Extended quasiparticle approximation for relativistic electrons in plasmas

Starting with Dyson equations for the path-ordered Green’s function, it is shown that the correlation functions for relativistic electrons (positrons) in a weakly coupled non-equilibrium plasmas can be decomposed into sharply peaked quasiparticle parts and off-shell parts in a rather general form....

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2006
Автори:	Morozov, V.G., Ropke, G.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2006
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/121373
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Extended quasiparticle approximation for relativistic electrons in plasmas / V.G. Morozov, G. Ropke // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 473–484. — Бібліогр.: 14 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-121373
record_format	dspace
spelling	irk-123456789-1213732017-06-15T03:03:06Z Extended quasiparticle approximation for relativistic electrons in plasmas Morozov, V.G. Ropke, G. Starting with Dyson equations for the path-ordered Green’s function, it is shown that the correlation functions for relativistic electrons (positrons) in a weakly coupled non-equilibrium plasmas can be decomposed into sharply peaked quasiparticle parts and off-shell parts in a rather general form. To leading order in the electromagnetic coupling constant, this decomposition yields the extended quasiparticle approximation for the correlation functions, which can be used for the first principle calculation of the radiation scattering rates in QED plasmas. Виходячи з рiвняння Дайсона для впорядкованої по контуру функцiї Грiна, показано, що кореляцiйнi функцiї для релятивiстських електронiв (позитронiв) у слабозв’язанiй нерiвноважнiй плазмi можна розкласти на квазiчастинковi частини та короткочасовi (off-shell) частини у досить загальнiй формi. З точнiстю до головного порядку по електромагнiтнiй константi взаємодiї цей розклад дає розширене квазiчастинкове наближення для кореляцiйних функцiй, яке можна використати для розрахункiв з перших принципiв швидкостi розсiяння у КЕД плазмi. 2006 Article Extended quasiparticle approximation for relativistic electrons in plasmas / V.G. Morozov, G. Ropke // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 473–484. — Бібліогр.: 14 назв. — англ. 1607-324X PACS: 05.30.-d, 52.25.Dg, 52.27.-h, 52.27.Ny DOI:10.5488/CMP.9.3.473 http://dspace.nbuv.gov.ua/handle/123456789/121373 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Starting with Dyson equations for the path-ordered Green’s function, it is shown that the correlation functions for relativistic electrons (positrons) in a weakly coupled non-equilibrium plasmas can be decomposed into sharply peaked quasiparticle parts and off-shell parts in a rather general form. To leading order in the electromagnetic coupling constant, this decomposition yields the extended quasiparticle approximation for the correlation functions, which can be used for the first principle calculation of the radiation scattering rates in QED plasmas.
format	Article
author	Morozov, V.G. Ropke, G.
spellingShingle	Morozov, V.G. Ropke, G. Extended quasiparticle approximation for relativistic electrons in plasmas Condensed Matter Physics
author_facet	Morozov, V.G. Ropke, G.
author_sort	Morozov, V.G.
title	Extended quasiparticle approximation for relativistic electrons in plasmas
title_short	Extended quasiparticle approximation for relativistic electrons in plasmas
title_full	Extended quasiparticle approximation for relativistic electrons in plasmas
title_fullStr	Extended quasiparticle approximation for relativistic electrons in plasmas
title_full_unstemmed	Extended quasiparticle approximation for relativistic electrons in plasmas
title_sort	extended quasiparticle approximation for relativistic electrons in plasmas
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/121373
citation_txt	Extended quasiparticle approximation for relativistic electrons in plasmas / V.G. Morozov, G. Ropke // Condensed Matter Physics. — 2006. — Т. 9, № 3(47). — С. 473–484. — Бібліогр.: 14 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT morozovvg extendedquasiparticleapproximationforrelativisticelectronsinplasmas AT ropkeg extendedquasiparticleapproximationforrelativisticelectronsinplasmas
first_indexed	2025-07-08T19:44:22Z
last_indexed	2025-07-08T19:44:22Z
_version_	1837109203713392640
