Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory

On the basis of Bogolyubov reduced description method and quasirelativistic quantum electrodynamics, the kinetics of an electromagnetic field in an equilibrium plasma has been constructed. The calculation is carried out in the Hamilton gauge up to the second order of a generalized perturbation theor...

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Дата:	2010
Автори:	Sokolovsky, A.I., Stupka, A.A., Chelbaevsky, Z.Yu.
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Мова:	English
Опубліковано:	Відділення фізики і астрономії НАН України 2010
Теми:	Плазма і гази
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/13282
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Цитувати:	Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory / A.I. Sokolovsky, A.A. Stupka, Z.Yu. Chelbaevsky // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 20-28. — Бібліогр.: 12 назв. — англ.

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spelling	irk-123456789-132822010-11-04T14:34:47Z Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory Sokolovsky, A.I. Stupka, A.A. Chelbaevsky, Z.Yu. Плазма і гази On the basis of Bogolyubov reduced description method and quasirelativistic quantum electrodynamics, the kinetics of an electromagnetic field in an equilibrium plasma has been constructed. The calculation is carried out in the Hamilton gauge up to the second order of a generalized perturbation theory in interaction. Following Bogolyubov in his theory of superfluidity, the leading contribution to the Hamilton operator of the field is chosen with an additional term depending on the interaction. This allows us to discuss the kinetics of the field in the terms of photons in the plasma and plasmons. On the basis of the obtained material equation supplementing the Maxwell equations, plane electromagnetic waves have been considered. For the case of the Maxwell plasma, the obtained spectra and the attenuation coefficients give results which coincide with those in the standard theory. However, the developed approach allows one to avoid some difficulties of that theory. The method of construction of an effective Hamilton operator of the electromagnetic field in the plasma is proposed. On this basis, we have performed the renormalization of quasiparticle spectra which coincide finally with the spectra of waves in the system. На основi методу скороченого опису Боголюбова i квазiрелятивiстської квантової електродинамiки побудовано кинетику електромагнiтного поля в рiвноважнiй плазмi. Обчислення проводиться в калiбровцi Гамiльтона до другого порядку узагальненої теорiї збурень за взаємодiєю. Наслiдуючи Боголюбова в його теорiї надплинностi, основний внесок в оператор Гамiльтона поля вибирається з додатком, який залежить вiд взаємодiї. Це дозволяє обговорювати кiнетику електромагнiтного поля в термiнах фотонiв у плазмi та плазмонiв. На основi отриманого матерiального рiвняння додаткового до рiвнянь Максвелла розглянуто плоскi електромагнiтнi хвилi. Для випадку максвеллiвської плазми отриманi закони дисперсiї та декременти згасання хвиль дають результати, якi збiгаються зi стандартною теорiєю. Однак розвинутий пiдхiд дозволяє позбутися деяких труднощiв цiєї теорiї. Запропоновано метод побудови ефективного оператора Гамiльтона електромагнiтного поля у плазмi. На цiй основi виконано перенормування спектрiв квазiчастинок, якi у пiдсумку збiгаються зi спектрами хвиль у системi. 2010 Article Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory / A.I. Sokolovsky, A.A. Stupka, Z.Yu. Chelbaevsky // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 20-28. — Бібліогр.: 12 назв. — англ. 2071-0194 PACS 52.35.-g http://dspace.nbuv.gov.ua/handle/123456789/13282 en Відділення фізики і астрономії НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Плазма і гази Плазма і гази
spellingShingle	Плазма і гази Плазма і гази Sokolovsky, A.I. Stupka, A.A. Chelbaevsky, Z.Yu. Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory
description	On the basis of Bogolyubov reduced description method and quasirelativistic quantum electrodynamics, the kinetics of an electromagnetic field in an equilibrium plasma has been constructed. The calculation is carried out in the Hamilton gauge up to the second order of a generalized perturbation theory in interaction. Following Bogolyubov in his theory of superfluidity, the leading contribution to the Hamilton operator of the field is chosen with an additional term depending on the interaction. This allows us to discuss the kinetics of the field in the terms of photons in the plasma and plasmons. On the basis of the obtained material equation supplementing the Maxwell equations, plane electromagnetic waves have been considered. For the case of the Maxwell plasma, the obtained spectra and the attenuation coefficients give results which coincide with those in the standard theory. However, the developed approach allows one to avoid some difficulties of that theory. The method of construction of an effective Hamilton operator of the electromagnetic field in the plasma is proposed. On this basis, we have performed the renormalization of quasiparticle spectra which coincide finally with the spectra of waves in the system.
format	Article
author	Sokolovsky, A.I. Stupka, A.A. Chelbaevsky, Z.Yu.
author_facet	Sokolovsky, A.I. Stupka, A.A. Chelbaevsky, Z.Yu.
author_sort	Sokolovsky, A.I.
title	Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory
title_short	Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory
title_full	Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory
title_fullStr	Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory
title_full_unstemmed	Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory
title_sort	dispersion relations for waves in plasma and bogolyubov ideas in many-body theory
publisher	Відділення фізики і астрономії НАН України
publishDate	2010
topic_facet	Плазма і гази
url	http://dspace.nbuv.gov.ua/handle/123456789/13282
citation_txt	Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory / A.I. Sokolovsky, A.A. Stupka, Z.Yu. Chelbaevsky // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 20-28. — Бібліогр.: 12 назв. — англ.
