fermion-antifermion pair creation (annihilation). Creation of boson pairs In this subsection we derive the kinetic equation with source term for the case of pairs of charged bosons in a strong electric field. The Klein-Gordon equation reads µ µ (∂ + ieA )(∂µ + ieAµ ) + m 2 φ(x) = 0 . (4.36) The solution of the Klein-Gordon equation in the presence of the electric field defined by the vector potential Aµ = (0, 0, 0, A(t)) is taken in the form [24] (±) φp̄ (x) = [2ω(p)]−1/2 eix̄p̄ g (±) (p̄, t) , (4.37) where the functions g (±) (p̄, t) satisfy the oscillator-type equation with a variable frequency g̈ (±) (p̄, t) + ω 2 (p̄, t) g (±) (p̄, t) = 0 . (4.38) Solutions of Eq. (4.38) for positive and negative frequencies are defined by their asymptotic behavior at t0 = t → −∞ similarly to Eq. (4.7). The field operator in the in-state is defined as φ(x) = Z (−) (+) d3 p [ φp̄ (x) ap̄ (t0 ) + φp̄ (x) b+ vp (t0 ) ] . (4.39) The diagonalization of the hamiltonian corresponding to the instantaneous stateis achieved by the transition to the quasiparticle representation. The Bogoliubov transformation for creation and annihilation operators of quasiparticles has the following form ap̄ (t) = αp̄ (t) ap̄ (t0 ) + βp̄ (t) b+ −p̄ (t0 ) , (4.40) b−p̄ (t) = α−p̄ (t) bp̄ (t0 ) + β−p̄ (t) a+ −p̄ (t0 ) with the condition |αp̄ (t)|2 − |βp̄ (t)|2 = 1 . (4.41) The derivation of the Bogoliubov coefficients α and β is similar to that given in the previous subsection. We obtain the equations of motion for the coefficients of the canonical transformation (4.40) as follows ω̇(p̄, t) ∗ βp̄ (t) e2iΘ(p̄,t) , 2ω(p̄, t) ω̇(p̄, t) ∗ αp̄ (t) e2iΘ(p̄,t) . β̇p̄ (t) = 2ω(p̄, t) α̇p̄ (t) = 131 (4.42) (4.43) Following the derivation for the case of fermion production, we arrive at the final result for the source term in the bosonic case 0 eE(t)ε⊥ Z t 0 0 0 eE(t )ε⊥ 1 + 2F (P̄ , t ) cos 2[Θ(p̄, t) − Θ(p̄, t )] , dt 2 S(+) (P̄ , t) = 2ω 2 (p̄, t) −∞ ω (p̄, t0 ) (4.44) which differs from the fermion case just by the sign in front of the distribution function due to the different statistics of the produced particles. 4.1.3 Discussion of the source term Properties of the source term We can combine the results for fermions (4.35) and for bosons (4.44) into a single kinetic equation dF(±) (P̄ , t) ∂F(±) (P̄ , t) ∂F(±) (P̄ , t) = S(±) (P̄ , t) . = + eE(t) dt ∂t ∂P3 (4.45) Here, the upper (lower) sign corresponds to the Bose-Einstein (Fermi-Dirac) statistics. Based on microscopic dynamics, these kinetic equations are exact within the approximation of a time-dependent homogeneous electric field and the neglect of collisions. The Schwinger source terms (4.35) and (4.44) are characterized by the following features: 1. The kinetic equations (4.45) are of non-Markovian type due to the explicit dependence of the source terms on the whole pre-history via the statistical factor 1 ± 2F (P̄ , t) for fermions or bosons, respectively. The memory effect is expected to lead to a modification of particle pair creation as compared to the (Markovian) low-density limit, where the statistical factor is absent. 2. The difference of the dynamical phases, Θ(p̄, t)−Θ(p̄, t0 ), under the integrals (4.35) and (4.44) generates high frequency oscillations. 3. The appearance of such a source term leads to entropy production due to pair creation (see also Rau[6]) and therefore the time reversal symmetry should be violated, but it does not result in any monotonic entropy increase (in absence of collisions). 4. The source term and the distribution functions have a non-trivial momentum dependence resulting in the fact that particles are produced not only at rest as assumed in previous studies, e.g. Ref.[19]. 5. In the low density limit and in the simple case of a constant electric field we reproduce the pair production rate given by Schwinger’s formula cl −3 S = lim (2π) g t→+∞ Z e2 E 2 d P S(P̄ , t) = exp 4π 3 3 132 πm2 − |eE| . (4.46) As noted above, Rau[6] found that the source term has a non-Markovian behaviour by deriving the production rate within a projector method. In the limit of a constant field our results agree with Ref.[6]. In our approach the electric field is treated as a general time dependent field and hence there is no a priori limitation to constant fields. However our result allows to explore the influence of any time-dependent electric field on the pair creation process. It is important to note that in general this time dependence should be given by a selfconsistent solution of the coupled field equations, namely the Dirac (Klein-Gordon) equation and the Maxwell equation. This would incorporate back reactions as mentioned in the introduction. However, the solution of such a system of equations is beyond the scope of this work. Herein we will restrict ourselves to the study of some features of the new source term. Finally we remark that the source term is characterized by two time scales: the memory time ε⊥ (4.47) τmem ∼ eE and the production interval τprod = 1/ < S(±) > , (4.48) with < S(±) > denoting the time averaged production rate. As long as E << m2 /e < ε2⊥ /e, the particle creation process is Markovian: τmem << τprod . This results for constant fields agree with those of Rau[6]. 30.0 20.0 S/ε 10.0 0.0 -10.0 1.0 S/ε 0.0 -1.0 -2.0 -3.0 -1.0 1.0 3.0 p|| /ε Figure 4.1: The pair production rate as a function of parallel momentum for a constant, strong field (upper plot: Ẽ0 = 1.5) and a weak field (lower plot: Ẽ0 = 0.5) at t̃ = 0. 133 Numerical results In order to study the new source term, we consider two different cases, namely a constant field and a time dependent electric field. For the numerical evaluation we start with Eq. (4.35) assuming a dilute system, F = 0, and 0 S(p̃k , t̃) Ẽ(t̃) Z t̃ 0 Ẽ(t̃ ) d t̃ cos 2[Θ(p̃k , t̃) − Θ(p̃k , t̃0 )] , = ε⊥ 2ω̃ 2 (p̃k , t̃) −∞ ω̃ 2 (p̃k , t̃0 ) (4.49) where we have introduced dimensionless variables S̃(p̃k , t̃) = Ẽ(t̃) = eE(t̃)/ε2⊥ , p̃k = pk /ε⊥ , t̃ = tε⊥ , (4.50) ω̃ = ω/ε⊥ . (4.51) This notation is convenient to distinguish the weak field (Ẽ < 1) and strong field (Ẽ > 1) limits. A particularly simple result is obtained if we assume a constant field, Ã(t̃) = A(t̃)/ε⊥ = t̃Ẽ0 /e , (4.52) where the electric field does not depend on time and the energy is given as ω̃02 (p̃k , t̃) = 1 + (p̃k − Ẽ0 t̃)2 . (4.53) For the source term we obtain S̃(p̃k , t̃) = Ẽ02 2ω̃02 (p̃k , t̃) Z t̃ −∞ dt̃0 1 cos 2 ω̃02 (p̃k , t̃0 ) Z t̃ t̃0 dt̃00 ω̃0 (p̃k , t̃00 ) . (4.54) In Fig. 4.1 we plot the particle production rate as function of the parallel momentum pk for a weak constant field and a strong field, respectively. The rates are normalized to be of the order of one for Ẽ0 = 0.5 at zero values of both momentum and time. The production rate is positive when particles are produced with positive momenta. Pairs with negative momenta are moving against the field and hence get annihilated, clearly to be seen in the negative production rate. These results mainly agree with those of Ref.[6], since no time dependence was considered. Note that using the prefactor and field strengths chosen by Rau, we reproduce exactly the results given in Ref.[6]. In considering the time dependence of the source term, we go beyond the analysis of Ref.[6]. In Fig. 4.2, we display the time dependence of the production rate at zero momentum. The maximum of the production rate is concentrated around zero and shows an oscillating behaviour for large times. Indeed it is possible to write Eq. (4.54) in terms of the Airy function because the constant field provides an analytical solution for the dynamical phase difference using the energy given in Eq. (4.53). The production of pairs for strong fields is larger 134 30.0 20.0 S/ε 10.0 0.0 -10.0 1.0 S/ε 0.0 -1.0 -2.0 -3.0 2.0 7.0 tε Figure 4.2: The pair production rate as a function of time for a constant, strong field (upper plot: Ẽ0 = 1.5) and a weak field (lower plot: Ẽ0 = 0.5) at zero parallel momentum. Figure 4.3: The pair production rate, S̃(p̃k , t̃), as a function of time and parallel momentum for a time dependent weak electric field charactarized by Ẽ0 = 0.5, σ̃ = 1 and τ = 0 . All plotted values are dimensionless as described in the text. 135 Figure 4.4: The pair production rate, S̃(p̃k , t̃), as a function of time and parallel momentum for a time dependent strong electric field charactarized by Ẽ0 = 1.5, σ̃ = 1 and τ = 0 . All plotted values are dimensionless as described in the text. than that of weak fields, and the typical time period of the oscillations is smaller. The situation changes if we allow the electric field to be time dependent. We assume a Gaussian field at the dimensionsless time τ with the width σ̃ = σε⊥ , Ẽ(t̃) = Ẽ0 e−(t̃−τ ) 2 /σ̃ 2 (4.55) and obtain √ π Ã(t̃) = −Ẽ0 σ Erf[(t̃ − τ )/σ̃] − Erf[(−τ − t0 )/σ̃] . 2 The occuring error function is defined as Z z 2 2 Erf(z) = √ e−x dx . π 0 (4.56) (4.57) Using this Ansatz for the field strength in Eq. (4.49) we obtain the numerical results plotted in Figs. 4.3 and 4.4. Therein all occuring values are dimensionsless. The electric field is non zero within a certain width σ̃ = 1 for τ = 0 around t̃ = 0. Therefore the oscillations for times beyond the time where the electrical field is finite are damped out. The pair production rate is peaked around small momenta for both weak and strong fields. For strong fields the distribution is shifted to positive momenta but remains still close to small parallel momenta. It is important to point out that the production of particles happens not only at rest (pk = 0) what is assumed in many works also addressing the back reaction problem, e.g.[19]. In contrary we find a non-trivial momentum dependence of the pair creation rate depending on the field strength and on time. 136 4.1.4 Summary We have derived a quantum kinetic equation within a consistent field theoretical treatment which contains the creation of particle-antiparticle pairs in a timedependent homogeneous electric field. For both fermions and bosons we obtain a source term providing a kinetic equation of non-Markovian character. The source term is characterized by a pair production rate that contains a time integration over the evolution of the distribution function and therefore involves memory effects. For the simple case of a constant electric field in low density limit and Markovian approximation, we analytically and numerically reproduce the results of earlier works[6]. Since our approach is not restricted to constant fields, we have explored the dependence of the source term on a time dependent (model) electric field with a Gaussian shape. Within these two different Ansätze for the field, we have performed investigations of the time structure of the source term. The production of pairs does not happen at rest only. We observe a non-trivial momentum dependence of the source term depending on the field strength and on time. The particle production source is dominated by two time scales: the memory time and the production time. The numerical results mainly show oscillations due to the dynamical phase and urge the need to include the Maxwell equation to determine the electric field by physical boundary conditions (back reactions), work in this direction is in progress. Furthermore, it would be of great interest to extend this approach to the QCD case to explore the impact of a non-Markovian source term on the pre-equilibrium physics in ultrarelativistic heavy-ion collisions. 