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pair creation by a constant electric field. N.B. Narozhny. W. Heisenberg, H. Euler , Z. Phys. 98 , 714 (1936). linear Maxwell equations. em fields (photons) do not interact. nonlinear correction to Maxwell equations. photons do interact!. elastic photon-photon scattering.

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W. Heisenberg, H. Euler, Z. Phys. 98, 714 (1936)

linear Maxwell equations

em fields (photons) do not interact

nonlinear correction to Maxwell equations

photons do interact!

elastic photon-photon

scattering



J. Schwinger, Phys. Rev., 82, 664 (1951)

The probability for vacuum to remain vacuum is

(R.Feynman)




Probability of pair creation

N.Narozhny, A.Nikishov, Yad.Fiz, 11, 1072(1970)

4-potential for a constant electric field:

Solutions of a wave equation cannot be characterized by a definite sign of frequency


There exists a complete and orthonormal set of solutions

Describes particles and antiparticles at

There exists another set of solutions

Describes particles and antiparticles at

There exist Bogolyubov relations between the sets


The Feynman Green function (fermions)

Matrix element for pair production


only this time interval effectively contributes to for given

classical turn point

formation length

and time


The relative probability of pair creation with quantum numbers

The absolute probability of pair creation with quantum numbers

the probability that pair with quantum

numbers was not created






Invariant characteristics of electromagnetic field numbers

1. Pairs can be created in an electromagnetic field with

2. Pairs cannot be created in an electromagnetic field with

For case 1 we can always find a reference frame with

For case 2 we can always find a reference frame with

J.Schwinger (1951):

Pairs can be created in an electromagnetic field with

independently of

If , we can always find a reference frame with

Pairs can be created if


For arbitrary static and uniform em field with numbers

If the field is not static and uniform but

and are space and time scale of variation of the field


THANK YOU numbers

FOR ATTENTION


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