fulltext	Condensed Matter Physics 2006, Vol. 9, No 3(47), pp. 473–484 Extended quasiparticle approximation for relativistic electrons in plasmas V.G.Morozov1, G.Röpke2 1 Moscow State Institute of Radioengineering, Electronics and Automation, 117454 Vernadsky Prospect 78, Moscow, Russia, 2 Department of Physics, Rostock University, 18051 Rostock, Germany Received April 17, 2006, in final form June 6, 2006 Starting with Dyson equations for the path-ordered Green’s function, it is shown that the correlation functi- ons for relativistic electrons (positrons) in a weakly coupled non-equilibrium plasmas can be decomposed into sharply peaked quasiparticle parts and off-shell parts in a rather general form. To leading order in the electromagnetic coupling constant, this decomposition yields the extended quasiparticle approximation for the correlation functions, which can be used for the first principle calculation of the radiation scattering rates in QED plasmas. Key words: Many-particle QED, nonequilibrium Green’s functions, kinetic theory of relativistic plasmas PACS: 05.30.-d, 52.25.Dg, 52.27.-h, 52.27.Ny 1. Introduction There is an undeniable interest in studying QED phenomena in plasmas. Originally it was motivated by astrophysical problems, but recent progress in laser-plasma experiments shows that a significant number of particles in laboratory plasmas may have ultrarelativistic velocities. For instance, laser pulses with intensities 1019 – 1020 W/cm −2 are now available to drive intense electron beams with energies 10 – 300 MeV [1,2]. Several difficulties are encountered when one attempts to consider QED processes in plasmas starting from first principles. Apart from the well-known vacuum divergencies and other field theo- retical aspects, here appear problems which are characteristic of any statistical theory of nonequilib- rium phenomena in many-particle systems. A general Green’s function approach to QED plasmas was developed by Bezzerides and DuBois [3]. Within the lowest weak coupling approximation, they were able to derive a covariant particle kinetic equation including electron-electron (electron- positron) collisions and Cherenkov emission/absorption of plasmons. It should be noted, however, that the inclusion of radiative processes involving transverse photons (say, Bremsstrahlung, pair creation by hard photons, and Compton scattering) lies beyond the simplest pole (quasiparticle) approximation for the electron correlation functions used in [3]. The essential point is that in de- scribing such processes, the off-shell parts of the field and particle correlation functions must be treated consistently. These terms describe virtual processes and contribute to the scattering rates. To solve the problem of off-shell contributions to the correlation functions, the so-called “extended quasiparticle approximation” was proposed in the context of the non-relativistic Green’s function method [4,5], where the off-shell corrections can be found to lowest order in the quasiparticle damping width. Unfortunately, this scheme becomes very complicated in the case of QED plasmas since one has to deal with matrix correlation functions and propagators in spinor space. A more general approach to the same problem was developed by Špička and Lipavský [6,7]. In principle, it allows one to go beyond the simplest version of the extended quasiparticle approximation by taking the collisional broadening of the quasiparticle spectral function into account. Another important point is that the main features of this approach are generalizable to relativistic systems. c© V.G.Morozov, G.Röpke 473 V.G.Morozov, G.Röpke In this paper, following essentially the scheme of [6,7], we formulate the extended quasiparticle representation for the correlation functions of relativistic electrons, which can serve as the first step toward Green’s function theory of radiation processes in QED plasmas. A few remarks about notation are appropriate here. We use the system of units with c = ~ = 1 and the “rationalized” Lorentz-Heaviside units for electromagnetic field, i.e., the Coulomb interaction is written as qq′/4πr. The signature of the metric tensor gµν is (+,−,−,−). Cartesian components of three-dimensional vectors are denoted by subscripts: Vi, where i = 1, 2, 3. With this convention, a four-vector has the components V µ = (V 0, Vi) and Vµ = (V0,−Vi). For Dirac γ-matrices the common notation γµ = (γ0, γi) is used. Summation over repeated spatial (Latin) and space-time (Greek) indices is implied. Our convention for the matrix Green’s functions on the time-loop Schwinger-Keldysh contour follows Botermans and Malfliet [8]. 