Kinetic theory of radiation in non-equilibrium relativistic plasmas

Annals of Physics 324 (2009) 1261–1302

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Kinetic theory of radiation in non-equilibrium relativistic plasmas V.G. Morozov a, G. Röpke b,* a

Moscow State Institute of Radioengineering, Electronics, and Automation (Technical University), Vernadsky Prospect 78, 119454 Moscow, Russia b University of Rostock, FB Physik, Universitätsplatz 3, D-18051 Rostock, Germany

a r t i c l e

i n f o

Article history: Received 3 February 2009 Accepted 10 February 2009 Available online 20 February 2009

PACS: 52.20.j 52.25Dg 52.38Ph 52.25Os 52.27Ny Keywords: Many-particle QED Non-equilibrium Green’s functions Relativistic plasmas

a b s t r a c t Many-particle QED is applied to kinetic theory of radiative processes in many-component plasmas with relativistic electrons and non-relativistic heavy particles. Within the framework of non-equilibrium Green’s function technique, transport and massshell equations for fluctuations of the electromagnetic field are obtained. We show that the transverse field correlation functions can be decomposed into sharply peaked (non-Lorentzian) parts that describe resonant (propagating) photons and off-shell parts corresponding to virtual photons in plasmas. Analogous decompositions are found for the longitudinal field correlation functions and the correlation functions of relativistic electrons. As a novel result a kinetic equation for the resonant photons with a finite spectral width is derived. The off-shell parts of the particle and field correlation functions are shown to be essential to calculate the local radiating power in relativistic plasmas and recover the results of vacuum QED. The influence of plasma effects and collisional broadening of the relativistic quasiparticle spectral function on radiative processes is discussed. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Current developments in ultra-short high-intensity laser technology and recent progress in laser– plasma experiments have regenerated interest in theory of relativistic plasmas. At focused intensities of 1019 –1020 W=cm2 , the electrons in the laser channels are accelerated up to multi-MeV energies * Corresponding author. Tel.: +49 381 4986940; fax: +49 381 4986942. E-mail addresses: [email protected] (V.G. Morozov), [email protected] (G. Röpke). 0003-4916/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2009.02.001

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[1–4]. It is expected that in near future the focal laser intensities will reach 1021 –1022 W=cm2 , so that electron energies are predicted to exceed 1 GeV. In the presence of dense laser-generated particle beams, the plasma is to be considered as a manycomponent non-equilibrium medium composed of highly relativistic beam electrons, relativistic (or non-relativistic) electrons in the plasma return current, and non-relativistic heavy particles.1 Up to now theoretical investigations of such systems were concerned mainly with semi-macroscopic mechanisms for acceleration of self-injected electrons in a plasma channel [8–10], collective stopping, ion heating [11], and collective beam-plasma instabilities [12–14]. Note, however, that a consistent theory of many other transport phenomena should be based on many-particle non-equilibrium QED which also takes account of characteristic plasma properties. In particular, a problem of great interest is photon kinetics, because radiative processes contribute to stopping power for highly relativistic electron beams, and they are interesting in themselves as an example of fundamental QED processes in a medium [15]. It should also be noted that the measurement of angular distribution of c rays in laser–plasma experiments has been found to be a powerful diagnostic tool [16]. It is important to recognize that the standard ‘‘golden rule” approach of relativistic kinetic theory [17] cannot be applied to the laser–plasma medium, since the very concepts of ‘‘collisions” and ‘‘asymptotic free states” are not straightforward due to collective plasma effects. There is also another non-trivial problem related to the kinetic description of radiation in a non-equilibrium plasma. The point is that transverse wave excitations may be associated with two quite different states of photon modes. First, virtual photons are responsible for the interaction between particles and cannot be detected as real photons outside the system. Second, for sufficiently high frequencies, propagation of weakly damped resonant photons is possible. These photons behave as well-defined quasiparticles and contribute to the plasma radiation. Clearly, the above physical arguments must be supplemented by a consistent mathematical prescription for separating the contributions of virtual and resonant photons in transport equations. The main difficulty is that, in general, characteristics of both the virtual and resonant (propagating) photons involve medium corrections. A systematic treatment of the photon degrees of freedom in non-equilibrium plasmas is still a challenging problem. At present, the most highly developed methods for treating non-equilibrium processes in manyparticle systems from first principles are the density matrix method [18–20] and the real-time Green’s function method [21–23]. Of the two methods, the former is especially suited for systems in which many-particle correlations are important. On the other hand, the Green function method turns out to be efficient in the cases where the description of transport processes does not go far beyond the quasiparticle picture. Since in the laser–plasma interaction experiments the plasma may be considered to be weakly coupled and the dynamics of many-particle correlations (bound states, etc.) is of minor importance for the radiative phenomena considered here, the Green function formalism seems to be the most direct way to transport equations for particles and radiation including relativistic and plasma effects. A systematic approach to non-equilibrium QED plasmas based on the Green function technique was developed by Bezzerides and DuBois [24]. Within the weak-coupling approximation, they were able to derive a covariant particle kinetic equation involving electron–electron collisions and Cherenkov emission and absorption of plasmons. Note, however, that the kinetics of transverse photons and radiative phenomena are mainly related to higher-order processes, like bremsstrahlung or Compton scattering. Although these fundamental processes have been studied in great detail within the framework of vacuum QED, they have not been studied to sufficient generality and clarity in many-particle non-equilibrium QED when plasma effects are of importance. Therefore, the purpose of the analysis in this paper is to perform the still missing formulation of photon kinetics which includes relativistic and plasma effects, and, where appropriate, recovers the conventional results of vacuum QED for radiative processes. The paper is outlined as follows. In Section 2, the essential features of the real-time Green’s function formalism for many-component relativistic plasmas are summarized. For the problems under

1 In laser–plasma experiments protons and ions with energies in the MeV region can be produced [5–7], but their velocities are much less than the speed of light.

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consideration, there is a preferred frame of reference, the rest frame of the system, in which the heavy particles are non-relativistic. It is thus convenient to choose the Coulomb gauge, allowing a description of electromagnetic fluctuations in terms of the longitudinal and transverse (photon) modes. The polarization matrix and the particle self-energies on the time-loop Schwinger-Keldysh contour are expressed in terms of Green’s functions and the vertex functions which are found in the leading approximation for weakly coupled relativistic plasmas. In Section 3, the kinetic description of the transverse wave fluctuations in plasmas is discussed. Dyson’s equation for the transverse field Green’s function is used to derive the Kadanoff–Baym equations and then the transport and mass-shell equations for the Wigner transformed correlation functions in the limit of slow macroscopic space–time variations. The analysis of the drift term in the transport equation shows that the transverse field correlation functions are naturally decomposed into ‘‘resonant” and ‘‘off-shell” parts which, in the case of small damping, may be regarded as contributions from the propagating and virtual photons, respectively. The resonant parts are sharply peaked near the effective photon frequencies and have smaller wings than a Lorentzian. This observation allows the derivation of a kinetic equation for the local energy–momentum photon distribution which is related to the resonant parts of the transverse field correlation functions. An essential point is that the transport and mass-shell equations lead to the same photon kinetic equation. This fact attests to the self-consistency of the approach. In the zero width limit for the resonant spectral function, the kinetic equation derived in the present work reduces to the well-known kinetic equation for the ‘‘on-shell” photon distribution function [25]. Section 4 is concerned with the relativistic electron propagators and correlation functions which are required for calculating the photon emission and absorption rates. As in the case of transverse photons, the structure of the relativistic transport equation leads to the decomposition of the electron correlation functions into sharply peaked ‘‘quasiparticle” parts and ‘‘off-shell” parts. To leading order in the field fluctuations, the full electron correlation functions are expressed in terms of their quasiparticle parts. This representation is analogous to the so-called ‘‘extended quasiparticle approximation” that was previously discussed by several workers [26–29] in a context of non-relativistic kinetic theory. As shown later on, the off-shell parts of the electron correlation functions play an extremely important role in computing radiation effects. In Section 5, we consider the transverse polarization functions which are the key quantities to calculate the collision terms in the photon kinetic equation. By classification of diagrams for the polarization matrix on the Schwinger-Keldysh contour, relevant contributions to the photon emission and absorption rates in weakly coupled plasmas are selected. It is important to note that the final expressions for the transverse polarization functions contain terms coming from both the vertex corrections and the off-shell parts of the particle correlation functions. The results of Section 5 are used in Section 6 to discuss the contributions of various scattering processes to the local radiating power with special attention paid to the influence of plasma effects. It is shown that Cherenkov radiation processes, which are energetically forbidden in a collisionless plasma, occur if the collisional broadening of the quasiparticle spectral function is taken into account. The physical interpretation of radiative processes is closely related to the decomposition of the field correlation functions into resonant and off-shell parts. For instance, the contribution of the resonant parts of the transverse field correlation functions to the radiating power may be interpreted as Compton scattering while the contribution of the off-shell parts corresponds to a relativistic plasma effect— the scattering of electrons on current fluctuations. The contribution to the radiating power arising from interaction between the relativistic electrons and the longitudinal field fluctuations can be divided into several terms corresponding to the electron scattering by resonant plasmons (the Compton conversion effect), by the off-resonant electron charge fluctuations, and by ions (bremsstrahlung). It is important to note that the transition probabilities for bremsstrahlung and Compton scattering known from vacuum QED are recovered within the present many-particle approach if all plasma effects are removed. Finally, in Section 7, we discuss the results and possible extensions of the theory. Some special questions are considered in the appendices. Throughout the paper, we use the system of units with c ¼ h  ¼ 1 and the Heaviside units for electromagnetic field, i.e., the Coulomb force is written as qq0 =4pr2 . Although we work in the Coulomb

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gauge and in the rest frame of the system, many formulas are represented more compactly in the relativistic four-notation. The signature of the metric tensor g lm is ðþ; ; ; Þ. Summation over repeated Lorentz (Greek) and space (Latin) indices is understood. Our convention for the matrix Green’s functions on the time-loop Schwinger-Keldysh contour follows Botermans and Malfliet [23]. 2. Green’s function formalism in Coulomb gauge 2.1. Basic Green’s functions We start with some notations and definitions. Following the quantum many-particle formulation in terms of Green’s functions developed by Bezzerides and DuBois [24], we assume that the system wasperturbed from its initial state by some prescribed c-number external four-current r; tÞ ¼ .ðextÞ ;~ J ðextÞ , where .ðextÞ ð~ r; tÞ is the external charge density and ~ J ðextÞ ð~ r; tÞ is the exterJ ðextÞl ð~ nal current density. Note, however, that the introduction of these quantities is only a trick to define correlation functions and propagators for particles and electromagnetic fluctuations in the system. Since we are not interested in the effects of initial correlations which die out after a few collisions, the initial time t 0 will be taken in the remote past, i.e., the limit t 0 ! 1 will be assumed. To describe statistical properties of the system, we introduce field and particle Green’s functions defined on the time-loop Schwinger-Keldysh contour C which runs from 1 to þ1 along the chronological branch C þ and then backwards along the antichronological branch C  [23]. rk Þ indicate that tk lies on the contour C, while From now on, the underlined variables k ¼ ðtk ;~ rk Þ is used for space–time variables. Integrals along the contour are underthe notation ðkÞ ¼ ðtk ;~ stood as

Z

d1 Fð1Þ ¼

Z

1

d1½Fð1þ Þ  Fð1 Þ;

ð2:1Þ

1

where Fð1 Þ stands for functions with time arguments taken on the branches C  of the contour.2 For any function Fð1 2Þ, we introduce the canonical form [23]

 Fð1 2Þ ¼

Fð1þ 2þ Þ Fð1þ 2 Þ Fð1 2þ Þ Fð1 2 Þ

 ¼

F < ð12Þ þ F þ ð12Þ

F < ð12Þ

F > ð12Þ

F < ð12Þ  F  ð12Þ

! ð2:2Þ

with the space–time ‘‘correlation functions” F ? ð12Þ and the retarded/advanced functions (usually called the ‘‘propagators”)

F  ð12Þ ¼ F d ð12Þ  hððt1  t2 ÞÞfF > ð12Þ  F < ð12Þg;

ð2:3Þ

where hðxÞ is the step function and F d ð12Þ is a singular part of F  ð12Þ. A possibility of singularities in the retarded/advanced functions for equal time-variables on the contour C was discussed in detail by Danielewicz [22]. In our case, such singular terms appear in the propagators for longitudinal field fluctuations (see Eq. (2.13)). Note the useful relation that follows from the above definitions:

F > ð12Þ  F < ð12Þ ¼ F þ ð12Þ  F  ð12Þ:

ð2:4Þ

It is convenient to treat the external charge and current on different branches of the contour C as indeb in the Heisenpendent quantities and formally define the ensemble average Oð1Þ for any operator Oð1Þ berg picture as [24]

D Oð1Þ ¼

E b I ð1Þg T C fS O hSi

;

ð2:5Þ

2 Sometimes the integration rule is defined with the plus sign in Eq. (2.1). Then it is necessary to introduce the sign factor g ¼ 1 for each argument on the contour [20,24]. We will use the convention (2.1) (see also [23]) which makes formulas more compact. One only has to keep in mind that the delta function satisfies dðt 1  t2 Þ ¼  dðt1  t 2 Þ on the branches C  .

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b I is the operator in the interaction picture and T C is the path-ordering operator on the contour where O C. The evolution operator S describes the interaction with the external current. With the four-potential b b l ð~ ^ ~ r; tÞ ¼ ð/; AÞ, the evolution operator is written as operator A

 Z  b l ð1ÞJ ðextÞ ð1Þ : S ¼ T C exp i d1 A l I

ð2:6Þ

At the end of calculations, the physical limit is implied: .ðextÞ ð1þ Þ ¼ .ðextÞ ð1 Þ and ~ J ðextÞ ð1þ Þ ¼ ~ J ðextÞ ð1 Þ. In this limit hSi ¼ 1, so that the quantity (2.5) coincides with the conventional ensemble average of a Heisenberg operator. The field Green’s functions are defined as functional derivatives of the averaged four-potential,

  b l ð1Þi  /ð1Þ; ~ Al ð1Þ ¼ h A Að1Þ ;

ð2:7Þ

with respect to the external current:

dAl ð1Þ

Dlm ð1 2Þ ¼

dJ ðextÞ m ð2Þ

ð2:8Þ

:

The most important Green’s functions are

D00 ð1 2Þ ¼

d/ð1Þ ; d.ðextÞ ð2Þ

Dij ð12Þ ¼

dAi ð1Þ ðextÞ

dJ j

ð2Þ

;

i; j ¼ 1; 2; 3:

ð2:9Þ

They characterize the longitudinal and transverse fluctuations of the electromagnetic field, respectively. The components D0i and Di0 describe the direct coupling between longitudinal and transverse modes. Recalling Eq. (2.5) for averages on the contour C, it is easy to show that in the physical limit

D E b i ð1ÞD A b j ð2Þ Dij ð1 2Þ ¼ i T C D A

ð2:10Þ

b i ð1Þ ¼ A b i ð1Þ  Ai ð1Þ. In evaluating D00 ð1 2Þ with Eq. (2.5), one must take account of the relation with D A which is valid in the Coulomb gauge:

^ ¼ /ð1Þ

Z

 ^ ð2Þ þ .ðextÞ ð2Þ ; d2 Vð1  2Þ .

ð2:11Þ

^ ð1Þ is the induced charge density operator, and where .

Vð1  2Þ ¼

1 dðt1  t 2 Þ: 4pj~ r2 j r1  ~

ð2:12Þ

^ I ð1Þ=d.ext ð2Þ ¼ Vð1  2Þ, so that in the physical limit the longitudinal field From Eq. (2.11) follows d/ Green’s function takes the form

D E ^ D/ð2Þ ^ D00 ð1 2Þ ¼ i T C D/ð1Þ þ Vð1  2Þ

ð2:13Þ

^ ^ with D/ð1Þ ¼ /ð1Þ  /ð1Þ. We now introduce the path-ordered Green’s functions for particles. Since our main interest is in problems where electrons may be relativistic, the corresponding Green’s function is defined in terms of the Dirac field operators:

  I ð2Þ hSi: Gð1 2Þ ¼ i T C ½SwI ð1Þw

ð2:14Þ

Note that each of the contour components of Gð1 2Þ is a 4  4 spinor matrix. Finally, the non-relativistic Green’s functions for heavy particles (protons and ions) are defined as

 G B ð1 2Þ ¼ i T C ½SWBI ð1ÞWyBI ð2Þ hSi;

ð2:15Þ y ~ B ðr; tÞ

where the index ‘‘B” labels the particle species. The operators WB ð~ r; tÞ and W obey Fermi or Bose commutation rules for equal time arguments. For definiteness, ions will be treated as fermions. The ion subsystem is assumed to be non-degenerate, so that the final results will not depend on this

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assumption. On the other hand, the Fermi statistics is natural for protons which may thus be regarded as one of the ion species. 2.2. Equations of motion for Green’s functions A convenient starting point in deriving equations of motion for the field Green’s functions is the set of Maxwell’s equations for the averaged four-vector potential (2.7). In the Coulomb gauge and Heaviside’s units, these equations read

Að1Þ ¼ ~ 1~ J T ð1Þ þ ~ J ðextÞ ð1Þ;

r21 /ð1Þ ¼ .ð1Þ þ .ðextÞ ð1Þ; 2

ð2:16Þ

where  ¼ r  o =ot is the wave operator, and ~ J T ð1Þ is the induced transverse current density. Using the four-component notation, Eq. (2.16) are summarized as 2

2

l

D k ð1ÞAk ð1Þ ¼ J l ð1Þ þ J ðextÞl ð1Þ;

ð2:17Þ



 where J ð1Þ ¼ .ð1Þ;~ J T ð1Þ and l

0

r21

B 0 B Dlm ð1Þ ¼ B @ 0 0

1

0

0

0

1

0

0

1

0 C C C: 0 A

0

0

ð2:18Þ

1

Equations of motion for the matrix Green’s function (2.8) can now be obtained by taking the functional derivatives of Eq. (2.17) with respect to J ðextÞ m . Here, one point needs to be made. Since in the Coulomb b ~ ~ A ¼ 0, only the transverse part of the external current enters the evolution operator gauge we have r ðextÞ ~ ~ (2.6). Because of the condition r J ðextÞ ð1Þ ¼ 0, the components J ð1Þ cannot be treated as indepeni

dent variables. It is therefore convenient to define the functional derivative with respect to any trans~T ð1Þ as verse field V

d dV Ti ð1Þ

!