137 Bibliography [1] N.B. Naroshni and A.I. Nikishov, Yad. Fiz. 11 (1970) 1072 (Sov. J. Nucl. Phys. 11 (1970) 596); V.S. Popov and M.S. Marinov, Yad. Fiz. 16 (1972) 809 (Sov. J. Nucl. Phys. 16 (1974) 449); V.S. Popov, Zh. Eksp. Teor. Fiz. 62 (1972) 1248 (Sov. Phys. JETP 35 (1972) 659); M.S. Marinov and V.S. Popov, Fort. d. Phys. 25 (1977) 373. [2] A. Bialas and W. Czyż, Phys. Rev. D 30 (1984) 2371; 31 (1985) 198; Z. Phys. C 28 (1985) 255; Nucl. Phys. B 267 (1985) 242; Acta Phys. Pol. B17 (1986) 635. [3] K. Kajantie and T. Matsui, Phys. Lett. B 164 (1985) 373. [4] G. Gatoff, A.K. Kerman, and T. Matsui, Phys. Rev. D 36 (1987) 114. [5] F. Cooper, J.M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky, Phys. Rev. D 48 (1993) 190. [6] J. Rau, Phys.Rev. D 50 (1994) 6911. [7] R.S. Bhalerao and V. Ravishankar, Phys.Lett. B 409 (1997) 38. [8] W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer-Verlag, Berlin, 1985). [9] A.A. Grib, S.G. Mamaev, and V.M. Mostepanenko, Vacuum quantum effects in strong external fields, (Atomizdat, Moscow, 1988). [10] S. Nussinov, Phys. Rev. Lett. 34 (1975) 1286. [11] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys. Rep. 97 (1993) 31. [12] N.K. Glendenning and T. Matsui, Phys. Rev. D 28 (1983) 2890. [13] F. Sauter, Z. Phys. 69 (1931) 742. [14] W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714. 138 [15] J. Schwinger, Phys. Rev. 82 (1951) 664. [16] F. Cooper and E. Mottola, Phys. Rev. D 40 (1989) 456. [17] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola, Phys. Rev. Lett. 67 (1991) 2427. [18] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola, Phys. Rev. D 45 (1992) 4659. [19] Y. Kluger, J.M. Eisenberg, and B. Svetitsky, Int. J. Mod. Phys. E 2 (1993) 333 (here further references may be found). [20] F. Cooper, S. Habib, Y. Kluger, E. Mottola, J.P. Paz, and P.R. Anderson, Phys. Rev. D 50 (1994) 2848. [21] J.M. Eisenberg, Phys. Rev. D 51 (1995) 1938. [22] Ch. Best and J.M. Eisenberg, Phys. Rev. D 47 (1993) 4639. [23] J. Rau and B. Müller, Phys. Rep. 272 (1996) 1. [24] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964). 139 4.2 Time-dependent masses: The Inertial Mechanism This lecture is devoted to construct the kinetic theory for vacuum creation of particles with the masses depending on time. We shall name for short such mechanism as inertial. A microscopic foundations of that dependence can be different. The Higgs mechanism leads to the most popular models of such class, when corresponding mean fields depend on time. It can consider rather general quantum field models with non-polynomial interactions, where the separation of non-stationary mean fields forms the time depending mass [1]. A well known example of this kind is the Witten – Di Veccia – Veneziano model [2, 3], where the mean field conception was analyzed in the work [4]. The model of the NambuJona-Lasinio [5] and σ-model [6] are the other examples of such kind, where the hadron masses are defined by intensity of a quark condensate. Its evolution can be described at the hydrodynamical [7] or kinetic level [8]. The particle mass can depend on the many-particle interaction in hot and dense nonstationary matter [9, 10, 11]. The field dependence on the mass is a general factor determining its time evolution in all these cases (F-class models). The conformal invariance of the scalar-tensor gravitational theory establishes a time dependence of the particle mass by means of the conformal multiplier [12, 13]. Such theories should be referred to other class (C-class). In the F-class theories, the vacuum particle creation admits a well known interpretation based on the simplified vacuum tunneling model in an external field [14, 15, 16]. The C-class models are not admit the similar interpretation. On the phenomenological level, however, both classes have the uniform mathematical description. It will be shown in Sec. 4.2.1-4.2.3 for the scalar, fermion and massive vector boson QFT, correspondingly. The kinetic theory construction will be based on the oscillator (holomorphic) representation (OR) [17], which is the most economical method for the nonpertrubative description (in comparison with the Bogoliubov’s method of the canonical transformation [18]) for the vacuum particle creation under the action of a time-dependent strong fields. This approach leads at once to the quasiparticle representation (QPR) with diagonal operator forms in the momentum space for the dynamical variable set and allows to get rather easily the Heisenberg-type equations of motion for the creation and annihilation operators. An important feature of the time dependent Fock representation is the necessity of agreement of commutation (anti-commutation) relations with the equations of motion. This circumstance can bring to the non-canonical quantization rules ( such example will be considered in Sect.4.1). On this OR basis it is possible to derive directly the corresponding kinetic equations (KE) using the well known methods [19]. Some particular results are published in the works [1, 20]. The kinetic theory for the scalar, spinor and massive vector fields is constructed in the Sect. 2,3 and 4, respectively. The main 140 attention will be spared to the particle creation in the conformal cosmological models (Sec. 4.2.4). It was shown there, that the choice of the equation of state (EoS) of the Universe allows to obtain, in principle, the observed number densities of matter and photons and, possibly, dark matter. The basic problem here is the description of vacuum particle creation which must be self-consistent with EoS. This problem remains outside of the framework of the present article. Finally, in the Sect.6 it is considered an other class of scalar QFT systems with non-monotonic self-interaction potentials admitting decomposition of the field amplitude into the quasiclassical space homogeneous time-dependent background field and the fluctuation part. In this case, the particle mass is defined by intensity of quasiclassical field. As an example, it is analyzed here the self-interaction potentials of the simplest polynomial type and a non-trivial case (the pseudoscalar sector of the Witten – Di Veccia – Veneziano model). It is showed, that the relevant definition of vacuum states allows to avoid of the tachyonic modes rise. The main purpose of this work is the summarizing of all known relevant results and to separate unsolved problems which will be listed shortly in Sec. 4.2.6. We use the metric g µν = diag(1, −1, −1, −1) and the natural system h̄ = c = 1. 4.2.1 Scalar field Oscillator and quasiparticle representations Let consider the simplest case of the scalar real field with the time dependent mass m(t) and the equation of motion [∂µ ∂ µ + m2 (t)]ϕ(x) = 0. (4.58) The corresponding Lagrange function is 1 1 L = ∂µ φ ∂ µ φ − m2 (t)φ2 . 2 2 (4.59) The assumption about the space homogeneity of the system allows to look for the solution of Eq. (4.58) in a discrete momentum space in the following form 1 X ipx e ϕ(p, t), ϕ(x) = √ V p (4.60) where V = L3 and pi = (2π/L)ni with the integer ni for each i = 1, 2, 3. The thermodynamical limit can be fulfilled in the resulting equations. Then the oscillator-type equation of motion follows from the Eqs. (4.58) and decomposition (4.60) ϕ̈(±) + ω 2 (p, t)ϕ(±) (p, t) = 0, (4.61) 141 where ω(p, t) is the time-dependent one-particle energy, ω(p, t) = q m2 (t) + p2 . (4.62) The symbols (±) correspond to the positive and negative frequency solutions of Eq. (4.60) defined by its free asymptotics in the infinite past (future) [18], ϕ(±) (p, t → ∓∞) ∼ e±iω∓ t , where ω∓ = q (4.63) m2∓ + p2 are defined by asymptotics of mass m∓ = lim m(t). t→∓∞ (4.64) The asymptotics (4.63) correspond to the in(out) - states and are necessary for definition in(out) - vacuums. This requirement, however, can be broken in cosmology [21]. We suppose here that such asymptotics exist and the relevant vacuum states will be denoted by |0i without indices ”in” or ”out” , which can be defined from context. In the considered class of the problems, the classification of the states on frequency sign turns out impossible for arbitrary time moment. According to general analysis of the work [72], it leads to instability of vacuum state at the action period of the inertial mechanism and vacuum particle creation. Then it is possible to talk about quasiparticle excitations at the period of evolution of the system (ṁ(t) 6= 0) (QR) and to describe their vacuum creation and annihilation over the vacuum state |0i. The conception of ’quasiparticle’ plays the central role in QFT in the presence of strong time dependent quasiclassical external fields [18, 72]. In the considered conditions, it is the nearest realization of the articleconception in the standard Fock representation of QFT, where an external field can be taken into account on the basis of perturbative approach. Thus, QPR corresponds to a possibility to write in diagonal form the row of commutative operators of physical (observable) values (the complete QPR) or, as minimum, the Hamiltonian only (the uncomplete QPR in the case of several observables). It is natural, the question is about the operators having the quadratic form in the Fock representation. Hence, interaction between the field constituents and self-interaction are not taken into account here. It corresponds to the non-dissipative approximation in the kinetic theory [73]. The transition to QPR can be realized by different ways. The traditional method is based on time dependent canonical Bogoliubov’s transformation [18]. The alternative approach is based on the OR, which leads at once to QRP [17]. In the considered case the transition to OR is fulfilled by means of the substitution m± → m(t) in the dispersion law of the corresponding free field decompositions (4.60) n o 1 1 X q ϕ(x) = √ eipx a(−) (p, t) + a(+) (−p, t) (4.65) V p 2ω(p, t) 142 and the corresponds generalized momenta i X π(x) = − √ V p s o ω(p, t) ipx n (−) a (p, t) − a(+) (−p, t) , e 2 (4.66) where a(±) (p, t) are the positive and negative frequency amplitudes corresponding to the asymptotic (4.63). The decompositions (4.65) and (4.66) are postulated essentially and they are the basis of a nonperturbative dynamics, which will be discussed below. The substitution of the decompositions (4.65) and (4.66) into the Hamiltonian H(t) = 1 2 Z n d3 x π 2 (x) + [∇ϕ(x)]2 + m2 (t)ϕ2 (x) o (4.67) leads simultaneously to a diagonal form, which corresponds to the QPR, H(t) = n o 1X ω(p, t) a(+) (p, t)a(−) (p, t) + a(−) (p, t)a(+) (p, t) . 