THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING RELATIVISTIC TRANSPORT EQUATIONS FOR PLASMAS,... Ercument Ozizmir A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Nuclear Engineering 1962 June, 1962 IP-571

Doctoral Committee: Professor Richard Ko Osborn, Chairman Professor Kenneth M. Case Professor Charles L. Dolph Professor Chihiro Kikuchi Professor Paul Fo Zweifel

ACKNOWLEDGEMENTS The author wishes to express his deep gratitude to Professor R. K. Osborn, the chairman of his doctoral committee, who suggested this problem to him, and generously provided his invaluable guidance and assistance throughout the course of this investigation, The author is also grateful to Professor Chihiro Kikuchi for his valuable criticism on the presentation which brought about a considerable improvement~ He is grateful to the Horace H, Rackham School of Graduate Studies for awarding him a Rackham Pre-Doctoral Fellowship, to the Radiation Laboratory, Department of Electrical Engineering, University of Michigan and to the Conductron Corporation, Ann Arbor, for their supports through employment which provided the necessary financial support during his graduate program at the Nuclear Engineering Department of the University of Michigan. Finally, the author wishes to express his appreciation to the Turkish Government for sending him to the United States for further study and to the International Cooperation Administration for its financial support during the early part of his stay in this country, ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.. o io o o e........ O.......................... i I. INTRODUCTION...,,.,.,.. O.....o...... 1 II. BASTIC FORMULATION., O o O O O............................. 8 A. Dynamical Aspects o...................O.................. 8 B. Statistical Aspects.... o.... e.........o...........0 o 1(o III. DERIVATION OF THE TRANSPORT EQUATIONS IN THE "SELF-CONSISTENT FIELD" APPROXIMATION......................................... 19 A. Connection with the Usual Vlasov Equation.............. 26 IV. DERIVATION OF THE RELATIVISTIC BOLTZMANN EQUATION............. 29 A Derivation of the "Master" Equation....... o............ 29 B. Deduction of the Boltzmann Equation from the "Master" Equation.o o................o o o o o o o o o o o o o o o................................ o 34 V. SOME ASPECTS OF EQUILIBRIUM....,.....o o..,....o.... o......... 43 VIo BOLTZMANN-VLASOV EQUATION FOR A RELATIVISTIC PLASMA.......O.o 50 VIIo CONCLUSIONS o... o o........ o o o...... o.................................. 59 APPENDICES'A INiVARIANCE OF f(x,p) UNDER LORENTZ TRANSFORMATIONS o.......... 62,B DETAILS OF THE DERIVATION OF THE SELF-CONSISTENT FIELD EQUATIONS................................................... 64 C GAUGE INVARIANCE OF (x, t)...... o o....... o. o...... 71 D CONTRIBUTIONS TO (dFe/at) OF COMPTON SCATTERING AND PAIR PRODUCT N o... o o..o o.. o o o. o o................... o o... o o o 2o o E A QUANTUM DISPERSION RELATION FOR TRANSVERSE OSCILLATIONS IN A RELATIVISTIC PLASMA..................................... 77 BIBLOGRAPHYo o o o o. o o o o o o.o o o o o o oo o o o o. o o........ 79 iii

I, INTRODUCTION In recent years increasing attention has been given to high temperature plasmas due to their importance in connection with controlled thermo-nuclear research. It is estimated that a minimum temperature of roughly 20 kev would be required for a self sustaining device employing the D-D reaction, Minimum temperature for a D-He device would be even higher (- 1.00 kev)o Consideration of plasmas of still higher temperatures is not beyond possibility. Since the rest mass of an electron is, 500 kev, one sees that in an operating thermo-nuclear device there would be a considerable number of electrons with kinetic energies comparable to the rest energy so that a relativistic description for electrons would be required. It is, therefore, of interest to formulate and study the balance relations appropriate to such high temperature systems. Several questions which have received no detailed consideration in the study of plasmas present themselves in the context of very high mean energy systems. In particular, the implications of the processes of pair creation and annihilation on irreversible behaviour and the nature of the thermodynamic state require careful consideration. We shall pay particular attention to these aspects in the present studyo In contrast to the non-relativistic case, it appears that the statistical mechanics of the irreversible processes in relativistic systems has been studied to a very limited extent, One of the earlier works is due to Eckart,( who has given a macroscopic relativistic theory of the simple fluido A relativistic form of the Boltzmann equation for a simple gas was first presented by Taub(2), He derived the hydrodynamical equations and -.7L

-2showed that some of the ad hoc restrictionsimposed upon the macroscopic theory to make it consistent with the special theory of relativity were furnished from this Boltzmann formulation of the theory. A comprehensive, phenomenological description of the equilibrium properties of the relativistic gas was given by Synge). A relativistic form of the collisionless Boltzmann equation was derived and its invariance under Lorentz transformations was proved by (4) Clemmow and Willson ) A systematic approach to the derivation of transport equations for spinless charged particles in the presence of external (5, 6) electromagnetic fields was developed by Klimontovich5 His approach (7) is, essentially, a direct generalization of Bogoliubov's well-known method(7 to the relativistic case. He obtained a chain of equations for the re.la tivistic distributions and showed that the first approximat.ion to this chai.n is the covariant form of the Vlasov equation(5), In the next order the retarded interactions of the charged particles are taken into account. By (6) studying the pair correlation function, Klimontovich derived the relativistic form of the Landau equation an equation of the Fokker-Planck type. "Exact" relativistic Fokker-Planck coefficients for a uniform plasma (8) including radiation were derived by Simon o It should be noted that all of the attempts mentioned above are based upon classical rather than quantum considerations. Nevertheless, one expects that the classical theory would be satisfactory for the description of phenomena in which specific quantum effects are not important. However^ there are several phenomena which can be investigated only through a quantum, mechanical formulation of the problem. For instance, exchange effects which

-3arise due to the indistinguishibility of the particles are specific quantum effects. Creation and annihilation of particles cannot be described classically and therefore require a quantum treatmento The difficulties faced by the classical theory of radiation in predicting the black-body spectrum is well-known. In fact, there seems to be no satisfactory way of treating a gas of photons within the classical formalism. In Syngevs classical treatment3) of the photon gas, for example, where photons were considered as material particles with vanishing rest mass, one is led, as was expected, to an equilibrium distribution which is different than that of the black-body spectrum. It is also true that calculationsof the transition probabilities for many elementary interactions are most naturally carried out using the methods of the quantum mechanics, It is, therefore, important to find out in what way the calculated transition probabilities for certain processes enter the transport equations which describe systems of particles. To our knowledge, no derivation of transport equations including the short range interactions has been given for relativistic plasmas. Another point could be mentioned in favor of a quantum mechani.cal formulation of the transport problem. We are here concerning ourselves wioth aspects of many body systems which depend crucially on electron dynamics. Therefore., it would seem most desirable that the statistical description be based upon the best microscopic theory of the electron, ioeo Dirac's theory. Then, for example, effects on the system arising from intrinsic magnetic moment of the electron will be accounted for both relativisticaly or nonsrel.a tivistically. Furthermore, it is the second-quantized version of Dirac's theory that provides a natural framework for the description of processes in

-4volving the creation and destruction of electrons and positrons and their interactions with radiation fields. Finally, within this framework,the effects on the system due to particle statistics (Fermi-Dirac or BoseEinstein) are naturally accounted for. It is, therefore, our feeling that our understanding of the relativistic (as well as any) plasmas would be incomplete before an appropriate quantum mechanical description of such systems could be achieved and that any attempt which might shed some light upon this problem would be of interesto In the present study, we attempt in a systematic, self-contained manner to formulate transport equations appropriate to relativistic plasmas based upon a quantum mechanical formalism, In quantum mechanics, the description of a system. of many particles is conveniently achieved by the use of the second quantized formalism. In systems where the number of particles is conserved this is nothing but an alternative expression of the many-particle Schrodinger theory. However, in the relativistic systems where actual creation and annihilation of particles may occur it is necessary to account for the physical phenomena. Therefore, we shall employ this formalism in the present study. In the non-relativistic approximation, the second-quantized formalism has already been used both to derive the Boltzmann-Tehling-Uhlenbeck equation(), and to obtain kinetic equations for the particle and photon transport in a fully ionized plasma(lO) We shall attempt in the present study to generalize the latter work to the relativistic case, Another aspect of the problem under consideration is the introduction of the statistical concepts into the description of a system consisting of huge numbers of degrees of freedom consistent with the principles of quan

~5tum mechanicso In classical statistical mechanics, for the description of a system of N particles one introduces a distribution function which depends on the coordinates and the momenta of the particles and which satisfies the classical Liouville equation, A quantum mechanical distribution function (QMYDF)X as a function of the coordinates and the momenta of the particles, was first introduced by Wigner(11) This function; in contrast to the classical distribution function, is not an observable quantity due to the impossibility of the simultaneous measurement of the coordinates and the momenta in quantum mechanicso However, it has striking similarities to the classical distribution function. For instance, the equation satisfied by QMDF reduces to the Liouville equation as hi -O Including higher order terms in n one obtains quantum corrections to the classical equation, It should be stressed that although the QMDF itself is not an observable it can be used as an appropriate weight function for calculating the averages of the obervables pertaining to the system0 A detailed study of Wigner's QMDF is given by Moyal(12) and others. We shall introduce a relativistic generalization of the Wigner distribution function appropriate to the study of spin 1/2 particles described by the Dirac equationo In Chapter II, we introduce the formalism which we adopt for the dynamical and the statistical description of the system0 Dynamical aspects are described via the conventional methods of the quantum. field theory, For the statistical description we use the aforementioned invariant distribution functiono Physical meaning and some properties of the distribution are discussed0

-6In Chapter III, the equation satisfied by the invariant distribution function is derived in the self-consistent field approximationo As E - O this equation. reduces to the covariant form of the Vlasov equation. 4 5) 3 o To the first order in ~ it includes the spin effects on the transport. We derive a coupled set of equations for the invariant distri2 bution function and the spin distributionsignoring the terms of 0( ) The meaning of the higher order terms is not evident since to higher orders the resulting equations are no longer invariant under gauge transformationso In Chapter IV, we derive a Boltzmann equation appropriate to a system of electrons, positrons, and photons in the absence of external electromagnetic fields.1n Chapter V, we discuss some aspects of equilibrium. An H-theorem is proved for the system. in which creation and destruction of the particles may occuro Equilibriumr distributions appropriate to this system, are displayed In Chapter VI, we sketch a derivation of a Bolltzmann-Vlasov equation within the forrmalism developed in this studyo Chapter VII is devoted to the conclusions. Notation We adopt the summation convention unless explicitly stated otherwise, Repeated Greek indices indicate summation from. I to 4~ Repeated Latin indices indicate summation from 1 to 35 A three-vector is indicated with a bar underneath (eog A); a four-vector is indicated with no additional sign. The four-vector A = {A; (a = 1.2,534) has the spatial components A = {Aj}? (j = 2?3) and the imaginary time component A4 - A0 o. Thus,

-7for example, p ~ x - pjxj and p ~ x = p1xX = p ~ x + p4X4 = p ~ x POX0o Dirac matrices, y7, satisfy the commutation relations [y,Z ]+ = 2b We define y5 = YiY2Y3Y4 o We use the following explicit representation of the Dirac matrices j io. 0/ ) J where oa are the two by two Pauli matrices. J Commutator brackets are defined by [ A B ] = AB + BA, where A, B are any two operators.