Z

d10 dTij ð1  10 Þ

d dV Tj ð10 Þ

ð2:19Þ

;

where

dTij ð1  2Þ ¼ dðt1  t2 ÞdTij ð~ r1  ~ r 2 Þ; and

dTij ð~ rÞ

¼

Z

3

d k ð2pÞ3

e

i~ k~ r

ki kj dij  2 j~ kj

ð2:20Þ

! ð2:21Þ

is the transverse delta function. Performing now functional differentiation d=dV Tj ð10 Þ in Eq. (2.19), all components V Tj ð10 Þ can be regarded as independent variables. Let us return to Eq. (2.17) and take the functional derivative of both sides with respect to J ðextÞ m ð2Þ. lm Noting that dJ ðextÞl ð1Þ=dJ ðextÞ m ð2Þ ¼ d ð1  2Þ, where lm

d ð1  2Þ ¼ dlm ð1  2Þ ¼

dð1  2Þ

0

0

dTij ð1  2Þ

!

;

ð2:22Þ

and using the chain rule3

dJ l ð1Þ ðextÞ

dJ m

ð2Þ

¼

dJ l ð1Þ dAk ð10 Þ dAk ð10 Þ dJ ðextÞ m ð2Þ

;

we obtain the equation of motion for the field Green’s function 3

From now on, except as otherwise noted, integration over ‘‘primed” variables is understood.

ð2:23Þ

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l

l

Dk ð1ÞDkm ð1 2Þ ¼ dlm ð1  2Þ þ Pk ð1 10 ÞDkm ð10 2Þ

ð2:24Þ

with the polarization matrix

Plm ð1 2Þ ¼

dJ l ð1Þ dAm ð2Þ

ð2:25Þ

:

Physically, this matrix characterizes the system response to variations of the total electromagnetic field. The calculation of Plm for a many-component relativistic plasma will be detailed later. The equation of motion for the electron Green’s function follows directly from the Dirac equation for wð1Þ on the contour C

ði

 e ð1Þ  mÞwð1Þ ¼ 0:

ð2:26Þ

Here and in what follows, we use the conventional abbreviation Recalling the definition (2.14), we can write

¼

cl a

l

for any four-vector al .

D h iE. b l ð1Þw ð1Þw  I ð2Þ  mÞGð1 2Þ þ iecl T C S A hSi ¼ dð1  2Þ: I I

ði

ð2:27Þ

Strictly speaking, the delta function on the right-hand side should be multiplied by the unit spinor matrix I. For brevity, this matrix will be usually (but not always) omitted. The second term in Eq. (2.27) can be further transformed by noting that

dGð1 2Þ ðextÞ

dJ l

ð1Þ

D h iE. l b l ð1Þw ð1Þw  I ð2Þ ¼  T C SA hSi þ iA ð1ÞGð1 2Þ I I

ð2:28Þ

which is a consequence of Eq. (2.6). Using the matrix identity

dFð1 2Þ ¼ Fð1 10 ÞdF 1 ð10 20 ÞFð20 2Þ;

ð2:29Þ

Eq. (2.27) is manipulated to Dyson’s equation

 i

  e ð1Þ  m Gð1 2Þ  Rð1 10 ÞGð10 2Þ ¼ dð1  2Þ

ð2:30Þ

with the matrix self-energy

Rð1 2Þ ¼ iecl Gð1 10 Þ

dG1 ð10 2Þ dJ ðextÞ l ð1Þ

:

ð2:31Þ

Let us also write down the adjoint of Eq. (2.30), which will be needed in the following:

 Gð1 2Þ i

  e ð2Þ  m ¼ dð1  2Þ þ Gð1 10 ÞRð10 2Þ:

ð2:32Þ

We would like to mention that the very existence of the inverse Green’s function G1 ð1 2Þ on the timeloop contour C and, consequently, the existence of Dyson’s equation is not a trivial fact. It can be justified only if the initial density operator admits Wick’s decomposition of correlation functions. In the context of the real-time Green’s function formalism, this implies a factorization of higher-order Green’s functions into products of one-particle Green’s functions in the limit t0 ! 1 [23,30]. This mathematical limit should be interpreted in the coarse-grained sense, i.e., jt 0 j should be a time long compared to some microscopic ‘‘interaction time” but short compared to a macroscopic ‘‘relaxation time scale”. The above boundary condition thus implies that there are no bound states and other long-lived many-particle correlations in the system. Otherwise, in order to have a Dyson equation for Green’s functions, it is necessary to modify the form of the contour C [19,31]. In the case of a weakly coupled relativistic plasma considered in the present paper, the boundary condition of weakening of initial correlations is well founded, so that one may use expression (2.31) for the electron selfenergy. Non-relativistic equations of motion for the ion field operators WB ð1Þ in the Heisenberg picture have exactly the form of the one-particle Schrödinger equation where the four-potential Al ð1Þ is reb l ð1Þ [25]. Since in all practical applications the ion transverse placed by the corresponding operator A

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current is very small compared to the electron transverse current, we shall neglect the direct interaction between ions and the transverse field ~ A. Then we have instead of Eq. (2.27) the equation

! D h iE. o r21 b I ð1ÞWBI ð1ÞWy ð2Þ G B ð1 2Þ þ ieB T C S / i þ hSi ¼ dð1  2Þ; BI ot 1 2mB

ð2:33Þ

where mB and eB are the ion mass and charge. The second term in this equation can be transformed in the same way as the second term in Eq. (2.27) to derive Dyson’s equation for the heavy particles (no summation over B)

! o r21 i þ  eB /ð1Þ G B ð1 2Þ  RB ð1 10 ÞG B ð10 2Þ ¼ dð1  2Þ; ot 1 2mB

ð2:34Þ

where

RB ð1 2Þ ¼ ieB G B ð1 10 Þ

0 dG 1 B ð1 2Þ ðextÞ d. ð1Þ

ð2:35Þ

are the ion self-energies. 2.3. Vertex functions A convenient way to analyze the polarization matrix and the particle self-energies is to express them in terms of vertex functions. The electron four-vertex is defined as

Cl ð1 2; 3Þ ¼ 

dG1 ð1 2Þ : dAl ð3Þ

ð2:36Þ

It is evident that each component of Cl is a 4  4 spinor matrix. Using the chain rule

dG1 ð1 2Þ ðextÞ

dJ l

ð3Þ

¼

dG1 ð1 2Þ dAm ð10 Þ ¼ Cm ð1 2; 10 ÞDml ð10 3Þ; dAm ð10 Þ dJ ðextÞ ð3Þ l

the electron self-energy (2.31) can be rewritten in the form ml 0 0 0 0 0 0 0 0 Rð1 2Þ ¼ iCð0Þ l ð1 1 ; 4 ÞGð1 2 ÞCm ð2 2; 3 ÞD ð3 4 Þ;

ð2:37Þ

m Cð0Þ l ð1 2; 3Þ ¼ edð1  2Þdlm ð1  3Þc :

ð2:38Þ

where

Equation (2.37) relates the electron self-energy to the four-vertex Cl . Another relation between these quantities follows from Dyson’s Eq. (2.30). Writing

 G1 ð1 2Þ ¼ i

  e ð1Þ  m dð1  2Þ  Rð1 2Þ

and recalling the definition (2.36) of the four-vertex, we find immediately that

Cl ð1 2; 3Þ ¼ Cð0Þ l ð1 2; 3Þ þ

dRð1 2Þ : dAl ð3Þ

ð2:39Þ

It is thus seen that Cð0Þ l is the bare four-vertex, i.e. the vertex in the absence of field fluctuations. The ion four-vertices are defined as

CBl ð1 2; 3Þ ¼ 

dG 1 B ð1 2Þ : dAl ð3Þ

ð2:40Þ

The arguments then go along the same way as for the electron self-energy and Eq. (2.35) is transformed into

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

ml 0 0 0 0 0 0 0 0 RB ð1 2Þ ¼ iCð0Þ Bl ð1 1 ; 4 ÞG B ð1 2 ÞCBm ð2 2; 3 ÞD ð3 4 Þ

1269

ð2:41Þ

with the bare vertices

Cð0Þ Bl ð1 2; 3Þ ¼ eB dð1  2Þdl0 ð1  3Þ:

ð2:42Þ

Using the ion Dyson’s Eq. (2.34), it is a simple matter to see that the analogue of Eq. (2.39) reads

CBl ð1 2; 3Þ ¼ Cð0Þ Bl ð1 2; 3Þ þ

dRB ð1 2Þ : dAl ð3Þ

ð2:43Þ

To write the polarization matrix (2.25) in terms of the Green functions and vertices, we need an expression for the induced four-current J l ð1Þ ¼ hbJ l ð1Þi. The component J 0 ð1Þ is the induced charge density. It can be written as the sum of the electron and ion contributions:

 c0 wð1Þi þ J 0 ð1Þ ¼ ehwð1Þ

X

eB trS hWyB ð1ÞWB ð1Þi

ð2:44Þ

B

with trS denoting the trace over spin indices. Since the transverse ion current is neglected, we have

 0 Þcj wð10 Þi: J Ti ð1Þ ¼ edTij ð1  10 Þhwð1

ð2:45Þ

Thus, in terms of Green’s functions (2.14) and (2.15),

X

 J l ð1Þ ¼ iedlk ð1  10 ÞtrD ck Gð10 10þ Þ  idl0 eB trS G B ð1 1þ Þ;

ð2:46Þ

B

where trD stands for the trace over the Dirac spinor indices, and the notation 10þ shows that the time 0 t 0þ 1 is taken infinitesimally later on the contour C than t 1 . The polarization matrix (2.25) can now be obtained by differentiating Eq. (2.46) with respect to Am ð2Þ and then using the identity (2.29) to write dG=dAm and dG B =dAm in terms of the vertices. A little algebra gives

h

i

0 0 0 0 0 0 0 0 Plm ð1 2Þ ¼  itrD Cð0Þ l ð1 2 ; 1ÞGð2 3 ÞCm ð3 4 ; 2ÞGð4 1 Þ

i

X

h i ð0Þ trS CBl ð10 20 ; 1ÞG B ð20 30 ÞCBm ð30 40 ; 2ÞG B ð40 10 Þ :

ð2:47Þ

B

The above formalism provides a basis for studying various processes in a many-component plasma with relativistic electrons. It is remarkable that the polarization matrix (2.47) involves the same vertex functions as the particle self-energies (2.37) and (2.41). Thus, any approximation for the vertex functions leads to the corresponding self-consistent approximation for the polarization functions and the self-energies in terms of Green’s functions. The method we use for calculating the vertex functions is based on Eqs. (2.39) and (2.43). Neglecting the last term in Eq. (2.39), i.e. replacing everywhere Cl by the bare vertex Cð0Þ l , we recover the Bezzerides–DuBois approximation [24] in kinetic theory of electron–positron relativistic plasmas. However, this simplest approximation is unable to describe radiative processes, so that the last terms in Eqs. (2.39) and (2.43) must be taken into consideration. Recalling expressions (2.37) and (2.41) for the self-energies, we get the following equations for the vertices:

Cl ð1 2; 3Þ ¼ Cð0Þ l ð1 2; 3Þ ð0Þ

0

þ iCk ð1 10 ; 40 ÞGð10 100 ÞCl ð100 200 ; 3ÞGð200 20 ÞCk0 ð20 2; 30 ÞDk k ð30 40 Þ i 0 d h ð0Þ þ iCk ð1 10 ; 40 ÞGð10 20 Þ l Ck0 ð20 2; 30 ÞDk k ð30 40 Þ ; dA ð3Þ

ð2:48aÞ

CBl ð1 2; 3Þ ¼ Cð0Þ Bl ð1 2; 3Þ ð0Þ

0

þ iCBk ð1 10 ; 40 ÞG B ð10 100 ÞCBl ð100 200 ; 3ÞG B ð200 20 ÞCBk0 ð20 2; 30 ÞDk k ð30 40 Þ i 0 d h ð0Þ þ iCBk ð1 10 ; 40 ÞG B ð10 20 Þ l CBk0 ð20 2; 30 ÞDk k ð30 40 Þ : dA ð3Þ

ð2:48bÞ

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

Although these equations are too complicated to solve them in a general form, approximate solutions can be found for weakly coupled plasmas where collisional interaction is taken into account to lowest orders. It should be noted that a simple asymptotic expansion of vacuum electrodynamics in powers hc is not appropriate even for weakly coupled plasmas due to colof the fine structure constant a ¼ e2 = lective effects (polarization and screening). It is therefore natural to work with the full Green’s functions G, G B , and Dlm , which contain polarization and screening to all orders in a. As discussed by Bezzerides and DuBois [24], the weak-coupling approximation for a plasma should be regarded as an expansion in terms of the field Green’s functions Dlm which play the role of intensity measures for fluctuations of the electromagnetic field, rather than being an expansion in a. This scheme applies if several conditions are fulfilled. The first condition reads kpl  1, where kpl ¼ 1=ðne r3sc Þ is the plasma parameter. Here, ne is the electron number density and rsc is the screening (Debye) length. The second hvÞ  1, where v a characteristic particle velocity, ensures the validity of the Born condition, e2 =ð approximation for scattering processes. The above conditions are usually met for relativistic plasmas. Finally, it is assumed that the plasma modes are not excited considerably above their local equilibrium level. This condition is not satisfied for a strongly turbulent regime. In what follows we shall assume that the system can be described within the weak-coupling approximation. As can be seen from Eqs. (2.48), the derivatives dC=dAl and dCB =dAl are at least of first order in the field Green’s functions D. On the other hand, using the identity (2.29) and Eq. (2.24), we may write symbolically

dD dD1 dP D D l D: l D dAl dA dA Thus, for a weakly coupled plasma, the last terms in Eqs. (2.48) are small compared to other terms on the right-hand sides. This suggests a self-consistent approximation in which the last terms in Eqs. (2.48) are neglected. The resulting equations for the vertices are still rather complicated but more tractable for specific problems.4 Another way is to solve Eqs. (2.48) by iteration. In this paper, we restrict our analysis to the first iteration of Eqs. (2.48), which means that the vertices in the second terms on the right-hand sides are replaced by the bare vertices and the last terms are neglected. This approximation can be represented graphically by the Feynman diagrams shown in Fig. 1. This approximation for the vertices generates the corresponding self-energies (2.37), (2.41), and the polarization matrix (2.47). They are summarized in Fig. 2. Once expressions for the self-energies and the polarization matrix are specified, Eqs. (2.24), (2.30), and (2.33), together with Maxwell’s equations for the mean electromagnetic field Al ð1Þ, provide a closed and self-consistent description of the particle dynamics and the field fluctuations in manycomponent weakly coupled plasmas with relativistic electrons (positrons). It is, of course, clear that these equations are prohibitively difficult to solve, so that one has to introduce further reasonable approximations, depending on the character of the problem. As already noted, our prime interest is with photon kinetics in a non-equilibrium relativistic plasma. The kinetic description of radiative processes is adequate if the medium is approximately isotropic on the scale of the characteristic photon wavelength and therefore the transverse and longitudinal electromagnetic fluctuations do not mix.5 In our subsequent discussion we shall assume this condition to be satisfied, so that the field Green’s function Dlm and the polarization matrix Plm will be taken in block form

 Dlm ð1 2Þ ¼

Dð1 2Þ

0

0

Dij ð1 2Þ

 ;



Plm ð1 2Þ ¼

Pð1 2Þ

0

0

Pij ð1 2Þ

 :

ð2:49Þ

Here, D  D00 and P  P00 are the longitudinal (plasmon) components, whereas Dij and Pij are the transverse (photon) components.

4

Analogous equations for non-relativistic plasmas were discussed, e.g., by DuBois [25]. The direct coupling between transverse and longitudinal field fluctuations described by the components D0i and P0i may be important for long-wavelength and low-frequency modes in the presence of electron beams and their return plasma currents [12,14]. 5

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

µ (1

2 ; 3) =

Bµ

1271

(1 2 ; 3) =

Fig. 1. Lowest order diagrams for the vertices. The first terms are the bare vertices, straight and doubled lines denote, kr respectively, G and G B . Dashed lines denote iD .