2 p (4.68) The equations of motion for the field amplitudes a(±) (p, t) can be obtained now from the minimal action principle [17] or on the basis of the Hamilton equations ∂H ∂H φ̇ = = π, π̇ = − = 4φ − m2 (t)φ. (4.69) ∂π ∂φ The relevant equations of motion can be obtained from here with taking into account of the decompositions (8) and (9), ȧ(±) (p, t) = 1 ∆(p, t)a(∓) (−p, t) ± iω(p, t)a(±) (p, t), 2 (4.70) where the overdot denotes the time derivative and ∆(p, t) = ω̇(p, t) m(t)ṁ(t) = 2 ω(p, t) ω (p, t) (4.71) is the transition amplitude between states with positive and negative energies. Then the canonical commutation relation [ϕ(x), π(x0 )]t=t0 = iδ(x − x0 ) (4.72) with the decompositions (4.65) and (4.66) form the standard commutation relations for the time dependent creation and annihilation operators, h i a(−) (p, t), a(+) (p0 , t) = δpp0 (4.73) etc. That relation is consistent obviously with the equations of motion (4.70). 143 The corresponding Hamilton operator (4.68) will be equal H(t) = X ω(p, t) a p (+) (p, t)a (−) 1 (p, t) + 2 = H0 (t) + hν (t), (4.74) where H0 (t) = X ω(p, t)a(+) (p, t)a(−) (p, t) (4.75) p and hν (t) = 1X ω(p, t). 2 p (4.76) Thus, vacuum energy of zero oscillations (”zitterbewegung” ) depends on time. The equation of motion (4.70) can be rewritten as the Heisenberg-type equation h i 1 (4.77) ȧ(±) (p, t) = ∆(p, t)a(∓) (−p, t) + i H(t), a(±) (p, t) . 2 These equations (or Eqs. (4.70)) in the instantaneous QPR are the basis for nonperturbative derivation of KE describing the processes of the scalar particles creation and annihilation under action of the inertial mechanism. Kinetic equation The basic object of the kinetic theory is the quasiparticle distribution function which is for the space homogeneous case L3 f (p, t) = h0|a(+) (p, t)a(−) (p, t)|0i, (4.78) where |0i = |0iin is the initial (t → −∞) vacuum state. Differentiating the distribution function (4.78) with respect to time, we obtain using the equations of motion (4.70) n o 1 f˙(p, t) = ∆(p, t) f (+) (p, t) + f (−) (p, t) , 2 (4.79) where the auxiliary one-particle correlation functions are introduced L3 f (+) (p, t) = h0|a(+) (p, t)a(+) (−p, t)|0i, L3 f (−) (p, t) = h0|a(−) (−p, t)a(−) (p, t)|0i. (4.80) The equation of motion for the functions (4.80) can be obtained by analogy with the equation (4.79). We write it out in the integral form f (±) (p, t) = Zt 0 dt0 ∆(p, t0 ) [1 + 2f (p, t0 )] e±2iθ(p;t,t ) , t0 144 (4.81) where the initial conditions have been used f (±) (p, t0 ) = 0. According to Eq. (4.79), it means the statement of the Cauchy problem f (p, t0 ) = 0. This condition should be considered together with the asymptotic relations (4.63) and (4.64). Finally, the dynamical phase in Eq.(4.81) is equal 0 θ(p; t, t ) = Zt dτ ω(p, τ ). (4.82) t0 The substitution of Eqs.(4.81) in Eq.(4.79) brings to the resulting KE (we write this equation in the thermodynamical limit V → ∞ at the fixed particle density) 1 f˙(p, t) = ∆(p, t) 2 Zt dt0 ∆(p, t) [1 + 2f (p, t0 )] cos [2θ(p; t, t0 )] , (4.83) t0 where the source term in right side describing a variation of particle number with given momentum as the result of vacuum creation and annihilation processes under the action of the inertial mechanism, ∆(p, t) is defined by Eq.(4.71). The non-Markovian KE (4.83) has the same form that KE for the Schwinger mechanism of pair creation in an electric field [19]. This equation had been investigated in detail in the works [22, 24] in application to description of pre-equilibrium evolution of quark-gluon plasma generated at collisions of ultrarelativistic heavy ions. The case of the scalar QED has considered in [17] for the electric field of the arbitrary polarization. As it follows from Eq.(4.71), in the framework of the inertial mechanism the particle production rate is defined by the velocity of mass change ξ(t) = 1 dm(t) , m(t) d(m0 t) (4.84) where m0 is the characteristic mass fixing the time scale. In particular, the adiabatic process (ξ(t) 1) is not accompanied by vacuum particle creation. Let us underline that KE (4.83) is valid in the framework of two basis suppositions: a) the particles (or antiparticles) are absent in the in-state; b) the collisionless approximation is used (i.e. the corresponding dissipative processes are not taken into account). In the low density approximation f (p, t) 1 KE (4.83) brings to the solution [22, 27] 2 t Z 1 f (p, t) = dt0 ∆(p, t0 ) exp [2iθ(p; t, t0 )] ≥ 0. 4 t0 145 (4.85) The KE (4.83) can be transformed to a system of ordinary differential equations with zero initial conditions 1 f˙ = ∆u, 2 u̇ = ∆(1 + 2f ) − 2ωv, v̇ = 2ωu, (4.86) which is convenient for numerical analysis. This equation system has the first integral (1 + 2f )2 − u2 − v 2 = 1, (4.87) according to it the phase trajectories are located on two-cavity hyperboloid with top coordinates f = u = v = 0 (physical branch) and f = −1, u = v = 0 (nonphysical one). Excluding the function f from Eqs. (4.86), we obtain the two dimensional dynamical system with the equations of motion √ u̇ = ∆ 1 + u2 + v 2 − 2ωv, v̇ = 2ωu, (4.88) having non-Hamilton structure. The functions f (p, t) and u(p, t) have the direct physical meaning (the last function describes the vacuum polarization effects, see below Sect.2.3) and are invariants relatively of the time inversion t → −t while the auxiliary function v(p, t) changes its sign. Thus, KE (4.83) is invariant at the time inversion. Observable variables and regularization The KE (4.83) describes the vacuum quasiparticle excitations existing during an external influence (ṁ(t) 6= 0, in the considered case). After switch off of this action, some density of real (residual) particles and antiparticles are remained. In absence of an interaction between the system constituents, the real particles are ”on-shell” one with the dispersion laws ω± of the free particles with the masses m± (4.64) while the quasiparticles are ”off-shell” one with the dispersion law (4.62). In general case, the on-shell condition m(t) − m 0 m0 1 (4.89) (m0 is some characteristic mass, e.g., m0 = m± ), is not connected directly with < 1, where ξ(t) the condition of the effectiveness of vacuum particle creation, ξ(t) ∼ is defined by Eq.(4.84). Differently, the presence of high frequencies in the function m(t) is necessary for vacuum creation and is not contradict to the on-shell condition (4.89). In principle, the KEs of such type are intended for description of evolution both real particles and quasiparticles. In particular, the distribution function of the residual particles is fout (p) = lim f (p, t). The simple formula t→∞ 146 for fout (p) follows from Eq. (4.85) in the low density approximation. However, the presence of the fast oscillated multiplier in the source term in r.h.s. of KE (4.83) brings to huge volume of numerical calculations which make impossible the research of the system evolution for rather great scales after the initial moment of switch on an external forces. The corresponding large scaling methods of calculations based on the KE (4.83) are not worked out at presence. The estimation of some properties of residual particle - antiparticle plasma as the result of a limited pulse of an external field action can be fulfilled on the basis of the imaginary time method [68]. The distribution function is the basic observable characteristic of system. The densities of other observable variables are some integrals in the momentum space containing the distribution function and the auxiliary functions u(p, t), which describe the effects of vacuum polarization. The simplest variable of such type is the total density of particles and antiparticles considered as the quasiparticles. We have in the thermodynamical limit L → ∞ ntot (t) = Z [dp]f (p, t), (4.90) where [dp] = (2π)−3 d3 p. The other important characteristics of the system are the energy density and pressure, which can be obtained as the average value of the energy-momentum tensor corresponding to the Lagrange density (4.127), Tµν = ∂µ φ∂ν φ − gµν L. (4.91) As the result we have [18] ε = h0|T00 |0i = 3p = ε − Z Z [dp] ωf, (4.92) 1 m2 [dp] (f + u) + ωu . ω 2 (4.93) The last two terms in the Eq.(4.93) represent the contribution of vacuum polarization. These equations produce the time-dependent p-ε EoS of the system. Finally, the entropy density can be introduced S(t) = − Z [dp][f ln f − (1 + f ) ln (1 + f )]. (4.94) It does not conserve (Ṡ(t) 6= 0) even in the considered non-dissipative approximation because the system is open (the mass change is defined by external causes). The direct proof of the convergence of (4.90), (4.92), (4.93), (4.94) integrals is complicated here because of an absence of an explicit form of these functions. Therefore, the method of the asymptotic expansions in power series of the inverse momentum p−n (N-wave regularization technique) [25] is used usually (the other 147 approach is based on WKB approximation [26]). The applicability of this method in the considered case can raise the doubts because of the researched functions are quickly oscillating at p → ∞. Let us show it by means of the explicit asymptotic solutions of the system (4.86) by |p| m. It is assumed that the integral (4.90) is convergent at any time. Then the function f (p, t) must decrease at p → ∞ and hence f (p, t) 1 in that region. This inequality corresponds to the low density approximation (4.85), where the KE solution can be written in explicit form Zt 2 1 0 0 0 0 ∞ (4.95) f (p, t) = dt m(t )ṁ(t ) exp [2ip (t − t )] , 4 4p t0 p = |p| → ∞ and 1 u (p, t) + iv (p, t) = 2 p ∞ ∞ Zt t0 dt0 m(t0 )ṁ(t0 ) exp [2ip (t − t0 )], (4.96) because δ ∞ (p, t) = m(t)ṁ(t)/p2 according to Eq. (4.71). Thus, asymptotic solution is indeed some quickly oscillating function (this fact was observed first in the work [20] in the case of massive vector bosons, see Sect. 4.2.3). Such behaviour corresponds to quasiparticle interpretation of vacuum exitations under action of the inertial mechanism. The real (observed) particles are the result of the evolution to the moment, when ṁ = 0 and the out-vacuum state is realized. The asymptotics (4.95) and (4.96) can be changed for other (non-inertial) mechanisms of vacuum particle creation ( e.g., in the case of harmonic ”laser” electric field [28, 29]. The regularizing procedure of the Pauli-Villars type is based on the substraction of the corresponding counterterms into the integrals of the type (4.90), (4.92), (4.93), (4.94) . These counterterms can be obtained by the change p2 → p2 + M 2 in the denominators of the asymptotics (4.95) and (4.96), fR = f − f M , uR = u − u M . (4.97) If the regularizing mass M m(t) can be chosen rather large, M Λ (Λ is ”the computer cut-off parameter”), influence of the conuterterms on the results of numerical calculations is negligible small. 