2. The electron Green’s function in QED plasmas As is usually done in the standard Green’s function formalism (see, e.g., [8,9]), we introduce the time-loop Schwinger-Keldysh contour C shown in figure 1. - -s st 0 � C− C+ tmax t Figure 1. The time-loop contour C with the chronological branch (C+) and the anti-chronological branch (C − ). In the following t0 → −∞ and tmax → ∞. From now on, an underlined variable (k) = (t k, rk) implies that t k lies on the contour C, while the notation (k) = (tk, rk) is used for space-time variables. Integrals along C are understood as ∫ C d1F (1) = ∞∫ −∞ d1F (1+) − ∞∫ −∞ d1F (1−), (1) where F (1±) stands for functions with time arguments on the branches C±. For any function F (1 2) we introduce the canonical form [8] F (1 2) = ( F (1+2+) F (1+2−) F (1−2+) F (1−2−) ) = ( F< + F+ F< F> F< − F− ) (2) with space-time “correlation functions” F ≷ (12) and retarded/advanced functions F±(12) = F0(12) δ(t1 − t2) ± θ ( ± (t1 − t2) ) { F>(12) − F<(12) } , (3) where F0(12) is a singular contribution1, and θ(x) is the step function. Note the useful relation F>(12) − F<(12) = F+(12) − F−(12), (4) which follows from the above definitions. The electron (positron) Green’s function in a QED plasma is defined as [3] G(1 2) = −i 〈 TC ψ(1) ψ̄(2) 〉 , (5) where ψ(1) and ψ̄(1) = ψ†(1)γ0 are Dirac fields. The path-ordering operator TC on the contour C includes the usual sign convention for permutations of Fermi operators. Neglecting the effects of initial correlations which die out after a few collisions, the initial time t0 in figure 1 will be taken 1In plasma physics, singular terms of this type enter the longitudinal field propagators [10]. 474 Extended quasiparticle approximation in the remote past, i.e., the limit t0 → −∞ will be assumed. In this case the ensemble average in (5) may be calculated with some initial density operator which admits Wick’s decomposition [8]. Most applications of the Green’s function method to plasma theory are based on the fact that G(1 2) obeys Dyson equations [3] ( i 6∂1 − e 6A(1) −m ) G(1 2) = I δ(1 − 2) + Σ(1 1′)G(1′ 2), (6a) G(1 2) ( − i ← 6∂2 −e 6A(2) −m ) = I δ(1 − 2) +G(1 1′)Σ(1′ 2), (6b) where Aµ(1) = ( φ(1),A(1) ) is the four-vector potential of the mean electromagnetic field, I is the identity spinor matrix, δ(1 − 2) is the delta function on the contour C, and Σ(1 2) is the matrix self-energy. We use the conventional notation 6∂ = γµ∂µ and 6a = γµaµ for any four-vector aµ. To simplify formulas, here and in what follows integration over “primed” variables is implied. Using the canonical form (2) of G(1 2), equations (6) can be reduced into equations for the retarded/advanced matrix propagators (i 6∂1 − e 6A(1) −m)G±(12) = I δ(1 − 2) + Σ±(11′)G±(1′2), (7a) G±(12) ( −i ← 6∂2 −e 6A(2) −m ) = I δ(1 − 2) +G±(11′)Σ±(1′2), (7b) and the Kadanoff-Baym (KB) equations for the correlation functions ( i 6∂1 − e 6A(1) −m ) G ≷ (12) = Σ+(11′)G ≷ (1′2) + Σ ≷ (11′)G−(1′2), (8a) G ≷ (12) ( − i ← 6∂2 −e 6A(2) −m ) = G ≷ (11′)Σ−(1′2) +G+(11′)Σ ≷ (1′2). (8b) Equations (7) and (8) are the starting point in the Green’s function approach to QED plasmas [3]. 3. Quasiparticle and off-shell parts of correlation functions 3.1. Wigner representation It is convenient to transform equations (7) and (8) to the four-dimensional Wigner representa- tion which is defined for any space-time function F (12) = F (x1, x2) as F (X, p) = ∫ d4x eip·x F (X + x/2,X − x/2) , (1) where p · x = pµxµ = p0t − p · r. Kinetic description of QED plasmas is based on the assump- tion that variations in the variable X are slow relative to the scale of 1/E, 1/\|p\|, where E and \|p\| are respectively some characteristic particle energy and momentum. Thus, going over to the Wigner representation, the variability of x may be treated perturbatively by expanding all func- tions in powers of x. This leads to the so-called “gradient approximations”. We shall restrict our consideration to linear corrections in X-gradients. Then, under the Wigner transformation (1), F1(11′)F2(1 ′2) −→ F1(X, p)F2(X, p) − i 2 {F1(X, p), F2(X, p)} , (2) where {F1(X, p), F2(X, p)} = ∂F1 ∂Xµ ∂F2 ∂pµ − ∂F1 ∂pµ ∂F2 ∂Xµ (3) is the four-dimensional Poisson bracket [3]. It should be noted that some care is required when dealing with spinor-dependent (matrix) quantities since, in general, they do not commute and, consequently, their Poisson brackets do not have the symmetry properties that greatly simplify the 475 V.G.Morozov, G.Röpke analysis when Green’s functions and self-energies are scalars. Nevertheless, in the following it will prove convenient to use the identities A { A−1, A } = { A,A−1 } A, (4) tr {A,B} = −tr {B,A} , (5) tr {A,B} = −tr { A−1, ABA } , (6) which are valid for any matrices A(X, p), B(X, p) and follow directly from the definition (3) of the Poisson bracket. 