IIo BASIC FORMULATION We consider a system of interacting electrons, positrons and photons in the presence of external electromagnetic fields, This is taken. for instance, as an idealization of a fully ionized plasma at high temperatureso We assume that the temperature is high enough to require a relativistic description for electrons but not for the nuclei in the system. Our object herein is to derive transport equations which describe the system under consideration. A Dynamical Aspects The dynamical description of the system will be based upon a Lagrangian for the system which is commnonly adopted in. the relativistic quantum field theory of the interacting electrons and which is discussed eliswhere*o The Lagrangian density is given by L(x) = Le + Ly + LI where Le is the Lagrangian density for the free electron-positron field2 Ly is the Lagrangian density for the free electromagnetic field and L- is the Lagrangian density for the interactions. Explicit expressions for these are**nc 2 r~i L = - - N i Y^y1 7 x mc N2 Sr i 6- 6 il L 8Nr N ax J6 j11Ar (2i LIA -, (J iec N I 7 \ ) (2l) See any book on the relativistic quantum field theoryo CGS Gaussian units are used throughout, -8

r9where + (and T = 4 t4) are four component wave operators describing the electron-positron field and A A + Ae A' stands for the internal electromagnetic fields which are quantizedo Ae stands for the external electromagnetic potentials which are presumed to be given and which satisfy the Lorentz condition, namely, a__- 0 (2o2) The symbol N denotes the usual "normal product" which,acting upon a product of creation and destruction operators, re-orders them so that annihilation operators operate first with a factor of minus one being introduced for each transposition of the anticommuting field operators during the reordering. The fieldequations in the Heisenberg representation are readily obtained using the variational principle with respect to V, i and A1 and are given by d mc ie' 7- ++ -! | y A 9 7 ^ ^ c y.. AP. DA'. 7. 8x'', 4 (2.5) where ax ax D k- -

1.0The physically realizable states, o, of the system satisfy the condition that 0 9 iD) = 0 o (2.4) a6 Bo Statist ical Aspects A statistical description of the system wll_ be obtained by the use of an invariant phase-space distribution fIunctiono This will be a relativistic generalization of the Wigner distribution''" appropriate to the study of the spin ~ particles described by the Dirac equationo However, before we introduce this function it seems desirable to review briefly the use of the phase-space distributions within the context of the quantum. theory mainly to establish the terminologyo For simplicity, let us consider a system of N.identical., non-,I interacting par'ticles in a static external potentia.. T.he Sch.rodin.ger equa- tion describing this system. i given by iN -. XNooo 2, _ t + 2} 2,5) w.h.,ere T is the wave funct.ion for the system from. whbch all the possible physical information about the system may be obtained. If. is given at a certain time9 then one, in principle, can determine it at any later time using the above equation. Then^ using W, one could calculate all the observable quantities pertaining to the systemo Specification of T at a given timen wol.d reu.ire te nowd requrge of a the knset of constants of motion whi.ch coul.d in principle, be obtained performing a set of compatjble meas"

-11 ] urements on the systemo However, it is usually assumed that for reasons of complexity this information about the system is not available to us One then sets up an ensemble to represent the physical system under considera.tiono In the present example, each of the systems in the ensemble would have the same number of particleso It is assumed that the state of each of the systems in the ensemble may be specified by giving the complete set of constants of motiono Then, one assumes that the average behaviour of the systems in the ensemble corresponds to the behaviour of the system under consideration. The expected value of an operator 0 is defined by < 0 > = wi d3xl.od3xN * (xl, xNt) 0 1i (x 000xN ), (2,6) where i is the wave function describing the i-th system in the ensemble and wi is its statistical weight' 7), and i runs over the systems in the ensemble o Now, Wigner introduces a function of both coordinates and momenta of the particles by, -N o pd3zi o od, z e 5N l N -2i (p zN 0 i wi (xlzl z, xN zN t). (xl+zl NxN +zN, (2 7) It is easily seen that the integral of fN over coordinates or momenta gives the ensemble averages of the usual probabilities in momentum or coordi

-12nate space, respectively, i.e., fN (xl...,xN; ple.,pNyt) dipl... d3pN - zwi yi(xl,-.,xN), (2.8) Nf. (x12. 1 pl N 0 x,NpNt) d3xl d3xN Wi |i(l tewNaefnt)c| o (2i9) where (pl o. pNti ) is the wave function in the omentm space Properties (8) and (9) must, of course, be required of a sensible phase-space distribution. There is, however, a seemingly disturbing feature of the Wigner distribution. Although it is a real quantity it is not positivedefiniteo Therefore, it could not be interpreted as an observable probability distribution in the phase-space. In fact, one should not expect the existence of a physical probability distribution (in the fine-grained sense) which is a function of the eigenvalues of a set of non-commuting operators (here coordinates and momenta). It should be emphasized that one is usually interested in the phase-space distributions not for their own sake but for the purpose of employing them as appropriate weight functions to calculate the expected values of the observable quantities. It is in this sense that Wigner distribution proves to be a useful quantity. To be specific, let us consider the quantity xT - d3loood~x3 dl apop~N zr fN (( pO 2 p nt) (2o10)

-13which is precisely what we would have written down if we wished to calculate classically the total kinetic energy of the system. Using the definition (7) it is easily seen that Equation (10) may be written as T = E wi 3d x lo.d3xN.i (x1,,xN t)r (X N t) J i,,,,,X: o~lz - ~ 1 Xi(xl,,__ Y J r=l -2m 6x 6xj J ~9 (2.11) which is, by definition (6), the quantum-statistical average of the kinetic energy of the system. This suggests that quantum-statistical averages of the physical quantities pertaining to the system may be calculated by averagingwith respect to fN, the corresponding classical quantities. The equation satisfied by f is found making use of Equation (5) and is given by ft + N r Pf + Z { m - U(xa) ~sin fN - (2012) 6t Cr=1 m axa -2 6x Ux; j j j where the sin function must be understood in terms of its power series expansion. It is seen that as a ->0 Equation (12) becomes identical with the Liouville equation for the corresponding classical system, namely, 6f I a 6p r (x) CUN 1 N + E. i - = 0 at +a = m ax a x ap = ~ Higher order terms in h give quantum corrections to the classical Liouville equation, We should point out that for calculating many quantities of interest the full knowledge of fN is not required but one can use the reduced distribution functions which are obtained by integrating fN over

-14all but a few coordinates of interest. Reduced s-particle ( 1 < s < N ) distribution functions are defined by f (xl,. o ~.xs; p S..apt) = d3x +l d3x d3pS+l..d3pN f (xl _ xN; p. t) It is easily seen that the kinetic energy of the system may be expressed as m ~r 2 T J d3x dp 2m fl(x pt) 0 The equation satisfied by fl(x9pt) is found by integrating Equation (12) over coordinates of particles 2,oooN as - + L+- - -- U(x) sin ( ) fl 0 (2,13) 6t m oxj -i 2 x Pj - A problem of considerably more interest, however, is the case in which particles are interacting via a two-body potential. It is found in that case that the equation satisfied by fl involves the doublet density, f2, the equation satisfied by f2 involves the triplet density, f3, etc. This gives one a quantum analog of Bogoliubov's chain equationso So long as we are interested in but a few reduced distribution functions, a reformulation of the above problem using the second-quantized formalism* seems to be more convenient in that only the relevant distribuFor a detailed discussion of this formalism the reader is referred to Reference 13) Chapter VIo

-15tions enter the discussiono In this formalism, the state function i(t) T. of the system satisfies, in the Schrodinger representation, the equation Th-= dix~,(x) ( - ) + d _ _ it ()( ) 2At(x) + x U(x) X (\x}T) f (t) (21l4) where *,(x),' (x) are operators satisfying the commutation relations [ (X), tt(x )]_ = 5(x-x) for bosons, [ |(x), i+(x')]+ = 6(x-x') for fermions, The singlet distribution is defined by f1('xpgt) = d(h)"- 3 z e -P Z. Wi(.i(t), x - z; x + z) Ti(t) f?^(x~p~t =i ~ \ "3 Si \ / where'p(x; x ) = t(x)>(x ) is the operator for the number density at x The equation satisfied by fl(xgp,t) may be obtained by using Equation (14) and the commutation relations to yield Equation (13) without recourse to the equation satisfied by the N-particle distribution, As we see, however, nothing new is gained by using the second-quantized formalism in this particular problem. So long as no actual creation and annihilation occurs in the system, this formalism is simply an alternative expression of the N-particle Schrodinger theory. However, in systems in which creation and annihilation of particles occurs so that the total number of a particular kind of particles in the system is no longer a constant of motion, this formalism is necessary.

