We present here the response theory for the pair plasma DIELECTRIC RESPONSE OF in the presence of a magnetic field. The mode structure in PARTICLE-ANTIPARTICLE this more complicated case must be unravelled before the emiss- ion from pulsars can be completely understood. Although there PLASMAS IN A MAGNETIC FIELD have been many theories of the pulsar emission mechanism, as detailed in Manchester and Taylor [4] and Irvine [5], no comp- N.E. FRANKEL, U.C. HINES, V. KOWALENKO lete investigation of the behaviour of the pulsar magnetosphe- School of Physics, University of Melbourne, re, consisting of dense relativistic electrons and positrons 12 Farkville, Victoria, Australia in magnetic fields of the order of 10 G, is possible until the properties of the pair plasma as a problem in plasma phys- 1. ABSTRACT ics have been elucidated. The. first treatment of the magnet- ized pair plasma is due to Svetozarova and Tsytovich [6] and We have considered the longitudinal dielectric response represents an extension of the work of reference [2]. of an ultra-degenerate relativistic plasma composed of elect- As a final 'way out' astrophysical application of the rons and positrons. We have used the relativistic Hartree physics of pair plasmas, mention may be made of the late lep- self-consistent field method to investigate the dispersion ton era of the early universe where the tenperature (circa relations and damping parameters of such a plasma in the pre- 1011 °K) is still above the pair threshold. sence of a magnetic field. These properties must be studied Electron-positron plasmas are also beginning to come in the various regimes appropriate for a relativistic plasma into serious consideration in the laboratory. Tsytovich and as detailed by Tsytovich and Jancovici. Although it is hop- Wharton [7] have discussed the possibility of creating a pair ed that this work will yield new insight into certain astro- plasma by means of intense relativistic electron beams and physical phenomena (such as pulsars), it is interesting to they suggest a variety of experiments involving linear and note that laboratory electron-positron plasmas may be a thing nonlinear wave propagation which it would be important to of the immediate future as a result of suggested new experi- carry out. ments using an intense relativistic electron beam. For the relativistic electron-positron plasma the class- ic papers are those of Tsytovich [2] and Jancovici [1]. Tsytovich has used a relativistic quantum propagator approach 2. INTRODUCTION to develop both longitudinal and transverse response functions, although the transverse interaction has not been considered. In recent years it has become clear that relativistic Later Svetozarova and Tsytovich extended this work to include plasma physics is an essential tool for astrophysics, quite the presence of an external magnetic field. apart from its increasing relevance for large scale fusion Jancovici used a quasi-boson Hamiltonian and included devices. Much of the work completed so far has teen of an not only the particle ring diagrams (taking account of the extremely formal, field theoretic nature, with the direct Coulomb interaction between particles) but photon ring diag- astrophysical applications being potential rather than actual. rams as «ell. For the work of Tsytovich and ourselves the Alternatively, astrophysicists have studied the accretion photons are included implicitly either in the field theory rates for electrons and positrons from a basically heuristic formalism or in the structure of the Dirac Hamiltonian. Jan- viewpoint. covici1 s results tend not to hold for conditions such that There has so far been very little work on the direct the approximate results of Tsytovich are valid, and where the study of plasma response theory for pair systems. This pap- two sets of results do overlap, they do not always agree. er is concerned with the development of such a response theory Following this pioneering work, the subject remained for an electron-positron plasma. It seems to us that the dormant for a number of years until interest in the phenomena direct calculation of the modes of oscillation and of the dam- mentioned earlier sparked a new surge of development during ping parameters needed for astrophysical work can best be the last decade. In 1971 Bezzeride» and Dubois (S] initi- carried through by this technique. ated a formal investigation of a covariant-framework for For the pair plasma in the absence of a magnetic field treating ultrarelativistic plasmas. They employed a combin- the astrophysical motivation comes from the fact that the ation of non-equilibrium statistical mechanics and quantum interior of dense white dwarf stars corresponds to conditions electrodynamics. Although this approach is extremely rigor- just slightly above pair threshold. This field-free electron- ous, it does not lead readily to numerical results on the mode positron plasma has been studied by Jancovici (1), Tsytovich structure of the plasma. [2) and Cusing the response theory approach) Delsante and Somewhat before our own work on relativistic plasmas, Frankel [3]. Bakshi, Cover and Kalman returned to the problem using much 353 the same methods as Tsytovich. Their particular concern to licit difference in interpretation of the following begin with was the vacuum polarization in the presence of expression: ^ j^f ^fc^. <4«fcj+ (3*
The second quantized form of the Hamiltonian is ident- ical in form to that given by Delsante and Frankel [ 3] , & -<) 354 and the essence of the present problem consists in the imp- With these solutions of the free-particle Oirac equation, the various matrix elements appearing in equation C7) can be }_ evaluated as follows: p ) ***** I V.s' ) *
'...(10) This completes the work for the zero field case. In equation (7) n+refers to the equilibrium Fermi-Dir- ac distribution function for the positrons and n" for the el- ectrons. The first term is the electron gas result of Dels- ante and Frankel. This would be tJie complete expression if the positrons were switched off. Likewise, the second term &_ Non~Zero Magn-.-tir refers to the positrons alone. The third and fourth terms refer to spontaneous emission and absorption processes occurr- The only modification required in the previous work ing in an electron-positron plasma. Tsytovich refers to when there is an external magnetic field concerns the matr- these as the pair production terms. These, mixed terms do ix elements. The new matrix elements may be readily eval- not vanish when TF = ri" - 0. In order to overcome this uated (see e.g. Johnson and Lippmann [11]). We designate problem the responce function has to be renormalized (renor re- the matrix elements in the order in which they occur in alization of the vacuum), which has the effect of taking the equation (10) by I, , I,. , I, and I» respectively. real part of the vacuum or 1-terms to zero. When a magnetic field is present these matrix elements We now consider explicit results for the field-free and become: magnetic field cases. I- - i A External Field Zero.
