Z. Phys. A - Atomic Nuclei 336, 317 320 (1990)

Zeitschrift for

PhysikA

Atomic Nuclei 9 Springer-Verlag 1990

The electron-positron system at short distances A.O. Barut* International Centre for Theoretical Physics,Trieste, Italy Received July 13, 1989 In answer to a recent paper we discuss the application of a relativistic two-body equation and its mass-conjugate first to ortho- and para-positronium and then its extrapolation to short distances where self energy terms dominate leading to a phase transition in QED and possible new (e + e-)-resonances. PACS: 03.65.Ge; 11.10.Qr; 12.20.DS

In a recent paper in this journal, Geiger, Reinhardt, Miiller and Greiner [1] have discussed the new relativistic two-body equation [2] for the electron-positron system and the idea of a possible (e + e-)-resonance at short distances due to nonperturbative self energy effects in order to account for the observed sharp peaks in the spectrum of positrons and electrons emitted back to back in heavy ion collisions. These authors rederive the 16 radial equations and obtain equivalent results to previous investigations [3]. But they make then a number of hasty and sweeping conclusions which are not justified and should be carefully examined. As these authors also indicate, the full nonperturbative treatment of the relativistic two-body problem is a very complex problem and the short-distance behaviour of quantum electrodynamics is still a largely unexplored domain. However, we think that progress can be made with the new covariant nonperturbative twobody equation, including all the self energy terms, and with the correct interpretation of negative-energy states. These points were not fully considered in [1]. There are two separate issues under discussion in the conclusions of Geiger et al. [1]. (i) The test of the new relativistic two-body equation in the known case of positronium. Of course, we must require that any new equation first reproduces the known results at low energies, or at distances of the order of Bohr radius. * Permanent address: Department of Physics, University of Colorado, Boulder, CO 80309, USA

(ii) The second issue is the extrapolation of the equation and its solution to higher energies, or to distances of the order of the Compton wave length of the electron, (and eventually down to distances of the order of the classical radius of the electron for the purposes of the particle physics). I should like now to respond to [1] with the following results: (1) The covariant two-body equation derived from field theory [2] can account for the spectrum of both ortho- and para-positronium if the negative energy states are properly and correctly interpreted (in contrast to the statement made in [1]). (2) The self energy terms in the two-body equation give rise to an anomalous magnetic moment-Pauli coupling alone only in the limit of weak two-body or external forces. And in this limit only, is the value of the anomalous magnetic moment equals ~/2 re, and the form factor associated with it is that use in [1]. (3) There are other potentials coming from self-energy terms in the wave equation, an electric potential, and a complex "spontaneous emission potential". For example, the large s-wave Lamb shift is due to nonmagnetic potentials. (4) The value and the form factor of the anomalous magnetic moment changes drastically for large external fields, in our case at short distances, and must be evaluated self-consistently. (5) There should be a new regime, a phase transition, for the (e + e-)-system at short distances, in which the nonlinear self-energy term in the action dominates over the mutual Coulomb interactions, and giving rise to new soliton-like self focussing configurations [5]. These states in turn depend on the external environment of the heavy ion collisions. But the details of this transition cannot yet be calculated. We now elaborate these points in detail. First of all there is some misunderstanding concerning the nature of the theory underlying our two-body equation. Looking from the perspective of perturbative QED the authors of [1] call our approach "semiclassical

318 approximation". Their terminology is misleading in identifying the new approach with some earlier different approximations. Rather the new approach is " d u a l " [43 to perturbative QED and not an approximation to it. In the perturbative QED there are no self-energy terms but a separate quantized radiation field is introduced; in our case there is no separate quantized field but the full self-energy is included from the beginning. Much work done in recent years have shown that the self energy approach to quantum electrodynamics fully reproduces all the radiative processes of QED [4]. One begins with the coupled Maxwell-Dirac Lagrangian for two (or more) spinor fields. The Maxwell-field can be eliminated by solving the electromagnetic potentials in terms of the current of the particles (one of the field equations).

A~,(x) = ~d y D (x - y)j, (y)

terms dominate over the mutual Coulomb interactions. We shall now discuss these two different regimes. One the usual positronium region, the other the short-distance super-or "micro"-positronium regions. At atomic distances self energies are of the order of mc~(Ze) 4 (Lamb shift, anomalous magnetic moment and spontaneous emission), compared to the mutual energy which is of order of me 2. At short distances the situation is quite different. In order to see this we write the Hamiltonian form of (1) for the choice nu = (1000) { ~ I ( P l - el A1)-t- fit ml +o:2"(P2-e2A2)+fl2mz+el

+e2 V2}-~b(xl, Xz)=E~b(Xx, x2)

{[Tu(Pl, - ei A~i)) - mi] | 7" n + 7" n | [7 u(P2, - ei A~2)) - m2] } ~ (xl x2) = 0 A(1)= A(2) = !