3.2. The electron propagators In the Wigner representation, equations (7) for the propagators read (g±)−1(X, p)G±(X, p) = I + i 2 { (g±)−1(X, p), G±(X, p) } , (7a) G±(X, p)(g±)−1(X, p) = I + i 2 { G±(X, p), (g±)−1(X, p) } , (7b) where g±(X, p) = 1 6Π(X, p) −m− Σ±(X, p) (8) are the local propagators, and Πµ(X, p) = pµ − eAµ(X). To first order in X-gradients, from equations (7) one readily derives two “explicit” expressions: G±(X, p) = g± + i 2 g± { (g±)−1, g± } , (9a) G±(X, p) = g± + i 2 { g±, (g±)−1 } g±. (9b) These are actually equivalent to each other due to the identity (4). Note the appearance of the gradient terms in equations (9), which are absent in the case that propagators are scalar quantities and hence { (g±)−1, g± } = { g±, (g±)−1 } = 0. The gradient corrections to the propagators may be neglected in evaluating local quantities, say, collision terms in a kinetic equation. The local propagators (8) have in general a very complicated spinor structure due to the presence of the matrix self-energies Σ±(X, p) which may be decomposed into the scalar (S), vector (V ), pseudo-scalar (P ), axial-vector (A), and tensor (T ) components according to [11] Σ±(X, p) = I Σ±(S) + γµΣ±µ (V ) + γ5Σ ± (P ) + γ5γµΣ±µ (A) + 1 2 σµνΣ±µν (T ) . (10) The situation is improved a great deal, however, for equal probabilities of the spin polarization. In this case a reasonable approximation for the self-energies is [8,12] Σ±(X, p) = I Σ±(S)(X, p) + γµΣ±µ (V )(X, p), (11) where ( Σ+ (S) )∗ = Σ−(S), ( Σ+ µ (V ) )∗ = Σ−µ (V ). (12) Expression (8) then takes the form g±(X, p) = 6P± +M± (P±) 2 − (M±) 2 , P±µ = Πµ − Σ±µ (V ), M± = m+ Σ±(S). (13) 476 Extended quasiparticle approximation 3.3. Decomposition of correlation functions Let us now turn to the KB equations (8) in the Wigner representation. Using the transformation rule (2) and keeping only first-order terms in X-gradients, we get (arguments X and p are omitted in all functions for brevity) 1 2 { (g+)−1, G ≷ } − 1 2 { Σ ≷ , g− } = i ( Σ ≷ G− − (g+)−1G ≷ ) , (14a) 1 2 { G ≷ , (g−)−1 } − 1 2 { g+,Σ ≷ } = i ( G+Σ ≷ −G ≷ (g−)−1 ) . (14b) Here one must be careful to distinguish between the full propagators G±(X, p) and the local propagators g±(X, p) in the right-hand sides since equations (9) contain the gradient corrections. The generalized transport equations for the correlation functions G ≷ (X, p) are obtained by taking the difference of equations (14). With expressions (8) for the local propagators and relations G>(X, p) −G<(X, p) = G+(X, p) −G−(X, p), (15a) Σ>(X, p) − Σ<(X, p) = Σ+(X, p) − Σ−(X, p), (15b) a simple algebra gives 1 2 ({ (g+)−1, G ≷ } − { G ≷ , (g−)−1 }) + 1 2 ({ g+,Σ ≷ } − { Σ ≷ , g− }) = − [ g,Σ ≷] − − [ 6Π −m− σ,G ≷ ] − + i 2 ( [ Σ>, G< ] + − [ Σ<, G> ] + ) , (16) where [A,B]∓ = AB∓BA stands for the commutator/anticommutator of spinor matrices, and we have introduced the designations g = 1 2 ( G+ +G− ) , σ = 1 2 ( Σ+ + Σ− ) . (17) By multiplying equation (16) with the matrices I, γµ, γ5, γ5γµ, σµν , and then taking the trace of all these equations, one finds in general a very complicated set of coupled equations for the components of the correlation functions defined through the spinor decomposition analogous to equation (10). Let us consider one of these transport equations, which is obtained from equation (16) by taking the trace in both sides. The trace of the second drift term is conveniently rearranged as follows: tr ({ g+,Σ ≷ } − { Σ ≷ , g− }) = tr ({ g+ + g−,Σ ≷ }) = −tr ({ (g+)−1, g+Σ ≷ g+ } + { (g−)−1, g−Σ ≷ g− }) , where we have used identities (5) and (6). Then the trace of equation (16) may be put in the form tr { 6Π −m− σ,G ≷ − 1 2 ( g+Σ ≷ g+ + g−Σ ≷ g− )} + 1 4 tr { ∆Σ, g+ Σ ≷ g+ − g−Σ ≷ g− } = i tr ( Σ>G< − Σ<G> ) , (18) where ∆Σ = Σ+ − Σ−. Let us now write G ≷ (X, p) as G ≷ (X, p) = G̃ ≷ + 1 2 ( g+ Σ ≷ g+ + g−Σ ≷ g− ) , (19) 477 V.G.Morozov, G.Röpke where G̃ ≷ (X, p) are