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first_indexed	2025-07-02T15:13:18Z
last_indexed	2025-07-02T15:13:18Z
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fulltext	PLASMAS AND GASES 20 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 DISPERSION RELATIONS FOR WAVES IN PLASMA AND BOGOLYUBOV IDEAS IN MANY-BODY THEORY A.I. SOKOLOVSKY, A.A. STUPKA, Z.YU. CHELBAEVSKY Oles’ Honchar Dnipropetrovs’k National University (72, Gagarin Ave., 49010, Dnipropetrovs’k, Ukraine; e-mail: alexsokolovsky@ mail. ru ) PACS 52.35.-g c©2010 On the basis of Bogolyubov reduced description method and quasirel- ativistic quantum electrodynamics, the kinetics of an electromag- netic field in an equilibrium plasma has been constructed. The cal- culation is carried out in the Hamilton gauge up to the second or- der of a generalized perturbation theory in interaction. Following Bogolyubov in his theory of superfluidity, the leading contribution to the Hamilton operator of the field is chosen with an additional term depending on the interaction. This allows us to discuss the kinetics of the field in the terms of photons in the plasma and plasmons. On the basis of the obtained material equation sup- plementing the Maxwell equations, plane electromagnetic waves have been considered. For the case of the Maxwell plasma, the ob- tained spectra and the attenuation coefficients give results which coincide with those in the standard theory. However, the devel- oped approach allows one to avoid some difficulties of that theory. The method of construction of an effective Hamilton operator of the electromagnetic field in the plasma is proposed. On this ba- sis, we have performed the renormalization of quasiparticle spectra which coincide finally with the spectra of waves in the system. 1. Introduction The modern quasirelativistic theory of electromagnetic (EM) processes in a plasma medium is usually based on the introduction of the effective direct Coulomb inter- action between charged particles (see, for example, [1]). This can be achieved in the Coulomb gauge, in which the scalar potential ϕ is equal to the Coulomb one ϕc, and the vector potential An is a transversal field divA = 0. In fact, the Coulomb interaction is introduced instead of the longitudinal part of the vector potential An. This approach leads to some difficulties in the consideration of longitudinal freedom degrees of the system. They can be avoided in the Hamilton gauge, in which the scalar po- tential ϕ = 0 and the electromagnetic field is described by a vector potential with transversal and longitudinal parts. In this gauge, charged particles interact with one another only through the EM field. Then the transver- sal part of the vector potential describes EM waves, and the longitudinal part describes plasma oscillations. In quantum theory, this leads to photons in the medium and plasmons. In this paper, we use the Hamilton gauge and build the kinetics of an EM field in the equilibrium plasma medium (bath) which describes the field by the average values of electric field and vector potential. In a certain sense in such an approach, we move in the reverse direction as compared with paper [2] (see also [3]). In that work, a system of particles which interact by Coulomb forces was investigated. Instead of the long-distance parts of the Coulomb interaction, they introduced an additional EM field which is described by a longitudinal vector potential and corresponds to plasma oscillations in the system. Our investigation is based on the Bogolyubov reduced description method of nonequilibrium states [4] (see the review in [1]) and quasirelativistic quantum electrody- namics. 2. Bogolyubov’s Reduced Description Method An arbitrary state of the system is described by the statistical operator (SO) ρ(t) that satisfies the Liouville equation ρ̇(t) = − i ~ [Ĥ, ρ(t)] ≡ Lρ(t); Ĥ = Ĥ0 + Ĥint (1) (Ĥ is the Hamilton operator, L is the Liouville opera- tor; and Ĥ0 is the main contribution to Ĥ). According to Bogolyubov [4], in the presence of a few character- istic times in the system, its evolution passes through the corresponding stages. At each stage, it is possible to DISPERSION RELATIONS FOR WAVES IN PLASMA describe the system by a relatively small set of reduced- description parameters ηa(t) (RDPs) which are average values calculated