-16It is seen that in the definition of the expected value of an operator given by Equation (6) two different averaging processes are involvedo One is the quantum averaging with respect to the state of the i-th system in the ensemble and the other is the statistical averaging with respect to the systems in the ensemble. The point of view is taken sometimes that in the quantum mechanical formulation of the transport problem. the concept of an ensemble need not be introduced at allo It was argued by van Kampen(l4) that one pure quantum mechanical state corresponded to a (15) classical ensemble, Osborn has derived transport equations for particles and photons in plasmas without introducing an ensemble and has argued that the probabilistic nature of the problem was provided by the quantum mechanics itself. We now proceed with the statistical formulation of our problem. 44 in 5 where d z d'z dzo' and 0 denotes the state vector describing the system in the Heisenberg representation. It is intended that f(xp) could be used for calculating the averages of certain physical quantities pertaining to the system in a fully covariant manner. It will be illustrative at this point to derive an expression which relates the quantum expectation of the When the system is represented by an ensemble we have f(x p) (z) d z e P/Z wi( ijN {(x~z) |(x+z) }i) o J i \ L J1

-17charge four-current to f(x,p). This will also enable us to give a direct physical interpretation of the quantity f(x,p). The quantum expectation of the charge four-current density is given by J (x) = ( n j A ) (2.16) where j = iec N T T(x) y,(x) It follows from the first two of Equations (5) that j can be written as t = Jo + jp where n et a C - e A "SE^; N ^ o^ ^ ^, (2.18) - 2m Vt N tv 1 and where v - 2i [ 7y Yv ]_ This particular decomposition of the total four-current operator into the "orbital" and the "polarization" fourcurrents as indicated by the superscripts o and p, respectively, was, apparently, first given by Gordon (see Reference 16, po343)o One may show that both j~ and jP satisfy the continuity equations t Vt i ii t t t t t By this decomposition we separate from the total current the part due to spin which we do not expect to express solely in terms of f(x,p)

-18Defining'(x) (, An ) - Ae + ( A) J (x) (D, ji) 9 it is a straightforward matter to show that the quantum expectation of the orbital four-current may be expressed as P, J~0(x) = p p ( c, ) f(x,p) + gx) (29) where gjx) e2 N{A }) N{ ) The term g (x) appears as a correlation term in the expression for the orbital four-currento We also note that f(xp) transforms like a scalar under Lorentz transformations as, will be shown in Appendix Ao

IIIo DERIVATION OF THE TRANSPORT EQUATIONS FOR A RELATIVISTIC PLASMA IN THE "SELF-CONSISTENT FIELD" APPROXIMATION We shall derive here an equation satisfied by f(xgp) o To do this, we take the first two of Equations (2,3) and let x - x + z in the first one and let x - x ~ z in the second one to obtain r(x+z) mec ie Y r x + ~- A (x+z) = \AV(x+z) *(x+z). (3m1) / Kx- z) mc - ie - YY *(x-z>!x-Z) YI- A ix-z>(3o2) Sxi' Yl - ) - 7(x-z) y= - (x-z) (x-z 2) We multiply Equation (l) from the left by T(xzz) and Equation (2) from the right by *kx+z) and add the resulting equations to find (3 - 3) 0 4 -r2ip-z/'T Multiplying (5) by ( i) e and integrating over z and taking the expectation with respect to 0 one finds + if c t) d z e T/(x-z)iy7 A(x+z)-A,(x-z) (x+z) = x.~ iV.c'z (354) where 7(p) (-) dz (2ip z/ (x -) i x 1(x+z) The method used here is the same as that of obtaining the continuity equation in the x-space if one takes z = 0 O ~19

-20We note that if g(x) is a function of x, then one may write e -2ip.z/n g(x+z) + g(x-z) 2 -2ip-z/f cos - g (X) (1113 - x 2i e sisin K 2 gp xx Using the relation (5) and recalling that A = Ae + A 9 Equation (4) may be written as d + 2e Ae sin( ) 6arx -c A 2 ax p (3o6) - (d) d4zz e ~2ipz O (x-z) =AH (x+z)-Al(x-z)x+z) = 0 Equation (6) may be further simplified in the "self-consistent field" approximation, that is, if one assumes (9T() Axix-+z)) (xx+z) x+z ) (5.7) ~(x A- tx+z) ) (OA x-Z) _YP(x+z) ) O) This amounts to ignoring the correlations between the electron-positron field and the quantized internal fields. Consequently, we are also ignoring here the correlations between the particles since they interact via the electromagnetic fields. Equation (7) expresses this approximation on a particular term. We shall make similar approximations on other terms in the followingo We wish to note here that the correlation term gj(x) in

-21the expression for the orbital four-current given by Equation (2.19) also vanishes in this approximation. Employing the approximation indicated by Equation (7) in Equation (6) we obtain a + 2e sin (xp —) (x ) = 0 (3.8) 6x Tac p 2 6x dp We now derive an expression which relates 2 (x9p) to f(x,p) This may be obtained from Equations (1), (2) in the following wayo Multiply Equation (1) from the left by r(x-z)y and Equation (2) by v 7y ~(x+z) from the right; then subtract the latter equation from the first one, Making use of the relations 79 7, ]+ = 2 &,[. v ]- Y 2i o.v' one obtains after some manipulations, F { x-z (XZZ)( i (x )a v t(x+z) 2mc +. m |(x-z) 7?v x+z) = ie (x-z)J A(x+z) + Av(x-z) } t(x+z) e- e + -T-c (x-z) av j AI(x+z) - A. (x-z) (x+z) (59) We apply to Equation (9) precisely the same operations and manipulations that we have used to derive Equation (8) from Equation (3).

-22The result is ia/,, e A3,, 0 ^ \},, (10) mc C (x,p) = P P cosy f(xp) P) C V 2 f 7 P) 4-. - + e sin. - a- ).~' (x,p) \2 x c k t 2 ax pp PVV where. (xp' ) =_ (~t) 4 4z e-2ip~z/n (I$D(x-z)~ a (x+z)$ Keeping only the terms to the first order infi one obtains from. Equations (8), (.0) the following equations: / o e o^,, e - (o 4 aC o -ap-) t 1X P) = 5, P 3011) mc)>(xp) = (pV,)f(x,p)+|( + (2p +P.) Introduce a change of variables from {x, p} to (x, r} by the relation p - I: r and define f(xTt) by f(xop) d4x d4r = f(x,p)d4x d4p and similarly for ixjt) and ~(xj) o Noting that under this change of variables { e + a } fa p c ( x) V } x c ox aP ax. c 6xV ax6 oI la.;a~V -1 ~iI 1 V

-23and defining F _P -I, (3153) vv a6v 6 we find from Equations (10), (11) -+ eF 6- iY x, r)H =0 (3 14) mc U(xr Tvf( + J+ F-( )i V~~~~~~~~~~~~p MC7 (X,, 3) vT1 (',) +1 ^ +1 (X T1~x" * 3ol5) V 2 ~.x c I f, 1"8~ b-. P Inserting Equation (15) into Equation (14) and using the antisymietry of Fv, and Ci it follows that {< - ^< ^, ^iJ t-'-.^ i~ a e~ ~ eh 6aF 6F-CEX1 kt + var, c POV (xX1j 4c 6x 1 By definition (15) one finds that ^. + +. (3.17) 6x c6x ^x v |j.a Using Equation (17), Equation (16) reduces to* It should be noted that if terms of 0(fn ) and higher were kept, the equation for f(xIT) is no longer invariant under gauge transformations, that is, the transformations defined by A *A' A 6-A where A satisfies the equation EA = 0o That this should be expected may be seen from the very definition of f(xri), as will. be shown in Appendix Co

-24aF a(]T 6f e 6f eh iv fv e,.8) y( - + - F It + 0 V X c V[ 6 4cX T x (,8 where f = f(x, T), V =. V(xd ) We note that, without the lasttermn Equation.1.8) corresponds to f4 p) the usual Vlasov equation in its covariant form\'5), The last term accounts for the interaction of the spin moments of the particles with the electromagnetic fields. It is observed, however, that with this additional term, Equation (1.8) is not simply an equation for f but involves the quantity (V o In order to obtain a closed set of equations we must find an equation satisfied by d7 We shall not enter any more of the derivations here since they are somewhat cumbersome. The details are given in Appendix Bo However, we shall quote some of the results. We have found it convenient to introduce the quantity S (xp) = ()4 jdz e 2poz/(,N (x-z)1 57 z ix+z). ) We note that 0 may be expressed in terms of S as (see Appendix B, Equation (3) ) -c t(xT) = icvp pS(x,it) + 0(.) ) where c vpa is the completely antisymmetic Levi-Civita tensor density, and S (x9 r) is defined by SL(xp)d4x d4r = S (x,p)ad4x d4p

-25Using Equation (1.9), Equation (18) may be written as 3T6f +e 6f eh VP. v. 0.2 J x + - F~,^ + 7:-7 E -~ - = 0o,!3020) V 6x c "- M 2-c^ x2 - VXCT 5C ~ + - I where = - C2 p Fpr o We find a subsidiary condition to Equation (20) as (see Appendix B, Equation (20) ),2+ m2c2) f(x9 r) e- F+, jSx ) = ~ o2 ) Finally, the transport equation satisfied by S (x, ) and its subsidiary condition are given by (Appendix B, Equations (25) and (24) ) 6 e e P F t } SV- F S * 522):.B2:+ m2o~) S., = 0, (3.25) whrere the terms of 0(.~i) are ignored. Equations (20) and (22) form the coupled set of equations between f(x.7) and S (x:<t) which we had aimed to obtain. Colpled with these equations are the M'axwell's equations which are satisfied between the expected values of the corresponding field operatorso It follows from equations (2o35) (2~4) and (2o21) that for the internal fields we have aX __:____ T i, \ 0`24) dFi+ dx.