some new spinor functions. Substituting this expression into equation (18), it is easy to see that the second term does not contribute to the right-hand side, and we get a transport equation where only the G̃ ≷ (X, p) appear: tr { 6Π −m− σ, G̃ ≷ } + 1 4 tr { ∆Σ, g+ Σ ≷ g+ − g− Σ ≷ g− } = i tr ( Σ> G̃< − Σ< G̃> ) . (20) Decomposition (19) of the correlation functions is closely similar to that proposed previously by Špička and Lipavský [6,7] in non-relativistic kinetic theory. These authors argued that, in the case of small damping, the terms like G̃ ≷ (X, p) may be interpreted as “quasiparticle” parts which are sharply peaked near the mass shell, while the additional term represents off-shell (short-time) parts of correlation functions. Within this interpretation, a transport equation for G̃<(X, p) can be transformed into a quasiparticle kinetic equation. On the other hand, the off-shell parts of the correlation functions contribute to scattering cross sections [6,7]. It is reasonable to assume that the representation (19) of the electron correlation functions in QED plasmas has the same meaning as in the non-relativistic case. Some arguments that support this assumption will be given below. 3.4. The full and quasiparticle spectral functions In order to exhibit the content of equation (19), it is instructive to consider the spectral prop- erties of G̃ ≷ (X, p). We recall that the full spectral function in spinor space is defined as [8,12] A(X, p) = i ( G>(X, p) −G<(X, p) ) = i ( G+(X, p) −G−(X, p) ) . (21) In the local approximation, G±(X, p) = g±(X, p), we may use expression (8) to rewrite this spectral function in the form A(X, p) = i g+∆Σ g−, (22) where ∆Σ(X, p) = Σ+(X, p) − Σ−(X, p). (23) We now introduce the quasiparticle spectral function associated with spinor matrices G̃ ≷ (X, p): Ã(X, p) = i ( G̃>(X, p) − G̃<(X, p) ) . (24) With the aid of equations (19) we obtain the relation between Ã(X, p) and A(X, p): Ã = A− i 2 ( g+∆Σ g+ + g−∆Σ g− ) . (25) Recalling equation (22), a little algebra leads to Ã(X, p) = − i 2 g+∆Σ g+∆Σ g−∆Σ g−. (26) To illustrate some essential differences between spectral functions (22) and (26), let us consider the zero damping limit. In this special case the local propagators (8) may be replaced by the free particle propagators2 g±0 (X, p) = 1 6Π −m± i6ε , (27) where εµ = (ε, 0, 0, 0), ε → +0. The corresponding retarded/advanced self-energies are Σ± = ∓iεγ0. After some spinor algebra the prelimit expressions for the spectral functions (22) and (26) 2Strictly speaking, even in the zero damping limit there are corrections to the particle energies coming from the real parts of the retarded/advanced self-energies. However, in a relativistic plasma, which is always weakly coupled, these corrections are very small and may be neglected. 478 Extended quasiparticle approximation are found to be A(X, p) = 4Π0Γ( Π2 0 − E2 p )2 + (Π0Γ) 2 [ 6Π +m+ E2 p −Π2 0 2Π0 γ0 ] , (28) Ã(X, p) = 4 (Π0Γ) 3 [( Π2 0 − E2 p )2 + (Π0Γ) 2 ]2 [ E2 p +Π2 0 2Π2 0 ( 6Π +m) + E2 p −Π2 0 2Π0 ( 1 + E2 p −Π2 0 4Π2 0 ) γ0 ] , (29) where Ep(X) = √ Π2 +m2 and Γ = 2ε is the infinitesimally small damping width. From the above expressions two important observations can be made. First, one easily verifies that both spectral functions have the same limiting form: lim Γ→0 A = lim Γ→0 Ã = 2π η(Π0) δ(Π 2 −m2) ( 6Π +m) , (30) where η(Π0) = Π0/\|Π0\| . Second, it is seen that the prefactor in Ã approaches the delta function faster than the prefactor in A. Thus, in the case of small damping, it is reasonable to suggest that the term G̃ ≷ in equation (19) may be expressed to close approximation by a singular mass-shell form, while the last term may be regarded as the off-shell contribution to the correlation functions. 3.5. The pole approximation We now want to find a way of expressing G̃ ≷ (X, p) in terms of the electron and positron distribution functions, which is required to convert equation (16) into a kinetic equation and calculate the scattering rates in a physically transparent form. Let us first consider the hermicity properties of G̃ ≷ (X, p) and Ã(X, p). Note that the correla- tion functions G ≷ (X, p), propagators G±(X, p), and the self-energies, as spinor matrices, satisfy relations [ G ≷ (X, p) ]† = −γ0G ≷ (X, p)γ0, [ Σ ≷ (X, p) ]† = −γ0Σ ≷ (X, p)γ0, (31a) [G+(X, p)] † = γ0G−(X, p)γ0, [ Σ+(X, p) ]† = γ0Σ−(X, p)γ0, (31b) which follow from the definition of these quantities. Recalling (8), it is easy to see that the local propagators g±(X, p) satisfy the same relations as G±(X, p). Then one derives from equation (19) [ G̃ ≷ (X, p) ]† = −γ0G̃ ≷ (X, p)γ0. (32) This immediately leads to the following property of the quasiparticle spectral function (24): [ Ã(X, p) ]† = γ0Ã(X, p)γ0. (33) We now introduce the distribution functions in spinor space, F ≷ (X, p), through G̃ ≷ (X, p) = ∓ i 2 ( ÃF ≷ + F ≷ Ã ) , F>(X, p) + F<(X, p) = I. (34) Relation (24) is automatically satisfied, whereas equations (32) and (33) require that [ F ≷ (X, p) ]† = γ0F ≷ (X, p)γ0. (35) In particular, from this property it follows that in the spinor decomposition of F ≷ (X, p) [cf. (10)] all coefficients are real. 479 V.G.Morozov, G.Röpke The spinor structure of F ≷ (X, p) is in general very complicated. For a weakly coupled plasma, however, reasonable approximations may be introduced by starting from the case of non-interacting particles, where G ≷ (X, p) = G̃ ≷ (X, p) and the correlation functions are expressed in terms of the electron (positron) distribution functions by using the plane-wave expansion of the field opera- tors [11]. Another way of arriving at the same result is to consider the sum of KB equations (14) in the collisionless and slow variation limit [3]: [( 6Π −m ) , G̃ ≷ ] + = 0. (36) Then, using equations (30), (34), and the spinor decomposition of F̃ ≷ , it can be shown that F ≷ (X, p) = 1 2 ∑ s (I + γ56s )F ≷ s (X, p) (37) with sµ satisfying sµsµ = −1 and sµΠµ = 0. In the rest frame sµ = (0, s), where the unit vector s represents the direction of spin. As discussed by Bezzerides and DuBois [3], the quantities F ≷ s (X, p) are related directly to the electron (positron) distribution functions for different spin states3. In the case of equal probabilities of the spin polarization equation (37) reduces to F ≷ (X, p) = I F ≷ (X, p), (38) where F ≷ (X, p) = 1 2 ∑ s F ≷ s (X, p), F>(X, p) + F<(X, p) = 1. (39) Then the correlation functions in the collisionless limit4 are given by G̃ ≷ (X, p) = ∓ 2πi η(Π0) δ(Π2 −m2) ( 6Π +m)F ≷ (X, p). (40) Before going further, one remark is appropriate here. The appearance of the vector potential Aµ(X) in equation (40) and other formulas is due to the fact that the electron Green’s function (5) and, consequently, the propagators G±(12) and the correlation functions G ≷ (12) are not invariant under gauge transformation of the mean electromagnetic field. In principle, we could work from the beginning with the gauge invariant Green’s function defined as G(1 2) = G(1 2) exp [ ie ∫ 1 2 Aµ(x) dxµ ] , (41) where xµ = (t, r) and integration over r is performed along a straight line connecting the points r2 and r1. It is an easy matter, however, to pass to the gauge invariant functions, G±(X, p) and G ≷ (X, p), in the Wigner representation. Within the first-order gradient approximation the corresponding relations are G±(X, p) = G±(X, p+ eA(X)), G ≷ (X, p) = G ≷ (X, p+ eA(X)). (42) For instance, from equation (40) we see that the gauge invariant quasiparticle correlation functions in the collisionless limit are given by G̃ ≷ (X, p) = ∓ 2πi η(p0) δ(p2 −m2) ( 6p+m) f ≷ (X, p), (43) where f ≷ (X, p) = F ≷ (X, p+ eA(X)) are the gauge invariant distribution functions. 3Note that expressions for the correlation functions in the collisionless limit given in [3] are not quite correct. They correspond to the form � G ≷ = ∓ i � AF ≷ and, if matrices F ≷ are non-diagonal, violate the property (32) in contrast to the symmetric representation (34). 4We recall that in the case being considered � G ≷ = G ≷ . 480 Extended quasiparticle approximation Since in a weakly coupled (nearly collisionless) plasma the difference between the free particle energy and the quasiparticle energy may be neglected, the expression (40) is a reasonable ap- proximation for the first (quasiparticle) term in equation (19). The pole approximation (40) was previously proposed by Bezzerides and DuBois [3], although for the full correlation functions G ≷ , so that the off-shell term in equation (19) was missing. 