with the SO ρ(t) ηa(t) = Spρ(t)η̂a. (2) We assume that the considered reduced description takes place at t ≥ 0. According to the Bogolyubov func- tional hypothesis, the solution of Eq. (1) has the struc- ture ρ(t) = ρ(η(t)), (3) where the SO ρ(η) does not depend on the initial value of the SO ρ(t). The RDPs ηa(t) satisfy the equation η̇a(t) = La(η(t)), La(η) ≡ −Spρ(η)Lη̂a. (4) The SO ρ(η) of the system is a solution of the equations∑ a ∂ρ(η) ∂ηa La(η) = Lρ(η), Spρ(η)η̂a = ηa. (5) Here, the first equation is the Liouville one (1) at the stage of reduced description, and the second is the defi- nition of RDPs. According to Bogolyubov [4], Eqs. (5) have at least two solutions for ρ(η). One has to add a boundary con- dition written in the terms of the evolution of the system in the natural direction of time to these equations. As a boundary condition, we chose the functional hypothesis in the zero approximation in interaction written for an arbitrary initial state ρ0 of the system. We take into ac- count that, for an arbitrary initial state ρ0, the system has a statistical operator of the form ρ(η(t)) only at long times t � τ0 (τ0 depends on the SO ρ0). Therefore, in the zero approximation in interaction (i.e. for the evo- lution with the main contribution Ĥ0 to the Hamilton operator Ĥ), we have etL0ρ0−−−−−→ t�τ0 etL0ρ(0)(η(0)(0)) = ρ(0)(η(0)(t)) (6) (the Liouville operator L0 corresponds to the Hamilto- nian Ĥ0; here, η(0) a (t) depends on ρ0). The SO ρ(0)(η) is the leading contribution to ρ(η) and is a solution of Eqs. (5) in the zero approximation in interaction. The parameters η(0) a (t) satisfy the equation η̇(0) a (t) = L(0) a (η(0)(t)), L(0) a (η) ≡ −Spρ(0)(η)L0η̂a, (7) which follows from (4). In our investigation, the op- erators η̂a of the reduced-description parameters ηa(t) satisfy the Peletminsky–Yatsenko condition [1] L0η̂a = −i ∑ a cabη̂b (8) (cab are some coefficients). Relation (7) yields obviously η(0) a (t) = ∑ b eitcab η (0) b (0). (9) It is possible to find the initial condition η (0) a (0) from (6), by multiplying it by η̂a and taking trace Sp of both sides of this formula η(0) a (0) = Spρ0η̂a. (10) Then condition (6) can be written in the form etL0ρ0−−−−−→ t�τ0 etL0ρ(0)(Spρ0η̂) = ρ(0)(eitcSpρ0η̂). (11) The SO ρ(0)(η) satisfies the equations ∑ a,b ∂ρ(0)(η) ∂ηa i cabηb = L0ρ (0)(η), Spρ(0)(η)η̂a = ηa. (12) Condition (11) is the functional hypothesis taken in the zero approximation of perturbation theory for an arbi- trary initial state ρ0. Following [1], we call it the ergodic relation. The evolution in (11) with the Liouville oper- ator L0 of free evolution cannot lead the system to an equilibrium; therefore, the SO ρ(0) is a quasiequilibrium one. From relation (11), it is possible to get the necessary boundary condition for Eqs. (5), by replacing ρ0 by ρ (η) and η by e−itcη. We have lim τ→+∞ eτL0ρ ( e−iτcη ) = ρ(0) (η) . (13) Writing this relation in an integral form and taking the Liouville equation (5) into account, we get [1] the inte- gral equation for ρ(η): ρ(η) = ρ(0)(η) + +∞∫ 0 dτeτL0{Lintρ(η)− − ∑ a ∂ρ(η) ∂ηa L̃a(η)}η→e−iτcη, (14) where the function L̃a(η) is defined by the formula L̃a (η) = −Spρ (η)Lintη̂a (15) (the Liouville operator Lint corresponds to Ĥint). This equation is solvable within perturbation theory in inter- action (first, it was obtained by Peletminsky and Yat- senko (see [1])). ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 21 A.I. SOKOLOVSKY, A.A. STUPKA, Z.Yu. CHELBAEVSKY 3. Kinetics of an EM Field in the Equilibrium Plasma We neglect the influence of an EM field on the equilib- rium plasma subsystem, by considering only low-energy processes. The Hamilton operator of the system is cho- sen here in the form Ĥ = Ĥs + Ĥb + Ĥ1 + Ĥ2, (16) where Ĥs = 1 8π ∫ d3x{Ê2(x) + B̂2(x)}, Ĥ1 = −1 c ∫ dxÂn (x) ĵn (x), Ĥ2 = 1 2c2 ∫ dxÂ2 (x) χ̂ (x). (17) In the Hamilton gauge, the Hamilton operator of the bath Ĥb is the operator of kinetic energy of particles. Therefore, the average values of plasma operators can be calculated by the Wick–Bloch–de Dominicis theorem. The operators of the EM field in (17) are given by the formulas B̂n = rotnÂ, Ên = − 1 c ^̇An, and the vector po- tential Ȧn(x) has longitudinal and transversal compo- nents. The interaction between plasma particles is real- ized only through the EM field. The operator