(1 2) = B (1

2) =

i

(1 2) =

Fig. 2. Lowest order diagrams for the electron self-energy (first line), the ion self-energies (second line), and the polarization matrix. The ion contribution to Plm is obtained as a sum of the last two diagrams over the species index B.

3. Kinetic equation for photons in plasmas 3.1. Transport and mass-shell equations With the assumption of the block structure (2.49) for Dlm and Plm , the equation of motion for the photon Green’s functions Dij ð1 2Þ is obtained from Eq. (2.24) in the form

1 Dij ¼ dTij ð1  2Þ þ Pik ð1 10 ÞDkj ð10 2Þ:

ð3:1Þ

We shall also need the adjoint of this equation which reads

2 Dij ð1 2Þ ¼ dTij ð1  2Þ þ Dik ð1 10 ÞPkj ð10 2Þ:

ð3:2Þ

The first step toward a photon kinetic equation is to rewrite the above equations in terms of the ?  canonical components (2.2) of Dij ð1 2Þ, which will be denoted as dij ð12Þ and dij ð12Þ. Without entering into details of algebraic manipulations described, e.g., in the paper by Botermans and Malfliet [23], let us write down the so-called Kadanoff–Baym (KB) equations for the transverse correlation functions 

ð3:3aÞ

? 0 kj ð1 2Þ;

ð3:3bÞ

0 0 1 dij ð12Þ ¼ pþik ð110 Þdkj ð10 2Þ þ p? ik ð11 Þdkj ð1 2Þ; ?

? 2 dij ð12Þ

?

¼

? dik ð110 Þ

p

0  kj ð1 2Þ

þ

þ dik ð110 Þ

p

and the equations for the propagators

1 dij ð12Þ ¼ dTij ð1  2Þ þ pik ð110 Þdkj ð10 2Þ;





ð3:4aÞ

 2 dij ð12Þ

0  kj ð1 2Þ;

ð3:4bÞ

? ij

¼

dTij ð1

 2Þ þ

 dik ð110 Þ

p

where p and p are the canonical components of Pij , and the primed space–time variables are integrated according to the rule

Z

 ij

d10    ¼

Z

1

1

0

dt 1

Z

3

d~ r 01   

1272

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302 

?

Note that the space–time correlation functions dij ð12Þ and the propagators dij ð12Þ enjoy symmetry properties which are useful in working with Eqs. (3.3) and (3.4): <

>

h



h

dij ð12Þ ¼ dji ð21Þ; þ

dij ð12Þ ¼ dji ð21Þ;

<

i

þ

i

dij ð12Þ dij ð12Þ

<

¼ dji ð21Þ;

ð3:5aÞ

þ

¼ dij ð12Þ:

ð3:5bÞ

b i are Hermitian operators.6 These properties follows directly from the fact that A The phase space description of photon dynamics is achieved by using the Wigner representation which is defined for any Fð12Þ  Fðx1 ; x2 Þ as

FðX; kÞ ¼

Z

4

d x eikx F ðX þ x=2; X  x=2Þ;

ð3:6Þ

l 0 where k  x ¼ k xl ¼ k t  ~ k ~ r. Within the framework of kinetic theory, the variations of the field RÞ are assumed to Green’s functions and the polarization matrix in the space–time variable X l ¼ ðT; ~ be slow on the scales of k and 1=x, where k and x are, respectively, some characteristic radiation wavelength and frequency. Therefore, going over to the Wigner representation in Eqs. (3.3) and (3.4), we shall keep terms only to first order in X-gradients.7 Formal manipulations are simplified by using the first-order transformation rule [23]

i F 1 ð110 ÞF 2 ð10 2Þ ! F 1 ðX; kÞF 2 ðX; kÞ  fF 1 ðX; kÞ; F 2 ðX; kÞg; 2

ð3:7Þ

where

fF 1 ðX; kÞ; F 2 ðX; kÞg ¼

oF 1 oF 2 oF 1 oF 2  oX l okl okl oX l

ð3:8Þ

is the four-dimensional Poisson bracket. After some algebra, Eqs. (3.3) and (3.4) reduce to the equa?  tions of motion for dij ðX; kÞ and dij ðX; kÞ (the arguments X and k are omitted to save writing):

  2 l o ? k þ ik d oX l ij   2 l o ? k  ik l dij oX   2 l o  k þ ik l dij oX   2 l o  k  ik d oX l ij

i  n þ ? o n ?  o pin ; dnj þ pin ; dnj ; 2 i  n ?  o n þ ? o ? þ ? ¼ din pnj þ din pnj  din ; pnj þ din ; pnj ; 2 ki kj i n  o  ¼ dij  2 þ pin dnj  pin ; dnj ; ~ 2 jkj ki kj i n  o  ¼ dij  2 þ din pnj  d ;p ; 2 in nj j~ kj 

?

¼ pþin dnj þ p? in dnj 

ð3:9Þ ð3:10Þ ð3:11Þ ð3:12Þ

2 l 2 where k ¼ k kl ¼ k0  j~ kj2 . l 0 kÞ as the photon four-momentum In the context of kinetic theory, it is natural to interpret k ¼ ðk ; ~ at the space–time point X. However, here one is faced with a difficulty which is a consequence of the Wigner transformation (3.6). To explain this point, let us consider some transverse tensor T ij ð12Þ satisfying

r1i T ij ð12Þ ¼ 0;

r2j T ij ð12Þ ¼ 0:

ð3:13Þ

Note that the components of Dij ð1 2Þ and Pij ð1 2Þ satisfy these relations due to the gauge constraint b ~ ~ r A ¼ 0. In terms of the Wigner transforms, Eq. (3.13) read

  1 o i þ ik T ij ðX; kÞ ¼ 0; 2 oRi

  1 o j  ik T ij ðX; kÞ ¼ 0: 2 oRj

ð3:14Þ

6 Using Eqs. (3.3) and (3.4), it can be verified that the polarization matrices pij ð12Þ and p ij ð12Þ have the same symmetry ?  properties as dij ð12Þ and dij ð12Þ. 7 This gradient expansion scheme is a usual way for deriving kinetic equations in the Green function formalism [21–24].

?

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302 i

1273

j

Thus, in an inhomogeneous medium, k T ij ðX; kÞ–0 and k T ij ðX; kÞ–0. In particular, we conclude that ?  k. However, in dij ðX; kÞ and dij ðX; kÞ in Eqs. (3.9)–(3.12) contain longitudinal parts with respect to ~ Appendix A it is shown that, up to first-order X-gradients, the energy flux of the radiation field < is completely determined by the transverse part of dij ðX; kÞ, which, for any tensor T ij ðX; kÞ, is defined as

T ?ij ðX; kÞ ¼ D?im ð~ kÞ T mn ðX; kÞD?nj ð~ kÞ;

ð3:15Þ

i j D?ij ð~ kÞ ¼ dij  k k =j~ kj2

ð3:16Þ

where

is the transverse projector. Since the longitudinal parts of the field correlation functions do not contribute to the observable energy flux, it is reasonable to eliminate them in Eqs. (3.9)–(3.12). This can be done in the following way. Let us first show that any tensor T ij ðX; kÞ satisfying Eq. (3.14) can be expressed in terms of its transverse part (3.15). We start from the identity (the arguments X and k are omitted for brevity) i m

T ij ¼ T ?ij þ

n j

i j

kk k k kk m n T mn D?nj þ D?im T mn þ 4 k T mn k ; j~ kj2 j~ kj2 j~ kj

i j ~2 m n which follows directly from the obvious relation dij ¼ D? ij þ k k =jkj . The terms k T mn and k T mn can then be eliminated with the help of Eq. (3.14). After a simple algebra, we obtain

 k k o2 T i o  i j mn m k T mj  k T im  m n: 2 4j~ kj4 oR oR 2j~ kj oR i j

T ij ¼ T ?ij þ

Solving this equation by iteration, one can find T as a series in derivatives of T ? . Keeping only first-order gradients yields

T ij ðX; kÞ ¼ T ?ij ðX; kÞ þ

! ? ? i i oT mj j oT im : k  k oRm oRm 2j~ kj2

ð3:17Þ

We next consider the contraction of two tensors, T ij ðX; kÞ and Q ij ðX; kÞ, each of which satisfies (3.14). Using Eq. (3.17), we find, up to first-order corrections,

T il ðX; kÞQ lj ðX; kÞ ¼ T ?il ðX; kÞQ ?lj ðX; kÞ þ

 ?  ? i i oT ml j ? oQ lm ? k Q  k T : il oRm lj oRm 2j~ kj2

ð3:18Þ

Formulas (3.17) and (3.18) allow one to rewrite Eqs. (3.9)–(3.12) in terms of the transverse parts of the correlation functions, photon propagators, and polarization matrices. Since the Poisson brackets are already first order in X-gradients, the corresponding matrices may be replaced by their transverse parts. Then, acting on both sides of Eqs. (3.9)–(3.12) on the left and right by the projector (3.16), the desired equations can easily be obtained. Note that in this way the peculiar gradient terms, such as the last term in Eq. (3.18), are eliminated.8 We will not write down the resulting equations because it is more convenient to deal with the ?  equivalent transport and mass-shell equations for the transverse parts of dij ðX; kÞ and dij ðX; kÞ. The transport equations are obtained by taking differences of Eqs. Eqs. (3.9)–(3.12) and then eliminating the longitudinal parts of all matrices as described above. In condensed matrix notation, the resulting transport equations read

8 From now on all d- and p-matrices in the Wigner representation are understood to be transverse with respect to ~ k, although this will not always be written explicitly.

1274

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

n

2

?

k ;d

o



 1 n

pþ ; d?

o?

þ



p? ; d

?

 n o?  ? ? þ  d ; p  d ; p?

2 i 

þ 

i h þ ?  < > ¼  ðd þ d Þ; p?  þ p> ; d þ  p< ; d þ ; ðp þ p Þ; d  2 n o 1 n o? n o?  h i 2      k ;d  p ;d  d ; p ; ¼ i p ; d  2

ð3:19Þ ð3:20Þ

where ½A; B ¼ AB  BA are the anticommutators/commutators of matrices, and the superscript ? denotes the transverse part of a tensor. Taking sums of Eqs. (3.9)–(3.12) instead of differences, one obtains the mass-shell equations that do not contain the dominant drift terms:

  i n þ ? o?  ?  ? n ?  o?  þ ? ? 2 ? k d þ p ;d þ p ;d þ d ;p þ d ;p 4   i

þ 

1 h þ ?  < >  ðd þ d Þ; p? þ þ p> ; d   p< ; d  ; ¼ ðp þ p Þ; d þ 4 n o? n o?  i 1 h  i 2  p ; d þ d ; p kÞ þ p ;d : ¼ D? ð~ k d þ þ 4 2 ?

ð3:21Þ ð3:22Þ

2

In vacuum, Eq. (3.21) reduces to the mass-shell condition dij ðkÞ / dðk Þ. By analogy with vacuum QED, it is convenient to go over in Eqs. (3.19)–(3.22) to a representation using photon polarization states. It seems natural to introduce these states through a set of eigenvectors of some Hermitian transverse tensor T ? ij ðX; kÞ. The properties of photon modes are closely related to symmetry properties of the polarization tensor pþ ij ð12Þ which determine the medium response to the transverse electromagnetic field [25]. This tensor can be chosen to define the photon polarization states. In the Wigner representation, the tensor pþ ij ðX; kÞ can always be written as a sum

pþij ðX; kÞ ¼ Re pþij ðX; kÞ þ i Im pþij ðX; kÞ;

ð3:23Þ

þ þ  where Re pþ ij and Im pij are Hermitian. Using relation ½pij ð12Þ ¼ pji ð21Þ; we find that

 1 þ p ðX; kÞ þ pij ðX; kÞ ; 2 ij  1 þ Im pþij ðX; kÞ ¼ pij ðX; kÞ  pij ðX; kÞ : 2i

Re pþij ðX; kÞ ¼

ð3:24Þ

Let ~ s ðX; kÞ be the eigenvectors of Re pþij ðX; kÞ. Since in general these eigenvectors are complex, we may write

Re pþij ðX; kÞ ¼

X

si ðX; kÞps ðX; kÞ sj ðX; kÞ;

ð3:25Þ

s

where ps ðX; kÞ, (s ¼ 1; 2), are real eigenvalues. This does not mean, however, that the full tensors pij ðX; kÞ, as well as p?ij ðX; kÞ, d?ij ðX; kÞ, and dij ðX; kÞ, are diagonalized by the same set of eigenvectors. In an anisotropic medium, the non-zero off-diagonal components of these tensors describe coupling between different polarization states. The tensor Re pþ ij ðX; kÞ for a weakly coupled relativistic plasma is considered in Appendix E. It is shown that effects of anisotropy on the photon polarization states are inessential for jk0 j  xe , where xe is the electron plasma frequency (see Eq. (6.4)). In what follows this condition is assumed to be fulfilled. Note also that, for a weakly coupled plasma, the photon polarization states can be defined through the eigenvectors of the tensor Re pþ ij ðX; kÞ in which only real parts of the elements are retained. The corresponding eigenvectors are real and satisfy

~ s ðX; kÞ  ~s0 ðX; kÞ ¼ dss0 ;

X s

si ðX; kÞsj ðX; kÞ ¼ dij 

ki kj : j~ kj2

ð3:26Þ ?



?

In this principal-axis representation, off-diagonal components of the tensors bij ðX; kÞ ¼ fdij ; dij ; pij ; p ij g are assumed to be small for jk0 j  xe , so that all these tensors can be expressed in terms of their diagonal components, bs ðX; kÞ, as

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

bij ðX; kÞ ¼

X

si ðX; kÞbs ðX; kÞsj ðX; kÞ:

1275

ð3:27Þ

s

Calculating the diagonal projections of matrix equations (3.19) and (3.21) on the polarization vectors ~ s , one obtains the transport and mass-shell equations for the field correlation functions in the principal-axis representation:

n o    2 ? þ < > ¼ i p>s ds  p
ð3:28aÞ ð3:28bÞ

The same procedure applied to Eqs. (3.20) and (3.22) yields

n o 2  k  ps ; ds ¼ 0;

  2  k  ps ds ¼ 1:

ð3:29Þ

Note that in the mass-shell equation for the photon propagators the gradient corrections cancel each other. A few remarks should be made about Eqs. (3.28). Eq. (3.28a) may be regarded as a particular case of the gauge-invariant transport equation derived by Bezzerides and DuBois [24] if the polarization states are chosen in accordance with the Coulomb gauge. The mass-shell equation (3.28b) was ignored in [24]. There is no reason, however, to do this because the field correlation functions must satisfy both equations. The mass-shell equation is in a sense a constraint for approximations in the transport equation. 3.2. Resonant and virtual photons in plasmas Let us turn first to Eq. (3.29). From the second (mass-shell) equation one readily finds the ‘‘explicit” expression for the photon propagators:

1



ds ðX; kÞ ¼

2

k  Re pþs ðX; kÞ  i k0 Cs ðX; kÞ

;

ð3:30Þ

where þ Cs ðX; kÞ ¼ k1 0 Im ps ðX; kÞ

ð3:31Þ

is the k-dependent damping width for the photon mode. It is clear that the first of Eq. (3.29) is automatically satisfied due to the identity fA; f ðAÞg ¼ 0. We now consider Eqs. (3.28). From the point of view of kinetic theory the physical meaning of these ? equations remains to be seen because the correlation functions ds ðX; kÞ involve contributions from the resonant (propagating) and virtual photons. Since a kinetic equation describes only the resonant photons, there is a need to pick out the corresponding terms from the field correlation functions. Our analysis of this problem parallels the approach discussed previously by Špicˇka and Lipavsky´ in the context of non-relativistic solid state physics [27,28]. Recalling the definition (3.8) of the four-dimensional Poisson bracket, it is easy to see that 2 ? fk  Re pþ s ; ds g has the structure of the drift term in a kinetic equation for quasiparticles with ener2 gies given by the solution of the dispersion equation k  Re pþ s ¼ 0. Analogous observations serve as a starting point for derivation of Boltzmann-type quasiparticle kinetic equations from transport equations for correlation functions. In most derivations, the peculiar terms like the second term on the left-hand side of Eq. (3.28a) are ignored. But, as was first noted by Botermans and Malfliet [23] (see also [27,28]), such terms contribute to the drift. To show this in our case, we introduce the notation 2 D ¼ k  Re pþs  ik0 Cs and use Eq. (3.30) to write

  1 þ ¼ Re ds ; p? s 2

( ) ( )   1 1 1 p?s 1 p?s þ  ? ; ps ¼  þ D ; þ 2  D ;  2 ; 2 2 Dþ D ðD Þ ðD Þ



1276

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

or

n     þ 2 o n  þ 2 o þ 2 þ k0 Cs ; p? ; ¼  k  Re pþs ; p? Re ds ; p? s s Re ds s Im ds