-26We can express J (x) in terms of f(xn) and (x9jxt) as J'(x) md1d4n,r'f(x, r) + 5m aX d4r u xjt), >5o26) where the correlations are ignored, Ao Connection with the TUsual'Vlasov Equation We have noted previously that as h ->0 our Equation:118) corresponds to the usual Vlasov equation, However, there is still a slight difference because in Equation (18) all four momentum variables inr appear as independent variables whereas in the conventional form the distribution function depends only on the first three components o As M -~0? Equation (18) and (21) reduces to' x + c F _ =0, ('527) - ax: c' aL + m ) f = 0 30 o28) Equation.28) expresses the fact that f(x9 t) vanishes except when (r2 + m2c2) = 0 which is to be expected. In other words, it implies that f(xr,) must be of the form. f(x.) = E ('F- )Fe(x~) + 6( 0+ -)Fp (x,) (29) where E = m2.712c2 and Fe Fp are arbitrary functions of their arguments o

-27Inserting this into Equation (27) and integrating with respect El E to 0 oTer the infinr;Isimal intervals about n = and T... one obtains o O C O C E, Fe tj Fe E Fe __ r-4- eJ-[lH] E(,E (55) -T + -- + e [rxH] + -2 Ej - 0- ~ (3~0 ) mrc at m axj c -- J mc a mc2 at m axj J me J M.c2 a where Fe = Fe(x,), p = Fp(x ) e - el It is clear that Fe and Fp are interpretable as distribution functions for electrons and positrons, respectively. That is, Fe(x,c)d3x d3jt is the expected number of electrons in d3x about x, in d3n about n at time t In the covariant language (, f(x Js) K|nldO/mc Id4 (5332) determines the probability that the world-line of a particle intersects the hypersurface element dO' and that the four-momentum has a value in d4-. about A If one chooses dO in the direction of the time axis, then Equation (32) reduces to f(x, ) (r43/imc) d3x d43, (35o3) which is interpreted as the expected number of electrons in d3x about x, in d4A about n at time t o The expectation of any quantity

-28Q(xg) is, therefore, defined by < Q >t) = Q(x,) f(x,T)(r4/tim.c)d3x d4 o (3534) (t)

IVo DERIVATION OF THE RELATIVISTIC BOLTZMANN EQUATION We shall consider here an infinite, homogenous plasma consisting of electrons, positrons and photons in the absence of any external fieldso The derivation will proceed in two stages: lo Derivation of a Pauli type equation (Master equation). 2. Deduction of the Boltzmann equation from the Pauli equationo Ao Derivation of the "Master" Equation In the Schrodinger representation the time-dependent state vector of the system satisfies the equation i t) > Ft) > = Ft) > (4o1) where H is the Hamiltonian appropriate to the system. The Hamiltonian can be written in the form H -Ho + (H, -4.2) where H0 is the Hamiltonian of the free-fields and HI is the interaction Hamiltoniano In the present problem, HI is responsible for the self interactions as we.L as the interactions between particles. Therefore, one must be careful in the application of the perturbation theory. However, we shall not be concerned with these questions here since they have been discussed elsewhereo For a detailed discussion of the formal theory of scattering and its applications in the quantum field theory the reader is referred to reference (13) =29

~30The density operator p(t) is defined for a pure-state by p(t) = |F(t)> < F(t)| (4o3) If the state of the system is represented by an ensemble of the systems, instead of this one has p(t) - Z w IFi(t)> < Fi(t)I, (4 4) where i runs over the systems in the ensemble and w, indicatestheir statistical weightso One finds that in either case the density operator satisfies the equation 1t t = [ H. pit)] a (4.5) We denote by |n> a complete set of state vectors in the occupation-number space of the unperturbed Hamiltonian which has the properties that <n m> = 6nm Z |n> <n| - I (4.6) n We expand the state IF(t)> into this set of states as IF(t) > Z cn(t) In> o (4)7) n Thus, the density operator may be written as p(t) = Z cn(t) c(t) In> <ml for a pure-state, nm

A(t) = Z t wicn(t) (t) > <ml for a mixed-state. (4~8) nm i The density matrix in the In> representation is given by Pnm(t) = <n(t) M >| n(t) cm(t) for a pure-state, (409) = W ic(t) c* (t) for a mixed-state so that the diagonal elements of the density matrix, pnn(t). gives the probability of finding the system in the state In> at time t We shall be interested in finding an expression for the time rate of change of the diagonal elements of the density matrixo Using the first one of the properties (7) one finds from Equation. (5) that Pnn = f. 1m { <n|HI^(t) In> }(4.10) nn P Let us consider the "incoming" wave eigenstates, In+>, which satisfy the Lippmann-,Schwinger equation (see Reference 13., po355), namely, In+ = in> + lim 1 - H1 In+> (4ol.l) -, 0+ En-Ho-+i 1 One verifies that H In+> EnIn+> <n+ Im+>= bnm Z In+> <n+l I Z I > < I > (4.12) n where I|> denotes the set of bound states of the total Hamiltonian H

-32It is expected that the bound states would have only a small effect on the transport in a relativistic plasma. Thus5 ignoring the bound states of the Hamiltonian, one may take E n.+> n+ ^ I o n Using the set In > we write Equation (10) as 2 rm at Pnnt) _- ~ Im pnlHI E m> <n lPt) \p+> <p+ln (401.3) In view of Equation (4o11) we write <m+(t) p+ + mp + am (4o14) which also defines am o Note that amp vanishes as HT- 0 o We shall ignore the contribution of these terms in the present analysis. Finally, we employ the so-called "random phase approximation". One argues that the major contribution to the time rate of change of pnn,(t comes from the diagonal elements of the density matrix, and that the effects of the off diagonal elements tend to cancel out when averaged over small intervals of time, When the system is represented by an ensemble, it is presumed that the ensemble is set up with "random a priori phases" such that off diagonal elements of the density matrix vanish at time t In any case, this quick elimination of the off diagonal elements is far from being satisfactoryo Nevertheless, this assumption is usually made in the derivation of the transport equations and a detailed analysis of this point is beyond the limits of the present studyo Ignoring the contribution

3553 of the off diagonal elements one finds at Pnnt)' m(t) Wm- n (4 ) The quantity mn s2 Im <nHilm+> nm+ln>,(4.6) appearing in Equation (15) is equal to the increase per unit time of the probability that a system initially in the state In> will. be found in ni> (see Reference.1.3 po324) and may, alternatively, be written as 2 2iv 1241 wmn =- 6 5rn {RJ + - (Em-En) Rnml2 (41) where'i H1 R:= H. + np pm +. l (4o18) ] — nm p Em-Ep+ ic It may be shown that (see Reference 13, po325) w. - m,...W. ~.m (h4.19) wm Im w n 4.l) n which expresses the fact that the rate of transition out of the state m> must be compensated by a decrease in the amplitude of the state Im> o Using Equation (1.9) one finds from Equation (15) that Pnn(t).. {Pmm(t)Wm n " Pnn(t)wnm} (20) mn.

4 where YT2V 5,Em En) IRnrl2 Equation J(0) is commonly referred to as Pauli or "master" equation. B. Deduction of the Boiltzmann Equation from the "Mast-+er" Equation'or convenience, we shall normalize the system, in a large but finit;e box of volume V with the periodic boundary conditions, We denote by N (k) the operator representing the number of particles of i-th kind in volume V with momentum k, The subscript i refers here to electrons or positrons of spin up or down and to photons of either polarization., Ni(k) has the property that it commutes with the Hamiltonian of the freefields, i eo [ N(k') H0 0 (4.21) The set of states In> which has been introduced previously diagonalizes both H0 and Nr(k) simultaneously so that in addition to properties (6) one has Ni(k) In> = n.(k) In>, (4,22) where n. (k) is the number of particles of i-th kind with momentum k in the state In> We define fi(kt)' t ((t), Ni(k) F(t) ), (4 23) so that fi(kjt) gives the expected number of particles of the i-th kind

-35with momentum k, per unit volume, at time t. Using the expansion (7) one finds fi(kt) = V P(t) n(k), (424) where Pn(t) Pnn(t). It follows from Equations (20) and (24) that.. =t Pn (t) rn, m {mi(k) - ni(k)} (4.25) 6t n m tnm m~n where rn m - Vn = V 5(Em-En) |Rmn12 (n m) (4.26) It should be emphazised that the normalization in a finite box of volume V is introduced only for convenience; we shall be interested, eventually, in the limit of the Equation (25) as V -4 0 The scattering amplitude, S mn (see Reference 13, po323) is related to Rn by Smn = 2ii (EmEn) Rmn (m t n) (4.27) Noting that +T/2 2 lim i(EmEn)t/T dt 2iti (Em-E), Tim e T-4 oo /2 one obtains an alternative expression for rn m as r =lim 1 IS |2 n —m V- oo VT mnlVT T-oo

-36where rn_ m is interpreted as the transition probability per unit spacetime volume that a system initially in the state In> will be found in the state Im>. In the following we shall restrict ourselves to the second order* perturbation analysis for the calculation of the transition probabilities. We hope it is understood at this point that our purpose is not to re-calculate the already calculated transition probabilities for several collision processes but to indicate the manner in which they enter into a Boltzmann type equation. The S-Matrix, in this approximation, is given by S(2) 1 (e ) /dx42d4x1 T {N(y7 A )l 21 1c 2 IJL where T denotes Wick's chronological product,which acting on a set of operators re-orders them such that the earlier time operators operate first with a factor of minus one being introduced for each transposition of the anticommuting fermion operators during the re-ordering. In the present case, no change in sign will result since the fermion operators appear in pairs in each term. The momentum space expansions of the fields in terms of their * This is the first non-vanishing order in the present analysis. For a systematic and concise discussion of the processes described by S(2)the reader is referred to Reference 18, Chapter 14o

-37positive and negative frequency parts are given by ((x) = T (= ),cs(p) ) Cp) e p /, p s=l H(x)) 1 m cmE2 1/2 2tt( u (r+2)) e-iq'x/ t (- (x) V _ E — r( u (q) e 1TV q q r=l A~(x)/, 1 2 c2 1/2 2 (s) -ikpx/ A ( =~ ct (px) _ Z (p ) Zam(k) e (k) -(+) i me 2 /22 (r+2) iqox/ l (x) C= -- Z /q)2 d (q ) e V qEq r=l -.(+)/ 1 y 2ntc2 1/2 4 (m) ik -x A (+V(x) I ( -~)_ am(k) c (k) e kV k - - -- (-) nc2 1/2 t (m) -ikx A()(x) F=, (2 ) am(k) E (k) e, (4.28) -I.,V k - m=l - where the terms have the same meanings as in Reference 18, and cX = c|k| Boltzmann Equation for Electrons The processes which effect the electron transport to -ne secona order in perturbation analysis may be summarized as follows: 1. Electron-electron scattering, 2. Electon-photon scattering, 30 Electron-positron scattering, 4. Two quantum pair annihilation or creation, 5o Pair annihilation or creation in the field of an electron or positron.