In calculating the matrix elements for zero field we use the standard plane wave solutions of the Dirac equation. We quote them first for electrons of momentum p with spin up where (s - 1) and spin down (s = 2):
M - ... (8) and then for the positrons, also for the '.wo spin possibilit- and ies:
«• ...(11) o with similar expressions for the other three I's (9) which we have insufficient space to quote here. 355 Xn equation (11) These results are in gene.ral agreement with the work of Tsytovich [2]. The conclusion is that the dan-ping in both cases is extremely weak. and H is a Hermite polynomial. The single-particle energy There is an additional regime where the damping can be found. Tsytovich refers to this as pair-production damp- £ ,, ing. It only occurs in the ultra-relativistic region and The matrix elements calculated in this way may now be requires that the dispersion curve for the longitudinal oscillations must intersect the curve «"• = 4ml + ql . introduced into equation (7) to yield the longitudinal diel- 1 ectric response function. In the next section we go on to To find whether a solution does indeed exist for this cond- consider the made structure of the plasma both with and with- ition on c<3 , we introduce ««»* - 4i*t q* into the gener- out the external field. al response function and take the small q limit to obtain
5. THE MODES Noting that p^ is the Fermi momentum, we can set the above expression equal to zero to obtain the density at which the real part of the response function will vanish. A External Field Zero. These densities are such that frfo >J»S4» . Let qs and «Oi represent the solution to the dispersion relation where As this paper is concerned mainly with the magnetic *>î = 4 m*- + qj . Standard procedures may now be used field case, there is no space to give the full dispersion to obtain the damping constant for this regime: relation for zero field. He shall later present the full response function for the magnetic field case. Here we ...(20) shall be content to look at the dispersion and damping in a few limiting cases of interest. Non-Zero Magnetic Field. We begin this section by giving first the real part L We consider first the region uxq. By solving the of the response function correct to order q : dispersion relation ...(13) we get the solution ** + '**. ...(14) Now by introducing tO-o>*+ AW and q = Wj+ Aq , i.e. pert- urbing about the solution in q and oJ space, the imaginary part of the dielectric function may be obtained from the complete expression for£(q,<4). This leads to the follow- ing egression for the dairying constant: " i£w
.(15)
where we have used the general expression
y » -l^etj.w)/.^ &£($,«) .(16) Again, by solving Re 6 (co , <0+4q) - 0, t Hi'y* i»o» we obtain:
(17) for the dispersion relation and the result
.(18) 358 for the damping constant.
_J It should be emphasized that everything done so far refers to the situation where £ is parallel to the magnetic field. Thus the general procedures of sections 3 and 4 need to be repeated for q perpendicular to the external magnetic field. This will prove more difficult since alth- ough the particle contribution (without the 1-tenns) seems tractable, we are uncertain about the procedure for determin- ing the vacuum contribution in this case. A search for these modes must await the solution of this renormalization procedure either by ourselves or others. Finally, for the investigation of electromagnetic wave propagation in the plasma we must include the effect of the transverse photons. It is our intention to work out the full dielectric tensor using the plasma conductivity and to include some relevant quantum electrodynamic effects. Some formal results for the propagating modes have been given by ' "^ L fc ~f ***• "i a number of Russian authors whom we shall reference in a later paper.
_ j; cf -^SW83JT + ^- _« i \ 1 7. REFERENCES
...(21) tl] JANCOVICI, B., Nuovo Cimento 25 (1962), 428. (2] TSYTOVTCH, V.N., Sov.Phys.JETP 13 (1961), 1249. [3] DELSANTE,A.E.,FRANKEL,N.E.,Ann.Phys.(N.Y.)125 (1980), As in the field-free case treated earlier, a rencrmal- 135. ization procedure has been used to obtain this result. As [4] MANCHESTER,R.N.,TAYLOR, J.H.,Pulsars,Freeman, 1977. has been pointed out by Cover, Kalman and Bafrshi [9] , the [S) IRVTNE, J.H., Neutron Stars, O.U.P-, 1978. vacuum cannot be omitted in determining the real part of the 16] SVETOZAROVA,G.I., TSÏTOVICH,V.N.,IZV.VyS3h.Ucheb. response function when a magnetic field is present. In Zaved. Radiofiz. £ (1962), 6S8. obtaining equation (21) we have used the renormalization [7] TSYTOVTCH,V.II.,WHARTON,C.B., Comn.P.Phys.4_( 1978)91. procedure recomnended by the above authors. [S] BEZZESIDES/B., DUBOIS,D.F.,Ann.Phys.(N.Y.)70(1972)10. 191 BAKSHI,P.,COVER,R.A., KMJRK.G. .Phys.Rev.I^tt.33(1974) 1113; Phys.Bev.D 14 (1976)2532; Phys.Rev.D20(1979) 3015. 6. DISCUSSION (101 SAKURAI,J.j., Advanced Quantum Mechanics, Addison- Wesley (1967). Analytical studies of equation (21) for the response [11] JOHNSON,M.H.,LIPPMWm,B.A., Phys. Rev.76(1949) ,828. function are in progress. Although the expression is extremely complicated, it is hoped that the mode st ucture for certain regime! can nevertheless be extracted. Bakshi, Cover and Kalman [9] have already investigated the modes to lowest order for the plasma, i.e. the qz « 0 limit. Their work should be a useful guide in dealing with our own result (correct to order q2).
357 J