ð3:32Þ

where the first term dominates in the case of small damping. Let us now define new functions e d s? ðX; kÞ through the relation

h 2 i ? þ ? : ds ðX; kÞ ¼ e d? s ðX; kÞ þ ps ðX; kÞRe ds ðX; kÞ

ð3:33Þ

Inserting expressions (3.32) and (3.33) into Eqs. (3.28), we obtain the transport and mass-shell equations for e d? s :

n

o n    þ 2 o 2 ¼ i p>s e k  Re pþs ; e d s? þ k0 Cs ; p? d s ; s Im ds n o n n  þ 2 o  þ 2 o 2 k0 Cs ; e þ k  Re pþs ; p? þ 2 k 0 Cs ; p ? d? s s Im ds s Re ds     þ 4 2 e   ðk0 Cs Þ2 p? : ¼ 2 k  Re pþs d? s  2 ds s

ð3:34aÞ

ð3:34bÞ

We wish to emphasize that the contributions from the last term of Eq. (3.33) to the right-hand side of Eq. (3.34a) cancel identically. Before going any further, it is instructive to consider the spectral properties of e d? s ðX; kÞ. First we recall the conventional full spectral function which is defined in terms of correlation functions or propagators (see, e.g., [23]). For the photon modes, the full spectral function is given by

 >  þ <  as ðX; kÞ ¼ i ds  ds ¼ i ds  ds :

ð3:35Þ

In the first gradient approximation, as ðX; kÞ is obtained from Eq. (3.30):

2k0 Cs as ðX; kÞ ¼  : 2 2 2 k  Re pþs þ ðk0 Cs Þ

ð3:36Þ

We now introduce the quantity

  e a s ðX; kÞ ¼ i e d >s  e d
ð3:37Þ

which will be referred to as the resonant spectral function. Using Eqs. (3.33), (3.36), and relations

p>s  p
ð3:38Þ

we find 3

4ðk0 Cs Þ 2 : 2 2 2 k  Re pþs þ ðk0 Cs Þ

e a s ðX; kÞ ¼ 

ð3:39Þ

When this expression is compared with (3.36), two important observations can be made. First, it is easy to check that both spectral functions take the same form in the zero damping limit:

  2 a s ¼ 2pgðk0 Þd k  Re pþs ; lim as ¼ lim e

Cs !0

Cs !0

ð3:40Þ

where gðk0 Þ ¼ k0 =jk0 j. Second, for a finite Cs , the resonant spectral function (3.39) falls off faster than the full spectral function (3.36). Thus, for weakly damped field excitations, the first term in Eq. (3.33) 2 dominates in the vicinity of the photon mass-shell (k Re pþ s ), while the second term dominates in 2 þ 2 a s has a stronger peak and smaller the off-shell region where it falls as ðk  Re ps Þ . In other words, e wings than as . We see that physically the first term in Eq. (3.33) may be regarded as the ‘‘resonant” part of the field correlation functions. The second term represents the ‘‘off-shell” part which may be identified as the contribution of virtual photons9. 9 We would like to emphasize that this interpretation makes sense only in the case of small damping. Formally, Eq. (3.33) in itself is not related to any interpretation.

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

1277

It is interesting to note that the spectral function (3.39) also occurs in the self-consistent calculation of thermodynamic quantities of equilibrium QED plasmas. As shown by Vanderheyden and Baym [32], the resonant spectral function rather than the Lorentzian-like full spectral function (3.36) determines the contribution of photons to the equilibrium entropy. That is why in [32] e a s was called ‘‘the entropy spectral function”. We have seen, however, that this function has a deeper physical meaning; it characterizes the spectral properties of resonant (propagating) photons in a medium. Therefore, we prefer to call e a s the resonant spectral function. 3.3. Kinetic equation for resonant photons On the basis of the above considerations, it is reasonable to formulate a kinetic description of resonant photons in terms of two functions N ? ðX; kÞ defined through the relations

e d
e d >s ðX; kÞ ¼ i e a s ðX; kÞN >s ðX; kÞ;

ð3:41Þ

where

N>s ðX; kÞ  N
ð3:42Þ

It is also convenient to use the representation

Ns ðX; kÞ ¼ hðk0 ½1 þ N s ðX; kÞ  hðk0 ÞNs ðX; kÞ;

ð3:43Þ

which serve as the definition of the local photon distribution function N s ðX; kÞ in four-dimensional phase space. We will now convert Eq. (3.34a) into a kinetic equation for N ? s ðX; kÞ. It suffices to consider only the e> equation for e d< s since the transport equation for d s leads to the same kinetic equation. Substituting e e< e < d< s ¼ i a s N s into Eq. (3.34a) for d s and calculating the first Poisson bracket on the left-hand side, 2 þ e we obtain a peculiar drift term N < s fk  Re ps ; a s g. Here, we shall follow Botermans and Malfliet [23] who have shown how this term can be eliminated by another peculiar term coming from the second Poisson bracket10. Since Eq. (3.34a) itself is correct to first-order X-gradients, in calculating Poisson brackets the polarization functions p? s may be expressed in terms of the correlation functions from the local balance relation

p>s ed s ¼ 0:

ð3:44Þ

With the help of Eqs. (3.37), (3.38), and (3.41) we find that

p?s ¼ 2ik0 Cs N?s :

ð3:45Þ

This expression for p can now be used to calculate the second Poisson bracket in Eq. (3.34a). A straightforward algebra then leads to < s

e as

" n

# o k2  Re pþ    2 s ¼ 0: k  Re pþs ; Ns Ns k 0 Cs

ð3:46Þ

The desired kinetic equation is obtained by setting the expression in square brackets equal to zero. We now apply the same procedure to the mass-shell equation (3.34b). As a result of simple manipulations, we get

  2 a s ½. . . ¼ 0 k  Re pþs e

ð3:47Þ

with the same expression in square brackets as in Eq. (3.46). We see that in the present approach the mass-shell equation is consistent with the kinetic equation.

10 The original arguments presented by Botermans and Malfliet refer to non-relativistic systems, but they are in essence applicable to any kinetic equation derived from the transport equations.

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For weakly damped photons, e a s is a sharply peaked function of k0 near the effective photon frekÞ, which are solutions of the dispersion equation quencies, xs ðX; ~ 2

k  Re pþs ðX; kÞ ¼ 0:

ð3:48Þ

For definiteness, it will be assumed that the photon frequencies are positive solutions. If k0 ¼ xs ðX; ~ kÞ

þ is such a solution, then, using the property ½pþ s ðkÞ ¼ ps ðkÞ, it is easy to see that the corresponding kÞ. A more detailed discussion of Eq. (3.48) is given in Appendix E. In negative solution is k0 ¼ xs ðX; ~ the small damping limit, the resonant spectral function may be approximated as (see Eq. (3.40))

  2 e a s ðX; kÞ ¼ 2pgðk0 Þd k  Re pþs :

ð3:49Þ

Then, integrating Eq. (3.46) over k0 > 0, we arrive at the kinetic equation



 o oxs o oxs o kÞ ¼ IðemissÞ ðX; ~ kÞ  IðabsÞ ðX; ~ kÞ  þ  ns ðX; ~  s s oT R o~ R o~ o~ k o~ k

ð3:50Þ

with the on-shell photon distribution function

  ns ðX; ~ kÞ ¼ N s ðX; kÞ

k0 ¼xs ðX;~ kÞ

:

ð3:51Þ

The first and the second terms on the right-hand side of Eq. (3.50) are, respectively, the photon emission and absorption rates:

h i IðemissÞ ðX; ~ kÞ ¼ iZ s ðX; ~ kÞp
ð3:52aÞ

kÞ IsðabsÞ ðX; ~

ð3:52bÞ

¼ iZ s ðX; ~ kÞp>s ðX; ~ kÞns ðX; ~ kÞ;

where



p?s ðX; ~kÞ ¼ p?s ðX; kÞk0 ¼xs ðX;~kÞ ;

ð3:53Þ

and Z s is given by

~ Z 1 s ðX; kÞ ¼

 o  2 k  Re pþs ðX; kÞ  : ok0 k0 ¼xs ðX;~ kÞ

ð3:54Þ

The photon kinetic equation (3.50) with emission and absorption rates (3.52) was derived many years ago by DuBois [25] for non-relativistic plasmas. The generalization to relativistic plasmas is obvious until the polarization functions p? s ðX; kÞ are specified. There was a reason, however, to discuss here the derivation of the photon kinetic equation. As we have seen, the kinetic equation is only related to the resonant parts of the transverse field correlation functions while their off-shell parts are represented by the last term in Eq. (3.33). We shall see shortly that these off-shell parts must be taken into account in calculating the emission and absorption rates. Note in this connection that the photon distribution function is usu? ally introduced through the relations ds ¼ ias N ? s with the full spectral function as ðX; kÞ which is then ? taken in the singular form (3.49) (see, e.g., [25]). In doing so, the off-shell corrections to ds are missing. 4. Electron propagators and correlation functions As seen from Eqs. (3.52), the transverse polarization functions p? s ðX; kÞ play a key role in computing radiation effects. Before going into a detailed analysis of these quantities, we will discuss some features of the electron Green’s function Gð1 2Þ and its components, the propagators G ð12Þ and the correlation functions G? ð12Þ, which are the necessary ingredients to calculate the polarization matrix (2.47). 4.1. Electron propagators Using Dyson’s equations (2.30) and (2.32) together with the canonical representation (2.2) of the electron Green’s function Gð1 2Þ and the self-energy Rð1 2Þ, one can derive the equations of motion for the retarded/advanced electron propagators:

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

 i

  e ð1Þ  m G ð12Þ ¼ dð1  2Þ þ R ð110 ÞG ð10 2Þ;   G ð12Þ i  e ð2Þ  m ¼ dð1  2Þ þ G ð110 ÞR ð10 2Þ:

1279

ð4:1aÞ ð4:1bÞ

The derivation follows a standard way (see, e.g., [23]), so that we do not repeat the algebra here. Going over to the Wigner representation and keeping only linear terms in the gradient expansion, we obtain

ðg  Þ1 G ¼ 1 þ

i n  1  o ðg Þ ; G ; 2

G ðg  Þ1 ¼ 1 þ

i n   1 o ; G ; ðg Þ 2

ð4:2Þ

where G ¼ G ðX; pÞ, and we have introduced the local propagators

g  ðX; pÞ ¼

1 ðX; pÞ  m  R ðX; pÞ

ð4:3Þ

with the kinematic momentum Pl ðX; pÞ ¼ pl  eAl ðXÞ. Up to first-order terms in X-gradients, we find from Eq. (4.2) two formally different ‘‘explicit” expressions for the propagators:

G ðX; pÞ ¼ g  þ

i  n  1  o g ðg Þ ; g ; 2

G ðX; pÞ ¼ g  þ

i n   1 o  g ; ðg Þ g : 2

ð4:4Þ

It is easy to verify, however, that these expressions are equivalent to each other due to the identity

o n o n A A1 ; A ¼ A; A1 A;

ð4:5Þ

which holds for any matrix AðX; pÞ and follows directly from the definition (3.8) of the Poisson bracket. The first-order gradient terms in Eqs. (4.4) arise due to the matrix structure of the relativistic propagators (4.3). These terms are zero for scalar propagators since, in the latter case, fðg  Þ1 ; g  g ¼ fg  ; ðg  Þ1 g ¼ 0. Note, however, that the gradient corrections to the electron propaga? tors may be neglected when evaluating local quantities, say, the polarization functions ps ðX; kÞ that enter the photon emission and absorption rates. The local propagators (4.3) have in general a rather complicated spinor structure due to the presence of the matrix self-energies R ðX; pÞ. Any spinor-dependent quantity Q can be expanded in a complete basis in spinor space as [33] l

l

Q ¼ IQ ðSÞ þ cl Q ðVÞ þ c5 Q ðPÞ þ c5 cl Q ðAÞ þ l

1 rlm Q lðTÞm ; 2 l

ð4:6Þ

lm

where I is the unit matrix, and Q ðSÞ , Q ðVÞ , Q ðPÞ , Q ðAÞ , Q ðTÞ are the scalar, vector, pseudo-scalar, axial-vector, and tensor components. The decomposition (4.6) can in principle be obtained for the local propagators (4.3) in terms of the components of the self-energies. It is clear, however, that deriving the propagators along this way is generally rather complicated. The situation becomes simpler for equal probabilities of the spin polarization, when a reasonable approximation is to retain only the first two terms in the decomposition (4.6) of the self-energies for relativistic fermions.11 The local propagators (4.3) then become

g  ðX; pÞ ¼

þ M ðP Þ2  ðM  Þ2

ð4:7Þ

with l

P l ¼ Pl  RðVÞ ;

M  ¼ m þ RðSÞ :

ð4:8Þ

A very simplified version of this approximation, which is used sometimes in thermal field theory, is the ansatz [35–37]

R ðX; pÞ ¼ iðCp =2Þc0 ; 11

For nuclear matter an analogous approximation was discussed, e.g., in [23,34].

ð4:9Þ

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where Cp is an adjustable ‘‘spectral width parameter”. 4.2. Quasiparticle and off-shell parts of correlation functions We now consider the electron (positron) correlation functions G? . Following the standard scheme [23], we obtain from Eqs. (2.30) and (2.32) the KB equations for the correlation functions:



  e ð1Þ  m G? ð12Þ ¼ Rþ ð110 ÞG? ð10 2Þ þ R? ð110 Þ G ð10 2Þ;   G? ð12Þ i  e ð2Þ  m ¼ G? ð110 ÞR ð10 2Þ þ Gþ ð110 ÞR? ð10 2Þ: i

ð4:10aÞ ð4:10bÞ

As in the case of transverse photons, it is natural to expect that the electron correlation functions G? ðX; pÞ contain sharply peaked ‘‘quasiparticle” parts and off-shell parts. To separate these contributions, we have to analyze drift terms in the transport equations which follow from the KB equations (4.10). This is detailed in Appendix B (see also [38]). In the local form, the decomposition of the electron correlation functions reads (cf. Eq. (3.33) for photons)

 e ? ðX; pÞ þ 1 g þ ðX; pÞ R? ðX; pÞ g þ ðX; pÞ þ g  ðX; pÞ R? ðX; pÞ g  ðX; pÞ : G? ðX; pÞ ¼ G 2

ð4:11Þ

e ? ðX; pÞ may be interpreted as the quasiparticle parts of the electron correlation funcTo show that G tions, it is instructive to consider their spectral properties. The full spectral function in spinor space is defined as [23,34]

 AðX; pÞ ¼ iðG> ðX; pÞ  G< ðX; pÞÞ ¼ i Gþ ðX; pÞ  G ðX; pÞ :

ð4:12Þ

In the local approximation G ðX; pÞ ¼ g  ðX; pÞ. Then, recalling Eq. (4.3), we find that þ

AðX; pÞ ¼ ig DRg  ;

ð4:13Þ

where DRðX; pÞ ¼ Rþ ðX; pÞ  R ðX; pÞ. We now introduce the quasiparticle spectral function associated e ? ðX; pÞ: with G

  e > ðX; pÞ  G e < ðX; pÞ : e AðX; pÞ ¼ i G

ð4:14Þ

Using Eqs. (4.11), (4.13), and the relation R>  R< ¼ Rþ  R , a simple algebra gives

i e AðX; pÞ ¼  g þ DRg þ DRg  DRg  : 2

ð4:15Þ

e To gain some insight into an important difference between the spectral functions AðX; pÞ and AðX; pÞ, let us consider the zero damping limit where the propagators (4.3) reduce to

1 mi

g 0 ðX; pÞ ¼

ð4:16Þ

with el ¼ ðe; 0; 0; 0Þ, e ! þ0. The corresponding retarded/advanced self-energies are R ¼ iec0 , so that DR ¼ iCc0 , where C ¼ 2e is the infinitesimally small spectral width. Then some spinor algebra in Eqs. (4.13) and (4.15) leads to the following prelimit expressions:

2P0 C AðX; pÞ ¼  2 2 P0  E 2p þ ðP0 CÞ2

"

# E 2p  P20 0 þmþ c ; 2P0

" 2 E p þ P20  4ðP0 CÞ3 2 2 2P20 P20  E 2p þ ðP0 CÞ2

e AðX; pÞ ¼ 

! #  E 2  P2 E 2p  P20 0 p 0 þm þ 1þ c ; 2P0 4P20

ð4:17Þ

ð4:18Þ

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 2 þ m2 . In the limit C ! 0 both spectral functions take the same form where E p ðXÞ ¼ P

e ¼ lim A ¼ 2pgðP0 ÞdðP2  m2 Þ lim A C!0

C!0



 þm ;

ð4:19Þ

where gðP0 Þ ¼ P0 =jP0 j . Note, however, that for finite but small C, the quasiparticle spectral function (4.18) falls off faster than the full spectral function (4.17). In other words, the ‘‘collisional broadening” of the quasiparticle spectral function is considerably smaller than that of the full spectral function. This has much in common with the properties of the resonant and the full spectral functions for the transverse field fluctuations, Eqs. (3.36) and (3.39). 4.3. Distribution functions of electrons and positrons In analogy with the kinetic description of resonant photons in Section 3.2, it is reasonable to introe ? ðX; pÞ and the quasiduce the electron (positron) distribution functions through relations between G particle spectral function (4.15). Before we go into a discussion of this point, we will briefly touch upon e ? ðX; pÞ and AðX; e pÞ. Note that the electron correlation, propagators, and the hermicity properties of G self-energies, as spinor matrices, satisfy the relations

? y

? y G ðX; pÞ ¼ c0 G? ðX; pÞc0 ; R ðX; pÞ ¼ c0 R? ðX; pÞc0 ;

 y

 y R ðX; pÞ ¼ c0 R ðX; pÞc0 ; G ðX; pÞ ¼ c0 G ðX; pÞc0 ;

ð4:20aÞ ð4:20bÞ

which follow directly from the definition of these quantities. Recalling (4.3), it is easy to see that the local propagators g  ðX; pÞ satisfy the same relations as G ðX; pÞ. Then one derives from Eq. (4.11)

h

e ? ðX; pÞ G

iy

e ? ðX; pÞc0 : ¼ c0 G

ð4:21Þ

This immediately leads to the following property of the quasiparticle spectral function (4.14):

h

e AðX; pÞ

iy

e pÞc0 : ¼ c0 AðX;

ð4:22Þ

We now introduce the distribution functions in spinor space, F ? ðX; pÞ, by the relation

  e ? ðX; pÞ ¼ i AðX; e e pÞF ? ðX; pÞ þ F ? ðX; pÞ AðX; pÞ : G 2

ð4:23Þ

Assuming

F > ðX; pÞ þ F < ðX; pÞ ¼ I;

ð4:24Þ

it is easy to verify that Eq. (4.14) is satisfied, whereas Eqs. (4.21) and (4.22) require that

? y F ðX; pÞ ¼ c0 F ? ðX; pÞc0 :

ð4:25Þ

This property ensures that all components of F ? ðX; pÞ in the expansion (4.6) are real. The spinor structure of F ? ðX; pÞ can be specified completely in the zero damping limit where ? e ? ðX; pÞ, and the quasiparticle spectral function is taken in the limiting form (4.19). Then, G ðX; pÞ ¼ G as shown by Bezzerides and DuBois [24], in the case of equal probabilities of the spin polarization,12 the matrices F ? ðX; pÞ are diagonal:

F ? ðX; pÞ ¼ IF ? ðX; pÞ;

F > ðX; pÞ þ F < ðX; pÞ ¼ 1;

ð4:26Þ

so that one has

e ? ðX; pÞ ¼ 2pigðP0 ÞdðP2  m2 Þ G



 þ m F ? ðX; pÞ:

ð4:27Þ

12 In what follows we restrict our consideration to this case. More general distribution functions of relativistic fermions are discussed, e.g., in [23,24,34].