-38To illustrate the method of deducing the Boltzmann equation from Equation (25) we shall consider in some detail the first two of the above processes. Electron-Electron Scattering (Moller Scattering) The part of S(2) which describes this process is given by (2)= 1 e(4 42 N 2( 4vv 4)X2( -) e-e 2", ftcj-2 _ __ x_ where Av(x2)A (x1) = 2rtic 5bDF(x2-Xl), DTF() =- d 2 k (2n)4 J~k ieDefining, r(e-e) m (2) In>2 n -,m VT-l e[e'<m -e one can show that F(e-e) (2_ 2 2 A2e-e n — * m V E E 1E E 2 Li E2 El S2. si ~s2 sl (4(4.29).{l-ne(l')}{l-ne(2' )}ne(2) ne(l)<ml..,ne(l')+l,..,ne(2')+l,..,ne(2)-1, O o,ne (1)-, o, > where ne(i) n me(pi si), 6p denotes the four-dimensional Kronecker

-39delta and Ae-e Ae-e (2';1' 2;1) ( u(s2)(p)) (S(Pi) up s. ((S )) (p _p )2! 1 (4.30) ( (p) u(S2) )) (S2)(Yu( l)(p)) (p2-Pl) 2 It may be shown that A'-' possesses the following symmetry properties: A-e(2';11 2;1) = - Ae-e(1';2' 2;1) Ae-e(2;1 | 21') (431) It follows from Equation (25) that the contribution to the electron transport due to the electron-electron collisions is given by afe (p,s,t) = e Pn(t) ree {me(Ps) - ne(p,s)} (4.52) 6t nPm n n-, m

-40Substitution of (29) into Equation (32) yields __e = (2ire 2f) 2 Z Pn(t) ZPIt,P EiEE 2P1-P-PI s i, s{1s Ae(2:;l' p,s;l) 2 {{lne(Ps)ne({l- l)}ne(2')ne(l') {l-ne(2)}{1-ne(l')} ne(p,s)ne(l) (455) In obtaining this equation we have carried out the summation over the states Im> and made use of the properties given by (31) We now approximate the average of the products of the occupation numbers, ne, by the product of the averages, namely, we assume?' Pn(t)ne(i)~o~ne(j)o.. - { { Pn(t)ne(i)}.oc Pn(t)ne(j)}o~ O n n n (4n34) Making use of this approximation in Equation (33) one obtains ( fe) (e — fe 42e V E Ae-e(21;l I p,s;l) 2 at 2 PV4 t P2P PP-P-p P1 {1-Vfe} {-Vfe()} Vf(2') V fe(2l') 48 m c - 1-Vf (2')} 1-Vf (l') V Vf (1) E - (4.35) e he J e e E E E E wEh2e 1Ep fp where f- f (p,st)

-41Transition to the continuum variables in the momentum space is achieved by defining 7d3 fe(Ps,t) = Fe(p,s,t) d3p, (4.36) pcd3p and noting that one must replace 7Z 1 -_ - V d5p ped3p (2rti)t V 6 ~ (2i)3 &(p) T &E (2n) 6(Ep) (4.37) p In the continuumvariables, Equation (35) becomes Ep aFe(p,st) 1 (2e2)2 p2, d5 3 p m3c6 _ _, ~ o3p'pl- 2epI mI06 mcz 6t s2S l1 2, s E'E1 1 mc2 ot 2 5(p+pT p -p l) 6(E +E -E -E1) Ae-e(2';' p,s;l) (2+P- -l 21 1' p 1 o {{PFe(P's't)} {P-Fe(1)} Fe(2') Fe(I') - {pFe(2')} {p-Fe(l')} Fe(p,s,t) Fe(l), (438) where p _ (2ir5)"-3 If we assume isotropic spin distributions, that is, Fe(Ps,t) =2 Fe(p,t), (s = 1,2)

-42then Ep 6Fe (p,t) 1 (2e2)2 ~ 2d3p~d3p, dnp m3c6 mc- Et) - =2 -)2 P E'' __ _ 6(p, + p -P p - pl) mc6 -t 2 4 1 2 1 1 2E2 E - ~1 - * 6 (E2, + E1, - Ep - E1) Be-e (pp P_,P) {{2p - Fe(Pt)} {2p - Fe(l)} Fe(2') Fe(l') { 2p - Fe(2')} {2p - Fe(l')} Fe(p,t) Fe(l), (49) where (Pt- -L L P0 (2) = s Ae-e(2';1' I p;1) It is noted that for the explicit calculation of the collision terms we have specialized to a particular frame of reference by working in the Schrodinger representation. However, the covariance of the resulting Equation (38) may be seen if one observes that for a homogenous Ep 6 L 6 d~p system 2 = - - and that is an invariant, and also that mc2 t m e x p rsd Mc x11 Ep JAe-e 2 is expressed explicitly in a covariant form by Equation (30). A similar derivation concerning the contributions to the electron transport due to the electron-photon scattering and the two quantum pair creation and annihilation will be given in Appendix D.

V. SOME ASPECTS OF EQUILIBRIUM Previously, we have derived an equation describing the electron transport which has the form 6Fe(p,s,t) = - Fe=t ( Fe/6t) + e +.4 ~ 6t e-e c,s, In this section we shall discuss briefly the implications of this equation for equilibrium systemso In particular, we shall prove an H-theorem for electron-photon systems. For the present, we ignore the positrons so that only the terms describing electron-electron and electron-photon scattering need be considered, Before proceeding to the proof of an H-theorem, we need a transport equation for photons also, This, however, may be written down immediately in view of the derivation leading to Equation (D-4). One has dX (_K,m,t) at = (aX/at) + ~, where c s = (ec)2P Z,, d3p'd3p d3' E b(p' +' - P - K) (-) m = (e~c)242 du c.S. s}sm' sf' ~f EpEp, - - * & (Ep.-KI'-Ep -~) i Ac Sop,'m'l 2 e6 (E W-s.T-E -i) ACSo_(p',s';;',m'jps;Km)l {p-Fe}{P+X}F1X' - {p-Fe}{p+X'} FeX (5.1) where of_ 5fo, K _ is o Equation (1) gives the time rate of change of the -43

-44photon distribution due to the collisions with electrons. Corresponding expression to this, in the non-relativistic approximation for electrons, has been derived previously ) Let us define a quantity S(t) by ) S(t) - k Z d3p { Fe ln(Fe/P-Fe) - p In (p/pFe) -k d3k X In (X/p + X) + p In (p/p+X) (5.2) where Fe - Fe(ps,t) X X (Kmt) Using Equations (4o38), (D-4) and (1) it is a straightforward matter to show that dS/dt > 0, and that dS/dt - 0 if and only if the foliowing conditions are satisfied simultaneously2 that is9 Fe(2) F e! n Fe(2') in F —l- + In F +n F( + (5 3) p-Fe () p-Fe(2) p'Fe(v) pFe(2?) for all si s2 si s 1. 2 provided 2. +22 =i (5.4) E +E (515) 1 2' 2

_45and Fe X Fe Xi in - + in - - = in + in (506) p -Fe P+X p-Fe p+X' for all sm,s Ym =1,2 provided p' + r' (5 7) P + K = P + s (5o7) E + 5 = E + o (5o8) P P' It is interesting to note that an examination of the quantity d(t) shows dt that it is a relativistically invariant quantity. Therefore9 if it vanishes in one frame of reference, then it vanishes in any other Lorentz frameo If we call S(t) the entropy per unit volume we may state that the time rate of change of the entropy per unit volume is the same in all Lorent z frame s One finds that the Equations (3) and (4) under the constraints (4), (5). (7) and (8) admit the solutions F (-ps) = - (s 1)9 2) Be e +1K and X(:,m) =, (m=2,2) Cy e'Kl. where Bey Cy7 C are, so far, arbitrary constants independent of p In order to be interpretable as physical photon density, X(,nm) must be non-negative for all values of _ a This restricts a

-46to be a time like vector. Therefore, one can always make a Lorentz transformation so that time axis coincides with a o In this frame of reference, electron and photon densities are expressed as Fe(Es) = Be +' (s = 1,2) B e EP +1 X(Jm) = C e, (m = 1,2) where 3 is a positive constant. Note that non-negativeness of X(_,m) also implies that C > 1 o The common constant p is identified with l/kT where T is the temperature of the system. There remains, however, two more constants, namely, the constants Be and Cy to be determinedo The only additional knowledge is the number of electrons per unit volume in the system which determines the constant B o It is seen that in this hypothetical system which consists only of electrons and photons and which interact only through electron-electron and electron-photon scattering there is no way of determining the constant C o That means that equi7 librium photon distribution is not completely determined. However, this is not the actual system under consideration. We shallnow show that in the physicsl system which consists of electrons, positrons and photons this arbitrariness is removed by the consideration of all interactions which are described to the second order in the perturbation analysiso Let us consider the effect of the two-quantum pair creation and annihilation on the equilibrium distributions. The system now includes electrons, positrons and photonso The additional condition to be satisfied by the equilibrium distributions is found to be

-47Fe F X2 Xi in -- + n - = n -+ n -(59) p-Fe p-Fp p+X2 p+Xl (Fp Fp(q,r)) for all s,r,mi = 1,2, provided P + = K2 + _ l E + E =aY2 + ~ p q 12 Equilibrium distributions are Fe(PS) = B e"P +1' (s = 1,2) Fp(qr) = Ep 9 (r = 1,2) Bp e-p +1 X(K,m) =, (m = 1,2) (5010) C e -1 with Bep = C2. (5011) Finally, from consideration of the pair creation and annihilation in the field of an electron one finds that the equilibrium distribu — tions must also satisfy the condition Fe Fe(1) Fe(2) Fp ln - = ln - + In + In - p-F p-Fe(l1) p-F (2) P-Fp

-48for all s',si,r = 1,2 provided that P = i + P + q, Ep, = Ep, +Ep +E The distributions (10) satisfy this condition only if 2 B = BBp (5012) From Equations (11) and (12) and recalling that C > 1 one finds Cy = 1, BeBp= 1 o In the thermodynamical language this corresponds to the fact that the chemical potential of the photon gas is equal to zero and that the sum of the chemical potentials of the electrons and positrons vanish (see Reference 28, p.325). In the reference cited an identical conclusion is drawn strictly from thermodynamical considerations, First it was established that the chemical potential of the photon gas should vanish because of the requirement that the free energy of the photon gas should be minimum, for given temperature and volume, as a function of the number of photons in the system. Then, considering the pair creation (or annihilation) as a chemical reaction it was concluded that the sum of the chemical potentials of the electrons and the positrons must be equal to that of the photon gaso Here, strictly from a consideration of the kinetic equations describing the system we have shown that the system approaches to this equilibrium state irreversibly, It is noted that Equations (11) and (12) together with the knowledge of the net charge per unit volume in the system is sufficient

-49to determine all the constants CyBeBp for a given temperature. If the net charge associated with the electron-positron field* per cm3 is en then one can determine Be(or Bp) from the relation 2 2 Z d3p F (P s) - Z d3q F (q,r) = n, (513) s=l e r=l or, equivalently, 4it d - -_ - = — o (5ol4) 4 iI Be eEP +1 Be eP+EP +1 p 2p (5.14) The implication of this result is important particularly for very high temperature equilibrium systems (kT X mc2 or greater). One (28) finds that, while the density of the electron-positron pairs is negligibly small for non-relativistic systems, it already exceeds by a great deal the usual densities of atomic electrons for kT i mc2 For extreme relativistic temperatures, the density of the electron positron pairs increases with (kT/hc)3 Over all charge neutrality is assumed to be maintained by a uniform background of ions.

