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pair creation by a constant electric field

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pair creation by a constant electric field

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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

is associated with the pair creation in the electric field

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

The probability for vacuum to remain vacuum is

(R.Feynman)

Let be the probability of pairs creation

Let

actually

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

The probability for vacuum to remain vacuum

as it should be

The total absolute probability of creation of

1 pair:

2 pairs:

The average number of particles

The total absolute probability of a single pair creation

and only if

Invariant characteristics of electromagnetic field

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

If the field is not static and uniform but

and are space and time scale of variation of the field

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