4.2.2 Fermion field Quasiparticle representation The equations of motion of the fermion field with variable mass have the form [iγ µ ∂µ − m(t)]ψ(x) = 0, ψ̄(x)[iγ µ ∂µ + m(t)] = 0, 148 (4.98) where ψ̄ = ψ + γ 0 . The corresponding Hamiltonian is (k = 1, 2, 3) H(t) = i Z 3 + d x ψ ψ̇ = Z d3 x ψ̄{−iγ k ∂k + m(t)}ψ. (4.99) In the considered case, the system is space homogeneous and non-stationary. Therefore the transition to Fock space can be realized on the basis functions ϕ = exp(±ikx) and creation and annihilation operators become the time dependent ones. So, we have the following decompositions of the field functions in discrete momentum space 1 X X ipx (α) ψ(x) = √ {e aα (p, t)u(α) (p, t) + e−ipx b+ (p, t)}, α (p, t)v V p α=1,2 1 X X −ipx + {e aα (p, t)ū(α) (p, t) + eipx bα (p, t)v̄(α) (p, t)}. (4.100) ψ̄(x) = √ V p α=1,2 The OR is intended for a derivation of the motion equations for the creation and annihilation operators on the basis of the primary equations (4.98) and the free u, v - spinors as the basic functions with the natural substitution m → m(t). The following free-like equations for the spinors are postulated in OR as [γp − m(t)] u(p, t) = 0, [γp + m(t)] v(p, t) = 0, (4.101) where p0 = ω(p, t). These definitions create the set of the standard orthonormality conditions [30] m(t) δαβ , v̄ α (p, t) ω(p, t) m(t) δαβ , v (β) (p, t) = − ω(p, t) u(α)+ (p, t)u(β) (p, t) = v (α)+ (−p, t)v(β) (−p, t) = δαβ , ūα (p, t)v(β) (p, t) = ū(α) (p, t)γ 0 v(β) (−p, t) = 0. ūα (p, t)u(β) (p, t) = (4.102) The decompositions (4.100) and the relations (4.102) lead to the diagonal form of the Hamiltonian (4.99) at once (before second quantization) H(t) = X p,α + ω(p, t)[a+ α (p, t)aα (p, t) − bα (−p, t)bα (−p, t)]. (4.103) Such form of the Hamiltonian is necessary for interpretation of the time dependent operators a+ , a (and b+ , b) as creation and annihilation operators of quasiparticles (anti-quasiparticles). Thus, this way results to QPR. We will not take an interest in diagonalization of the spin operator. Therefore, such QPR can be named as incomplete one (see below Sec. 4.2.3). 149 Now, in order to get the equation of motion for the creation and annihilation operators in the OR, let us substitute the decomposition (4.100) in Eqs.(4.98) and use the relations (4.102). Then we obtain as the intermediate result the following closed system of equations of motion in the matrix form ȧ(p, t) + U(1) (p, t)a(p, t) + U(2) (p, t)b+ (−p, t) = −iω(p, t)a(p, t), + + ḃ(−p, t) + a+ (p, t)V(1) (p, t) − b(−p, t)V(2) (p, t) = −iω(p, t)b(−p,(4.104) t). The spinor constructions were introduced here αβ U(1) = u+(α) (p, t)u̇(β) (p, t), αβ V(1) = v +(α) (−p, t)u̇(β) (p, t), αβ U(2) = u+(α) (p, t)v̇(β) (−p, t), αβ V(2) = v +(α) (−p, t)u̇(β) (−p, t) . (4.105) The matrixes U(2) and V(1) describe transitions between states with the positive and negative energies and different spin while the matrixes U(1) and V(2) show the spin rotations only. The equations (4.104) are compatible with the standard anti-commutation relations 0 + 0 {aα (k, t), a+ β (k , t)} = {bα (k, t), bβ (k , t)} = δkk0 δαβ , (4.106) + that can be shown with the help of the relation U(1) = −U(1) . The material part of the Hamilton operator will be equal then: H0 (t) = X + ω(p, t)[a+ α (p, t)aα (p, t) + bα (p, t)bα (p, t)] . (4.107) p,α Let us write now the u, v - spinors in the explicit form, using the corresponding free spinors [31]: u+(1) (p, t) u+(2) (p, t) v +(1) (−p, t) v +(2) (−p, t) = = = = A(p) [ω+ , 0, p3 , p− ] , A(p) [0, ω+ , p+ , −p3 ] , A(p) [−p3 , −p− , ω+ , 0] , A(p) [−p+ , p3 , 0, ω+ ] , (4.108) where p± = p1 ± ip2 , ω+ = ω + m and A(p) = [2ωω+ ]−1/2 . The spin rotations matrixes (4.105) in the representation (4.108) are equal to zero, U(1) = V(2) = 0. (4.109) 150 The rest matrixes (4.105) are equal 11 22 U(2) (p, t) = −U(2) (p, t) = 12 U(2) (p, t) = ṁ(t)p− , 2ω 2 (p, t) ṁ(t)p3 , 2ω 2 (p, t) 21 U(2) (p, t) = ṁ(t)p+ 2ω 2 (p, t) (4.110) and in addition V(1) = −U(2) . Kinetic equation The equations of motion (4.104) are not contain the spin rotation matrices (4.109) and they are similar to Eqs.(4.70), therefore the KE derivation has not problems now. Let us introduce one-particle correlation function of electrons and positrons gαβ (p, t) = < 0|a+ β (p, t)aα (p, t)|0 >, g̃αβ (p, t) = < 0|bβ (−p, t)b+ α (−p, t)|0 > . (4.111) Differentiation the first of them over time leads to equation (we use the equations of motion (4.104) and the short notation U(2) = U ) ġ(p, t) = −U(p, t)g(+) (p, t) − g(−) (p, t)U(p, t), (4.112) where the auxiliary correlation functions were introduced (+) + gαβ (p, t) = < 0|a+ β (p, t)bα (−p, t)|0 >, (−) gαβ (p, t) = < 0|bβ (−p, t)aα (p, t)|0 > . (4.113) The subsequent transformations are distinguished from the KE derivation in the case of scalar particles (Sec. 4.2.1) by replacement of the commutator (4.73) in the anticommutator (4.106) and taking into account of matrix character of the equations of motion (4.104). As the result, we have ġ(p, t) = Zt t0 0 dt0 {U(p, t)[g̃(p, t0 )U(p, t0 ) − U(p, t0 )g(p, t0 )]e−2iθ(p,t ,t) + h.c.}. (4.114) The KE for the all component of the correlation function fαβ (p, t) (4.111) is not of physical interest by virtue of isotropy of the system. Hence, gαβ = f δαβ , where the distribution function of the fermi particles is introduced, f (p, t) = 1 X gαα (p, t). 2 α=1,2 151 (4.115) Further, it is easy to calculate T r U (p, t)U(p, t0 ) with help of the equalities (4.110). The corresponding KE for this distribution function has the standard form [19, 22]: f˙(p, t) = 2λ(p, t) Zt t0 dt0 λ(p, t0 )[1 − 2f (p, t0 )] cos[2θ(t, t0 )], (4.116) where the amplitude λ(p, t) is defined by the relation (p = |p|) ω̇p . ω(p, t)m(t) λ(p, t) = (4.117) The KEs (4.83) and (4.116) have the analogical form but differ by the statistical factors 1 ± 2f (the Boze enhancement or the fermi suppression) and by structure of the vacuum transition amplitudes (4.71) and (4.117). The corresponding system of ordinary differential equations has the following form: 1 f˙ = λu, u̇ = λ[1 − 2f ] − 2ωv, v̇ = 2ωu. (4.118) 2 This system possesses one first integral of motion (e.g., [33]) (1 − 2f )2 + v 2 + u2 = 1. (4.119) This relation represent the ellipsoid in the phase space of (f, u, v) variables. After excluding function f from (4.118), we obtain the dynamical system in d = 2 + 1 space √ v̇ = ±λ 1 − v 2 − u2 − 2ωu, u̇ = 2ωv. (4.120) This system is non-Hamiltonian as a consequence of non-unitary transition from the initial to the quasiparticale representation (see, Sect.2.3.). As it can proof easily, KE (4.116) is invariant in respect to time inversion. The equations analogous to (4.86) and (4.118) was obtained in the work [32] (see also [18]) for the conformal flat space-time. Observables and regularization The observable variables of spinor field are defined by following expressions: ntot (t) = 2 Z [dp]f (p, t), ε = h0|T00 |0i = 2 p(t) = 1 (ε + m2 3 Z S(t) = −2 Z Z (4.121) [dp]ωf, [dp]u), [dp][f ln f + (1 − f ) ln (1 − f )]. 152 (4.122) (4.123) (4.124) The same regularization procedure for calculation of the divergence integrals showed in Sect.2.3 for the scalar bosons can be realized here. The Eq.(4.85) for the distribution function in the low density approximation is valid here also after the replacement ∆(p, t) → λ(p, t). The asymptotic of the amplitude (4.117) is equal λ∞ (p, t) = ṁ(t)/p. Using Eq.(4.85) and the rules of Sect 2.3, it is easily to obtain the following expressions for the counterterms for the case of fermion fields: t Z 1 2 (p + M 2 )| dt0 ṁ(t0 ) exp [2ip(t − t0 )]|2 , fM (p, t) = 4 (4.125) t0 uM (p, t) = √ p2 1 + M2 Zt t0 dt0 ṁ(t0 ) cos [2p(t − t0 )]. (4.126) However these counterterms can be ignored by computer calculations. 4.2.3 Massive vector bosons Complete QPR The simplest version of quantum field theory of the massive vector bosons is based on the Lagrangian density [30] L(§) = −∂µ uν ∂ µ uν + m∈ (t)uν uν , (4.127) which corresponds to the equation of motion (∂µ ∂ µ + m2 (t))uν = 0 (4.128) with the additional exterior constraint ∂µ uµ = 0. (4.129) The alternative way is based on the Lagrangian [34] ∞ ∞ L(§) = − F µν Fµν + m∈ (t)uν uν , 4 ∈ (4.130) where Fµν = ∂ν uµ −∂µ uν . The constraint (4.129) is here a consequence of dynamical equations. The Lagrangians (4.127) and (4.130) bring to different energymomentum tensors, that can be of great importance for EoS. The transition to the QPR is carried out on the basis of the standard decomposition of the free field operators and momenta in the discrete momentum space with the replacement m → m(t) in the dispersion law (see Sect.4.2.1), uµ (x) = P √1 V p πµ (x) = − √iV √ Pq p 1 2ω(p,t) eip x n o aµ(−) (p, t) + a(+) µ (−p, t) , n o (−) 2ω(p, t) e−ip x a(+) µ (p, t) − aµ (−p, t) . 153 (4.131) The substitution of the field operators (4.131) into the Hamiltonian H=− Z dx πµ π µ + ∇uµ ∇uµ + m2 (t)uµ uµ (4.132) brings it at once to a diagonal form in Fock space, which corresponds to the QPR, H=− X ω(p, t) p (−)µ a(+) (p, t) µ (p, t)a + aµ(−) (−p, t)a(+)µ (−p, t) . (4.133) However, this quadratic form is not positively defined. In order to exclude the µ = 0 component with the help of the additional condition (4.129), it is necessary to derive the equations for the amplitudes a± . Substituting the decompositions (4.131) into the equation of motion (4.128), we find the Heisenberg equations of motion for the time-dependent creation and annihilation amplitudes 1 ∆(p, t)aµ(∓) (−p, t) ± iω(p, t)aµ(±) (p, t), (4.134) 2 where ∆(p, t) is defined by Eq.(4.71). The additional condition (4.129) may be transformed now with the help of Eqs. (4.134) to the following form (i = 1, 2, 3) ȧµ(±) (p, t) = (±) (±) ω(p, t)a0 (p, t) = pi ai (p, t). (4.135) These equations allow to exclude the µ = 0 component in the Hamiltonian (4.133) with the result H= X (+) (−) (−) (+) ω(p, t) ai (p, t)ai (p, t) + ai (−p, t)ai (−p, t) p,i,k 1 (−) (−) (+) (+) t) . pi ai (p, t)pk ak (p, t) + pi ai (−p, t)pk ak (−p,(4.136) − 2 ω (p, t) The next step is the additional diagonalization of the quadratic form (4.136) by means of the linear transformations [30] ω (±) (±) (±) a(±) (p, t) = Eα(±) (p, t) ≡ e1 α1 (p, t) + e2 α2 (p, t) + e3 α3 (p, t), m(t) (4.137) where {e1 (p), e2 (p), e3 (p)} is the local basis built on the vector e3 = p/|p|. These real unit vectors form the triad, eik ejk = eki ekj = δij , eik = (ei )k . (4.138) The presence of the factor ω/m in the non-unitary matrix E in Eq. (4.137) leads to a violation of the unitary equivalence between the a and α - representations. The transformation (4.137) establishes the positively defined Hamiltonian H= X p ω(p, t) (+) (−) αi (p, t)αi (p, t) + 154 (−) (+) αi (−p, t)αi (−p, t) . (4.139) The equations of motion for these new amplitudes follow from a combination of Eqs. (4.134) and (4.137): 1 (∓) (±) (±) ∆(p, t)αi (−p, t) ± iω(p, t)αi (p, t) + ηij (p, t)αj (p, t). 2 (4.140) The spin rotation matrix ηij is defined as (±) α̇i (p, t) = ηik (p, t) = −∆m δi3 δk3 , (4.141) where ∆m = −ṁ/m + ∆. This relation shows the distinguished role of the third component. Together with the Hamiltonian (4.133), the total momentum operator takes also the diagonal form. However, the spin operator Si = εijk Z dx uk πj + πj uk − uj πk − πk uj (4.142) has non-diagonal form in spin space in terms of the operator α(±) Sk = iεijk X (+) (−) αi (p, t)αj (p, t) p − (−) (+) αi (−p, t)αj (−p, t) , (4.143) εijk is the unit antisymmetric tensor. In particular, the spin projection to the p3 -axis is S3 = i X p + (+) (−) (+) (−) α1 (p, t)α2 (p, t) − α2 (p, t)α1 (p, t) (−) (+) α2 (−p, t)α1 (−p, t) − (−) (+) α1 (−p, t)α2 (−p, t) . (4.144) Thus, this representation can be named an incomplete quasiparticle one with non-fixed spin projection. The operator (4.144) can be diagonalized with a linear transformation to the new basis [30] (±) (±) (±) ci (p, t) = Rik αk (p, t), (4.145) 1 ∓i 0 1 ± R = √ ±i 1 √0 . 