VI. BOLTZMANN-VLASOV EQUATION FOR A RELATIVISTIC PLASMA In the derivation of the self-consistent field equations we have ignored the correlations completely. In this way, we were able to obtain the usual Vlasov's equation for a relativistic plasma with additional spin-dependent terms. It is usually assumed that the Vlasov equation, embodying in itself the self-consistent internal fields is an appropriate starting point for the study of the collective behaviour of plasmas. Starting with the Vlasov's equation [Equation (3.30) ]coupled with the Maxwell's equations, the linear oscillations in a relativistic plasma has been studied to some extent by several authors.(21-24) The absence of the correlation terms in the Vlasov's equation reflects itself in the fact that this equation does not have any indication of how the equilibrium is attained in the system and what the equilibrium distributions should be. On the other hand, employing a field-theoretic formalism we have also derived a set of Boltzmann type equations for a spatially homogenous,'relativistic plAsm&a in the absence of external fields, We have shown that, in the sense of certain approximations, these equations imply unique equilibrium distributions for the constituents of the plasma. A question now arises: what is the connection between the correlation terms that we have ignored in the derivation of the Vlasov equation and the collision terms of the Boltzmann equation? To acquire some insight into this question let us return to Equation (3~6) where -50

-51the correlations have not yet been discarded. We have + 2e Ae sin(t a a ) V ax fic P 2 ax p (61) e_ (Af)-4; d4z e2iPZ/(N{x-z)7 (A(+)A(x-z)) (x+z)}~)= 0. -tic We have seen in Chapter III that if one uses the "self-consistent" field approximation on the last term in Equation (1), then one is led to the Vlasov equation. We wish to investigate here in what sense one may incorporate the collision terms with the Vlasov's equation. We introduce, on physical grounds, a decomposition of the internal potentials into two parts(10) AI =A + A, (6.2) where it is presumed that AS is responsible for the "self-consistent" fields and Ac is responsible for the interactions which require a "collision" description. We shall employ the self-consistent field approximation in the same sense as before on all terms involving As but noton c AC Introducing this decomposition into Equation (1) one finds 4- -4 __p 2e'. 3_ ~x+ -c sin( -6) pa.4 +e -O sin(- a-) C/ = I, (6.3) 6x t-fie 2 6x ap where -- Ae + (,As), I - -()-4fd4z e 2ip Z/(ON{(x-z)7y,(Ac(x+z)-A(x-z))*(x+z)})o

-52To avoid additional complications we ignore all the terms of order Ti on the left hand side of Equation (1) so that a +e. - I o (6.4) 6x c X aXV Pv From Equation (3o10), ignoring again the terms of order Ti, we find mc2 (pv -e ~) f. (605) Changing variables from {x,p} to {xj} by the relation p _- e = t and inserting the relation (5) into Equation (4) one obtains Df(x,i) = I, (6~6) where D = 6 0 + e Fv4lr a6 6-~ ~1 c V 11 v e I em( ( j ) -4fd4z e-2i( + c )~z/T(,N{(x-_z)7 (A~(x+z) -A(x-z))t(x+z)}D). Integrating Equation (6) over the variable Ito and making use of Equation (3.29) it follows that {DeFe(xT) + DpFp(x,-)} = I, (6 7) where De = E{ X + F} = E/c EI X 6kt FjC i jIj 0 = Et - I+ e Fi j lt } Dp = E- { ax + T F O O = -ES/C, I = -() 3d3z e-2 i( )Z' ~ (~,N{(x-zt)7/(A^(x_ )t)-A^(xt))t(x+z,t)} ~ ).

-53Let us now examine the term I more closely. We first note that it may be expressed in the Schrodinger representation as I =- -()-3jd3z e-2i(+) z/t (F(t) N{t(x-z)7y (Ac(x+z)-A((x-z))*(x+Z)}F(t)). Using the commutation relations of the field operators one can show that I = i(F(t), [f(x,),HI] F(t)), (6.8) ifJ where C C HI - iefd3x N{t(x)7pA(X)()}, (x^,) _(n)3 jd3 z e 2i(+c ^ )2 / N{ft(xz)4(x+z)} Let us fix our attention to a volume V about the space point x. We define the coarse-grained distribution, Fe(x,_), by the relation F_ (x,) = 1 f d3x Fe(x,Jt). e - V xEV Assuming that Fe, Fp and Fjl are slowly varying over the dimensions of the volume V, one obtains by integrating Equation (7) over the volume V {DeFe(x,_) + DpFp(x,-_)} 1 1 (F(t), [ f d3x f(x,r),Hj]_ F(t))o ii V xeV (6.9) The right hand side of Equation (9) gives the effect on transport of the collisions which occur in the presence of the eene external and the self-consistent fieldso One notes that the self-consistent fields

-54are not really known until the whole problem is solved. Also, without an explicit knowledge of the external fields,the states of the particles which participate in the collisions would not be available. Therefore, we specialize here to a case where the external and the self-consistent fields do not effect the collisions appreciably. In other words, we assume that during the collisions we may consider the particles to be isolated from the rest of the system so that initial and final states of the particles may be taken as the free-particle solutions of the Dirac equation. Thus, using the expansions given by (4.28) with x4 = 0 (in the Schrodinger representation) and ignoring here the term compared to s for consistency,one finds that f d3x (x,t) ~ Ne(es) - Z Np(-I_,r), (6.10) xCV s=l,2 r=1,2 where Ne(,) C()Cs(), Np(,r) t= drr(~). It is noted that Ne(it,s) is the operator for the number of electrons in the volume V, with momentum _ and spin s; and Np is similarly for positrons. Inserting (10) into Equation (9) and identifying the contributions to the electron and positron transport separately we have DeFe( v(F(t) [ Ne(s) H] F(t) N), (6.11) ifn V r-2 P r-l,2 where Fv(t) is a state vector in the Schrodinger representation which approximately describes the system in volume V during the time of the

-55collision.* In the sense of these approximations, the state vector IFv(t) > satisfies the equation if - | Fv(t)> = Hv|Fv(t)>, 6t where Hv is the total Hamiltonian in V in the absence of the external and the self-consistent fields. Noting that Ne, Np commute with the Hamiltonian of the free-fields we may re-write Equations (11) and (12) as DeF-e(x,_) = OzFe (it,s,t) (613) s=1,2 at DpFp(x,-) E aFp (-_,r,t) (6.14) r=l,2 (1 where 1 Fe(_,s,t) =- (Fv(t), Ne(_,s)Fv(t)), Vp(N,r,t) = V(Fv(t), Np(t,r)Fv(t)) Equation (14) may also be written as DpFp(x,-) = E (,r' t), (6.15) P P r=l,2 6t where D' differs from De only by a change in the sign of e, as p would be expected. Explicit forms of the collision integrals arising from the right hand sides of Equations (13) and (15) have been given in Chapter IV. The assumption which is being made here is that the state vector IF(t)> may be factorized as IF(t)> | Fv(t)>lFo(t)>, where V and 0 refer to the volume V and to the outside of volume V, respectively, and states are normalized separately to unity. Since Ne, Np represent only the particles in V, |Fo(t)> may be omitted here.

-56These equations will be called Boltzmann-Vlasov equations since they include both the self consistent fields of the Vlasov equation and the collision integrals of the Boltzmann equation. They are coupled due to the collision terms describing creation and annihilation processes. In this section, we have given a systematic (although quite intuitive) derivation of a set of Boltzmann-Vlasov equations for electrons and positrons in a relativistic plasma taking pair creation and annihilation into account. Also, by this "unified" derivation we have shown the connection between the two problems considered separately in Chapters III and IV, namely the derivations of the Vlasov and the Boltzmann equations. Lastly, we have tried to indicate qualitatively the restrictions on the validity of the Boltzmann-Vlasov equations by indicating the assumptions needed. to be made in such a derivation. We wish to make a few more comments on the validity of a Boltzmann-Vlasov description of a plasma, not necessarily a relativistic plasmao If one considers the Vlasov equations (where all correlations are ignored) as the "lowest order" in an approximate description of a plasma, then the collision terms appear as corrections to the Vlasov equation. Objection, however, may be made to the validity of this description, especially for dilute plasmas. The point is that the collision terms describing the interactions between the charged particles are of binary character and in a dilute plasma these encounters would be quite rare, therefore, less important compared to the simultaneous interactions of many particles due to the infinite extension of the Coulomb interactions. Recently, this problem has been discussed successfully

-57for the non-relativistic systems interacting via the Coulomb forces by both Balescu(25) and Guernsey(26). Using e2n where n is the density of electrons, as a parameter of smallness and summing explicity the contributions to all orders in this parameter, Balescu was able to derive an equation appropriate to the description of a plasma which has also been obtained indepently by Guernsey. It was shown that(26) this equation could be cast into an equation of the Fokker-Planck type * An equation of this type may also be derived from the Boltzmann integral for charged particles by considering the collisionsof small momentum transfer and by introducing an appropriate cut-off procedure to avoid divergences. This equation is usually referred to as Landau Equation(27) The difference between the two FokkerPlanck type equations is reflected in the coefficients, Whereas the coefficients aj, bjk in the Landau equation are the first and second moments of the momentum change in a binary collision, the coefficients in the Fokker-Planck equation given by Guernsey do not have such simple interpretation and are of more general validityo In the latter case, the coefficients reflect the non-locality of the interactions. In this connectionthe work by Lenard(28) should also be mentioned. Starting with an integral equation proposed by Bogoliubov to determine the time development of the velocity distribution for a spatially uniform plasma, he has shown that the time derivative of the velocity distribution can be expressed, exactly, in terms of the distribution itselfo The resulting equation is again of the Fokker-Planck type, * By Fokker-Planck equation we mean here an equation of the form 3F 2 ( 1 2 at ap (ajF) + 2 p p JK' where coefficients aj, bjk themselves may depend on F o

-58He has further shown that by means of an excellent approximation it could be reduced to that derived by Landau. This indicates that binary collision description although restricted in general terms does contain sufficient information for many purposes. Finally, we note that our Equation (4.39) which describes the electron-electron collisiornfor relativistic systems could also be reduced into a Fokker-Planck form. Since the retarded interactions as well as the Coulomb interactions are accounted for in the transition probability appearing in this equation,we expect that the resulting Fokker-Planck equation (when specialized to non-degenerate systems) should correspond to that (6) derived by Klimontovich through the study of the pair-correlation function for a relativistic plasma.