"

e2

2 el

A l _ e 2 ~z

e, S d z d u D ( x i _ z ) ~ + ( z

A 2 _ e l ~1 2 r

e2SdzduD(xa_z)~b+(z,u)e2~(z,u) 2

'~ " n ~)# _1../[(2)self

(1)

In this equation, the spin matrices of the two-particles are always written as tensor products, for example Yu| 7", with the first factor always referring to particle 1, the second to particle 2. Here n~ is the normal to a space-like surface and r x = [ ( x . n ) 2 - x 2 ] 1/2 is the relativistic relative distance. Equation (1) contains also self energy terms which are non-linear. The self-potential are given by (with 7" n - 7, nU)

A(ul)seU(x O = jdz du D (xt - z) U~(z, u) Yu | 7" n 9 (z, u)

where D ( x - y ) tian. []

vie2

2

' u)~l~(z, u)

1 2 r

ea ~dzduD(xt_z)qb+(z, u)~(z, u) 2

v2_el 1

e2 ~ d z d u D ( x 2 _ z ) ~ + ( z ,u)~b(z,u).

2r

2

We see that self potentials are both of electric and magnetic nature. The magnetic part can be approximated by a Pauli-coupling term to lowest order in the external field [6-8], hence gives rise to an anomalous magnetic moment e--2n-n" However, if we use the full Green's

~)#~" K/ A(1)self r2 + ' ~ #

A(u2)selr(Xz)=~dzduD(xz-u)UP(z, u)7.n|

(3)

where

2 r

This expression we refer to as the self-field. We insert this into the Lagrangian together with any external fields A~,xt. For one body problem this results in nonlinear integro-differential equation, the nonlinear part being precisely the self-energy. For two (or more) body problems, we define a composite field ~(Xxx2) =Ol(xl)Oz(X2), reexpress the Lagrangian in terms of ~, and vary it to obtain an equation for ~. The covariant two-body equation that one so derives from the action of the coupled Maxwell-Dirac equations, including the full self-energy effects, is [2]

V1

u)

(2)

is the Green's function of d'Alember-

A full adequate understanding of this nonlinear equation will probably require some time. But we know from many recent work that cubic nonlinear equations of the Schr6dinger (or Dirac) type have non-perturbative soliton-like solutions. This is the regime where self energy

function of the external (Coulomb) field in the self energy both the magnitude of the anomalous magnetic moment and its form factor would be quite different than the values used in [1]. There are in addition complex parts of the self energy terms due to the use of causal Green's function and the relation 1 = P 1 + in 6 (x). These complex potentials comX

X

pletely account for the spontaneous emission [7-9]. Reference [1] thus takes into account only the anomalous magnetic moment contribution with its low energy value C~

=2nn" This procedure is rather good for atomic systems up to order es, although to order es the Lamb-shift corrections must also be included I-3, 7]. Concerning the applicability of the two-body wave equation to para-positronium and the treatment of negative energy states even without the self-energy terms, the situation is as follows: For any Dirac-type first quantized wave equation, even for the one-particle Dirac equation, we must properly interpret the negative energy states. In the one particle Coulomb case, it is easy to select the correct physical solutions from the unphysical ones and one does not worry too much about the negative energy solutions. In the two-body case however, the problem becomes

319 very subtle, namely in the parapositronium case. It is a difficulty discussed for a long time, since Breit [10] first considered a two-body equation. It arises most prominently for states for which one of the particles have positive, the other a negative energy [11]. In the physical interpretation we must make sure that negative energy states propagate backwards in time. One elegant way to take this into account in the first quantized theory is to use the "mass conjugate" Dirac equation for negative energy states. There are namely fi priori two Dirac equations, (7 P--m)~b r = 0, and its mass conjugate (Tp+rn)~bu=0,- both coming from the factorization of the Klein-Gordon operator. The negative energy solutions of ~bi coincide with the positive energy solutions of 0 u - Particles and antiparticles differ by an internal quantum number n, in this case the sign of mass, which is a general consequence of the algebraic 0(4, 2)-formulation of the Dirac equation [12]. In an external electromagnetic field, similarly, we can see that the negative energy solutions of [7 (P- e A ) - m] Oi = 0 are positive energy solutions of the minimally coupled second equation, namely [7 (P- eA) + m] ~bH = 0, i.e. a particle with opposite charge and opposite energy momentum when compared to ~bI. For the two-body equation (1) or (3), the mass conjugate equations are obtained precisely with the change m2 ~ - m 2 and e2 ~ - e 2 keeping mt and el unchanged. There is no violation of any "charge parity" in this, as claimed in [1]. The change of the sign of both masses and charges results again in the original equation. With this procedure it has been shown that positive energy solutions of the mass conjugate equation do indeed correctly account for the parapositronium levels and the authors of [1] agree with this conclusion but think that there is a "violation of charge parity". Since the procedure has been described very recently in several places [6] we will not repeat it here. The main point is that one must use 8 radial equations for (3) and another 8 radial equations for the mass conjugate equation to have the correct 16 equations to describe all the states of the two spin 89 Now we come to the extrapolation of (1) or (3) to shorter distances. It has been argued on physical grounds that the regime where self energy terms dominate, the self organization domain, [5], should be at much shorter distances, namely at r = s/m, the classical radius of the electron, rather than the region of heavy ion (e + e-)-resonances at r,,~ I/m, the Compton wave length. It can be seen by order of magnitude arguments that as the localization of the electron gets smaller and smaller, the self field produced by the electron increases, hence anomalous magnetic moment increases, and at r~e/m, it reaches the value of the order of/~o, while the expectation value of the normal magnetic moment decreases due to relativistic effects. In fact for the value of the anomalous magnetic moment equal to the normal magnetic moment, c~-#o , many models have been worked out and lead to resonances at distances of the order-one Fermi. One question remains for the heavy-ion problem and