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We would like to make one remark here about the kinematic four-momentum Pl ðXÞ ¼ pl  eAl ðXÞ that enters the above expression and other formulas. The appearance of the vector potential is due to the fact that the electron Green’s function (2.14) is not invariant under gauge transformation of the mean electromagnetic field. In principle, one could work from the beginning with the gaugeinvariant Green’s function

" Z

Gð1 2Þ ¼ Gð1 2Þ exp ie

1

# Al ðxÞdxl ;

ð4:28Þ

2

where xl ¼ ðt;~ rÞ and integration over ~ r is performed along a straight line connecting the points ~ r 2 and ~ r1 . In the leading gradient approximation, there are simple relations between the corresponding propagators and correlation functions:

G  ðX; pÞ ¼ g  ðX; p þ eAðXÞÞ;

G ? ðX; pÞ ¼ G? ðX; p þ eAðXÞÞ:

ð4:29Þ

In particular, it follows from Eq. (4.27) that the gauge invariant quasiparticle correlation functions in the collisionless approximation are given by

e ? ðX; pÞ ¼ 2pigðp0 Þdðp2  m2 Þ G



 þ m f ? ðX; pÞ;

ð4:30Þ

where

f ? ðX; pÞ ¼ F ? ðX; p þ eAðXÞÞ

ð4:31Þ

are gauge-invariant functions. As shown in [24], they are related to the on-shell distribution functions of electrons ðfe Þ and positrons ðfeþ Þ:

f < ðX; pÞjp0 ¼Ep ¼ fe ðX; ~ pÞ; >

pÞ; f ðX; pÞjp0 ¼Ep ¼ 1  fe ðX; ~

f < ðX; pÞjp0 ¼Ep ¼ 1  feþ ðX; ~ pÞ; f > ðX; pÞjp0 ¼Ep ¼ feþ ðX; ~ pÞ;

ð4:32Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Ep ¼ ~ p2 þ m2 . In working with the components of the non-invariant Green’s function Gð1 2Þ, one must be careful to distinguish between the kinematic (P) and the canonical (p) momenta in evaluating the drift terms in a kinetic equation for electrons, because the Poisson brackets contain derivatives of AðXÞ. On the other hand, for local quantities, such as the emission and absorption rates, the vector potential AðXÞ drops out from the final expressions as it must be due to gauge invariance. Formally, this can be achieved by the change of variables pi ! pi þ eAðXÞ in integrals over the electron four-momenta, or equivalently by setting AðXÞ ¼ 0 and replacing F ? ðX; pÞ ! f ? ðX; pÞ in the quasiparticle correlation functions. e Since for a weakly coupled relativistic plasma the spectral function AðX; pÞ is sharply peaked near the mass-shell, it is reasonable to assume that the spinor matrices F ? ðX; pÞ are related to the distribution functions of electrons (positrons) in just the same way as in the zero damping limit. Then, in the case of non-polarized particles, Eq. (4.23) reduces to

e ? ðX; pÞ ¼ i AðX; e pÞF ? ðX; pÞ: G

ð4:33Þ

Where appropriate, this ansatz allows one to recover the singular form (4.27) for the quasiparticle correlation functions. 4.4. Extended quasiparticle approximation for relativistic electrons e ? ðX; pÞ with the full correlation Neglecting the second (off-shell) term in Eq. (4.11), i.e. identifying G functions, we recover, in essence, the relativistic quasiparticle approximation used by Bezzerides and DuBois [24]. This approximation is sufficient to derive a particle kinetic equation in which the collision term involves electron–electron (electron–positron) scattering and Cherenkov emission/absorption of longitudinal plasma waves, but, as will be shown below, is inadequate to describe radiative processes (e.g., Compton scattering and bremsstrahlung). Therefore, the off-shell term in Eq. (4.11) has to be taken into account, at least to leading order in the field fluctuations.

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

1283

Let us first of all consider the self-energies R? ðX; pÞ appearing in Eq. (4.11). They can be calculated from the matrix self-energy Rð1 2Þ shown in Fig. 2. To leading order in the field Green’s functions, only the first diagram is retained. This gives

Rð1 2Þ ¼ ie2 dlr ð1  30 Þcr Gð1 2Þdmr0 ð2  40 Þcr Dml ð40 30 Þ: 0

?

ð4:34Þ

?

The self-energies R ð12Þ are the components R ð12Þ ¼ Rð1 2 Þ. In the local Wigner representation, we obtain

R? ðX; pÞ ¼ ie2

Z

4

d k ð2pÞ4

c^l ðkÞG? ðX; p þ kÞc^m ðkÞD7ml ðX; kÞ;

ð4:35Þ

where we have introduced the notation





c^l ðkÞ ¼ c0 ; D?ij ð~ kÞ cj :

ð4:36Þ ?

It is clear that the relation (4.11) is in fact an integral equation for G ðX; pÞ because the self-energies R? ðX; pÞ are functionals of the electron correlation functions. We recall, however, that the expression (4.35) itself is valid to first order in the field correlation functions. Therefore, within the same approxe ? ðX; pÞ. Then Eq. (4.11) reduces to imation, we may replace G? ðX; pÞ in Eq. (4.35) by G 2

e ? ðX; pÞ þ i e G? ðX; pÞ ¼ G 2

Z

4

d k 4

h þ þ ^l e ? ^m D7 ml ðkÞ g ðpÞc ðkÞ G ðp þ kÞc ðkÞg ðpÞ

ð2pÞ i l ? e ^ ^m ðkÞg  ðpÞ : þ g ðpÞc ðkÞ G ðp þ kÞc 

ð4:37Þ

Note that in non-relativistic kinetic theory an analogous decomposition of correlation functions into the quasiparticle and off-shell parts is called the ‘‘extended quasiparticle approximation” [26–28]. Expression (4.37) may thus be regarded as the extended quasiparticle approximation for relativistic electrons. We shall see later that the second term on the right-hand side of Eq. (4.37) plays an important role in calculating the photon emission and absorbtion rates. 5. Transverse polarization functions We are now in a position to perform calculations of the transverse polarization functions p? s ðX; kÞ determining the photon emission and absorption rates. The starting point is the polarization matrix Plm ð1 2Þ on the time-loop contour C, which is represented by the diagrams shown in Fig. 2. Recalling the definition (2.42) of the bare ion vertex, it is easy to verify that the last two diagrams do not contribute to Pij ð1 2Þ, so that the component p< ij ð12Þ ¼ Pij ð1þ 2 Þ is given by the diagrams shown in Fig. 3, where we have introduced the transverse bare vertex

Ci ð12; 3Þ ¼

 edð1  2ÞdTij ð1  3Þcj :

ð5:1Þ

The analogous representation for p> ij ð12Þ ¼ Pij ð1 2þ Þ is obvious. The next step is to find the contributions of the diagrams in Fig. 3 to p< s ðX; kÞ by Wigner transforming all functions and then using the diagonal principal-axis representation (3.27) for p< ij ðX; kÞ. Since the

Fig. 3. Space–time diagrams for ip< ij ð12Þ. The ± signs indicate the branches of the contour C.

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

polarization functions will be used for the calculation of the local radiating power, the X-gradient corrections to the Green functions can be omitted. Then, the contribution of the diagram in the first line of Fig. 3 to p< s ðX; kÞ is found to be

ps<ð1Þ ðX; kÞ ¼ ie2

Z

4

d p ð2pÞ

4

trD

n

o ðkÞG< ðX; pÞ ðkÞG> ðX; p  kÞ

ð5:2Þ

l with the polarization four-vectors defined as s ðX; kÞ ¼ ð0; ~ s ðX; kÞÞ. The computation of the diagrams in the second line of Fig. 3 requires a more elaborate analysis. To understand the physical meaning of different processes represented by these diagrams, it is convenient to use the canonical form (2.2) for the electron and field Green’s functions. Within the approximation where the direct coupling between transverse and longitudinal field fluctuations is neglected, each canonical component of Dlm ð1 2Þ is a block matrix: ?

Dlm ð12Þ ¼

!

D? ð12Þ

0

0

dij ð12Þ

?

;



Dlm ð12Þ ¼

D ð12Þ

0

0

dij ð12Þ



!

;

ð5:3Þ

?

where the correlation functions dij ð12Þ and D? ð12Þ characterize the degree of excitation of the field  fluctuations, D ð12Þ describes the screened Coulomb interaction, and dij ð12Þ are the photon propagators. In terms of the canonical components of Green’s functions, the second line of Fig. 3 generates ten space–time diagrams which differ from each other by the positioning of the electron correlation functions G? . In eight diagrams, both correlation functions appear at the left or right of the intermediate vertices. Examples are given in Fig. 4. Writing down explicit expressions, it can easily be verified that such diagrams give small corrections to the one-loop diagram in the first line of Fig. 3. For more details see Appendix C. In the following the contributions due to these diagrams will be neglected. The two remaining diagrams shown in Fig. 5 have an entirely different structure. They describe interactions between quantum states of incoming and outgoing particles. As we shall see later, these diagrams contribute to the rates of higher-order radiative processes (e.g., bremsstrahlung and Compton scattering). In the local Wigner form, the contribution of the diagrams in Fig. 5 to p< s ðX; kÞ is given by (the fixed argument X is omitted)

ps<ð2Þ ðkÞ ¼

Z

e4

4 0

4 Y

4

0

d pi d4 ðk  p2 þ p3 Þd4 ðk  p3 þ p4 Þd4 ðp1 þ p3  p2  p4 Þ ð2pÞ i¼1 h n o 0 0 ^k ðk0 Þg  ðp2 Þ ðkÞG> ðp3 Þc ^k ðk0 Þg  ðp4 Þ  D ðp4 Þ ; ðkÞg þ ðp1 Þc þ D>kk0 ðk ÞtrD 8

d k

ð5:4Þ

where we have used the notation (4.36). The analysis of diagrams corresponding to p>ij ð12Þ ¼ Pij ð1 2þ Þ follows along exactly the same lines and therefore will not be discussed here. The contributions to p> s ðX; kÞ are the same as Eqs. (5.2) and (5.4) with > and < signs interchanged.

Fig. 4. Corrections to the one-loop diagram for ip< ij ð12Þ. The symbol D denotes longitudinal or transverse components of Dlm ð12Þ.

Fig. 5. Diagrams for ip< ij ð12Þ contributing to higher-order radiative processes. The symbol D has the same meaning as in Fig. 4.

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Recalling Eq. (4.37), we can now express the relevant contributions to p< s ðX; kÞ in terms of the quasiparticle parts of the electron correlation functions. Since our consideration is restricted to the lowest order in the field fluctuations, the full electron correlation functions G? ðX; pÞ entering Eq. (5.4) are to e ? ðX; pÞ. Note, however, that in Eq. (5.2) one must keep first-order off-shell corrections be replaced by G to the electron correlation functions coming from the integral term in Eq. (4.37). Collecting all contri> butions to p< s ðX; kÞ, it is convenient to eliminate the field correlation functions Dlm ðX; kÞ with the help of the symmetry relation

D>lm ðX; kÞ ¼ D
ð5:5Þ

Then some algebra gives

p
Z

4

4

d p1 d p2 4

4

ð2pÞ ð2pÞ Z

4

4

d4 ðp1  p2  kÞtrD 4 0

d p1 d p2 d k 4

4

ð2pÞ ð2pÞ ð2pÞ

4

n

o e < ðp Þ ðkÞ G e > ðp Þ ðkÞ G 1 2

 0 0 0 0 < d4 p1 þ k  p2  k K kk s ðp1 ; p2 ; k; k ÞDkk0 ðk Þ;

ð5:6Þ

where 0

0

K kk s ðp1 ; p2 ; k; k Þ ¼

h 1 n 0 e < ðp Þc ^k ðk0 Þ G ^k ðk0 Þg þ ðp2 þ kÞ ðkÞ g þ ðp2 þ kÞc trD 1 2 i o 0 e < ðp Þc e > ðp Þ ^k ðk0 Þ G ^k ðk0 Þg  ðp2 þ kÞ ðkÞ G þg  ðp2 þ kÞc 1 2 1 n k 0 h þ e < ðp Þ ðkÞg þ ðp  kÞ ^ ðk Þ g ðp1  kÞ ðkÞ G þ trD c 1 1 2 i 0 o 0 e>  <  k e ^ ðk Þ G ðp2 Þ þg ðp1  kÞ ðkÞ G ðp1 Þ ðkÞg ðp1  kÞ c n o 0 e < ðp Þ ðkÞg þ ðp  kÞc e > ðp Þ ^k ðk0 Þ G ^k ðk0 Þ G ðkÞg þ ðp2 þ kÞc þ trD 1 1 2 n o 0 e < ðp Þc e > ðp Þ : ^k ðk0 Þg  ðp1  kÞ ðkÞ G ^k ðk0 Þg  ðp2 þ kÞ ðkÞ G þ trD c 1 2

ð5:7Þ

The expression (5.6) deserves some comments. Formally, it looks like an expansion in powers of e2 in vacuum electrodynamics, but, as has already been noted above, the situation is more complicated for the plasma case since medium effects must be included to all orders in e2 . These effects enter Eq. (5.6) e ? are corrected for multiple-scattering in three types. First, the quasiparticle correlation functions G effects through the spectral function (4.15). Second, the local propagators g  contain the quasiparticle damping described by the self-energies in Eq. (4.3). Finally, the correlation functions D< kk0 contain polarization effects and contributions from resonant excitations of the electromagnetic field (photons e ? in the first term of Eq. (5.6), i.e. taking and plasmons). Neglecting the collisional broadening of G these functions in the approximation (4.27), we recover the well-known expression for the transverse polarization function in a non-equilibrium QED plasma [24]. The second term in Eq. (5.6) is one of our main results. It may be worth recalling that the structure of the trace factor (5.7) in this term is closely related to the decomposition (4.11) of the electron correlation functions, which leads to the extended quasiparticle ansatz (4.37). In the next section, we will use the expression (5.6) to discuss radiative processes in relativistic plasmas. 6. Photon production: interpretation of elementary processes 6.1. Local radiating power The kinetic results (3.52) for the photon production and absorption rates imply the singular approximation (3.49) for the resonant spectral function e a s ðX; kÞ. In Appendix D, we derive a general expression for the local energy production associated with emission and absorption of resonant photons, in which the finite width of e a s ðX; kÞ is taken into account. Here we shall quote the result:

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

  X Z d4 k   oEðXÞ a s ðX; kÞ ps ðX; kÞN s ðX; kÞ ; ¼i hðk0 Þk0 e 4 oT phot ð2pÞ s

ð6:1Þ

where N s ðX; kÞ is the local photon distribution function in four-dimensional phase space defined by Eq. (3.43). The two integrals in Eq. (6.1) have a straightforward physical interpretation in terms of the radiated and absorbed power. In particular, the quantity in which we are interested here primarily is the differential radiating power associated with the photon production:

dRðX; kÞ 4

d k

¼

i 4

ð2pÞ

k0

X

e a s ðX; kÞp
ð6:2Þ

s

where k0 > 0 is implied. We observe that, for a fixed ~ k, the e a s ðX; kÞ is just the weight function that determines the emission profile. This supports the interpretation of e a s ðX; kÞ as the spectral function for propagating photons in a plasma. In the approximation (3.49), the differential radiating power can be expressed in terms of the three-momentum ~ k of the emitted photons by integrating Eq. (6.2) over x ¼ k0 :

h i dRðX; ~ kÞ i X ¼ xs ðX; ~ kÞZ s ðX; ~ kÞp
ð6:3Þ

with the quantities on the right-hand side defined earlier (see Section 3.3). Note that the expression (6.3) is consistent with the photon production rate (3.52a) derived from the kinetic equation. kÞ  xe , In most situations, the effective photon frequencies of interest satisfy the inequality xs ðX; ~ where

x2e ðXÞ ¼ 2e2

Z

3 d~ pÞ p f ðX; ~ ð2pÞ3 Ep

ð6:4Þ

is the relativistically corrected plasma frequency and f ðX; ~ pÞ  fe ðX; ~ pÞ is the electron distribution function. Then, as shown in Appendix E, a good approximation for the photon dispersion is



xðX; ~ kÞ  xs ðX; ~ kÞ ¼ ~ k2 þ x2e ðXÞ

1=2

:

ð6:5Þ

Physically, this implies local isotropy for the photon polarization states. Under the assumption that xðX; ~ kÞ  xe , Eq. (3.54) gives Z 1 ¼ 2xðX; ~ kÞ, so that Eq. (6.3) reduces to s

h i X dRðX; ~ kÞ i < ~ ~ ¼ p ðX; kÞ 1 þ n ðX; kÞ : s s 3 2ð2pÞ3 s k d~

ð6:6Þ

In the following, we shall restrict ourselves to this simple expression for the radiating power. It is not difficult, however, to generalize the results using Eq. (6.2). Since the width of the resonant spectral function is assumed to be small, even in the latter case N s ðX; kÞ can be identified with the on-shell phokÞ. ton distribution function ns ðX; ~ 6.2. Cherenkov emission in plasmas We begin with the contribution of the first term of Eq. (5.6) to the radiating power. Usually the corresponding process is referred to as Cherenkov radiation. Note, however, that the energy–momentum conserving Cherenkov emission of transverse photons in QED plasmas is kinematically forbidden [24] kj. Strictly because, for all expected densities, the transverse dispersion curve stays above the x ¼ j~ speaking, this is only true when the quasiparticle spectral function in Eq. (4.23) is approximated by the singular mass-shell expression (4.19), i.e., when medium effects are ignored. The collisional broade ening of the spectral function AðpÞ results in a statistical energy uncertainty for electrons (positrons), so that the emission of low-energy photons becomes possible.13 Assuming, for simplicity, equal prob13 A similar situation occurs in the theory of a quark–gluon plasma [35] where the thermal broadening of the quark spectral function is one of the mechanisms for the soft photon production.

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abilities for the photon polarizations and the particle spin states and using Eq. (4.33), we obtain the local radiating power in the ‘‘Cherenkov channel”14

dRðX; ~ kÞ 3~ d k

! Cher

h i Z d4 p 1 ¼ 8pe2 1 þ nðX; ~ kÞ ð2pÞ4 4



d p2 ð2pÞ4

d4 ðp1  p2  kÞCðp1 ; p2 ; kÞf < ðX; p1 Þf > ðX; p2 Þ;

ð6:7Þ

where nðX; ~ kÞ  ns ðX; ~ kÞ, k0 ¼ xð~ kÞ, and

Cðp1 ; p2 ; kÞ ¼

o 1X n e e trD ðkÞ Aðp 1 Þ ðkÞ Aðp2 Þ : 8 s

ð6:8Þ

The actual evaluation of expression (6.7) requires a knowledge of the quasiparticle spectral function e AðpÞ and a model for the distribution functions f ? ðX; pÞ in the plasma. Here, we choose a simple parametrization of the quasiparticle spectral function

 e AðpÞ ¼ SðpÞ

 þm

ð6:9Þ

with

SðpÞ ¼ h

 3 4 p0 Cp  2 i2 : 2 ðp2  m2 Þ þ p0 Cp

ð6:10Þ

This ansatz may be regarded as a generalization of Eq. (4.18) to the case of a finite but small spectral width parameter Cp  Ep . Within this approximation, the Cp is assumed to be taken at p2 ¼ m2 . Note that in the limit Cp ! 0 we recover from Eq. (6.9) the singular spectral function (4.19) (in the gaugeinvariant form). With Eq. (6.9) the trace in Eq. (6.8) is evaluated explicitly and we obtain

    ^ ^ Cðp1 ; p2 ; kÞ ¼ Sðp1 ÞSðp2 Þ p01 p02  m2  ~ p1  ~ k ~ p2  ~ k ;

ð6:11Þ

^ k ¼~ k=j~ kj. Under the conditions currently available in laser–plasma experiments, the positron where ~ contribution to the radiating power (6.7) is negligible. In that case, Sðp1 Þ and Sðp2 Þ are sharply peaked near p01 ¼ Ep1 and p02 ¼ Ep2 , so that f < ðp1 Þ and f > ðp2 Þ are expressed in terms of the electron distribution p1 Þ, f > ðp2 Þ ¼ 1  f ð~ p2 Þ. Then, since in real laboratory plasmas the subsysfunction f ð~ pÞ, i.e. f < ðp1 Þ ¼ f ð~ tem of relativistic electrons is non-degenerate, the blocking factor 1  f ð~ p2 Þ can be replaced by 1. Finally, it is clear that the dominant contribution to the integrals in Eq. (6.7) comes from jp01  Ep1 j K Cp1 and jp02  Ep2 j K Cp2 because the quasiparticle spectral function (6.9) has very small wings. The charkÞ K Cp  Ep and acteristic frequencies of emitted photons are thus expected to satisfy xð~ 1

1

xð~ kÞ K Cp2  Ep2 . This observation allows one to use for the function (6.11) the approximate expression

"

 2 # ^ ~ p1  ~ k C3p1 C3p2 p21  ~

Cðp1 ; p2 ; kÞ 

4Ep1

2 h

p01  Ep1

2

þ

1 2

Cp 1

2 i2 h

p02  Ep2

2

þ

1 2

Cp2

2 i2 :

ð6:12Þ

Here, the difference between the spectral width parameters Cp1 and Cp2 may be neglected. On the other hand, the difference between Ep1 and Ep2 in the denominator plays a crucial role. Indeed, in the limit Cp ! 0 we have Cðp1 ; p2 ; kÞ / dðp01  Ep1 Þdðp02  Ep2 Þ. Then it can easily be seen that the inte-

14 From now on Green’s functions, distribution functions, etc. are understood to be gauge-invariant, i.e., Al ðXÞ is set equal to zero as explained in Section 4.3.

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

gral in Eq. (6.7) vanishes due to the presence of the four-dimensional delta function and the dispersion relation (6.5) for photons. In the case of a finite quasiparticle spectral width, the stringent energy– momentum conservation is violated, so that, for sufficiently small k, the function Sðp1 Þ overlaps with Sðp2 Þ ¼ Sðp1  kÞ, which leads to a finite radiating power in the Cherenkov channel. Based on the above considerations, one can transform Eq. (6.7), after integration over p2 and p01 , to

dRðX; ~ kÞ 3~ d k

!

¼ Cher

e2 h

iZ 1 þ nð~ kÞ

3 d~ p

Kð~ p; ~ kÞf ð~ pÞ;

ð6:13Þ

 2  ~~ kÞ 1  E pxkð~kÞ 5C2p þ x2 ð~ p :   2 3 ~ ~ 2 2 p  k Ep Cp þ x2 ð~ kÞ 1  E xð~kÞ

ð6:14Þ

ð2pÞ3

ð2pÞ3

where



Kð~ p; ~ kÞ ¼

^ C3p ~ p ~ kÞ2 p2  ð~



p

Let us briefly mention some important properties of this function. Since in a plasma kÞÞ–0 for all possible values of ~ p and ~ k, it is clear that Kð~ p; ~ kÞ ! 0 in the limit ½1  ð~ p ~ kÞ=ðEp xð~ Cp ! 0 (more precisely, Cp =xð~ kÞ ! 0). The frequencies xð~ kÞ  Cp correspond to the hard photon emission. Although, for a finite Cp , in this region the radiating power is not zero, its contribution is kÞ  Cp ) the difficult to detect experimentally. On the other hand, in the low-frequency region (xð~ Kð~ p; ~ kÞ depends only weakly on the photon frequency and can be approximated as

  5 ^ ~ Kð~ p; ~ kÞ 2 p ~ kÞ2 ; p2  ð~ Ep Cp

kÞ  Cp Þ: ðxð~

ð6:15Þ

Finally, we note that for fixed ~ p and xð~ kÞ, the quantity Kð~ p; ~ kÞ Cp may be regarded as the angular diskÞ Cp is tribution of emitted photons. The behavior of this distribution in the frequency region xð~ illustrated in Fig. 6. Any further details of the radiating power predicted by Eq. (6.13) require explicit expressions for the electron distribution function f ðX; ~ pÞ and an estimate of the quasiparticle spectral width Cp for real experimental situations. 6.3. Interactions of electrons with transverse field fluctuations We now turn to the contribution of the second term of Eq. (5.6) to the local radiating power. According to Eqs. (5.3), the Wigner transformed field correlation functions are given by

0 Dlm ðX; kÞ ¼ @ ?

D? ðX; kÞ 0

P s

0

si ðkÞd?s ðX; kÞsj ðkÞ

1 A:

ð6:16Þ

Fig. 6. Angular distribution of emitted photons in the Cherenkov channel: (a) xð~ kÞ ¼ 0:1Cp ; (b) xð~ kÞ ¼ Cp ; (c) xð~ kÞ ¼ 10Cp . Here, h is the angle between the electron momentum ~ p and the photon momentum ~ k. The electron kinetic energy is assumed to be Ekin ¼ 10 MeV.

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

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We therefore have two different contributions to the radiating power arising from the interaction of electrons (positrons) with longitudinal and transverse fluctuations of the electromagnetic field. Let us first consider the interaction of electrons with transverse field fluctuations. As before, we shall assume equal probabilities for the particle spin states. Then, using the representation (4.33) for the quasiparticle correlation functions (in the gauge-invariant form) and the definition (4.36) of ^l ðkÞ, the corresponding contribution to the local radiating power can be written as matrices c

dRðX; ~ kÞ 3 k d~

! ¼ ipe4

Xh

1 þ ns ðX; ~ kÞ

i Z d4 p d4 p 1 2 ð2pÞ4 ð2pÞ4

s;s0

transv

4 0



d k

ð2pÞ4

 0 0 < 0 d4 p1 þ k  p2  k T ss0 ðp1 ; p2 ; k; k Þf < ðX; p1 Þf > ðX; p2 Þds0 ðX; k Þ; ð6:17Þ

where we have introduced the function

n h 0 0 T ss0 ðp1 ; p2 ; k; k Þ ¼ Re trD g þ ðp1 þ k Þ h þ trD g þ ðp1  kÞ h 0 þ2trD g þ ðp1 þ k Þ

i 0 e g þ ðp1 þ k Þ Aðp 2Þ i þ e e Aðp Aðp 1 Þ g ðp1  kÞ 2Þ io þ e e Aðp Aðp : 1 Þ g ðp1  kÞ 2Þ e Aðp 1Þ

ð6:18Þ

0

For brevity, we use the notation s ¼ s ðkÞ, s0 ¼ s0 ðk Þ. Before continuing, it is expedient to dwell briefly on the approximation in which all medium effects are completely ignored, i.e., the gaugeinvariant quasiparticle spectral function is taken in the form

 e AðpÞ ¼ 2pgðp0 Þdðp2  m2 Þ

 þm ;

ð6:19Þ

and the off-shell propagators are

1

0

g  ðp1 þ k Þ ¼

þ

m

;

g  ðp1  kÞ ¼

1 

m

:

ð6:20Þ

Then the function (6.18) reduces to 0

T ss0 ðp1 ; p2 ; k; k Þ ¼ ð2pÞ2 gðp01 Þgðp02 Þdðp21  m2 Þdðp22  m2 ÞtrD # (" 1 1 ð þ mÞ  þ  m þ m " # ) 1 1 ð þ mÞ : þ   m þ m ð0Þ

ð6:21Þ

The trace factor in this expression is just the same as in the cross-sections for Compton scattering (the Klein-Nishina formula) and for electron–positron annihilation known from vacuum QED [33]. It is important to emphasize that the cross sections of QED are correctly reproduced in many-particle Green’s function theory only if the vertex corrections in the first term of Eq. (2.47) as well as the off-shell parts of the electron correlation functions (4.11) are taken into account. Medium effects enter Eq. (6.18) through the collisional broadening of the propagators g  ðpÞ and the e quasiparticle spectral function AðpÞ. To analyze the role of these effects, let us go back to the decomposition (3.33) of the transverse field correlation functions into the resonant (photon) and off-shell 0 parts. Physically, the contribution of the resonant correlation function e d< s0 ðk Þ to the radiating power (6.17) can be ascribed to electron–positron annihilation and Compton scattering. As discussed above, the collisional effects are of importance when photon frequencies are comparable to the quasiparticle 0 0 spectral width Cp . For pair annihilation processes (p02 < 0), both k0 and jk0 j (k0 < 0) in Eq. (6.17) are much larger than Cp1 and Cp2 (provided Cp  Ep ), so that one can use approximation (6.21). On the

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302 0

other hand, the conditions k0 K Cp and (or) k0 K Cp can be satisfied for Compton scattering, and therefore the collisional broadening should be taken into account. The contribution of Compton scattering to the radiating power is determined by the resonant cor0 0 0 e< relation function e d< s0 ðX; k Þ with k0 > 0. Relations (3.41) and (3.43) can be used to express d s0 ðX; k Þ in 0 terms of the four phase–space photon distribution N s0 ðX; k Þ. Again, assuming equal probabilities for the photon polarizations and taking approximation (3.49) for the resonant spectral function, we have 0 for k0 > 0 0 e d
ip

xðX; ~ k0 Þ

  0 k0 Þ nðX; ~ k0 Þ: d k0  xðX; ~

ð6:22Þ

Then, neglecting as before the positron contribution to the radiating power and replacing the blocking factor 1  f ð~ p2 Þ by unity, we obtain from Eq. (6.17)

dRðX; ~ kÞ 3~ d k

! ¼ Comp

h

e4 3

4ð2pÞ

1 þ nð~ kÞ

iZ

3 0 d~ k

4

d p 4

ð2pÞ ð2pÞ3

0 Tðp; k; k Þf ð~ pÞnð~ k0 Þ;

ð6:23Þ

0 where k0 ¼ xð~ kÞ, k0 ¼ xð~ k0 Þ, and

0

Tðp; k; k Þ ¼

1 X 0 0 T ss0 ðp; p þ k  k; k; k Þ: xð~ k0 Þ s;s0

ð6:24Þ

For high-intensity laser–plasma experiments, an estimate of the effective temperature of the bulk plasma is T pl 100 eV [39]. Since the energies of the laser-generated electrons are in the MeV region, their collisions with ambient photons should be more correctly classified as the so-called inverse Compton scattering in which the energies of photons are increased by a factor of the order c2 , where c is the Lorentz factor of the electrons. The inverse Compton scattering was studied by many authors with a view toward astrophysical problems (see, e.g., [40–45]) and the thermal QCD [46], but without those collisional effects which are relevant if the characteristic energy of ambient photons is comparable to or smaller than the quasiparticle spectral width. The importance of the collisional broadening 0 in this case is evident from the fact that, in the limit k ! 0, the function (6.21) contains divergent terms. Note that there is another contribution to the radiating power associated with the second (off< 0 shell) part of the field correlation function ds0 ðk Þ given by Eq. (3.33). Since the transverse polarization effects are caused by current fluctuations, this contribution may be interpreted physically as coming from the scattering of electrons by the current fluctuations in the plasma. It should be remembered that the formula (6.17), as such, is valid to first order terms in the field correlation < 0 functions. Therefore, in calculating the off-shell part of ds0 ðk Þ, we have to retain only the first term in the transverse polarization function (5.6). As before, we neglect the positron contribution and assume equal probabilities for the electron spin states. Then, for a non-degenerate electron subsystem,

p
Z

4

d p ð2pÞ

4

trD

n

o e e  kÞ f ðX; ~ ðkÞ AðpÞ ðkÞ Aðp pÞ:

ð6:25Þ

Using this expression to calculate the off-shell term in Eq. (3.33), we obtain the following result for the local radiating power associated with the electron scattering by the current fluctuations:

dRðX; ~ kÞ 3 k d~

! ¼ curr: fl:

h

e6 2ð2pÞ

3

iZ 1 þ nð~ kÞ

4

4

d p0

d p 4

ð2pÞ ð2pÞ4

T ðp; p0 ; kÞf ð~ pÞf ð~ p0 Þ;