ĵn(x) is the operator of current, and the auxiliary operator χ̂(x) is expressed through the plasma frequency operator χ̂ (x) = ∑ a e2a ma ρ̂a (x) = Ω̂2 (x) /4π (18) (ρ̂a (x) is the operator of mass density; ma and ea are, respectively, the mass and the charge of particles of the a-th component of the bath (ea = zae, e > 0)). We chose the main contribution Ĥ0 to the Hamilton operator of the system Ĥ in the form Ĥ0 = Ĥf + Ĥb, Ĥf = 1 8π ∫ d3x{Ê2(x) + B̂2(x) + Ω2Â2(x)/c2}, (19) where Ĥf can be considered as the Hamilton operator of a free EM field in the bath Ĥf (Ω2 ≡ SpbwΩ̂2, i.e. Ω is the plasma frequency; w is the equilibrium SO of the bath, Spb is the trace over bath states). This under- standing of the Hamiltonian Ĥf follows from the proce- dure of canonical quantization. Canonical commutation relations have the form [Ân(x), Âl(x′)] = 0, [Ân(x), π̂l(x′)] = i~δnlδ(x− x′), [π̂n(x), π̂l(x′)] = 0, (20) where π̂n(x) = −Ên(x)/4πc are the generalized mo- menta corresponding to Ân(x) as generalized coordi- nates of the field. Usual steps lead to a representation of the vector potential operator through the creation and annihilation operators c+αk, cαk, Ân(x) = c ∑ αk ( 2π~ V ωαk )1/2 eαkn(cαk + c+α,−k)e ikx; [cαk, cα′k′ ] = 0, [cαk, c+α′k′ ] = δα,α′δkk′ (21) and transform the Hamilton operator Ĥf in the following way: Ĥf = ∑ αk ~ωαk ( c+αkcαk + 1 2 ) (22) (V is the volume of the system, and eαkn are vectors of polarization (α = 1, 2, 3)). Formulas (21), (22) introduce transversal (with α = 1, 2) and longitudinal (with α = 3) excitations in the system with the dispersion laws ωαk = ωk (α = 1, 2), ω3k = Ω; ωk ≡ (c2k2 + Ω2)1/2. (23) These excitations are photons in the medium and plas- mons, respectively (see another approach to their intro- duction in [5]). Therefore, using the operator (19) as the main contribution to the Hamilton operator of the sys- tem allows us to discuss all processes in it in the terms of photons in the medium and plasmons. Extracting the last term in the expression for Ĥf from operator Ĥ2, we form a new interaction between the introduced excita- tions and charged particles: Ĥ = Ĥ0 + Ĥint, Ĥint = Ĥ1 + Ĥ ′2, Ĥ ′2 = 1 8πc2 ∫ d3xÂ2(x){Ω̂2(x)− Ω2}; 22 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 DISPERSION RELATIONS FOR WAVES IN PLASMA Ĥ1 ∼ λ, Ĥ ′2 ∼ λ2 (24) (operator Ĥ1 is defined in (17), λ is a small parameter of the theory). Our introduction of the main Hamiltonian Ĥf corresponds to the role of processes with characteris- tic frequency Ω in plasma (see, for example, [6]). Pertur- bation theory with the Hamiltonian Ĥint as interaction gives results which can be obtained in usual perturbation theory with the summation of a class of its contribution. The same idea was proposed by Bogolyubov in his theory of superfluid Bose gas [7]. We construct a reduced description of the EM field in the equilibrium plasma by the average val- ues of the electric field and the vector potential ηa(t): En(x, t), An(x, t). At the reduced description, the SO ρ(η) ≡ ρ(E,A) satisfies relations of type (5): Spρ(E,A)Ên(x) = En(x), Spρ(E,A)Ân(x) = An(x). (25) According to (19) and (20), these RDPs satisfy the Peletminsky–Yatsenko condition (8) i ~ [Ĥ0, Ân(x)] = −c Ên(x), i ~ [Ĥ0, Ên(x)] = −c rotnrotÂ(x) + Ω2 c Ân(x), (26) and the kinetics of an EM field in the equilibrium plasma can be built on the basis of integral equation (14). The SO in the zero approximation, ρ(0)(η), can be written as ρ(0)(E,A) = wρq(E,A), Spρq(E,A)Ên(x) = En(x), Spρq(E,A)Ân(x) = An(x), (27) where w is the equilibrium SO of the bath, and ρq(E,A)) is the quasiequilibrium SO of the EM field. This expres- sion for ρ(0)(E,A) is in accordance with the Bogolyubov principle of the spatial weakening of correlations. In other terms, the ergodic relation (11) in the problem under study is the Bogolyubov condition of the complete correlation weakening [4] eτL0ρ0−−−−−→ τ�τ0 eτL0wρq(E,A). (28) This relation takes into account that the free evolution of the system brakes correlations between subsystems of the field and of the plasma (the arbitrary SO ρ0 describes a nonequilibrium EM field in the equilibrium plasma). Using the operator of free evolution etL0 in the above- presented formula is only a tool to express this and has no influence on the domain of applicability of the de- veloped theory. However, according