VIIo CONCLUSIONS In the present study, we have formulated transport equations appropriate to high temperature plasmas based upon second-quantized Dirac theory of electrons. For the study of the statistical aspects of the problem we have introduced an invariant phase space distribution, in analogy with the Wigner distribution function,(11) which is suitable to the study of spin 1/2 particles described by the Dirac equation. In Chapter III, we have derived a coupled set of equations between this function and the spin distributions which reduces to the covariant form of the Vlasov equation as f -> 0. In Chapter IV, we have derived a Boltzmann equation for electrons in a system consisting of electrons, positrons and photons which to our knowledge has not previously been given. We have used the S-Matrix formalism and restricted ourselves to the lowest non-vanishing order in the perturbation expansion. Employing the results of this chapter, in Chapter V we have discussed the equilibrium properties of this system. An H-theorem was proved and the equilibrium solutions were displayed. The results are in agreement with those obtained strictly from thermodynamical considerationso(28) This question also does not seem to have been examined previously from the point of view of the quantum electrodynamics. We have obtained here the kinetic equations describing the system and its approach to equilibrium based upon a field theoretic formalism and have shown that these equilibrium distributions are attained irreversibly. The kinetic equations for relativistic systems based upon a classical formalism had necessarily ignored such processes as pair creation and annihilation which,as we have shown -59

-6oin Chapter V,is necessary to consider for a complete description of the system. The interesting feature of this result was that for very high temperature systems (kT& mc2) the density of the electronpositron pairs in the system is very large compared to the usual densities of the atomic electrons in the system. This, for instance, would have a profound effect upon the propagation of the electromagnetic waves in such systems. In the study of the plasma oscillations in relativistic systems, reference is made sometimes(23,24) to extremerelativistic limits without a consideration of the electron-positron pairs which would be present in such systems. Therefore, caution would be necessary in interpreting these results from the point of view of actual physical systems at extreme relativistic temperatures. Finally, in Chapter VI, we have sketched the derivation of a Boltzmann-Vlasov equation for an electron-positron system taking pair creation and annihilation into account. We would like to point out that in the present study we have restricted ourselves with a lowest order description of the system in that the "dressing" of particles or photons by the collective effects of the medium were not taken into account. It has, however, been shown in the study of the photon transport in non-relativistic systems(l9) that the consideration of the first order collective effects in a plasma leads one to a modified black-body spectrum effected by the plasma oscillations. Therefore, a refinement on the present study from this point of view is suggestive. In Appendix E we present a

-61quantum dispersion relation for transverse oscillations in a relativistic plasmao In the non-relativistic limit, this reduces to that given by Bohm and Pines(29) with an additional term due to spin.

APPENDIX A INVARIANCE OF f(x,p) UNDER LORENTZ TRANSFORMATIONS Recall the definition f(x,p) = (Rfi)4 | d4z e-2ip.z/ (,N{fTx (x-z)&(x+z)}D), (A-1) where we have indicated the summation over the spinor indices explicitly, We wish to prove that under the simultaneous Lorentz transformations on x and p, namely, x - x' = Lx, p - p' = Lp, (A-2) the following statement holds true f'(x',p ) = f(x,p), (A-3) where f'(x' p') is given by ff(XI p') = ( ) 4 | d4z e-2iPZ/ ((,sN{f(x'-z),(xf+z)}v)), (A-4) where 0' denotes the transformed state-vectoro Proof: We first note that the transformed state vector may be written as* O( =U(L)D, (A-5) See, for instance, Reference 13, p. 651o -62

-65where U(L) is a unitary transformation such that U(L) 4(x) U-1(L) = S&l(L) 4p(Lx), (A-6) U(L) Ir(x) U-1(L) = j(Lx) Spa(Lx), (A-7) where Sp(L) is the spinor representation of the homogenous Lorentz group. Following statements follow from above and from each other and need no further comment. a(x'-z) 4r(x'+z) = T(Lx-z) +(Lx+z) =,(L(x-L-lz)) (L(x-L-lz)) =,(L(x-L- z)) S,,(L) S_(L) 7(L(x+L-1z)) - U(L) V(x-L1z) U-1(L)U(L) iV(x+L-lz) U 1(L) = U(L) T,(x-Lz) 4(x+L1lz) U-1(L). (A-8) From Equations (5) and (8) one finds f'(x p') = (T)-4 f d4z e"2iLPZ/' (0,N{p(x-L-lz) p(x+L-lz)})o. (A-9) -1 We note that Lp.z = p.L z. Then, we perform a change of variables from z — T = L-lz and note that the Jacobian of this transformation is unity owing to the fact that Det(L) = 1o It follows that f'(x'p') =() f d4T e2iP (),N{ e(x-T) i (x+T)}T). (A-10) From a comparison of Equations (10) and (1), Equation (3) follows. Q.EoDo

APPENDIX B DETAILS OF THE DERIVATION OF THE SELF-CONSISTENT FIELD EQUATIONS It is convenient for the present purpose to obtain from the Equations (2.3) a set of equations by multiplying the first one from the left and the second one from the right by the Dirac Matrices I, = = iY = T = i]5 The reason for the selection of these particular matrices is that they form a complete set in the algebra of the Dirac Matrices. To suppress the notation we define _ ie A(x), a ie (x) D1 = A(x), D - =Ac After simple manipulations one obtains the following set of equations: {7Dpi + M} r = 0, {7pD -M} =O, -.->. -> +- +{Dv - iVDila + M Yv} O1 = f {Dv + iCvD - M 7v} = ~ -4 -4 -- {sC,YIcr + i( vD, -.vD) + M,v} j = o (B-l) __ ^1D, —,( —,fS {So,ID + i(yDv - V ) - M Dv, } = ~, -- -> {T Dy +,75kLVDI + M SY} t = 0, 4- 44 {-T Dy + Y75,vDI - M Sv} = 0o -64

-65-4 {SDU + M T} 4r _ 0, r {sD, + M rT =0, where M = mc/T, sa,v, = 7(x), jv = t = T(x). We now take any pair of the Equations (1), let x - x + z in the first one and x -~x - z in the second one, then multiply the first one from the left by T(x-z) and the second one from the right by *(x+z). By adding or subtracting the resulting equations and introducing further the notation. 4-6- -~ 6-4 L 2M kD k' + D> 2iM - [D D, "^ ^ -L C. 2iM I-1 where D = - ie A(x+z), D = + ie A(x-z), one obtains f vTi> = 0, I {vIH + l} = 0, T {nI + CVP} = 0, (B-2) - { + - a^ - vv} 4 = 0, j {i soL, -II + (v,,, - Vi,)} = 0 T {i say HT - (V1t - vvyI1) + ~av} 4 = 0~

-66t {T H1 - iy5aLVnI} 4 = O, # {T II+ + iy75ag1V + sV} 4 = 0, {sH + T} 0, f SpH = 0, where 4 = T(x-z) 4, = Ir(x+z). We give a generic definition of phase-space distributions associated with the Dirac matrices by o (x,p) -< D > = ()i)- 4 d4z e-2iP'z/5(,N{r(x-z) D *(x+z)} -), (B-3) where D stands for any one of the matrices I, v a, va s, or T. Thus, we introduce the notation <- I > ~ f(x,p), < v > (xp), < C > vX,)9 < S, > S= S(Xp), < T > = T(x,p). We now obtain two relations which are useful in deriving the transport equations from Equations (2). Let us consider the quantities (xp) = (th)"4 J d4z e -2Pz/n ( MNJ x-z) D I: *(x+z)} 0). (B-4) Recalling the definitions of Hf and making use of the Equation ( 3.5) one obtains after straightforward manipulations that

-674- -4 Q p a1 + e A sin(. a a ) +&x 1 2 6x4 c 2 (x pp pm e e cos( T i ) (B-5) e- {} ((fi)-4 d4z e-2ip Z/(,N{T(x-z)D(A (x+z) + AL(x-z))ft(x+z)} 0). Employing the "self-consistent field" approximation in the sense that (O,Nj{(x-z)DA1(x'+z)(x+z)}) - (,A (x+z)O)(,N(x-z)D(x+z)} ), one gets from Equation (5) =- -4 me T 6 + IA, sin(Ta ) + 2 77t] c 2;x p. 1(X, P) -c m - -4 }) (xp), (B-6) where A, (x) =Ai + (, A Ignoring the terms of 0(t2) and changing variables from {x,p} to {xt} by the relation p e = t we obtain from Equation (6) h ( a + eF a) (B-7) 2mc 6xn,, c dv Q-(x,) - 2mc x V }t(X,), mc where Fv = ix - and 2 (x,r) is defined by the relation ) (x,) d4xd4 = (x,p) d4xd4p

-68We now multiply Equations (2) by (Ri)-4e2ip Z/1 and integrate over z and take the expectation with respect to Oo Using Equation (7) and the relations Sav 5 = sp pav', 1 = +:C5r v 2i Cvpoa P C Pa 9 which may be verified, we obtain the following set of equations: T7 = 0, (B-8) t; + mcf = O, (B-9) 19-P1 A tv =- 2 Tvf (B-10) v^ v ^T ^1 (B-ll) 1 - 1 = i 2 vpoTaSpy (B-12) mcV = -i ClvpaSpa+ 2 {T v -, (B-13)' |W ~T L V T =0,~ (B-14) mcSv = +v 2 v + (B-15) 2mc TS,+ =0, (B-16) IS = o, (B-17) where we have omitted the arguments (x,rT), and defined ^ a e Fa T —+ - -F ^ ax, c V1rp drt

-69It is observed that Equations (8) - (17) are not all independent. We have written them all because they make a convenient reference. Transport Equation for f(x,t) From Equations (8) and (11) one obtains T Vf + T-To T, 0 (B-18) Noting that 0V, = -(v c-ad.hat T T e FVI a V pV - c axp ao' it follows that A a 8F, 6i v alW f + T- a3 t 5 =0. (B-19) A P This is the same as Equation (3.18) in the main context. A subsidiary condition to Equation (19) is obtained by using Equations (9), (10), (11) and noting that [p T, Tv]_ = c FVl as (i2c2 + m2c4)f 1 e2ic Fv~v = 0. (B-20) Transport Equation for S,(x, j) From Equations (14) and (16) one finds VTPSP + mc T;v = 0. (B-21) Also, it follows from Equation (13) that mco = - S (B-22) Plv 1v P SI "

-70Combining Equations (21) and (22) one obtains Iv e F V (B-23) ~~LkL V c VkL~L which is the transport equation for S (x,n). Subsidiary condition to Equation (23) is obtained by combining the Equations (15), (17) and (22) as (nC2 + m2c2) Sy = 0. (B-24) It should be noted that we have ignored the terms of order -f in equations for the spin density since the spin dependent terms in Equations (19) and (20) appear already multiplied by fi.