ð6:26Þ

where

T ðp; p0 ; kÞ ¼

X Z d 4 k0 4

s;s0

ð2pÞ

 þ 0 2 n 0 0 T ss0 ðp; p þ k  k; k; k ÞRe ds0 ðk Þ trD

e 0Þ ðk Þ Aðp 0

o 0 e 0 0 ðk Þ Aðp kÞ : ð6:27Þ

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

1291

0

To calculate the functions Tðp; k; k Þ and T ðp; p0 ; kÞ which determine the scattering probabilities in Eqs. e (6.23) and (6.26), one has to specify the quasiparticle spectral function AðpÞ. For instance, such calculations can be performed using the parametrization (6.9) or some improved expressions based on a more detailed analysis of the relativistic electron self-energies R ðpÞ in plasmas. We leave, however, this special problem, as well as the discussion of Eqs. (6.23) and (6.26) for realistic electron distributions, to future work. 6.4. Interactions of electrons with longitudinal field fluctuations The analysis of the contribution to the radiating power coming from the interaction of electrons (positrons) with longitudinal field fluctuations proceeds exactly in parallel with that in the previous subsection. In the case of equal probabilities for the particle spin states, we now have instead of Eq. (6.17)

dRðX; ~ kÞ 3 k d~

! ¼ ipe4 longit

i Z d4 p d4 p Xh 1 2 1 þ ns ðX; ~ kÞ 4 4 ð2 p Þ ð2 p Þ s 4 0



d k

4

ð2pÞ

 0 0 0 d4 p1 þ k  p2  k Ls ðp1 ; p2 ; k; k Þf < ðX; p1 Þf > ðX; p2 ÞD< ðX; k Þ;

ð6:28Þ

where

n h i 0 0 0 0 þ e e Aðp Ls ðp1 ; p2 ; k; k Þ ¼ Re trD g þ ðp1 þ k Þc0 Aðp 1 Þc g ðp1 þ k Þ 2Þ h i þ 0e e þ trD c0 g þ ðp1  kÞ Aðp 1 Þ g ðp1  kÞc Aðp2 Þ h io 0 þ 0e e þ2trD g þ ðp1 þ k Þc0 Aðp : 1 Þ g ðp1  kÞc Aðp2 Þ

ð6:29Þ

In the simplest approximation where all medium effects are omitted, this function reduces to 0

2 0 0 2 2 2 2 Lð0Þ s ðp1 ; p2 ; k; k Þ ¼ ð2pÞ gðp1 Þgðp2 Þdðp1  m Þdðp2  m ÞtrD # (" 1 1 0 0  c þ c ð þ mÞ  m þ m " # ) 1 1 0 0 þ c ð þ mÞ :  c  m þ m

ð6:30Þ

Here, the trace factor is identical to the one in vacuum QED cross-sections for bremsstrahlung emission of photons (the Bethe-Heitler formula) and for the one-photon electron–positron annihilation [33]. Once again we see that the results of vacuum QED are correctly reproduced within the extended quasiparticle approximation formulated in Section 4.4. Neglecting the processes involving positrons and assuming equal probabilities for the photon polarizations, one obtains from Eq. (6.28) the expression where only the contribution of electron scattering by the longitudinal field fluctuations is taken into account:

dRðX; ~ kÞ 3~ d k

! ¼ longit

ie

4

h

2ð2pÞ3

iZ 1 þ nð~ kÞ

4

d p

4 0

d k

ð2pÞ4 ð2pÞ4

0

0

Lðp; k; k Þf ð~ pÞD< ðk Þ:

ð6:31Þ

Here, we have defined 0

Lðp; k; k Þ ¼

X

0

0

Ls ðp; p þ k  k; k; k Þ:

ð6:32Þ

s

Note that the field correlation function D< ðX; kÞ is closely related to charge fluctuations in a plasma. To show this, we use relation (2.11) which, in the local form, yields

^ ^ ð1Þ; r21 D/ð1Þ ¼ D.

ð6:33Þ

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^ ð1Þ  h. ^ ð1Þi is the operator for fluctuations of the induced charge density. Now, ^ ð1Þ ¼ . where D. recalling definition (2.13) of the longitudinal field Green’s function leads to

^ ð2ÞD. ^ ð1Þi: r21 r22 D< ð12Þ ¼ ihD.

ð6:34Þ

Finally, performing the Wigner transformation on both sides of this equation and neglecting the X-gradients, we find that

D< ðX; kÞ ¼ i

SðX; kÞ ; j~ kj4

ð6:35Þ

where

SðX; kÞ ¼

Z

4

^ ðX  x=2ÞD. ^ ðX þ x=2Þi d x eikx hD.

ð6:36Þ

is the local dynamic structure factor. There are several scattering processes that can contribute to Eq. (6.31) depending on the structure of the longitudinal field correlation function D< ðX; kÞ in different regions of the k space. Before discussing this point we will consider some important properties of this function which can be derived from the plasmon transport equation (F.7) given in Appendix F. If the space–time variations of the drift terms on the left-hand side of Eq. (F.7) are so slow that these terms are small compared with the emission/absorption terms on the right-hand side, a good approximation for the field correlation functions is obtained by neglecting the drift terms. We arrive at the local balance equation

P> ðX; kÞD< ðX; kÞ ¼ P< ðX; kÞD> ðX; kÞ;

ð6:37Þ

whence follows the well-known local equilibrium or the adiabatic approximation for the longitudinal field fluctuations [25]:

 þ 2 ?   D? ad ðX; kÞ ¼ P ðX; kÞ D ðX; kÞ :

ð6:38Þ

The adiabatic approximation is inapplicable in the presence of unstable plasma waves where the resonance approximation seems to be appropriate [47]. To include the resonance portion of the spectrum of plasma modes, Bezzerides and DuBois [24] proposed the following ansatz for the field correlation functions:

 2 D? ðX; kÞ ¼ DD? ðX; kÞ þ P? ðX; kÞDþ ðX; kÞ ; ?

ð6:39Þ

?

where DD ðX; kÞ is the contribution to D ðX; kÞ in excess of the local equilibrium value. In general, kÞ which are to be found as solutions there is a number of resonances with frequencies k0 ¼ xa ðX; ~ of the equation

~ k2  Re Pþ ðX; kÞ ¼ 0;

ð6:40Þ

where Pþ ðX; kÞ is the retarded component of the longitudinal polarization function. Assuming sharp resonances, it seems reasonable to approximate DD? ðX; kÞ by a sum of the delta functions corresponding to different unstable modes [24]. It should be noted, however, that the physical meaning of the term DDðX; kÞ is not quite clear. In particular, expression (6.39) leads to the kinetic equation for the occupation numbers of unstable modes which differs from the familiar plasmon kinetic equation in plasma turbulence theory [25,48]. Our above analysis of the photon kinetics and the formal similarity between the photon transport Eq. (3.28a) and the plasmon transport equation, Eq. (F.7), suggest another representation for the correlation functions D? ðX; kÞ. The same line of reasoning as in Section 3 leads to the following decomposition of D? ðX; kÞ into ‘‘resonant” and ‘‘off-shell” parts:

h i e ? ðX; kÞ þ P? ðX; kÞRe Dþ ðX; kÞ 2 ; D? ðX; kÞ ¼ D

ð6:41Þ

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1293

where the first term represents the resonant contribution from unstable plasma modes. Formally, Eqs. (6.41) and (6.39) are of course equivalent as long as no approximations are invoked. The exact relation e ? can easily be found: between DD? and D

 2 2P? Im Pþ ? e DD ¼ D     ; 2  2 2 ~ k2  Re Pþ þ Im Pþ ?

ð6:42Þ

e ? differ essentially from where we have used the expression (F.6) for Dþ ðX; kÞ. It is seen that DD? and D each other just in the vicinity of the strong resonances determined by Eq. (6.40). This means that Eqs. (6.41) and (6.39) give different prescriptions for picking out the contribution of the resonant plasma modes, i.e., they lead to somewhat different pictures of collisions involving plasmons. Physically, the use of the decomposition (6.41) rather than (6.39) is preferable since the former is closely related to the structure of the drift terms in the plasmon transport equation. In the off-resonant region we e ? 0 and DG? 0, so that both decompositions reduce to the adiabatic expression (6.38). have G By analogy with transverse photons, it is convenient to introduce the spectral function for the resonant plasma modes,

  e > ðX; kÞ  D e < ðX; kÞ ; e a L ðX; kÞ ¼ i D

ð6:43Þ

and the plasmon distribution functions N ðX; kÞ in the four-dimensional k space through the relations (cf. Eqs. (3.41)) ?

? e ? ðX; kÞ ¼ i e D a L ðX; kÞN ðX; kÞ;

>

<

N  N ¼ 1:

ð6:44Þ

Then it is easy to verify that

 3 4 Im Pþ e a L ðX; kÞ ¼    : 2  2 2 ~ k2  Re Pþ þ Im Pþ

ð6:45Þ

As in the case of transverse photons, the plasmon spectral function is non-Lorentzian. Let us turn back to the expression (6.31) for the local radiating power. Substituting here the field correlation function from Eq. (6.41), we obtain two terms which can be interpreted physically. The e ? represents the contribution of the Compton effect on plasmons, which may term arising from D be regarded as a conversion of longitudinal plasma waves into transverse electromagnetic waves. This relativistic effect was first studied by Galaitis and Tsytovich [49] (see also [50]) and has been discussed as a frequency boosting mechanism for laboratory beam-laser experiments [51,52]. The maximum frequency boost is xmax =xe c2 , where xe is the electron plasma frequency (6.4) and c is the Lorentz factor of the beam electrons. In laser–plasma systems, the Compton effect on plasma waves may thus be relevant when the energies of relativistic electrons reach the GeV region. Note that the transition probability determined by Eqs. (6.29) and (6.32) is very sensitive to the collisional broadening of the electron quasiparticle spectral function and the electron propagators. This feature of Compton conversion in plasmas deserves further investigation. To gain some insight into radiative processes associated with the second term in Eq. (6.41), we consider the longitudinal component Pð1 2Þ  P00 ð1 2Þ of the polarization matrix given by the diagrams in Fig. 2. The leading contributions to Pð1 2Þ come from those one-loop diagrams in which the full electron and ion Green’s functions are to be replaced by their quasiparticle parts. Then, in the local Wigner representation, we obtain

P< ðX; kÞ ¼ P
ð6:46Þ

with the electron and ion polarization functions

P
Z

4

d p 4

n o e < ðX; pÞc0 G e > ðX; p  kÞ ; trD c0 G

ð2pÞ Z 4 n o X d p e < ðX; pÞ G e > ðX; p  kÞ : P
ð6:47Þ ð6:48Þ

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As shown above, the Green functions can be expressed in terms of the quasiparticle spectral functions and the distribution functions, but we will not write down the corresponding obvious formulas here. The term P< el ðX; kÞ in the longitudinal polarization function (6.46) yields the contribution to the radiating power (6.31) from relativistic electron scattering on off-resonant fluctuations of the electron charge in the plasma. This effect is analogous to electron scattering on the current fluctuations discussed in the previous subsection. In the relativistic case, both effects are important while the former dominates in the non-relativistic limit. The ion term P< ion ðX; kÞ is associated with bremsstrahlung processes. As already stressed, the cross section for these processes reduces to the Bethe-Heitler crosssection in vacuum QED if all medium corrections are removed. In this section, we have considered the emission of photons. The analysis of absorption processes follows along exactly the same lines by using Eqs. (5.6) and (6.1) together with the symmetry relation p>s ðX; kÞ ¼ p
V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

1295

Appendix A. Energy flux of radiation field We start with the energy density operator of the radiation field in the Coulomb gauge and Heaviside’s units. Using the notation x ¼ ðt;~ rÞ for space–time points, we have [57]

 2 1 ~ ~b b b ðxÞ ¼ 1 ~ E rA ; P2 þ 2 2

ðA:1Þ

b b b where the transverse field operators ~ AðxÞ and ~ PðxÞ ¼ o~ AðxÞ=ot satisfy the canonical equal-time commutation relations

h

b i ðt;~ b j ðt;~ A r1 Þ; P r2 Þ

i 

¼ idTij ð~ r 2 Þ: r1  ~

ðA:2Þ

b jðxÞ, is defined through the equation The energy flux operator for the radiation field, ~

h i ^ b EM ðtÞ ¼ r ~ ~jðxÞ b ðxÞ; H i E

ðA:3Þ



with the Hamiltonian of the electromagnetic field

b EM ðtÞ ¼ H

Z

b ðxÞd3~ E r:

ðA:4Þ

Using Eq. (A.2) to compute the commutator in Eq. (A.3), we obtain

h i ^ji ðxÞ ¼ 1 imn b b n ðxÞ ; E m ðxÞ; B þ 2

ðA:5Þ

where ijk is the completely antisymmetric unit tensor (123 ¼ 1), and ½:; :þ stands for the anticommub tator. We have also introduced the transverse electric field operator ~ EðxÞ and the magnetic field operb ~ ator BðxÞ:

b o~ AðxÞ b ~ EðxÞ ¼  ; ot

b b ~ ~ ~ AðxÞ: BðxÞ ¼ r

ðA:6Þ

^ jðxÞi, are represented conveniently in the form The components of the average energy flux, ~ jðxÞ ¼ h~

D E  i 1 i b n ðxÞ ; E m ðxÞ; D B j ðxÞ ¼ ~ EðxÞ  ~ BðxÞ þ imn ½D b þ 2

ðA:7Þ

b b ~ E ¼~ E  E and where ~ E and ~ B are, respectively, the mean electric and magnetic fields, while D~ b ~b ~ ~ D B ¼ B  B are the operators of the field fluctuations. Physically, the first term in Eq. (A.7) is the energy flux associated with electromagnetic waves. The second term is the photon contribution which < can be expressed in terms of the correlation functions dij ðX; kÞ. With the aid of Eqs. (A.6) the photon l ~ energy flux at a space–time point X ¼ ðT; RÞ is transformed to

   Z 4 i imn njl d k 1 o 1 o 0 j <   þ ik þ ik dlm ðX; kÞ 2 2 oT 2 oRj ð2pÞ4     1 o 1 o 0 j < þ d  ik ðX; kÞ :  ik ml 2 oT 2 oRj

i

jphot ðXÞ ¼ 

ðA:8Þ

<

This expression can be simplified by using conditions (3.14) for dij ðX; kÞ and the identity 0 0

ijk i j k ¼ dii djj 0

0

 dij0 dji0 :

After some algebra which we omit, we find up to first-order X-gradients i

jphot ðXÞ ¼ i <

Z

where tr d ðX; kÞ ¼

4

d k ð2pÞ4 P

0 i

<

k k tr d ðX; kÞ þ

< j djj ðX; kÞ,

and

o oRj

M ij ðXÞ;

ðA:9Þ

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

M ij ðXÞ ¼

i 2

Z

4

d k ð2pÞ

0

4

k

n o < < dij ðX; kÞ  dji ðX; kÞ :

ðA:10Þ <

Strictly speaking, Eq. (A.9) contains the full tensor dij ðX; kÞ, and not just its transverse part with respect to ~ k. However, as follows directly from Eq. (3.17), we have up to first-order X-gradients

tr TðX; kÞ ¼ tr T ? ðX; kÞ

ðA:11Þ < dij ðX; kÞ

for any T ij ðX; kÞ which satisfies conditions (3.14). The full tensor in Eq. (A.9) can thus be replaced by its transverse part. Note that, in the slow variation case, the last term in Eq. (A.9) is very small compared to the first term. Moreover, it is easy to verify that Mij ¼ 0 in the diagonal princi< pal-axis representation (3.27) for the transverse correlation functions dij ðX; kÞ. We thus obtain the expression i

jphot ðXÞ ¼ i

XZ s

4

d k

0 i <

ð2pÞ4

k k ds ðX; kÞ

ðA:12Þ

which is valid up to first-order X-gradients. Appendix B. Decomposition of electron correlation functions We will show how the decomposition (4.11) follows from transport equations for the electron correlation functions. To a degree our consideration is similar to that given in Section 3.2 for photons. The starting point are the Wigner transformed KB Eqs. (4.10). Keeping only first-order terms in Xgradients, we obtain the set of equations (arguments X and p are omitted for brevity)

  1 n þ 1 ? o 1  ?   R ; G ¼ i R? G  ðg þ Þ1 G? ; ðg Þ ; G  2 2   1 n ?  1 o 1  þ ?   G ; ðg Þ G ; R ¼ i Gþ R?  G? ðg  Þ1 ; 2 2

ðB:1aÞ ðB:1bÞ

where g  ðX; pÞ are the local propagators (4.3). Note that, generally speaking, on the right-hand sides of these equations the full propagators G ðX; pÞ cannot be replaced by the local ones since Eq. (4.4) contain the gradient corrections. The transport equations for G? ðX; pÞ are derived by taking the difference of Eqs. (B.1). With expressions (4.3) for the local propagators and the relations

G>  G< ¼ Gþ  G ;

R>  R< ¼ Rþ  R ;