to Bogolyubov, the boundary condition for the Liouville equation must be written in the terms of the evolution in the natural di- rection of time. The necessary boundary condition for the Liouville equation (5) follows from (28) by the sub- stitution ρ0 → ρ(E,A). Finally, we note that, for the calculation of the right-hand sides La(η) of the equa- tions for RDPs up to the second order of perturbation theory, we do not need a specific expression for the SO ρq(E,A), and it is enough to use the last formulas from (27). To derive the evolution equation for the RDP, it is convenient to use the Schrödinger equations of motion for the operators of the EM field [1] ^̇En(x) = c rotnrotÂ(x)− 4πĴn(x), ^̇An(x) = −c Ên(x), (29) where the EM current Ĵn(x) is defined as Ĵn(x) ≡ ĵn(x)− 1 c Ân(x)χ̂(x). (30) Averaging this relation with the SO ρ(E,A) of the sys- tem gives the time equations for the RDPs Ėn(x, t) = c rotnrotA(x, t)− 4πJn(x,E(t), A(t)), Ȧn(x, t) = −cEn(x, t), (31) where the average current Jn(x,E,A) = Spρ(E,A)Ĵn(x) (32) is introduced. In fact, relations (31) are average Hamil- ton equations, because An(x, t) are the average gener- alized coordinates, and En(x, t) are proportional to the average generalized momentum of the field. To calculate the average current in (31), we need a solution of the in- tegral equation (14). The simple consideration gives [8] ρ(E,A) = wρq(E,A) + 1 c~ 0∫ −∞ dτ ∫ dx[Ân(x, τ)× ×ĵn(x, τ), wρq(E,A)] +O(λ2), (33) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 23 A.I. SOKOLOVSKY, A.A. STUPKA, Z.Yu. CHELBAEVSKY where the operators ĵn(x, τ) = e−τL0 ĵn(x), Ân(x, τ) = e−τL0Ân(x) (34) are introduced into the interaction picture. The longi- tudinal and transversal parts Âlnk ≡ Âmkk̃mk̃n, Âtnk ≡ Âmkδtmn; k̃n ≡ kn/k, δtmn ≡ δmn − k̃mk̃n. (35) of the Fourier components of the operator Ân(x, τ) are given by the formulas Âlnk(τ) = Âlnk cos(Ωτ)− c Ω Êlnk sin(Ωτ), Âtnk(τ) = Âtnk cos(ωkτ)− c ωk Êtnk sin(ωkτ). (36) They follow from relations (26) which give ¨̂ Alk(τ) + Ω2Âlk(τ) = 0, Âlk(0) = Âlk, ˙̂ Alk(0) = −c Êlk; ¨̂ Atk(τ) + ω2 kÂ t k(τ) = 0, Âtk(0) = Âtk, ˙̂ Atk(0) = −c Êtk. (37) The results of our calculation of the average EM cur- rent Jn(x,E,A) can be expressed through the retarded Green function of currents Gnl(x, t). It is defined by the formula Gnl(x, t) = − i ~ θ(t)Spbw[ĵn(x, t), ĵl(0)] = = ∫ d3k dω (2π)4 Gnl(k, ω)ei(kx−ωt). (38) In the considered problem, the plasma is an isotropic medium, and the function Gnl(k, ω) has the structure Gn l(k, ω) = Gt(k, ω)δtnl +Gl(k, ω)k̃nk̃l, (39) where the scalar functions Gt(k, ω), Gl(k, ω) are its transversal and longitudinal parts. In these terms, the calculation of the average current with the help of (32)- (36) gives Jn(x,E,A) = ∫ dx′σnl(x− x′)El(x′)+ + ∫ dx′λnl(x− x′)Al(x′) +O(λ3), (40) where the Fourier transformed functions σnl(x) and λnl(x) have the form σnl(k) = − ImGt (k, ωk) ωk δtnl − ImGl (k,Ω) Ω k̃nk̃l, λnl(k) = −1 c {χδnl + ReGt(k, ωk)δtnl+ +ReGl(k,Ω)k̃nk̃l} (41) (χ ≡ Spbwχ̂(0); Spbwĵn(0) = 0 ). The functions σnl(k) and λnl(k) are the conductivity and the magnetic sus- ceptibility of the equilibrium plasma and determine its EM properties, taking the spatial dispersion into ac- count. Equations (31) together with the material equa- tion (40) give a closed set of equations for an EM field in the plasma. However, they do not look like the usual Maxwell equations because of the presence of the lon- gitudinal part of the vector potential. This is a conse- quence of the absence of a time dispersion in the ma- terial equation (40). One can see this, by applying the standard procedure (see, for example, [1]) based on the time Fourier transformation, which gives this material equation as the Ohm law Jn(k, ω) = σnl(k, ω)El(k, ω), σnl(k, ω) ≡ σnl(k)− i c ω λnl(k) (42) (according to (31), An(k, ω) = −icEn(k, ω)/ω). 