2 2 0 0 (4.146) with the unitary matrix As a result, the new amplitudes c(±) correspond to the creation and annihilation operators of vector quasiparticles with the total energy, 3-momentum and spin 155 projection on the chosen direction, H(t) = X ω(p, t) p Π(t) = X p (+) (−) ci (p, t)ci (p, t) (+) (−) ci (p, t)ci (p, t) p S3 (t) = X p (+) − (−) + (−) (+) ci (−p, t)ci (−p, t) (−) (+) ci (−p, t)ci (−p, t) (−) , ,(4.147) (4.148) (+) c1 (p, t)c1 (p, t) − c1 (−p, t)c1 (−p, t) (−) (+) (+) (−) + c2 (−p, t)c2 (−p, t) − c2 (p, t)c2 (p, t) . (4.149) This representation can be named the complete quasiparticle one. The equations of motion for these amplitudes follow from Eqs. (4.140),(4.145) 1 (∓) (±) (±) (±) ∆(p, t)ci (−p, t) ± iω(p, t)ci (p, t) + ηij (p, t)cj (p, t), 2 (4.150) where the matrix ηij is fixed by Eq.(4.141). The transition to this representation from the initial one is defined by the combination of the transformations (4.137) and (4.145), (±) ċi (p, t) = c(±) (p, t) = U(±) (p, t)a(±) (p, t), √ with non-unitary operator (e(±) = (e1 ± ie2 )/ 2) (∓) e1 U (±) (p, t) = R(±) · E−1 (p, t) = e1(±) m e ω 31 (∓) e2 (±) e2 m e ω 32 (4.151) (∓) e3 (±) e3 . m e ω 33 (4.152) The quantization problem is solved while taking into account the equation of motion. The commutation relation has the non-canonical form (−) [ci (+) (p, t), cj (p0 , t)] = Qik (p, t)Qjk (p, t)δpp0 , (4.153) where the matrices Qil (p, t) are defined by the equations Q̇ij (p, t) = ηik (p, t)Qkj (p, t) (4.154) with the initial conditions lim Qij (p, t) = δij , t→−∞ (4.155) i.e. the commutation relation transform to the canonical form only in the asymptotic limit t → −∞. The relation (4.153) provides the definition of positive energy quasiparticle excitations with some time dependent energy reservoir of the vacuum. 156 Kinetic equations The standard procedure to derive the KE [19] is based on the Heisenberg-type equations of motion (4.134) or (4.150). Let us introduce the one-particle correlation functions of vector particles and antiparticles in the initial (a)-representation (−) Fµν (p, t) = h0|a(+) µ (p, t)aν (p, t)|0i, F̃µν (p, t) = h0|aµ(−) (−p, t)a(+) ν (−p, t)|0i, (4.156) where the averaging procedure is performed over the in-vacuum state [18]. Differentiating the first one with respect to time, we obtain n o 1 (−) ∆(p, t) F(+) (p, t) + F (p, t) , µν µν 2 where the auxiliary correlation functions are introduced as (4.157) Ḟµν (p, t) = (±) Fµν (p, t) = h0|aµ(±) (±p, t)aν(±) (±p, t)|0i. (4.158) The equations of motion for these functions can be obtained by analogy with Eq. (4.157). We write them out in the integral form (±) Fµν (p, t) 1 = 2 Zt h i 0 dt0 ∆(p, t0 ) Fµν (p, t0 ) + F̃µν (p, t0 ) e∓2iθ(p;t,t ) . −∞ (4.159) (±) The asymptotic condition Fµν (p, −∞) = 0 (the absence of quasi-particles in the initial time) has been introduced here. The substitution of Eq. (4.159) into Eq. (4.157) leads to the resulting KE 1 Ḟµν (p, t) = ∆(p, t) 2 Zt dt0 ∆(p, t0 )[Fµν (p, t0 ) + F̃µν (p, t0 )] cos[2 θ(p; t, t0 )]. −∞ (4.160) This KE is an almost natural generalization of the corresponding KE for scalar particles (Sec. 4.2.1). On this stage, however, there is a number of problems that are specific for the theory of massive bosons: the energy is not positively defined, the spin operator has a non-diagonal form in the space of spin states etc., see above. This circumstance hampers the physical interpretation of the distribution function (4.156). In order to overcome this difficulty, it is necessary to pass on to the complete QPR in which the system has well-defined values of energy, spin etc. The simplest way of derivation of the KE in this QPR is based on the application of the transformations (4.151) directly onto the KE (4.160). By analogy with the definitions (4.156), we introduce the correlation functions of vector particles and antiparticles in the complete QPR (+) (−) fik (p, t) = h0in |ci (p, t)ck (p, t)|0in i, (+) (−) f˜ik (p, t) = h0in |ci (−p, t)ck (−p, t)|0in i. 157 (4.161) They are connected with the primordial correlation functions (4.156) by relations of the type − fik (p, t) = U+ (4.162) in (p, t)Ukm (p, t)Fnm (p, t), where Fnm (p, t) is the ”spatial” part of the tensor function Fµν (p, t) (4.156) (m, n = 1, 2, 3). As a result, the KE (4.160) takes the following form: 1 f˙ik (p, t) = ∆(p, t) 2 Zt dt0 ∆(p, t0 )Mikjl (p, t, t0 )[fjl (p, t0 ) + f̃jl (p, t0 )] cos 2θ(p; t, t0 ) −∞ −∆m (p, t)[δi3 f3k (p, t) + δk3 fi3 (p, t)], (4.163) where ω(t0 ) m(t) ω(t0 ) m(t) h ⊥ ⊥ δi3 δk3 δj3 δl3 δ δ δ + δ δ δ + Mikjl (t, t ) = + i3 j3 k3 l3 kl ij ω(t) m(t0 ) ω(t) m(t0 ) (4.164) ⊥ and δik = δik − δi3 δk3 . As to was expected, the distribution functions fαβ (p, t) satisfy the same KE (4.160) for α = 1, 2. The feature of the complete QPR becomes apparent only in the components of tensor distribution function fik (p, t) that contain the preferred values of spin index i and (or) k = 3. Let us select the KE for the diagonal components of the correlation functions (4.161) having a direct physical meaning as the distribution functions of the transversal (i = 1, 2) 0 # ⊥ δij⊥ δkl 1 f˙i (p, t) = ∆(p, t) 2 Zt dt0 ∆(p, t0 )[1 + 2fi (p, t0 )] cos 2θ(p; t, t0 ), (4.165) −∞ and longitudinal components t 1 m2 (t) Z ω 2 (t0 ) ˙ f3 (p, t) = ∆(p, t) 2 dt0 ∆(p, t0 ) 2 0 [2f3 (p, t0 ) + Q(p, t0 )] cos 2θ(p; t, t0 ) 2 ω (t) m (t ) −∞ −2∆m (p, t)f3 (p, t) . (4.166) Here the shorthand notation fii = fi has been introduced for the diagonal components of the matrix correlation functions (4.161), and ∆= mṁ , ω2 ∆m = −∆ p2 . m2 (4.167) The longitudinal and transversal distribution functions are connected by the relation f3 (p, t) = Q(p, t)f1 (p, t), (4.168) 158 where Q(p, t) is the function occurring in the commutator of the creation and annihilation operators for the longitudinal bosons, (−) (+) [c3 (p, t), c3 (p0 , t)] = Q(p, t) δpp0 , " m(t) ω(t0 ) Q(p, t) = exp −2 ∆m (t0 )dt0 = m(t0 ) ω(t) t0 Z t #2 ,(4.169) where m(t0 ) and ω(t0 ) are corresponding initial values. Owing to Eq. (4.168), it is sufficient to solve the one equation (4.165). KE for transversal bosons is reduced from the integro-differential form to the system of ordinary differential equations as well as the previous cases (Sects. 2.2 and 3.2): 1 f˙k = ∆uk , 2 u̇k = ∆(1 + 2fk ) − 2ωvk , v̇k = 2ωuk . (4.170) The general initial condition for all diagonal components of the distribution function lim fk (t) = lim uk (t) = lim vk (t) = 0 (4.171) t→−∞ t→−∞ t→−∞ establishes the following requirement lim m(t) = m0 , t→−∞ or lim ṁ(t) = 0. t→−∞ (4.172) The main characteristic of the vacuum creation process is the total number density of vector bosons, 1 ntot (t) = 2 π Z∞ 0 1 p dp [2f1 (p, t) + f3 (p, t)] = 2 π 2 Z∞ p2 dp f1 (p, t)[2 + Q(p, t)], (4.173) 0 where isotropy of the system was taken into account, p = |p|. As it will be shown in Sec. 4.2.3, the integral (4.173) is convergent. We will consider below the case when the time dependence of vector boson mass is defined by the conformal evolution of the Universe. The equation set similar to the (4.170) was be obtained first in the work [50] in the framework of the alternative model Lagrangian (4.130) in FRW space-time. The quantization procedure is ordinary in this approach. It would be useful to compare the predictions of these two models in detail. EoS for the isotropic case The relations for the energy density and pressure can be obtained from the energymomentum tensor corresponding to the Lagrangian (4.127) Tµν = −∂µ uα ∂ν uα − ∂ν uα ∂µ uα − gµν L. 159 (4.174) Using the decompositions (4.131) and the relations, which are a consequence of the spatial homogeneity of the system (in the discrete momentum representation), (−) 0 h0|a(+) µ (p, t)aν (p , t)|0i = δpp0 Fµν (p, t), 0 h0|aµ(−) (−p, t)a(+) ν (−p , t)|0i = δpp0 F̃µν (p, t), (±) h0|aµ(±) (p, t)aν(±) (−p0 , t)|0i = δpp0 Fµν (p, t), (4.175) we obtain ε(t) = < T00 > = − p(t) = 1 < Tii > = 3 2 + (2p + 3m 2 Z [dp] ω(p, t)[Fµµ (p, t) + F̃µµ (p, t)], [dp] −p2 [Fµµ (p, t) + F̃µµ (p, t)] ω(p, t) Z )[F(+)µ (p, t) µ + Fµ(−)µ (p, t)] . (4.176) We perform now the series of the consecutive transformations of the functions Fµµ (p, t), F̃µµ (p, t) and Fµ(±)µ (p, t): the exclusion of the µ = 0 component (according to Eq. (4.135)) and the transition to the complete QR representation with the help of Eqs. (4.152). Thus, we arrive at Fµµ (p, t) = − Fµ(±)µ (p, t) = − 3 X l=1 3 X l=1 fl (p, t), F̃µµ (p, t) = − (±) fl (p, t) − 2 3 X f̃l (p, t), l=1 p2 (±) f (p, t), m2 3 (4.177) where fl (p, t), f˜l (p, t) and f (±) (p, t) are the corresponding functions in the complete QR representation. The substitution of these relations into Eqs. (4.176) and (4.176) leads to the following expressions: ε(t) = 2 1 p(t) = 3 3 Z X l=1 3 Z X l=1 [dp]ω(p, t)fl(p, t), [dp] 2p2 fl (p, t) − [2ω 2 (p, t) + m2 ] ω(p, t) ! i h i 2p2 h (+) (−) (+) (−) × f (p, t) + f3 (p, t) + fl (p, t) + fl (p, t) (4.178) . 3m2 3 Taking into account the isotropy of the system, we obtain the EoS for the massive vector boson gas Z ε(t) = 2 [dp]ω(2 + Q)f1 , p(t) = i 2 Z [dp] h 2 2p (2 + Q)f1 + δpvac (t), 3 ω 160 (4.179) where δpvac (t) is the contribution in the pressure induced by the vacuum polarization, 2 δpvac (t) = − 3 Z [dp] 2ω 2 + m2 ω " 1 p2 1+Q + 2 m2 !# u1 . (4.180) In order to prove the convergence of the integrals (4.179), (4.180) we investigate the asymptotic behavior of the solution to the equations system (4.170). This system can be solved exactly in the asymptotic limit p m for the case α = 1/2 (the parameter α is defined in the Sec. 5) m2 1 f˙ = H 2 u, 4tH p u̇ = m2H 1 (1 + 2f ) − 2pv, 2tH p2 v̇ = 2pu . (4.181) The asymptotic solution of Eqs. (4.181) with the initial conditions (4.171) is f (p, t) = v(p, t) sin2 p(t − t0 ) ∼ , (2p/m0 )3 16(p/m0 )6 u(p, t) ≈ sin 2p(t − t0 ) , 4(p/m0 )3 (4.182) 2/3 −1/3 where m0 = m(t0 ) = mH tH and t0 = 1/m0 . The numerical investigation of the general Eqs. (4.170) shows that the basic features of the solutions (4.182) for α = 1/2 are conserved also for other α > 0. It corresponds to the results of Sect.2.3. According to Eq. (4.182), the particle number and energy densities (Eqs. (4.173) and (4.179), respectively) are convergent, but the vacuum polarization contribution to the pressure (4.180) is divergent. Moreover, irrelevant fast vacuum oscillations of the pressure are observed here. Such a behavior of the pressure for a plasma created from vacuum is not a special feature of the present theory but is characteristic also for the models where an electron - positron plasma is created in strong, time-dependent electric fields as investigated in [35]. The standard regularization procedure of similar integrals with some unknown functions satisfying ordinary differential equations is based on the investigation of asymptotic decompositions of these functions in power series of the inverse momentum, 1/pn (see Sect.4.2.1). In the considered case, such a procedure is not effective because the solutions (4.181) has quickly oscillating factors (”zitterbewegung” ), the asymptotic decompositions of which lead to the secular terms. Therefore we regularize the pressure by a momentum cut-off at p = 10m0 for the numerical calculation and separate its stable part by the time averaging 1 < p >= (t − t0 ) Zt p(t)dt. (4.183) t0 The such ”coarse graining” procedure was proposed in [36] in order to exclude the ”zitterbewegung” from the description of vacuum particle creation. In reality, 161 0,30 n1 1,5 Density [ cm -3 ] Distribution function [ 10 -4 ] n3 0,24 0,18 0,12 0,06 α = 1/2 0,00 1,2 f1 f3 0,9 0,6 0,3 0,0 1,0x10 -11 3,0x10 -11 5,0x10 -11 0 1 2 3 4 p / m0 Time [ s ] Figure 4.5: Time evolution of number density of vector bosons with the initial condition m0 · t0 = 1 for α = 1/2 (left) and the corresponding momentum distribution at the time t t0 (right). the smoothing of these fast oscillations occurs due to dissipative processes that are not taken into account here (the first attempt of a derivation of the collision integral for the scalar quasiparticle gas in a strong electric field was made in the work [37]). 