that is the following. If at total energies of the order of 3 m, the self energy effects are not large enough to produce (e + e-)-resonances, can the very strong environmental fields of the heavy ions, modify the potentials in such a way as to facilitate the formation of resonances temporarily? Recently, several groups have postulated the existence of a new phase of Q E D due to strong external fields to account for the (e + e-) peaks in heavy ion collision [15-17]. This is a special case of our phase transition mechanism due to self energy terms in the 2-body problem. The strong fields acting on electrons and positrons are due not only to external ions, but also due to mutual interactions between localized electrons and positron themselves (which can even be stronger than the fields of ions). The origin of the phase transition in Q E D in our formulation is the self energy. It is already contained in the non-perturbative treatment of the theory and need not be postulated from outside. The fields of the moving heavy ions are time dependent and have probably rather complicated configurations. Their effect on the (e + e-) system is quantitatively difficult to evaluate. But qualitatively, it can be seen that a strong magnetic field, for example, can produce a broad confining tail in the 3P0 potential plotted in [1], since the potential of the magnetic field B is oscillator-like in the plane perpendicular to B. We conclude that the relativistic 2-body equation (1) and its mass conjugate is tested well for positronium, and allows us in principle to extrapolate it to short distances. In doing so, however, the nonlinear self energy terms must be treated self-consistently. Only part of the self energy can be written in the form of an effective Pauli-coupling, but the value of the anomalous magnetic moment and its form factor should be taken as a parameter until it can be calculated in a self consistent way depending on the external field and on the wave function of the bound state. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

References 1. Geiger, K., Reinhardt, J., Miiller, B., Greiner, W. : Z. Phys. A Atomic Nuclei 329, 77 (1988) 2. Barut, A.O. : In: Physics of strong fields. Greiner, W. (ed.), p. 585, 601. New York: Plenum Press 1987 3. Barut, A.O., Unal, N.: Physica 142A, 467, 488 (1987); J. Math. Phys. 27, 3055 (1986) 4. See the review Barut, A.O.: Phys. Scr. T21, 18 (1988); In: New frontiers of quantumelectrodynamics and quantumoptics. Barut, A.O. (ed.). New York: Plenum 1990 5. Barut, A.O.: Found. Phys. 17, 549 (1987) 6. Barut, A.O., Kraus, J.: Phys. Rev. DI6, 161 (1977); Found. Phys. 13, 189 (1983) 7. Barut, A.O., van Huele, J.F.: Phys. Rev. A32, 3187 (1985) 8. Barut, A.O., Dowling, J.: Z. Naturforsch. 44a, 1051 (1989) 9. Barut, A.O., Salamin, Y.: Phys. Rev. A 37, 2284(1988)

320 10. Breit, G.: Phys. Rev. 34, 553 (1929); 36, 383 (1930); 39, 616 (1932) 11. Brown, G.E., Ravenhall, D.G.: Proc. R. Soc. A208, 352 (1951); Childers, R.W.: Phys. Rev. D26, 2902 (1984); Krolikowski, W.: Phys. Rev. D29, 2414 (1984); For a review see Sucher, J.: In: Relativistic effects in atoms, molecules and solids. Malli, G. (ed.) New York: Plenum 1982; Int. J. Quant. Chem. 24, 3 (1984). 12. Barut, A.O.: Phys. Rev. Lett. 20, 893 (1968) 13. Barut, A.O.: Phys.Ser. 36, 493 (1987) 14. For a review see Barut, A.O.: In: Quantum electrodynamics of strong fields. Greiner, W. (ed.), p. 755. New York: Plenum 1983

15. Celenza, L.S., Misra, V.K., Shakin, C.M., Liu, K.F.: Phys. Rev. Lett. 57, 55 (1986) 16. Caldi, D.G., Chodos, A.: Phys. Rev. D36, 2876 (1987) 17. Ng, Y.Jack, Kikudi, Y.: Phys. Rev. D36, 2880 (1987) 18. Hirata, G.Y.S., Minakata, H.: Phys. Rev. D39, 2813 (1989) A.O. Barut Department of Physics University of Colorado Campus Box 390 Boulder, CO 80309-0390 USA