4. EM Waves in the Equilibrium Plasma The obtained equations (31) together with the material equation (40) can be divided into equations for longitu- dinal and transversal fields. In the terms of the Fourier components, the equations for the longitudinal field are Ėlk = −4π{σl(k)Elk + λl(k)Alk}, Ȧlk = −cElk (43) and give the following time equation for Elk: Ëlk + 4πσl(k)Ėlk − 4πcλl(k)Elk = 0 (here, σl(k) and λl(k) are, respectively, the longitudinal parts of the functions σnl(k) and λnl(k) from (41)). We find a solution of this equation in the form Elk ∼ e−itzl 24 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 DISPERSION RELATIONS FOR WAVES IN PLASMA that gives zl = ±ωl(k) − iγl(k), where ωl(k), γl(k) are the dispersion law and the damping rate for longitudinal waves ωl(k) = 2 √ −π2σl(k)2 − πcλl(k), γl(k) = 2πσl(k) (44) for −πσl(k)2 − cλl(k) ≥ 0. In the opposite case −πσl(k)2 − cλl(k) < 0, longitudinal waves do not ex- ist. The dispersion law ωl(k) corrects the frequency Ω of longitudinal oscillations described by the Hamilton oper- ator Ĥf (see (23)), which is considered here as the main contribution to the Hamilton operator of the system. In accordance with [2, 3], we deal here with plasmons. Equations for the transversal field have a structure similar to that of (43), Ėtk = k2cAtk − 4π{σt(k)Etk + λt(k)Atk}, Ȧtk = −cEtk (45) and give the following time equation for Etk: Ëtk + 4πσt(k)Ėtk + {(kc)2 − 4πcλt(k)}Etk = 0 (σt(k) and λt(k) are, respectively, the transversal parts of the functions σnl(k) and λnl(k) from (41)). We find a solution of this equation in the form Etk ∼ e−itzt that gives zt = ±ωt(k)−iγt(k), where ωt(k), γt(k) are the dis- persion law and the damping rate for transversal waves ωt(k) = √ (kc)2 − 4π2σt(k)2 − 4πcλt(k) γt(k) = 2πσt(k) (46) for (kc)2 − 4π2σt(k)2 − 4πcλt(k) ≥ 0. In the opposite case (kc)2−4π2σt(k)2−4πcλt(k) < 0, transversal waves do not exist. The dispersion law ωt(k) corrects the fre- quency of transversal oscillations ωk described by the Hamilton operator Ĥf (see (23)). In accordance with [2, 9], we deal here with photons in the plasma. This result coincides with our previous one obtained in the Coulomb gauge [8]. 5. Electromagnetic Waves in the Maxwell Plasma The developed theory allows one to avoid some difficul- ties inherent in the standard theory at the calculation of dispersion laws and damping rates of EM waves in the equilibrium plasma (see the standard approach, e.g., in [10]). Here, an analysis of the obtained results is given for the case of the Maxwell plasma, i.e. for the classical ideal gas of charged particles. The consideration is based on a spectral representation of the Green function (38) Gnl(k, ω) = 1 2π~ ∞∫ −∞ dω′ Inl(k, ω′) ω − ω′ + i0 (e~ω′/T − 1), (47) where the correlation function of the currents is defined by the formula Inl(x, t) = Spbwĵl(0)ĵn(x, t) (48) (see, for example, [1]). According to the definition of Maxwell plasma, its particles are completely described by the Maxwell distribution. The correlation function of this system Inl(k, t) can be written in the form Inl(k, t) = ∑ a e2a ∫ d3vvnvlfa(v)eikvt, fa(v) ≡ na ( ma 2πT )3/2 e− mav 2 2T . (49) Taking this expression and formula (47) into account, we obtain the transversal and longitudinal parts of the Green function Gt(k, ω) = ∑ a e2a 2k2T ∫ d3v {k2v2 − (kv)2}(kv)fa(v) ω − (kv) + i0 , Gl(k, ω) = ∑ a e2a k2T ∫ d3v (kv)3fa(v) ω − (kv) + i0 . (50) After the standard calculation [10], we obtain Gt(k, ω) = − ∑ a χaF ( ω kva √ 2 )− χ, Gl(k, ω) = − ∑ a χa( ω kva )2{1 + F ( ω kva √ 2 )} − χ, (51) where the function F (x) = x√ π +∞∫ −∞ dz e−z 2 z − x− i0 (52) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 25 A.I. SOKOLOVSKY, A.A. STUPKA, Z.Yu. CHELBAEVSKY was introduced (υ2 a ≡ T/ma, χa ≡ e2ana/ma; some authors [6, 9] prefer to use the function J+(x) = −F (x/ √ 2). Function (52) has the following properties: ReF (x) = −2x2 + 4 3 x4 +O(x6) (x� 1), ReF (x) = −1− 1 2x2 − 3 4x4 +O(x−6) (x� 1), ImF (x) = π1/2xe−x 2 (53) (see, for example, [10]). According to (51) and (53), the attenuation constants of EM waves in the Maxwell plasma are given by the exact expressions γl(k) = [π 8 ]1/2∑ a Ω2 a Ω [ Ω kυa ]3 e− 1 2 [ Ω kυa ]2 , γt(k) = [π 8 ]1/2∑ a Ω2 a ωk [ ωk kυa ]3 e− 1 2 [ ωk kυa ]2 (54) (here, Ω2 a ≡ 4πχa). In the developed nonrelativistic the- ory, υa � c, and the inequality kυa � ωk is true (see (23)). Therefore, γt(k) ≈ 0 in the considered theory. However, in a consequent relativistic theory, the damp- ing of transversal EM waves in a collisionless plasma is absent [10]. Note that, in the standard theory [1,10], the dispersion laws of EM waves in a plasma and their damping rates are calculated from the equations εt(k, zt(k)) zt(k)2 = c2k2 (zt(k) ≡ ωt(k)− iγt(k)), εl(k, zl(k)) = 0 (zl(k) ≡ ωl(k)− iγl(k)). (55) These formulas contain the