4.2.4 Application to conformal cosmology models The description of the vacuum creation of particles in the time dependent gravitational fields of cosmological models goes back to Refs. [38]- [41] and has been reviewed, e.g., in the monographs [21, 42, 57]. The specifics of our work consists in the consideration of vacuum generation of particles in the conditions of the early Universe in the framework of a conformal-invariant cosmological model [43], thus assuming, that the space-time is conformally flat and that the expansion of the Universe in the Einstein frame (with metric g̃µν ) with constant masses m̃ can be replaced by the change of masses in the Jordan frame (with metric gµν ) due to the evolution of the cosmological (scalar) dilaton background field [12, 13, 44]. This mass change is defined by the conformal factor Ω(x) of the conformal transformation g̃µν (x) = Ω2 (x)gµν . (4.184) As mass terms generally violate conformal invariance, a space-time dependent mass term 1 m̃ (4.185) m(x) = Ω(x) has been introduced which formally keeps the conformal invariance of the theory [13, 21]. In the important particular case of the isotropic FRW space-time, the conformal factor is equal to the scale factor, Ω(x) = a(t̃), and hence m(t̃) = 162 16 10 6 10 n1 ] n3 -3 2 10 0 10 -2 10 α = 1/2 -4 12 10 10 10 8 10 α = 1/3 6 10 10 -6 10 n1 14 10 n3 Final density [ cm Final density [ cm -3 ] 4 10 4 -18 10 -16 10 -14 10 -12 10 -10 10 10 -8 10 -22 10 -20 10 Initial time [ s ] -18 -16 10 10 Initial time [ s ] -14 10 -12 10 Figure 4.6: The dependence of final number density of vector bosons on the initial time t0 : α = 1/2 (left), α = 1/3 (right). a(t̃)mobs , where t̃ is ”the Einstein time” and mobs is the observable present-day mass. Such a dependence was used, e.g., in Ref. [18] for the FRW metric. On the other hand, the scaling factor a(t̃) is defined by the cosmic equation of state. For a barotropic fluid, this EoS has the form pph = (γ − 1)εph = c2s εph , (4.186) where pph and εph are phenomenological pressure and energy density (in the distinction from ”dynamical” p and ε, see Sect. 4.2.3), γ is the barotropic parameter, cs is the sound velocity. The solution of the Friedman equation for such EoS leads to the following scaling factor a(t̃) ∼ t̃ 2/3γ . (4.187) The kinetics of vacuum creation of massive vector bosons (Sect. 4.2.3) was constructed in the flat Jordan frame with the proper conformal time t, which is necessary to introduce now in the Eq. (4.187). The transition to the conformal time is defined by relation dt = dt̃/a(t̃). From this relation and Eq. (4.187) follows ! #3γ/(3γ−2) " 2 . (4.188) t t̃ ∼ 1 − 3γ The substitution of this relation into Eq. (4.187) establishes the mass evolution law in the terms of the conformal time m(t) = (t/tH )α mW , α= 2 , 3γ − 2 (4.189) where tH = [(1 + α)H]−1 is the scaling factor (the age of the Universe), H is the Hubble constant and mW = 80GeV is the W-boson mass. The values of the 163 6 4 ] -3 [ Gev * cm ] -3 [ Gev * cm 4 energy density pressure mean pressure 2 3 2 energy density mean pressure 1 0 0 -2 -1 -4 1,82x10 -10 1,84x10 -10 1,86x10 -10 1,88x10 -10 1,90x10 -2 -10 3,0x10 -10 6,0x10 -10 9,0x10 -10 1,2x10 -9 Time [ s ] Time [ s ] Figure 4.7: Time dependence of energy density and pressure at t t0 with initial condition mi · ti = 1 for α = 1/2. parameter α for some popular EoS are: γ = 2, α = 1/2 (stiff fluids); γ = 4/3, α = 1 (radiation); γ = 1, α = 2 (dust). Due to back reactions and dynamical mass generation during the cosmic evolution the detailed mass history remains to be worked out. The central question, however, is whether the number density of produced W-bosons could be of the same order as that of the CMB photons, nCMB ∼ 465 cm−3 . If this question could be answered positively, the vacuum pair creation of W-bosons from a timedependent scalar field (mass term) could be suggested as a mechanism for the generation of matter and radiation in the early Universe. The non-Abelian nature of the W-bosons could even imply consequences for the generation of the baryon (and lepton) asymmetry due to topological effects [45]. The numerical analysis of the Eqs. (4.170) for massive vector bosons is performed by a standard Runge-Kutta method on a one-dimensional momentum grid. As one can see from Fig. 4.5, the creation process ends very quickly and the particle density saturates at some final value. The momentum distribution of particles is formed also very early when m(t) ≈ m0 and frozen in such form so that later on for times t t0 most of the particles have very small momentum p m(t). The spectrum of created bosons is essentially non-equilibrium, hence we should continue further the analysis of relevant dissipative mechanisms and other observable manifestations of the non-equilibrium state (e.g., CMB photons; in this connection, see, for example, [46, 47]). The dependence of the corresponding final value of density on initial time is shown in Fig. 4.6. The final density n1 of the transversal vector bosons with spin projection ±1 reaches a maximum, when we let the initial time go to very early times, close to the birth of the Universe. However, in the same limit, the density f3 of the longitudinal particles with spin projection zero grows beyond all bounds. The choice of the EoS changes drastically the quantity of the created 164 0.2 t = 5 . 10 s -11 1.5 f(k) Density [10 44 -3 cm ] 2.0 0.1 1.0 0.5 0 2 4 t [1/m0] 6 8 0 Figure 4.8: The time dependence of fermion pair density for a mass evolution given by α = 1/2 and mq = 170 GeV. 10 k [m0] 20 Figure 4.9: The momentum distribution of fermions created by the inertial mechanism at t = 5 · 10−11 s. particles, thus giving values which are too small (α = 1/2) or too large (α = 1/3) in comparison with the observed CMB photon densities. In order to improve this model, we should use an improved EoS, assuming that the barotropic parameter γ characterizing the evolution of the particle masses can change during the time evolution. Such a time-dependence could be induced by the back-reaction of the created particles on the scalar field. Furthermore, we could use another spacetime model, e.g., the Kasner space-time [48] instead of the conformally flat de Sitter one. The main achievement relative to the earlier work [44] is that in the present approach, there is no divergence in the distribution function, thus we do not need to introduce some ambiguous regularization procedure. Fig. 4.7 shows that the mean pressure remains negative and its magnitude becomes negligible in comparison with the energy density. On large times the energy density grows but the pressure stays very small, p ' 0. The energy growth with the condition p ' 0 results in the conclusion, that the massive vector bosonanti-boson gas created from the vacuum is a cold one (see also Fig. 1). It can be seen directly from Eq. (4.179), that at large times ε(t) ' m(t)ntot (t), because of ω(t) ' m(t). Such EoS of the massive vector boson gas(ε 6= 0 and p ' 0) corresponds to dust-like matter [49], which would characterize the evolution of the Universe during those stages when the vector boson gas is the dominant component of its matter (energy) content. At a qualitative level, this conclusion is valid independently of the concrete choice of an EoS and, in particular, in the case of the dust-like EoS. It would be interesting to obtain a formula of the type (4.189) as a result of the solution of the Friedman equation with the EoS (4.179)-(4.180) (such a procedure represents the back reaction problem) and to investigate self-consistently the production of vector bosons in the Universe. Let 165 ] 35 10 30 10 25 10 20 10 15 10 10 Density [ cm -3 10 10 5 10 0 10 -15 10 -13 10 -11 10 -9 10 -7 10 -5 10 -3 10 -1 10 1 ∆m /m Figure 4.10: The final density of created particles for the conformal time dependence of mass (4.189) as function of the initial time t0 for α = 1/2. Figure 4.11: The final density of created particles at the linear change of the mass in time of order 1/m0 by m0 = 0.2 Gev. us remark, that vacuum creation of massive vector bosons in the FRW metric was be considered at first in the work [50]. In order to investigate the features of the fermions vacuum creation in the conformal model of the Universe, let us use the corresponding basic KE (4.116) and chose as an example the system of the heavy top quarks with mass mq = 170Gev (with the change mw → mq in Eq. (4.189). The values of the parameter α are the same as were taken for the vector boson. As one can see from Fig. 4.8, the creation process ends very quickly and the particle density saturates at some final value. According to Eq. (4.117), the creation of fermion pairs with zero momentum is forbidden, Fig. 4.9, in contrast to the pair creation in the electric field, where the p = 0 corresponds to the maximum of the distribution function [18]. The similar ”spectrum inversion” has the place for the massive vector bosons. This feature can be used as an observational tool to discriminate between the two different mechanisms of vacuum particle creation. Finally, Fig. 4.10 shows the dependence of the final density value on the initial time t0 for m(t) given by Eq. (4.189). The intensity of fermion creation at fast change of their mass is presented in Fig. 4.11. On the qualitative level, the same picture will have place for the vacuum creation of neutralino, which can be main component of dark matter (e.g., [51]). Some analogous problem in early cosmology was be considered in the work [52] on the basis of the original approach close to the Bialynicki-Birula-Gornicki-Rafelsky method of the covariant Wigner functions [58, 59]. 166 4.2.5 The systems with unstable vacuum Wording of the problem Here it will be considered the rather general mechanism of mass formation under action of some space homogeneous time dependent quasiclassical field. Separating of the quasiclassical background field is a standard procedure of different non-perturbative approaches in QFT [18, 21, 60, 71]. In the framework of this procedure the quantum fluctuations can be described by perturbation theory. There is a class of problems, in which the strong background field produces particles, that in turn influence on the background field (the back reaction problem). It is worthy to mention such problems as decay of disoriented chiral condensate [61], the resonant decay of CP-odd metastable states [4, 62], QGP preequilibrium evolution [19, 22, 23, 24], phase transition in systems with broken symmetry [63] etc. The construction of general kinetic theory of such kind for various potential types is shown in Sect. 4.2.5. We will derive the closed system of equations for the self-consistent description of back reaction (BR) problem, including the KE with non-perturbative source term describing particle creation in the quasiclassical background field and equation of motion for this background field. We use the OR to derive the KE. As an illustrative examples, in Subsect. 