ðB:2Þ

a simple algebra gives

1 n þ 1 ? o n ?  1 o 1  þ ?   ?   ðg Þ ; G  G ; ðg Þ g ;R  R ;g þ 2 2 h i

 i  m  r; G? þ ½R> ; G< þ  ½R< ; G> þ ; ¼  g; R?    2

ðB:3Þ

where ½A; B ¼ AB BA is the commutator/anticommutator of spinor matrices, and

g¼

1 þ G þ G ; 2

r¼

1 þ R þ R : 2

ðB:4Þ

With regard to its spinor structure, Eq. (B.3) is a very complicated 4  4 matrix equation. In principle, upon multiplying both sides of this equation by the Lorentz invariants I; cl ; c5 ; c5 cl ; rlm , and then taking the trace, one obtains coupled transport equations for the scalar, vector, pseudo-scalar, axial-vector, and tensor components of the correlation functions. It suffices for our purpose, however, to consider only one of these transport equations, which is obtained from Eq. (B.3) by taking the trace of both sides. A useful device for manipulating the traces of the drift terms are the matrix identities which follow directly from the definition (3.8) of the Poisson bracket:

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

n o trfA; Bg ¼ tr A1 ; ABA :

trfA; Bg ¼ trfB; Ag;

1297

ðB:5Þ

With the aid of these identities the traces of the drift terms in Eq. (B.3) are rearranged as

n o 1 n þ 1 ? o n ?  1 o  m  r; G ? ; trD ðg Þ ; G  G ; ðg Þ ¼ trD 2   1  þ ?   ?   1 trD g ; R  R ; g ¼ trD g þ þ g  ; R? 2 2 o n o 1 n ¼  trD ðg þ Þ1 ; g þ R? g þ þ ðg  Þ1 ; g  R? g  : 2 Then the trace of Eq. (B.3) can be written in the form    1 1  trD  m  r;G?  g þ R? g þ þ g  R? g  þ trD DR;g þ R? g þ  g  R? g  ¼ itrD ðR> G<  R< G> Þ; 2 4

ðB:6Þ þ



where DR ¼ R  R . For a weakly coupled plasma, the first term on the left-hand side of Eq. (B.6) e ? in Eq. dominates. It is seen that this term contains just that part of G? which is denoted as G e ? may be identified as the quasiparticle parts of (4.11). By analogy with photons, the quantities G the electron correlation functions, whereas the additional term in Eq. (4.11) represents the off-shell e ? discussed in Section 4. It is also parts. This interpretation is confirmed by the spectral properties of G important to observe here that the off-shell parts do not contribute to the collision term on the righte ?: hand of Eq. (B.6) and hence we get a transport equation which involves only G

trD

n

o     e ? þ 1 trD DR; g þ R? g þ  g  R? g  ¼ itrD R> G e <  R< G e> :  m  r; G 4

ðB:7Þ

This equation is analogous to the transport Eq. (3.34a) for resonant photons. Appendix C. Two-loop contributions to polarization functions Here we examine the contributions to polarization functions, which are generated by the second (two-loop) diagram for the polarization matrix Plm ð1 2Þ in Fig. 2. For convenience, we assign indices to the electron Green’s functions as shown in Fig. C.1. Then, recalling the expression (2.38) for the bare four-vertex, we have





0

Plm ð1 2Þ2-loop ¼ e4 Dkk ð60 30 ÞtrD dlr1 ð10  1Þcr1 G1 ð10 20 Þdk0 r2 ð20  30 Þcr2 G2 ð20 40 Þ dmr3 ð40  2Þ   cr3 G3 ð40 50 Þdkr4 ð50  60 Þcr4 G4 ð50 10 Þ :

ðC:1Þ

<

For the polarization functions Plm ð12Þ ¼ Plm ð1þ 2 Þ, this formula generates several terms which in0 volve different canonical components of Dkk and Gi . Let P < lm ð12Þ be one of these terms with some 0 space–time functions Dkk ð12Þ and G i ð12Þ. In the local Wigner form, we obtain (the fixed argument X is omitted for brevity)

P
Z

e4 ð2pÞ

8

4 0

d k

4 Y

4

0

0

0

d pi d4 ðk  p2 þ p3 Þ d4 ðk  p3 þ p4 Þ Dkk ðk Þd4 ðp1 þ p3  p2  p4 Þ

i¼1

  ^l ðkÞG 1 ðp1 Þc ^k0 ðk0 ÞG 2 ðp2 Þc ^m ðkÞ G 3 ðp3 Þ c ^k ðk0 ÞG 4 ðp4 Þ ;  trD c

Fig. C.1. Two-loop diagrams for iPlm ð1 2Þ.

ðC:2Þ

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

where we have used the notation (4.36). Let us now compare Eq. (C.2) with the one-loop contribution to P< lm ðX; kÞ that corresponds to the first diagram for Plm ð1 2Þ in Fig. 2:

 

P
1loop

¼ i

Z

e2 ð2pÞ

4

  4 4 ^l ðkÞ G< ðp1 Þ c ^m ðkÞG> ðp2 Þ : d p1 d p2 d4 ðk þ p2  p1 Þ  trD c

ðC:3Þ

A simple analysis based on Eq. (C.1) shows that each term (C.2) involves at least one electron correlation function G< (G 1 ðp1 Þ or G 2 ðp2 Þ), and at least one correlation function G> (G 3 ðp3 Þ or G 4 ðp4 Þ). As discussed in Section 4, the leading approximation for the two-loop contributions to the polarization e ? ðpÞ which are sharply peaked functions is obtained by replacing G? ðpÞ by their quasiparticle parts G about the mass-shell. It is clear that, due to the four-dimensional delta functions, the terms (C.2) with e > ðp Þ and G e < ðp Þ G e > ðp Þ correspond to the same scattering process as the one-loop e < ðp Þ G products G 1 4 2 3 ? e ? . For weakly coupled plasmas, such two-loop corrections term (C.3) in which G are replaced by G 15 e < ðp Þ G e > ðp Þ and G e < ðp Þ G e > ðp Þ describe may be neglected. On the other hand, the terms (C.2) with G 1 3 2 4 new scattering processes and therefore must be retained. The corresponding diagrams for the transverse polarization functions are shown in Fig. 5. Appendix D. Electromagnetic energy production In the notation of Appendix A, the average electromagnetic energy production in a unit volume due to the interaction with matter can be written as

Dh i E oEðxÞ b ðxÞ; Hint ðtÞ ; ¼ i E  ot

ðD:1Þ

b ðxÞi and where EðxÞ ¼ h E

Hint ðtÞ ¼ 

Z

b b 3 ~ d~ r AðxÞ  ~ J T ðxÞ

is the interaction Hamiltonian. Using the relations (A.2) to calculate the commutator in Eq. (D.1) gives

oEðxÞ ¼ ot

  b b ~ PðxÞ  ~ J T ðxÞ :

ðD:2Þ

We now identify x with the space–time point X ¼ ðT; ~ RÞ in the kinetic picture and rewrite Eq. (D.2) as

  oEðXÞ oEðXÞ ; ¼ ~ EðXÞ  ~ J T ðXÞ þ oT oT phot

ðD:3Þ

where ~ EðXÞ ¼ o~ AðXÞ=oT is the mean electric field, and ~ JT ðXÞ is the mean transverse current density. The second term in Eq. (D.3) is associated with photons:

  Z oEðXÞ 1 4 4 d x1 d x2 d4 ðx1  XÞd4 ðx2  XÞ ¼ oT phot 2 E D Ei o hD bT b i ð2Þ þ D A b i ð2ÞDbJ T ð1Þ ; D J i ð1ÞD A  i ot2

ðD:4Þ

b i ð2Þ ¼ A b i ðx2 Þ  Ai ðx2 Þ. The correlation functions in the above forwhere DbJ Ti ð1Þ ¼ bJ Ti ðx1 Þ  J Ti ðx1 Þ and D A mula can be written in terms of the field Green’s functions and the polarization functions by noting that

oE 1 D n bT dJ Ti ð1Þ b Ij ð2Þ ¼ i ðextÞj ¼ iPij0 ð1 10 Þ Dj0 j ð10 2Þ T C SD J Ii ð1Þ D A hSi ð2Þ dJ as follows directly from Eqs. (2.5) and (2.6). In the physical limit, one obtains

15

Examples of the corresponding diagrams for the transverse polarization functions are given in Fig. 4.

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V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

D

E

h i b i ð2Þ ¼ i p> ð110 Þd ð10 2Þ þ pþ ð110 Þd> ð10 2Þ ; DbJ Ti ð1ÞD A ji ji ij ij D E h i b i ð2ÞDbJ T ð1Þ ¼ i pþ ð110 Þd< ð10 2Þ þ p< ð110 Þd ð10 2Þ : DA ji ji i ij ij After inserting these expressions into Eq. (D.4), we use the rule (3.7) to express all functions in terms of their Wigner transforms, and then go over to the principal-axis representation (3.27). These manipulations give (on the right-hand side the X-dependence is not shown explicitly)

    Z 4

þ  >   oEðXÞ i X d k 1 o ¼ ps ðkÞ ds ðkÞ þ ds ðkÞ þ p
7

ds ðX; kÞ ¼ ds ðX; kÞ;  ds ðX; kÞ

¼

ds ðX; kÞ;

p?s ðX; kÞ ¼ p7s ðX; kÞ; ps ðX; kÞ ¼ p s ðX; kÞ;

we get the expression in which the integration is over k0 > 0:

  X Z d4 k

 oEðXÞ < > ¼ hðk0 Þk0 p>s ðX; kÞds ðX; kÞ  p
Recalling the decomposition (3.33), we see that here the field correlation functions ds ðX; kÞ can be replaced by their resonant parts e d? s ðX; kÞ. Then, using the relations (3.41) and (3.43) leads to Eq. (6.1). Appendix E. Photon dispersion The dispersion relation for photons follows from Eq. (3.48). To solve this equation, one needs an  explicit expression for Re pþ s ðX; kÞ. We start with the space–time functions pij ð12Þ which can be written in terms of the transverse polarization matrix Pij ð1 2Þ. Recalling the canonical notation (2.2) for functions on the contour C, we have

pij ð12Þ ¼ Pij ð1 2 Þ Pij ð1þ 2 Þ:

ðE:1Þ

Here, it is sufficient to retain only the contribution of the one-loop diagram (see Fig. 2) that dominates in the case of a weakly coupled plasma. Then the functions (E.1) are given by the space–time diagrams

pij ð12Þ ¼

:

ðE:2Þ

Going over to the local Wigner representation and using the first of Eq. (3.24), we obtain 2 Re pþij ðX; kÞ ¼ ie D?ii0 ð~ kÞD?jj0 ð~ kÞ

Z

4

d p ð2pÞ

4

n 0 o 0 trD ci ½gðX; p þ kÞ þ gðX; p  kÞcj G< ðX; pÞ ;

ðE:3Þ

~ where D? ii0 ðkÞ is the transverse projector (3.16), and

gðX; pÞ ¼

1 þ ½g ðX; pÞ þ g  ðX; pÞ: 2

Note that in Eq. (E.3) the off-shell part of G< ðX; pÞ must be dropped since it gives the contribution of the same order as the two-loop diagram in Fig. 2, which has been neglected. In other words, the full

1300

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

e < ðX; pÞ. To the leading approximation for a function G< ðX; pÞ is to be replaced by its quasiparticle part G  e < ðX; pÞ may be weakly coupled plasma, the local propagators g ðX; pÞ and the correlation functions G taken in the collisionless form, Eqs. (4.16) and (4.27). Then, by changing the integration variable p into p þ eAðXÞ, the polarization tensor (E.3) is expressed in terms of the gauge-invariant distribution functions (4.31). After some algebra, we have

Re pþij ðX; kÞ ¼ 8pe2 D?ij ð~ kÞ

Z

4

d p ð2pÞ4

" 1þ

# 4  k } gðp0 Þd p2  m2 f < ðX; pÞ  4 ðp  kÞ2  k4 =4

Z

d p

2  8pe2 k D?ii0 ð~ kÞD?jj0 ð~ kÞ

4

} 4

4

2

ð2pÞ ðp  kÞ  k =4

 pi0 pj0 gðp0 Þd p2  m2 f < ðX; pÞ;

ðE:4Þ

where } denotes the principal values of integrals. Two comments are relevant concerning the above expression. First, it should be noted that Re pþ ij ðX; kÞ contains a divergent vacuum term which is to be eliminated by applying the procedure of vacuum QED (for a discussion of this point see Bezzerides and DuBois [24]). In what follows we ignore this vacuum term, assuming the physical mass and charge throughout. Second, for experimental conditions available at present, the positron contribution to f < ðX; pÞ may be neglected in calculating Re pþ ij ðX; kÞ. Therefore, in Eq. (E.4) we shall make the replacement

 < d p0  Ep gðp Þd p  m f ðX; pÞ ¼ f ðX; ~ pÞ; 2Ep 0



2

2

ðE:5Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pÞ  fe ðX; ~ where Ep ¼ j~ pj2 þ m2 , and f ðX; ~ pÞ is the electron distribution function. The next step is to find eigenvectors and eigenvalues of the polarization tensor (E.4). In general, this is a rather complicated problem which requires a knowledge of the non-equilibrium particle distribution functions. Note, however, that our main interest is in the region of sufficiently high frequencies x ¼ jk0 j where the dispersion curve for transverse photons is close to the vacuum limit x ¼ j~ kj. Since 2 k 2 is small compared to x2 , a simple and reasonable approximain this region the quantity k ¼ x2  ~ 2 tion for the polarization tensor can be obtained from Eq. (E.4) by setting k ¼ 0. Taking also Eq. (E.5) into account, we obtain a locally isotropic tensor

Re pþij ðX; kÞ ¼ x2e ðXÞ dij 

ki kj j~ kj2

! ðE:6Þ

2 with xe ðXÞ defined in Eq. (6.4). In this case, Re pþ s ðX; kÞ ¼ xe ðXÞ, so that the dispersion Eq. (3.48) re2 duces to k  x2e ðXÞ ¼ 0, whence follows the expression (6.5) for the effective photon frequencies. If x2  x2e , the anisotropic corrections to the polarization tensor (E.4) are relatively small. The leading anisotropic contribution comes from the last term and is given by

  Re pþij ðX; kÞ

anisotr

2 ¼ 4pe2 k D?ii0 ð~ kÞD?jj0 ð~ kÞ

Z

4

d p

}

ð2pÞ4 ðp  kÞ

p 0p 0 2 i j

 d p0  Ep f ðX; ~ pÞ; Ep

ðE:7Þ

where we have used Eq. (E.5). Appendix F. Longitudinal field correlation functions We start with the equation of motion for the longitudinal field Green’s function Dð1 2Þ  D00 ð1 2Þ on the time-loop contour C. Recalling Eqs. (2.24) and (2.49), we have

r21 Dð1 2Þ ¼ dð1  2Þ þ Pð1 10 ÞDð10 2Þ:

ðF:1Þ

The adjoint of this equation reads

r22 Dð1 2Þ ¼ dð1  2Þ þ Dð1 10 ÞPð10 2Þ:

ðF:2Þ

Then, using the canonical form (2.2) of Dð1 2Þ and Pð1 2Þ, it is an easy matter to derive the equations for the retarded and advanced longitudinal ‘‘propagators”

V.G. Morozov, G. Röpke / Annals of Physics 324 (2009) 1261–1302

 r21 D ð12Þ ¼ dð1  2Þ þ P ð110 ÞD ð10 2Þ; r

2  2 D ð12Þ



0



1301

ðF:3aÞ

0

¼ dð1  2Þ þ D ð11 ÞP ð1 2Þ;

ðF:3bÞ

and the KB equations for the space–time correlation functions

 r21 D? ð12Þ ¼ Pþ ð110 ÞD? ð10 2Þ þ P? ð110 ÞD ð10 2Þ; r

2 ? 2 D ð12Þ

?

0



0

þ

0

?

0

¼ D ð11 ÞP ð1 2Þ þ D ð11 ÞP ð1 2Þ:

ðF:4aÞ ðF:4bÞ

The analysis of the above equations proceeds exactly in parallel with that for the transverse field fluctuations. By going over to the Wigner representation (3.6) and keeping only first-order terms in the Xgradients, the sum and difference of Eqs. (F.3) become

  ~ k2  P ðX; kÞ D ðX; kÞ ¼ 1;

n o ~ k2  P ðX; kÞ; D ðX; kÞ ¼ 0;

ðF:5Þ

whence

D ðX; kÞ ¼

1 : ~ 2 k  P ðX; kÞ

ðF:6Þ

In the Wigner representation, the KB Eqs. (F.4) are manipulated to

n o   ~ k2  Re Pþ ; D? þ Re Dþ ; P? ¼ iðP> D<  P< D> Þ;        Im Pþ ; D? þ Im Dþ ; P? ¼ 2 ~ k2  Re Pþ D?  jDþ j2 P? ;

ðF:7Þ ðF:8Þ

where the arguments X and k are omitted for brevity. Equation (F.7) is derived by taking the difference of Eqs. (F.4) and may be regarded as the transport equation for longitudinal field fluctuations. It is quite similar in structure to the plasmon transport equation in non-relativistic plasmas [25]. Eq. (F.8) is analogous to the mass-shell Eq. (3.28b) for transverse field fluctuations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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