transversal and longitudinal permittivities εt(k, ω) and εl(k, ω) of the plasma. In the self-consistent field approximation based on the Vlasov equation for the Maxwell plasma, they have the form [10] εt(k, ω) = 1 + ∑ a Ω2 a ω2 F ( ω kva √ 2 ), εl(k, ω) = 1 + ∑ a Ω2 a (kva)2 {1 + F ( ω kva √ 2 )}. (56) Equation (55) can be solved only approximately with re- spect to γl(k). The result obtained by this way [10] (the Landau damping rate γLl (k)) differs from the above ex- act formula (54) by the multiplier γLl (k) = e−3/2γl(k). However, the standard expression for γLl (k) is valid for kva � Ω. In this situation, the damping rate γLl (k) is exponentially small, and this multiplier is not important. Note also that there is a problem with the usual solu- tion of Eqs. (55) and (56) related to the substitution of complex values zt(k), zi(k) with negative imaginary part in the function F (x) from (52). The simple substitution brakes the rule of pole passing in the function F (x). In our approach, this problem does not exist, because we substitute only real values in this function (see formu- las (41), (44), (46), (51)). A different approach to the Landau damping theory is discussed in [11]. For small wave vectors, formulas (44), (46), and (53) give the dispersion laws of EM waves in the system ωl(k)2 = Ω2 + 3 ∑ a Ω2 a (kυa)2 Ω2 +O(k4) (kυa � Ω), ωt(k)2 = ω2 k + ∑ a Ω2 a (kυa)2 ω2 k +O(k4) (kυa � ωk) (57) which coincide with the known results [9], [10]. In the considered nonrelativistic case, υa � c, and the inequal- ity kυa � ωk is true always (see (23)). In the case of large wave vectors, longitudinal waves do not exist because, according to (53), the expression under root in (44) becomes negative. For transversal waves, a simplification of the dispersion relation (46) on the basis of formulas (53) is impossible due to the in- equality ωk > kυa. 6. Effective Hamilton Operator of an EM Field in the Equilibrium Plasma Here, we construct the effective Hamiltonian of an EM field in the equilibrium plasma Ĥef on the basis of the definition proposed in [12]. The SO of the field subsys- tem is given by the formula ρf(E,A) = S̃pbρ(E,A), (58) where S̃pb is the trace over states of the bath, which gives an operator in the space of EM field states (see details in [12]). Formula (33) for the SO of the system yields ρf(E,A) = ρq(E,A) +O(λ2). (59) 26 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 DISPERSION RELATIONS FOR WAVES IN PLASMA In accordance with (1), (19), and (24), the evolution equation for the SO ρf(E(t), A(t)) can be obtained from the identity ∂ρf(E(t), A(t)) ∂t = − i ~ [Ĥf , ρf(E(t), A(t))]− − i ~ S̃pb[Ĥint, ρ(E(t), A(t))]. (60) According to [12], the last term of this equation contains values of the type Âρf+ρfÂ + (Â is some operator), which give contributions to the effective Hamilton operator Ĥef of the form i~(Â − Â+)/2. The above-obtained results lead to the second-order contributions to Ĥef . Note that the relations S̃pb[Ĥ2, ρ(E,A)] = O(λ3), S̃pb[Ĥ1, ρ(E,A)] = = − i ~ 0∫ −∞ dτ S̃pb [ Ĥ1, [Ĥ1(τ), wρq(E,A)] ] +O(λ3) (61) are true. Using the Jacobi identity, it is easy to see that the last term below gives the commutator of an operator with the SO of the field ρf(E,A): S̃pb [ Ĥ1, [Ĥ1(τ), wρq] ] = 1 2 S̃pb [ Ĥ1, [Ĥ1(τ), wρq] ] + + 1 2 S̃pb [ Ĥ1(τ), [Ĥ1, wρq] ] + 1 2 S̃pb [ wρq, [Ĥ1(τ), Ĥ1] ] . (62) This identity allows us to rewrite Eq. (60) in the form ∂ρf(E(t), A(t)) ∂t = − i ~ [Ĥef , ρf(E(t), A(t))]+ +Ldisρf(E(t), A(t)) +O(λ3), (63) where the effective Hamilton operator of the EM field and the dissipative Liouville operator are introduced by formulas Ĥef = Ĥf − 1 2c ∫ dx{J (2) n (x, Â, Ê)Ân(x)+ +Ân(x)J (2) n (x, Â, Ê)}, Ldisρf = − i 2~ ∞∫ 0 dτ ( S̃pb [ Ĥ1, [Ĥ1(τ), ρfw] ] + + S̃pb [ Ĥ1(τ), [Ĥ1, ρfw] ]) (64) (J (2) n (x,E,A) is the second-order contribution to the function Jn(x,E,A) given by expression (40)). The effective Hamilton operator of an EM field in the medium Ĥef is a quadratic form in the operators of gen- eralized coordinates Ân(x) and momenta π̂n(x) of the field. By using the Bogolyubov transformation [7], it can be written in the form Ĥef = ∑ α,k ~ωα(k) ( c̃+αk c̃αk + 1 2 ) ; ωα(k) = ωt(k) (α = 1, 2), ω3(k) = ωl(k). (65) Here, the spectra ωt(k), ωl(k) coincide with the disper- sion laws of transversal and longitudinal EM waves in the plasma given by