4.2.5 one-component scalar theory with φ4 and double-well potential is considered. In these examples, we study some features of proposed approach. In particular, the problem of stable vacuum state definition and possibility to emerge tachyonic regimes is discussed. As an example of less trivial case, the pseudo-scalar sector of the Witten-De Veccia-Veneziano model will be considered. The similar analysis was done for some other models of such kind (e.g., [4, 62, 64, 65, 66]). In this section we follow to the work [1]. The System of Basic Equation Let us consider now the scalar field Lagrangian with a self-interaction potential V [Φ]: L[⊕] = 1 1 ∂µ Φ∂ µ Φ − m20 Φ2 − V [Φ], 2 2 (4.190) where m0 is the bare mass. In general case, the potential V [Φ] is an arbitrary continuous non-monotonous function with at least one minimum Vmin [Φ] (this is necessary for correct definition of vacuum state). It is assumed that the field Φ can be decomposed in the neighborhood of some Vmin into the quasi-classical space homogeneous time dependent background field φ0 (t) and fluctuation part φ(x) Φ(x) = φ0 (t) + φ(x). (4.191) 167 In accordance with definition of fluctuations, we have hφi = 0 and hΦi = φ0 , where symbol h. . .i denotes some averaging procedure. The background field φ0 (t) can be treated as quasi-classical one at the condition [67]: √ h̄c , (4.192) |φ̇0 | (c∆t)2 where ∆t is the characteristic time of the field averaging. We consider the case of rather small fluctuations in the neighborhood of the background field. Therefore, the potential energy expansion in powers of φ(x) can be performed 1 V [Φ] = V [φ0 ] + R1 φ + R2 φ2 + Vr [φ0 , φ], 2 (4.193) where V [φ0 ] Vmin R1 = R1 [φ0 ] = d2 V [φ0 ] dV [φ0 ] , R2 = R2 [φ0 ] = dφ0 dφ20 (4.194) and Vr [φ0 , φ] is a residual term containing the high orders to be neglected in current article (non-dissipative approximation). The decomposition (4.193) can be finite (for polynomial theories) or infinite. After field decomposition (4.191) the equation of motion dV [Φ] =0 dΦ (4.195) (−∂µ ∂ µ − m2 )φ = Q[φ0 , φ], (4.196) m2 (t) = m2 [φ0 ] = m20 + R2 [φ0 ] (4.197) ∂µ ∂ µ Φ + m20 Φ + can be rewritten in the following form: where is the time-dependent in-medium mass and Q[φ0 , φ] = φ̈0 + m20 φ0 + R1 [φ0 ] + Q2 [φ0 , φ], 1 dR2 [φ0 ] 2 φ . Q2 [φ0 , φ] = 2 dφ0 (4.198) (4.199) As a result of averaging Eq.(4.196), the equation of motion for background field is obtained φ̈0 + m20 φ0 + R1 [φ0 ]+ < Q2 [φ0 , φ] >= 0. (4.200) The time independence of the averaging procedure is taken into account. 168 The assumption about space-homogeneity means that the function < Q2 [φ0 , φ] > in Eq. (4.200) can only depend on time. As it follows from Eqs. (4.199) and (4.200), the source term in the right side of Eq. (4.196) is exclusively defined by the fluctuations, Q[φ0 , φ] = Q2 [φ0 , φ]− < Q2 [φ0 , φ] > . (4.201) On the other hand, the field function φ(x) in nonstationary situation allows the decomposition: φ(x) = Z [dk]{φ(+) (k, t)e−ıkx + φ(−) (k, t)eıkx }, (4.202) [dk] = (2π)(−3/2) d3 k, φ(±) (k, t) are the positive and negative frequency solutions of the equation of motion φ̈(±) (k, t) + ω 2 (k, t)φ(±) (k, t) = −Q[φ0 , φ| ± k], (4.203) ω 2 (k, t) = m2 (t) + k2 (4.204) where and Q[φ0 , φ|k] is the Fourier image of the function Q[φ0 , φ], Q[φ0 , φ] = Z [dk]Q[φ0 , φ|k]e−ıkx . (4.205) The function Q[φ0 , φ|k] contains non-linear contribution to the Eq. (4.203). (±) We suppose a finite limit lim φ(±) (k, t) = φ− (k) in the infinite past and ast→−∞ sume that the solutions φ(±) (k, t) become asymptotically free φ(±) (k, t) → e±iω− t , where ω− (k, t) = lim ω(k, t). The existence of the last limit is based on adiat→−∞ batic hypothesis about switching off of self-interaction in the Eq.(4.197). After the decompositions (4.191) and (4.193) the Hamiltonian density is H[Φ] = H[φ0 ] + H1 [φ0 , φ] + H2 [φ0 , φ] + Vr [φ0 , φ], (4.206) where H0 [φ0 ] is the background field Hamiltonian and H1 and H2 are the Hamiltonian functions of the first and second order with respect to the fluctuation field, 1 1 H[φ0 ] = H0 [φ0 ] + V [φ0 ] = φ̇20 + m20 φ20 + V [φ0 ], 2 2 2 2 H1 [φ0 , φ] = φ̇0 φ̇ + (m0 φ0 + R1 [φ0 ])φ, 1 2 1 1 H2 [φ0 , φ] = φ̇ + (∇φ)2 + m2 φ2 . 2 2 2 (4.207) (4.208) (4.209) One can built the Hilbert space on the system of basic functions φ(±) (k, t) defined by Eq. (4.203) and proceed to the Fock representation. After the decomposition (4.193), H2 (4.209) has non-diagonal form with respect to creation and 169 annihilation operators and hence does not allow quasi-particle interpretation [18] (Sect.2.1). In order to diagonalize Hamiltonian it is necessary to apply either the Bogoliubov transformation [18] or OR [17]. Following the idea of OR, let us write the free field decomposition of φ(x) and φ̇(x) in the absence of background field φ0 with the substitution the free particle energy ω(k) = (m20 + k2 )1/2 to the time dependent one (4.204) (see also Sect. 4.2.1) φ(x) = [dk] Z q Z 2ω(k, t) π(x) = ı [dk] q n 2ω(k, t) n 2 o a(+) (k, t)e−ıkx + a(−) (k, t)eıkx , (4.210) o a(+) (k, t)e−ıkx − a(−) (k, t)eıkx , (4.211) where π(x) plays a role of the canonical momentum. The remarkable feature of the decomposition (4.210) and (4.211) is fulfillment of the standard commutative relations [π(x), φ(x)]|t=t0 = −ıδ(x̄ − x̄0 ), (4.212) self-considered with the commutative relations for time dependent creation and annihilation operators [a(−) (k, t), a(+) (k, t)] = δ(k − k0 ) (4.213) (the rest commutators are equal to zero). It is not less important that the Hamiltonian (4.209) after the replacement φ̇ → π has the diagonal form, H2 [φ0 , φ] = Z d3 k ω(k, t)a(+) (k, t)a(−) (k, t) (4.214) and hence OR is the quasiparticle representation [17]. In order to obtain the equations of motion for the operators a(±) (k, t), let us write corresponding action with the Hamiltonian (4.207) – (4.209) (this way is alternative one for the Hamilton approach of the Sect. 4.2.1) S[φ] = Z d4 x {π φ̇ − H1 − H2 − Vr }. (4.215) After substitution here of decompositions (4.210) and (4.211), we get S[φ] = ı dt d3 k { [a(+) (k, t)ȧ(−) (k, t) − a(−) (k, t)ȧ(+) (k, t)] (4.216) 2 ω̇(k, t) (+) − [a (k, t)a(+) (−k, t) − a(−) (−k, t)a(+) (k, t)] (4.217) 2ω(k, t) 1 − ω(k, t)[a(+) (k, t)a(−) (k, t) + a(−) (k, t)a(+) (k, t)] − Vrk [φ0 , φ]} + (4.218) S1 [φ] , 2 Z 170 where S1 [φ] is the part of the action corresponding to the Hamiltonian (4.208) and Vk [φ0 , φ] is the Fourier image of the residual potential term. Variation with respect to a(±) (k, t) and subsequent transition to the occupation numbers representation lead to the Heisenberg-type equations of motion (k 6= 0) ȧ(±) (k, t) = W(k, t)a(∓) (k, t) + ı[H2 + Vr , a(±) (k, t)], (4.219) where W (k, t) = Ṙ2 [φ0 ] ω̇(k, t) = 2ω(k, t) 4ω 2 (4.220) is the vacuum transition amplitude of a particle creation. In the Eq. (4.219), the condensate contribution generated by the action part S1 [φ] is omitted because it corresponds to k = 0 (the connection mechanism of the condensate state with k = 0 and excitations is absent in the present model). For the first time, equations of the type (4.219) was obtained in the work [68] in QED in the framework of Bogoliubov time-dependent canonical transformations approach. They are the basis for nonperturbative derivation of KE. We will use now this procedure in the present statement of problem. Let us introduce the distribution function of quasiparticles (it is convenient to do in discrete momentum representation and subsequent transition to infinite volume of the system in the resulting KE, L3 → ∞) L3 f (k, t) =< in | a(+) (k, t)a(−) (k, t) | in >, (4.221) where the averaging procedure is carried out over the in-vacuum state. We will consider nondissipative approximation Vr [φ0 , φ] → 0. Using the method of the Sect. 4.2.1 and basic equation (4.219) it is not difficult to get the KE df (k, t) = 2W (k, t) dt Z t −∞ 0 0 Z dt W(k, t )[1 + 2f (k, t)]s 2 t t0 dτ ω(k, τ ) (4.222) where ω(k, t) is defined by Eq. (4.204). The KE (4.222) can be transformed to a system of ordinary differential equations, which is convenient for numerical analysis, f˙ = W u, u̇ = 2W (1 + 2f ) − 2ωv, v̇ = 2ωu. (4.223) To rewrite Eq. (4.200) for background field in nondissipative approximation, one has to calculate the averaging value < in | φ2 (x) | in >. Using Eq. (4.210) and the relations (in the limit L3 → ∞) < in | a(+) (k, t)a(−) (k0 , t) | in > = (2π)3 f (k, t)δ(k − k0 ), 171 (4.224) in the space homogeneous case one can obtain < in | φ2 (x) | in >= 1 Z d3 k [1 + 2f (k, t) + u(k, t)]. 2 ω(k, t) (4.225) Then Eq. (4.200) is φ̈0 + m20 φ0 1 dR2 + R1 [φ0 ] + 2 dφ0 Z 1 d3 k f (k, t) + u(k, t) = 0 ω(k, t) 2 (4.226) (the vacuum unit is omitted in the integral). The KE (4.222) (or the equivalent system of equations (4.223)) and Eq. (4.226) form the closed system of nonlinear equations describing of the BR problem. In the case of v[Φ0 , Φ] = 0, this system of equations is a direct nonperturbative consequence of the dynamics and the assumption (4.191). For the description of particle creation we will use the particle density (4.90) as well as background field energy Ecl and energy of created quasiparticles Eq 1 2 1 2 2 φ̇ + m φ + V (φ0 ), 2 0 2 0 0 Z d3 p = ω(k, t)f (p, t). (2π)3 Ecl = (4.227) Eq (4.228) The conservation of the full energy of the system can be shown analitically. The proof based on the differentiation of total energy Ecl + Eq with respect to time and taking into account (4.226) and relations (4.194),(4.197) and (4.220). The constructed formalism allows the consideration of the following initial problems at the time t = t0 : some initial excitation level of background field φ0 (t0 ) and φ̇0 (t0 ) are given with the additional condition either f (p, t0 ) = 0 or f (p, t0 ) 6= 0, where f (p, t0 ) is some initial plasma distribution. Examples Φ4 potential The separation of background field (4.191) in the potential 1 V [Φ] = λΦ4 , 4 λ>0 (4.229) R2 [φ0 ] = 3λφ20 , (4.230) leads to the follows decomposition coefficients R1 [φ0 ] = λφ30 , background field potential 1 V [φ0 ] = λφ40 4 (4.231) Vr [φ0 , φ] = λ(φ0 + φ/4)φ3 . (4.232) and residual potential 172 Thus, the time dependent quasiparticle mass of fluctuating field (4.197) is equal to m2 (t) = m20 + 3λφ20 . (4.233) If λ < 0 and the excitation level is large enough, it is possible that the tachyonic mode will arise, that corresponds to unstable state [69]. The mass (4.233) determines the vacuum transition amplitude (4.220) W (k, t) = λ 2φ0 φ̇0 . ω 2 (k, t) (4.234) KE (4.222) with this amplitude is correct in non-dissipative approximation which corresponds to the neglecting of the residual potential (4.232). Let us write also the equation of motion for the background field (4.226) in the same approximation: φ̈0 + M 2 (t)φ0 + λφ30 = 0 (4.235) with the corresponding mass equal to 2 M (t) = m20 + 3λ Z 1 d3 k f (k, t) + u(k) , ω(k, t) 2 (4.236) i.e. the mass of the condensate excitations is defined both the distribution of quasiparticles and vacuum polarization. In numerical calculations we apply zero initial conditions for the distribution function and nonzero one for the background field φ0 (t0 ) = 1.2. The choice of parameters (λ = 1 and m0 = 1) as well as initial conditions here and in the Sec. 4.2.5 is motivated by the desire to make a qualitative comparison between our work and [63], where authors offered the alternative method for description of quantum systems under strong background field action (so called CornwallJackiw-Tomboulis method [70]). As it can be seen on Fig.8 at the early evolution stage all energy is mainly concentrated in the field oscillation. For t < 50 we have a slow growing of number density. However it drastically increases at t ∼ 50 and after this time the quantum energy dominates over classical one. The case λ < 0 (absolutely unstable potential) is associated with tachyonic regime, that is realized for enough high excitation level, when the initial amplitude φ0 (t0 ) satisfies with condition m20 + 3λφ20 (t0 ) ≤ 0. The low excitation level m20 + 3λφ20 (t0 ) > 0 corresponds to pre-tachyonic regime. The vacuum particle creation in the pre-tachyonic region is characterized by developing instability, i.e. by (c) unrestricted growth at tendency φ0 (t) to the critical value φ0 = m0 (3|λ|)−1/2 . 