APPENDIX C GAUGE INVARIANCE OF f(xn) We show here that f(x,jt) is invariant under gauge transformation to order (h2). By definition, f(xi) = (rt)-4 f d4z e-2i( + c )z/t(T(x-z)(x+z ). (-l) change dummy variable z - T by the relation z = fT so that f(x,c) = n-4 f d4T e-2i( + e)'T(,'(X-hlT)>(X+tT) (). (C-2) gauge transformations of the second kind is defined by A, A = A + x i -, j = r e-ieA/fc (C-3) _- - ieA/n c tJr — i j = i"e' where A satisfies the equation liA = 0. Under these transformations, f(x,t) -,f'(x,x) = I-4 J d4T e 2i( + c)oT(,'(x-fT)t*(x+ftT) 0). (c-4) Inserting the relations (C-3) into (C-4) one finds that f (x,'t) = n-4 i d4T e-2 + c)T ) eiQ(,T(x-nT)>r(+lT) m), (C-5) where Q - /A(x+hT) - A(x-A T)) - 2i e T (c-6) easily seen that Q is of 0(i2). ~7?l

APPENDIX D aFe CONTRIBUTIONS TO ( -) OF COMPTON SCATTERING AND PAIR PRODUCTION lo Electron-Photon Scattering (Compton Scattering) The part of S(2) which describes this process is given by S(2) = + S, C.S. 1 2 where Si = (e)2 J d+)4xld4x2 y L s2 = (e)2 Jf d4xld4x2N{( -)7vA+)TA)x(+)))x},,, I and (x2) (x) - SF(x2-xl) iqx/i 2 4 e icx, SF(X) = (2d)4 fd i - imc ( 7L ) Defining cs.m < ml S1(2) In >12 rn- m VT Im 5c.s.n one can show that rc.S. = (4e2c)2 V mc 6p' k- p n-+ m V P p_'p kk 4Iw'(flEp- kAcEp SP2 ss mm - * {l-ne(pS')}{l+ny(km')}nT(km)ne(p,s) < m|.,ne(pYs') (D-l) + 1,..,ne(p,s) - 1,..;..,n7(ktm') + 1,..,ny(k,m) - 1,...>, -72

-75where n (k,m) denotes the number of photons of momentum k and polarization m in the state |n>, and AC. S' AC. S(p S;fikkm' p,s;tk,m) (m) -(S)(p) {(m )(k') (+t)-imc k) (D-2) + (m)) ( f) (m))(k') u(s)(p) AC.s. has the following symmetry property Ac. s. (ps s;Tikm'i ps;tmkm) = -A*(p,s;k_,ml ps s'; mk1m'). Note that in Equation (1) and thereafter sums over photon polarizations m, m' are restricted to 1,2. This is because the terms with m, m' - 3, 4 do not give any contribution due to the fact that initial and final states in > and. Im > do not contain(l8) any photons with polarizations 3 and 4. The rest of the derivation is quite similar to that of the Moller scatteringo Therefore, we shall be more brief here. One finds [Corresponding to Equation ( 4 35)] afe (post) (4te2c2 VT c m c4 4ve ) E E 3p'+Ilk'-p-fik 6t v py/ -Is. - 4ck'w EpEp, m m IAC'S~ (p's';Tik'm' Ip,s;ik,m)12 (D-3) {{l-Vfe}{l+Vf7} VfyVf - {l-Vfe}{l+Vfy} Vf7Vfe} where fy(k,m,t) 1 _ v Pn(t)ny(k,m)

-74Transition to the continuum variables is achieved by defining, in addition to (4.36) and (4.37), Z fy(k,m,t) = X(_mt)d3K, ked3k where K = fik, and noting that 1 - V d3n ked3k (2tff)3 One obtains, in the continuum variables, 6Fe (ps =t) (e2c)2 p2 Z f d3p'd3K'd3 m2c4 (p'+ p6t s'mm KOB' Ep tEp (E'+Z'-E-T') AC s (ps t;Km'|ps;,m) 2 (D-4) {{p-Fe}{p+X} FeX' - {p-Fe}{p+X'} FeX} where T = S, p (2f3i)-3If we assume isotropic distributions in spins and polarizations, that is, Fe(p,s,t) = 1 Fe(pt), (s=l,2) X(Km,t) = X(Kt), (m=l,2) then aFe (pt) = (e2c)2 f d3p'd3',d3 m2c4 (p p - E(E' +f'-E-a) - Bc's. (K'?E,,) (D-5) a {{2p-Fe}{2p+X} FeX' - {2p-Fe}{2p-X'} FeX},

-75where B (PSI!t IP) = I 1, IAc'S' (Ps';'m' Ips;Sm) |)!2. S,ssm,m 1 2o Two-Quantum Pair-Creation and Annihilation The parts of S(2) which describe these processes which are the inverse of each other are given by S(2) = (e )2 dxd4x2N {4 (-A( ) Y) } p. a. S(2) (e )2 Sp~ -()2 JJf d4x1d4x2N -(V(-)VA(+) ( ())} where *(x2)*t(x) = J_ d4 K ei' (X2 xl) / A similar analysis as before yields Fe(st) 1 (e2)2-2 7 f d3qd3 d31 m24 at -- 2 Wfl ^ 2 r,m2,ml'~21 EE S(i+ 7- -)5(-52+!-b-Ep-Ea) IAJ2 (D-6) {{P-Fe(Pst)}{p-Fp(qrt)} X (!2,m2,t) X (l ml t) -{p+X(,m2m29t)}{p+x(lm1lt)} Fe(ps,t)Fp(pst)}, where IAl2 IAP'aO12 = lAPoO I2,

-76and APao = AP.a. (2,m2;1_gmlIs;qr) = (r+2)() {~(m2)() (_)_ic (ml)() + / i)(!,) (i_} ):1 (m2 (m_2)} u(S)(p) AP C. = AP-C (,s;qrl2,m2;Klml) -(s)(p) {f(m2)(2) -M) im11) + (ml)(&) ( 1 ) (m2)(!2)} u(r+2)(q) (!)(l)_imc The symmetry relation between Ap)a' and AP'c' which has been used in deriving the Equation (6) is noted as AP c. (p s;9,rI _2 m2; Lml) = -A*P. a (2 m2;!Lmmll ips; qr).

APPENDIX E A QUANTUM DISPERSION RELATION FOR TRANSVERSE OSCILLATIONS IN A RELATIVISTIC PLASMA In the previous analysis we have assumed that photons propagate with the velocity c between successive events so that X = cl|k. It is known that(30) the energy-momentum relation for photons traveling in a medium differs from this relation due to the collective effects of the medium. We wish to present here a quantum dispersion relation which determines w as a function of k appropriate to a fully ionized relativistic plasma in the absence of any external fields. (31) We use here a procedure developed by Meadi3L) in the study of the quantum theory of the refractive index. The details of the calculations is of little interest since it only involves a straightforward perturbation calculation to second-order. We find that the modified energy-momentum relation for transverse photons (i.e., m = 1, 2) is determined by the dispersion relation ~ 2 2 c2k*km = 2 mc f Fe((p) (E-1) 2( (m)(k).p)2 - (Pq3 + m2 c 2((m) (k).p) - (P'q2 + m c) hkm q3 + (P'q3 + m2c2) hkmq2 - (p'q2 + m2c2) 2( (m)(k)p)2 + (P'q4 - m2c2) 2((m)(k).p) + (pql - m c2) hkm' q4 - (p q4 - m2c2) hkm 1ql + (p'q1 - m2c2) 4 rNe2 where e )(k) (/|k|;, km (k; icm/c), 2 smV _ 77-77

-78and _ = -(p+lt), 3 = p_+fk 2 = (p4k), 4 = -(p-ik) and where Fe(p) is normalized to unity. In the non-relativistic limit the relation (1) reduces to 2 ~21 12 (2{+((m) v')k2 2 (fl2)2 2 ck + rf + \, + (k2 2)2 = c +ep {1 + - d3v fe(v) 2-m (E-2): 2 tk2 2 (k_.-.z)2 (' ) It is observed that this relation differs from that given by Bohm and PinesO30) with the appearance of an additional term, (k/2m)2 in the nominator of the integrand. This additional term arises due to the inclusion of the spins of the electrons in the analysis. * After the work presented. here was completed,we found that in a recent article by P. Burt and H. Do Wahlquist 29) spin and exchange corrections to the plasma dispersion relations have been calculated by a different procedure. They have employed a quantum Vlasov equation based upon the non-relativistic Pauli equation in order to include the spin effects. Their result for the transverse dispersion relation including spin correction agrees with our Equation (E-2). Exchange corrections do not appear in our dispersion relation given by (E-l). In order to include the exchange effects in the transverse dispersion relation we must go to the fourth order in the perturbation analysis. In fact, the ex4hange correction given in the aforementioned paper is proportional to opew e

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