the formulas (44), (46), c̃+αk, c̃αk are new Bose operators of creation and annihilation with the usual commutation relations (21). The usual expression for the operators of vector potential Ân(x) in terms of operators (21) is also valid. The ex- pression for new Bose operators in terms of old ones (21) can be given too. So, in this section, we have performed a renormalization of quasiparticle spec- tra introduced by our choice of the leading contri- bution Ĥf (19) to the Hamilton operator of the EM field. Moreover, in the third approximation of per- turbation theory, the effective Hamiltonian will contain terms which describe the interaction between quasipar- ticles. 7. Conclusion Bogolyubov’s ideas in many-body theory allow one to build the kinetics of an electromagnetic field in an equilibrium plasma in the terms of photons in the medium and plasmons. The proposed approach al- lows one to avoid some difficulties related to the cal- culation of the dispersion laws and the damping rates of EM waves in the plasma. The effective Hamil- ton operator obtained in this paper for quasiparticles gives renormalized quasiparticle spectra which coincide with the spectra of EM waves. The consideration is based on the following Bogolyubov’s ideas in many- body theory: the method of reduced description of ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 27 A.I. SOKOLOVSKY, A.A. STUPKA, Z.Yu. CHELBAEVSKY nonequilibrium states (stages of evolution of a nonequi- librium system, the functional hypothesis, boundary condition to the Liouville equation), idea of the lead- ing contribution to the Hamilton operator of a sys- tem, and the principle of spatial correlation weaken- ing. This work is supported by the State Founda- tion for Fundamental Research of Ukraine (project No. 25.2/102). 1. A.I. Akhiezer and S.V. Peletminsky, Methods of Statisti- cal Physics (Pergamon Press, New York, 1981). 2. D. Bohm and D. Pines, Phys. Rev. 92, 609 (1953). 3. D. Bohm, in The Many-Body Problem, edited by C. De- Witt (Wiley, New York, 1959). 4. N.N. Bogoliubov, in Studies in Statistical Mechanics, I, edited by G.E. Uhlenbeck and J. de Boer (North-Holland, Amsterdam, 1962). 5. H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo field dynamics and condensed states (North-Holland, Am- sterdam, 1982). 6. A.F. Aleksandrov, L.S. Bogdankevich, and A.A. Rukhadze, Foundations of Plasma Electrodynamics (Vysshaya Shkola, Moscow, 1988) (in Russian). 7. N.N. Bogoliubov, Izv. AN SSSR 11, 77 (1947). 8. A.I. Sokolovsky and A.A. Stupka, Visnyk Dnipr. Univ. Fiz. Radioel. 10, 57 (2003). 9. A.G. Sitenko, Electromagnetic Fluctuations in Plasma (Acad. Press, New York, 1967). 10. E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics (Pergamon Press, Oxford, 1981). 11. V.P. Maslov and M.V. Fedoryuk, Mat. Sborn. 127(169), 445 (1985). 12. A.I. Sokolovsky and A.A. Stupka, VANT 6(2), 268 (2001). Received 05.10.09 ДИСПЕРСIЙНI СПIВВIДНОШЕННЯ ДЛЯ ХВИЛЬ У ПЛАЗМI ТА IДЕЇ БОГОЛЮБОВА У ТЕОРIЇ БАГАТЬОХ ТIЛ О.Й. Соколовський, А.А. Ступка, З.Ю. Челбаєвський Р е з ю м е На основi методу скороченого опису Боголюбова i квазiреляти- вiстської квантової електродинамiки побудовано кинетику еле- ктромагнiтного поля в рiвноважнiй плазмi. Обчислення прово- диться в калiбровцi Гамiльтона до другого порядку узагальне- ної теорiї збурень за взаємодiєю. Наслiдуючи Боголюбова в йо- го теорiї надплинностi, основний внесок в оператор Гамiльтона поля вибирається з додатком, який залежить вiд взаємодiї. Це дозволяє обговорювати кiнетику електромагнiтного поля в тер- мiнах фотонiв у плазмi та плазмонiв. На основi отриманого ма- терiального рiвняння додаткового до рiвнянь Максвелла роз- глянуто плоскi електромагнiтнi хвилi. Для випадку максвел- лiвської плазми отриманi закони дисперсiї та декременти зга- сання хвиль дають результати, якi збiгаються зi стандартною теорiєю. Однак розвинутий пiдхiд дозволяє позбутися деяких труднощiв цiєї теорiї. Запропоновано метод побудови ефектив- ного оператора Гамiльтона електромагнiтного поля у плазмi. На цiй основi виконано перенормування спектрiв квазiчасти- нок, якi у пiдсумку збiгаються зi спектрами хвиль у системi. 28 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1

Dispersion Relations for Waves in Plasma and Bogolyubov Ideas in Many-body Theory

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