173 Double well potential 1 1 V [Φ] = λΦ4 − µ2 Φ2 , 4 2 λ>0 (4.237) leads to the same equation (4.235) for background field with new mass (we set here m0 = 0 and µ2 > 0) 2 2 M (t) = −µ + 3λ Z 1 d3 k f (k, t) + u(k, t) . ω(t) 2 (4.238) The vacuum transition amplitude (4.220) is equal here W (k, t) = λ 3φ0 φ̇0 , 2ω 2 (k, t) (4.239) where now the quasiparticle frequency (4.204) contains the mass m2 (t) = −µ2 + 3λφ20 . (4.240) The neighborhood of the central point φ0 (t) = 0 is the instability region. In this region the group velocity vg = dω(k)/dk = k/ω(k) is either super-luminous (vg > 1 and Imω = 0 for k > kc , where kc is the root of the equation ω(k, t) = 0) or indefinite (Reω(k) = 0 for k < kc ). Thus, it is tachyonic region. √ Let us mark the minimum points of the potential (4.237) as Ψ± = ±µ/ λ = ±Ψ0 and to put new origin of frame of reference in one from these points, Φ = Ψ± + Ψ. We separate now background component φ0 from the field Ψ, i.e. Ψ = φ0 + φ. We omit the marks (±), which identify the belonging of the fields Ψ, φ0 andφ to the corresponding branch Ψ± . The rather small level of excitations in the vicinity of the stable √ √ points Ψ± will be considered below only, |φ(x)|, |φ(x)0 | << 2Ψ0 , where ± 2Ψ0 are the roots of the equation V [Ψ] = 0. Using the Eq.(4.195) and the methods of the Sect.6.2, it is easy to obtain the following system of equations of motion: φ̈0 + 2µ2 φ0 + 3λΨ± φ20 + λφ30 + 3λ(Ψ± + φ0 )hφ2 i + λhφ3 i = 0, −[∂µ ∂ µ + m2± (t)]φ + 3λ(Ψ± + φ0 )[hφ2 i − φ2 ] + λ[hφ3 i − φ3 ] = 0, (4.241) (4.242) where m2± (t) = 2µ2 + 3λφ0 (φ0 + 2Ψ± ). (4.243) In the accepted nondissipative approximation, we must keep in the Eq.(4.242) the linear terms only. That leads to KE (4.222) with the amplitude defined by time dependent mass (4.243) W± (k, t) = ω̇(k, t) 3λφ̇0 (φ0 + Ψ± ) = . 2ω(k, t) 2ω 2 (k, t) 174 (4.244) The mean values hφ2 i and hφ3 i are calculated either in minimal order of perturbative theory (for λ 1) or in RPA. We obtain the result (4.225) for hφ2 i and hφ3 i = 0. Thus, we have d3 k [2f (k, t) + v(k, t)] = 0. 2 ω(k, t) (4.245) Fig.9 present the numerical results for parameters set m = 1, λ = 1, φ0 (0) = 0.58, near to border of a tachyonic mode. The stationary regime is achieved faster, than in case of symmetric potential. Other formalism for the description of strong field problem for the quantum field system with the potential (4.237) is developed by J. Baacke et al ([63], and works cited therein). The general kinetic approach for the description of an arbitrary strong excited nonequilibrium states in the scalar QFT with selfinteraction admitting the existence of unstable vacuum states was developed in the present work. We restrict ourselves by the collisionless (non-dissipative) approximation. However the attempts to go beyond this approximation was done [37]. As the concrete example, φ4 and double-well potentials were investigated. It would be interested to research some other properties of the considered model, such as the features of excitation transitions between states with different vacuums (in the same space-time point). Apparently, it is possible both under √ rather high initial excitation level |φ0 (0) ≤ 2Ψ(0) and as a result of tunneling process through the central barrier. The last problem is interested especially in the generalized double-well potential model with non-degenerated vacuum states. φ̈0 + λφ0 (2 + 3φ0 )Ψ± + φ20 + 3λ(Ψ± + φ0 ) Z η-meson system We will use now the cumulative experience in order to consider more complicated quantum field system - η-meson sector of the WittenDiVecchia-Veneziano model [2]: i a fπ2 [θ − tr(ln U − ln U + )]2 , tr(∂µ U ∂µ U + ) + tr(M U + M U + ) − Lef f = 4 Nc 2 (4.246) which describes the low-energy dynamics of the nonet or the pseudoscalar mesons in the large Nc limit of QCD. We focus here our attention on the singlet state of this model with the following Lagrangian: η 1 a L = ∂µ η∂ µ η + f 2 µ2 cos (4.247) − η2, 2 f 2 q where f = 32 fπ , fπ = 92 MeV is the semi-leptonic pion decay constant, µ2 = 1 (m2π + 2m2K ) ' 0.171 GeV2 , mπ and mK are π- and K - meson masses, a = 3 m2η + m2η0 − 2m2K ' 0.726 GeV2 for zero temperature. The corresponding total Hamiltonian density is given by 1 1 1 H = H0 + Vin = η̇ 2 + (∇η)2 + aη 2 + Vin , (4.248) 2 2 2 175 Vin = 2f 2 µ2 sin2 η , 2f (4.249) where the constant addend f 2 µ2 was discarded. The Hamiltonian density (4.248), (4.249) can be reconstructed to the form [2] 1 H = H0 + Vin = η̇ 2 + 2 η Hin = 2f 2 µ2 sin2 − 2f 1 1 (∇η)2 + m20 η 2 + Hin , 2 2 !2 η , 2f (4.250) (4.251) where m20 = a + µ2 . The effective potential Hin is selected here so that its formal decomposition with respect to field function η(x) does not contain the corresponding square contribution that is associated with the mass term. However, the redefinition (4.250), (4.251) leads to absolutely unstable potential (Sec. 4.2.5) with corresponding tachyonic modes [2]. In our opinion, they have artificial character and can be eliminated by return to the original Hamilton density (4.248), (4.249), on the basis of which will be fulfilled below the investigation of excitations in the η- meson system. By analogy with Eq. (4.191), let us select the quasi-classical field η0 (t) and quantum fluctuation part φ(x), η(x) = η0 (t) + φ(x). (4.252) Now we can use the general formalism of Sec. 4.2.5 to η-meson system with Hamiltonian density (4.248), (4.249). The master equation (4.226) takes the form η0 η̈0 + aη0 + f µ sin f 2 !" 1 1− 2 2f Z 1 d3 k f+ v ω 2 # =0 (4.253) where ω(k, t) is quasi-particle energy (4.204) with the mass (4.197) m2 (t) = a + µ2 cos η0 . f (4.254) KE (4.222) or equivalent to it the equation system (4.223) is defined via the vacuum transition amplitude (4.220) that has now the form W (k, t) = − η0 µ2 η̇ sin . 0 4f ω 2 (k, t) f (4.255) In principle, the mass formula (4.254) contains a possibility of tachyonic modes appearance. However, the concrete values of the parameters a and µ2 in this model are such that m2 (t) > 0 for any amplitudes of quasi-classical field η0 (t) and, consequently, the pre-tachyonic regime is realized only (Sec. (4.2.5)). 176 4.2.6 Summary This last chapter of the lecture series had the aim to summarize the information about the kinetic description of vacuum particle creation as a result of inertial mechanism action stipulated on the phenomenological level by some time dependencies of particle masses. Three basic quantum field models were considered here: massive scalar, vector and spinor fields. The constructed kinetic theory was applied here (Sect. 4.2.4) upon to a conformal cosmology model for investigation of matter creating from vacuum in an early period of the Universe evolution. In particular, it was showed that the density of the produced vector bosons is sufficient for the explanation of the present density of CMB photons. The obtained results can be used for the subsequent investigation of the matter dynamics created from vacuum in the early Universe (the equation of state, the long wave-length acoustic excitations, the back-reaction problem etc.). We have considered here the single mechanism of mass change induced by the conformal expansion of the Universe with switching on the mass m0 (t0 ) at some arbitrary initial time t0 . For the construction of a more consistent theory, one should eventually take into account the inflation mechanism of mass generation acting during an earlier period the Universe evolution [53]-[56]. It would be interesting to consider the generation of particles of different masses and statistics. The equation (4.189) is valid for all particles independent of their inner symmetry. The chemical composition of created matter and the EoS of the Universe can be fixed rather precisely and can be subject to experimental verification. The interesting perspectives are opened also the results of the Sect. 4.2.5, where it was investigated some simplest quantum field models of scalar fields with different kinds of selfinteractions and the corresponding quasiclassical nonstationary condensate field (the problem of phase transition under restoration of a broken symmetry, particles tunneling between states with different vacuum etc). Of course, the considered examples are not exhaust of all variety of the systems, where the vacuum generation of particles is stipulated by the inertial mechanism. in particular, the large class of experimentally controlled models of meson and quark subsystems evolved in the vicinity of the relativistic phase transition had remained beyond our scope. Acknowledgement D.B. was supported by the Virtual Institute of the Helmholtz association under grant No. VH-VI-041. We are grateful to Prof. M.P. Da̧browski for discussion of some cosmological aspects of the present work. 177 1,5 1,0 1,0 0,8 0,5 n(t) φ0(t) 0,6 0,0 0,4 -0,5 0,2 -1,0 -1,5 0 20 40 60 80 0,0 100 0 20 40 t 60 80 100 t 1,2 8000 t = 40 t = 50 6000 0,8 f(p,t) E(t) E tot E cl E qu 4000 0,4 2000 0,0 0 20 40 60 80 100 0,00 t 0,05 0,10 0,15 0,20 0,25 p Figure 4.12: Time evolution for the symmetric Φ4 potential. Parameters are: m0 = 1, φ0 (0) = 1.2, λ = 1. From left to right: (a) evolution of the mean field; (b) evolution of the particle density; (c) evolution of the energy; (d) momentum spectrum of particles at times t = 40 and t = 50. 178 1,3 0,07 1,2 0,06 n(t) 1,1 0,05 φ 0(t) 1,0 0,04 0,9 0,03 0,8 0,02 0,7 0,01 0,6 0,5 0,00 0 20 40 60 80 100 0 20 t 40 60 t 140 0,08 120 0,00 f(p) 0,04 E qu E cl -0,04 t=40 t=10 100 80 E tot -0,08 60 -0,12 40 -0,16 20 -0,20 -0,24 0 20 40 0 0,00 60 t 0,25 0,50 0,75 1,00 p Figure 4.13: Time evolution for the bistable Φ4 potential (4.237). Parameters are: µ2 = 1, λ = 1, φ0 (0) = 0.58. 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