3.6. Extended quasiparticle approximation for the correlation functions Insertion of (40) into equation (20) leads to a kinetic equation for the electron (f−) and positron (f+) distribution functions defined as f<(X, p) ∣∣ p0=E p = f−(X,p), f<(X, p) ∣∣ p0=−E p = 1 − f+(X,−p), (44) where Ep = √ p2 +m2. Neglecting unessential self-energy corrections to the drift, we arrive at the kinetic equation which is formally the same as that derived by Bezzerides and DuBois [3]. The new feature is that now the additional off-shell parts of the correlation functions (19) contribute to the scattering cross sections in the collision term. To discuss this point we recall the general expression for the matrix self-energy Σ(1 2) (see, e.g., [13]). In the Coulomb gauge it can be written as Σ(1 2) = −γ0G(1 1′) ΓL(1′ 2 ; 2′)DL(2′ 1) − γiG(1 1′) ΓT j (1′ 2 ; 2′)DT ji(2 ′ 1), (45) where ΓL and ΓT j are, respectively, the longitudinal and transverse vertex functions, DL(1 2) is the longitudinal (plasmon) Green’s function DL(1 2) = i 〈 TC ∆φ̂(1)∆φ̂(2) 〉 − 1 4π\| r1 − r2\| δ ( t 1 − t 2 ) (46) with ∆φ̂(1) = φ̂(1) − φ(1), and DT ij(1 2) is the transverse photon Green’s function DT ij(1 2) = −i 〈 TC ∆Âi(1)∆Âj(2) 〉 (47) with ∆Âi(1) = Âi(1)−Ai(1). Using the canonical representation (2), equation (45) can be written for the components Σ ≷ (12). Since the resulting expression is rather complicated, we shall restrict the discussion to the case where the full vertex functions are replaced by the bare ones ΓL(0)(1 2 ; 3) = ie2 δ(1 − 2) δ(1 − 3) γ0, Γ T (0) i (1 2 ; 3) = −ie2 δ(1 − 2) δT ik(1 − 3) γk, (48) where δT ik(1 − 3) is the transverse delta function on the contour C: δT ij(1 − 2) = δ(t 1 − t 2) ∫ d3k (2π)3 eik·(r 1 −r 2 ) ( δij − kikj k2 ) . (49) With expressions (48) for the vertex functions, equation (45) becomes Σ(1 2) = −ie2γ0G(1 2) γ0DL(2 1) + ie2γiG(1 2) γj DT ≶ ji (2 1). (50) For the components Σ ≷ (12) this reads Σ ≷ (12) = −ie2γ0G ≷ (12) γ0DL ≶ (21) + ie2γiG ≷ (12) γj DT ≶ ji (21). (51) We see that equation (19) itself does not determine the correlation functions in terms of the distribution functions because the self-energies Σ ≷ depend on G ≷ . 481 V.G.Morozov, G.Röpke Let us now make the Wigner transformation in equation (51) and introduce the principal-axis representation for photon states through [3] DT ≷ ij (X, k) = ∑ s εsi(X, k) d ≷ s (X, k) εsj(X, k), (52) where εs(X, k), (s = 1, 2), are unit polarization vectors satisfying εs(X, k) · εs′(X, k) = δss′ , ∑ s εsi(X, k)εsj(X, k) = δij − kikj k2 . (53) Then equation (51) is transformed into5 Σ ≷ (X, p) = −i e2 ∫ d4p′ (2π)4 DL ≶ (X, p′ − p) γ0G ≷ (X, p′) γ0 + i e2 ∑ s ∫ d4p′ (2π)4 d ≶ s (X, p′ − p) 6εs(X, p ′ − p)G ≷ (X, p′) 6εs(X, p ′ − p), (54) where we have introduced the polarization four-vectors εµs = (0, εs). Insertion of (54) into equa- tion (19) leads to the integral equation for the electron correlation functions which may be approx- imately solved by iteration since the self-energies Σ ≷ (X, p) are proportional to the small coupling constant6. Following this way, one obtains the full correlation functions as some functionals of the quasiparticle correlation functions, or, with equation (34), as functionals of the distribution functions in spinor space. These functionals can then be used to obtain from equation (45) an analogous representation for the self-energies Σ ≷ (X, p). Note, however, that some care is required in order that the self-energies be calculated in a self-consistent fashion. The point is that the ver- tex functions in equation (45) contain higher-order terms in the coupling constant, which must be considered along the off-shell parts of G ≷ (X, p). Therefore, taking the vertex functions in the form (48), one may keep only the leading off-shell corrections to the correlation functions. Within this approximation, we obtain from equations (19) and (54) (the fixed argument X is omitted everywhere for brevity) G ≷ (p) = G̃ ≷ (p) − ie2 2 ∫ d4p′ (2π)4 DL ≶ (p′ − p) [ g+(p)γ0G̃ ≷ (p′)γ0g+(p) + g−(p)γ0G̃ ≷ (p′)γ0g−(p) ] + ie2 2 ∑ s ∫ d4p′ (2π)4 d ≶ s (p′ − p) [ g+(p) 6εs(p ′ − p) G̃ ≷ (p′) 6εs(p ′ − p)g+(p) + g−(p) 6εs(p ′ − p) G̃ ≷ (p′) 6εs(p ′ − p)g−(p) ] . (55) Keeping here only the first (quasiparticle) contribution and then taking the simplest pole form (40), we recover in fact for the self-energies (54) the expression which was used by Bezzerides and DuBois [3] to calculate the collision term in equation (20). As already mentioned in the introduction, this quasiparticle approximation is sufficient to describe electron-electron (electron- positron) scattering and Cherenkov emission/absorption of longitudinal plasma waves. Inclusion of the off-shell integral terms in equation (55) leads to the extended quasiparticle approximation 5In the self-energies we have omitted the gradient corrections which give irrelevant contributions to the drift term in the particle kinetic equation. 