Fortschritte der Physik / Progress of Physics: Band 25, Heft 7 1977 [Reprint 2021 ed.] 9783112519486, 9783112519479

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H E F T 7 • 1977 • B A N D 25

A K A D E M I E

- V E R L A G EVP 1 0 , - M 31728

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B E R L I N

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Fortschritte der Physik 25, 3 7 3 - 4 0 0 (1977)

Electron-Positron Pair Creation from Vacuum Induced by Variable Electric Field M . S. MAKINOV a n d V . S. POPOV

Institute for Theoretical and Experimental Physics, 117259 Moscow,

USSR

Abstract Problem is considered of spontaneous creation of electron-positron pairs from the vacuum induced by external electric field, that is homogeneous and depends on time in an arbitrary way. The Heisenberg equations of motion are obtained for the creation-annihilation operators. The solution is a linear canonical transformation. The problem is reduced to a set of differential equations for the second-order matrices determining this transformation. A consequence of the CP symmetry of the Dirac equation with an external electric field is that the e+e - pair is created from the vacuum in a state with total spin 1. The case when the variating electric field conserves its direction, is considered in more detail. In this case the equations are much simplified and may be reduced to the Riccati equation or to problem of oscillator with variable frequency, so the problem is equivalent to the one-dimensional quantal problem of a barrier penetration. Two approximate methods to calculate the pair creation probabilities are discussed: the quasiclassical approach and the antiadiabatical method, applicable for sharp variations of the external field. Numerical estimates are obtained for the number of e + e _ pairs produced by the field E(t) = E cos cot. Group-theoretical aspects of the problem are also considered.

I. Introduction

In recent years considerable work was undertaken in order to test the quantum electrodynamics. With this aim in view, various radiative corrections (the Lamb shift, the value of (g — 2) for electron and muon, level shifts in muonic atoms induced by the vacuum polarization etc.) were investigated with high precision [2, 2], However, there is a qualitative prediction of QED that has not been tested experimentally until now. The phenomenon in view is the electron-positron pair creation from the vacuum induced by strong external fields. Such a process may occur in the Coulomb field of colliding heavy nuclei with the total charge large enough, Z1 + Z2 > Zcr «a 170, as well as in a quasi-homogeneous electric field, produced in focusing of a highly intensive laser radiation. The theory of the e+e~ creation in Coulomb collisions of nuclei is now developed in some detail {one may see the papers [3—12], containing also further references). The cross section is a few dozen millibarn for the Uranium nuclei [5, 6]; its dependence on the energy and scattering angle of the incident nuclei isy calculated, as well as the energy spectrum of positrons. An experimental observation of this process will be possible when heavy-ion accelerators producing beams of actinide nuclei will be operated. The purpose of this paper is the theory of the pair creation induced by electric field. The pair creation from vacuum in a constant homogeneous electromagnetic field was considered by Schwinger in his famojis work [13]. The probability of the creation (in a unit 28

Zeitsehrift „Fortschritte del- Physik", Heft 7

374

M . S . MABINOV a n d V . S . P O P O V

volume and per a unit time) is determined by the imaginary part of Lagrangian of the electromagnetic field interacting with vacuum of charged particles. For the electric field m4 / E\2 °° 8 w = 2 Im L = (2, + 1) — _ £ % exp {-nnE^E), (1.1) (¿ny \E0/ n = 1 nwhere s is spin of the particles, = ( —)re_1 for bosons and fjn = 1 for fermions, E0 — m2c3/ eh is the characteristic field intensity, at which the nonlinear corrections to the Maxwell Lagrangian L0 = (E2 — H2)/8n become substantial. In fact, two spin values were considered, s = 0 and s = 1/2, as only in these two cases the electrodynamics is renormalizable. A property of Eq. (1.1) is that the expression increases at E > E0: iv ~ ( EjE0)2, in contradiction with the unitarity. However, there is no contradiction, because in very strong fields (E ~J® 0 ) the probability w is not just 2 Im L and the formula is invalid. An appearance of this difficulty is due to the idealization of the problem (the time of action of the constant field is supposed to be infinite). To deal with the problem correctly one has to switch off the field at oo, i.e. to consider a nonstationary situation. A number of works was done recently, where the results by Schwinger were expanded to the case of variable electric fields [14—26]. In this work we develop an approach to the problem of pair creaction, based on solution of the Heisenberg equations of motion for the particle annihilation-creation operators. This method is most general and enables one to consider a homogeneous electric field E(t) having an arbitrary time dependence. Certain results in this approach were obtained previously in the works [21, 26]. In Section I I the basic equations are obtained that determine the probability of the pair creation. The subject of Section I I I is the case of linear electric field. In Section IV the problem of the pair creation is related to the problem of oscillator with variable frequency. Section V describes the quasi-classical approximation for the probability iv, that is applicable under the conditions E E0, hm < mc2, that are actual for contemporary lasers. For this case, an explicit analytical expression is obtained for the probability at any external field E{t). The opposite case of an abrupt variation of the field (to TO) is considered in Section VI. Some concrete numerical estimates of numbers of e+e~ pairs depending on the intensity E and frequency co of the field are given in Section VII. The group-theoretical aspects of the problem and deductions of the quasiclassical and anti-adiabatical approximations are presented in the Appendices. II. The Heisenberg equations of motion The external field is assumed to be homogeneous, as the range of spatial variation of the laser field is much more than the Compton wave length1). The Dirac equation for particle of massTO;charge e and spin 1/2 in an external field E(t) has a solution (r, t) = ((t) exp (irp(t)),

f

p(t) =

t + e / E(t') dt',

(2.1)

- oo

where p(t) is momentum of the classical particle, p± = pit ¿oo), if = H =

1= ^ v ,

Vi = rj F V 1 ,

U

so that the matrices U1 and F x m a y be expressed via U and F . The following notation is useful:

V = r)V Trj

(2.14)

or in an explicit form

The final result is 2 ) 2

tt TtTX

(2.15)

) The CP invariance works also for scalar particles. In this case the matrix r is 2 x 2 and has the form r

= l

w

\v

N * - i » r = i,

U*J

while the same r mixes two pairs of the operators: (»CP).

tf(-P))

and

(6(-P).