6It should be noted that, in contrast to vacuum electrodynamics where one deals with an asymptotic expansion in powers of e2, the Green’s functions for particles and the field in plasmas contain polarization and multiple-scattering effects to all orders on e2. As discussed by Bezzerides and DuBois [3], the dimensionless strength of interactions is characterized by the plasma parameter which is always small for relativistic plasmas. Formally, in this case the expansion is performed in terms of the Green’s functions DL and DT which characterize the intensity of fluctuations of the electromagnetic field. 482 Extended quasiparticle approximation for the correlation functions, which is necessary for description of higher-order processes, such as Bremsstrahlung and Compton scattering in plasmas. This requires, however, a more detailed analysis of the vertex functions and the field Green’s functions, which will be done in future publications (see also [14]). 4. Discussion We have shown that the decomposition of the electron correlation functions in QED plasmas into the quasiparticle and off-shell parts can be done in a rather general form (19) which bears resemblance to the analogous decomposition in non-relativistic kinetic theory [6,7]. Note that equation (19) itself does not imply from the outset the pole approximation for the quasiparticle parts. Nevertheless, we have seen that the “collisional broadening” of the quasiparticle spectral function (26) is considerably smaller than that of the full spectral function (22). Therefore, at some stage the pole approximation for the quasiparticle terms is sufficient in order to calculate scattering rates in a kinetic equation. It should be emphasized, however, that even in the case of a weakly coupled plasma the pole form (40) is not always an adequate approximation for G̃ ≷ (X, p) since it can violate some important properties of the full correlation functions G ≷ (X, p), say, the sum rules. In such situations improved forms of the quasiparticle spectral function Ã(X, p) have to be used in the general representation (34). Acknowledgement This work was supported by “DFG-Sonderforschungsbereich 652” and U.S.CRDF–RF Ministry of Education Award VZ–010–0. References 1. Santala M.I.K. et al., Phys. Rev. Lett., 2001, 86, 1227. 2. Mangles S P D. et al., Phys. Rev. Lett., 2005, 94, 245001. 3. Bezzerides B., DuBois D.F., Ann. Phys. (N.Y.), 1972, 70, 10. 4. Köhler H.S., Malfliet R., Phys. Rev. C, 1993, 48, 1034. 5. Bornath Th., Kremp D., Kraeft W.D., Schlanges M., Phys. Rev. E, 1996, 54, 3274. 6. Špička V., Lipavský P., Phys. Rev. Lett., 1994, 73, 3439. 7. Špička V., Lipavský P., Phys. Rev. B, 1995, 52 (20), 14615. 8. Botermans W., Malfliet R., Phys. Rep., 1990, 198, 115. 9. Danielewicz P., Ann. Phys. (N.Y.), 1984, 152, 239. 10. DuBois D. F. – In: Lectures in Theoretical Physics, edited by W.E. Brittin. Gordon and Breach, New York, 1967, 469-619. 11. Itzykson C., Zuber J.-B. Quantum Field Theory. McGraw-Hill, New York, 1980. 12. Mrówczyński St., Heinz U., Ann. Phys. (N.Y.), 1994, 229, 1. 13. Bonitz M. Quantum Kinetic Theory. Teubner, Stuttgart-Leipzig, 1998. 14. Morozov V.G., Röpke G., J. Phys.: Conference Series, 2006, 35, 110. 483 V.G.Morozov, G.Röpke Розширене квазiчастинкове наближення для релятивiстських електронiв у плазмi В.Г.Морозов1, Г.Репке2 1 Московський державний iнститут радiотехнiки, електронiки та автоматики, просп. Вернадського 78, 117454 Москва, Росiя 2 Унiверситет м.Ростока, фiзичний факультет, 18051 Росток, Нiмеччина Отримано 17 квiтня 2006 р., в остаточному виглядi – 6 червня 2006 р. Виходячи з рiвняння Дайсона для впорядкованої по контуру функцiї Грiна, показано, що кореляцiйнi функцiї для релятивiстських електронiв (позитронiв) у слабозв’язанiй нерiвноважнiй плазмi можна розкласти на квазiчастинковi частини та короткочасовi (off-shell) частини у досить загальнiй формi. З точнiстю до головного порядку по електромагнiтнiй константi взаємодiї цей розклад дає розширене квазiчастинкове наближення для кореляцiйних функцiй, яке можна використати для розрахункiв з перших принципiв швидкостi розсiяння у КЕД плазмi. Ключовi слова: багаточастинкова КЕД, нерiвноважнi функцiї Грiна, кiнетична теорiя релятивiстської плазми PACS: 05.30.-d, 52.25.Dg, 52.27.-h, 52.27.Ny 484

Extended quasiparticle approximation for relativistic electrons in plasmas

Репозитарії

Схожі ресурси