Pair production as a process of classical fields

Suppose that we have two fields, F_1 and F_2. The hamiltonian of the system contains an interaction term which has a zero value if F_1 is zero or F_2 is zero.

Suppose that we have a wave (in other words, a particle) in F_1.

Can the wave in F_1 "create a particle" in F_2? In other words, can we get waves in F_2? Of course, we can.

                                           tense string

      F_2  -------------------------------------

                                |  rubber

                                |  band

      F_1  -----~~~~-------------------------

                    wave -->        tense string

In the diagram we have the two fields. The rubber band (if not loose) will transfer some wave energy from F_1 to F_2.

If we make a little wave in F_2, the interaction term of the rubber band saves more energy than we must consume to make the wave in F_2.

In pair production in QED, the creation of a pair probably reduces the field energy of the two colliding photons enough, so that the system total potential energy is lower when the pair exists. That is why the waves in the electromagnetic field produce a wave in the Dirac field.

This explanation of pair production is formulated with classical electromagnetic and Dirac fields.

When an observer measures the Dirac field, he will see two particles, that is, a real electron with a momentum p and a real positron with a momentum q.

The Schwinger effect

Julian Schwinger claimed that a strong static electric field will produce pairs. What does our crude classical field model say about the Schwinger effect?

A static electric field corresponds to a permanent depression in the string F_1 in the diagram above. The rubber band makes a static depression to the string F_2.

How could this produce waves in F_2?

If the electric field is produced by electrons, they are not static objects but must move according to quantum mechanics. Could this produce the movement which in turn could create pairs?

We have to figure out how to model the 1 / r field of an electron if we describe the electron as a Dirac wave packet.

In the Schrödinger equation we can treat two electrons as a single particle in 6 spatial dimensions, and add the Coulomb potential as an external potential which depends on the 6 spatial coordinates of the particle. Is there anything similar for a Dirac field?

Let us have a Feynman diagram of two colliding electrons. In the momentum space view, we do not have any knowledge of the location of the electrons in our 1 cubic meter area. The Fourier decomposition of the 1 / r potential, however, works in computing the scattering probability amplitudes. The decomposition looks very much the same wherever we place the pointlike source of the potential.

A single component of the Fourier decomposition looks like a time-independent sine wave potential to our electron. The potential will "refract" the Dirac wave of the electron, producing a small scattered Dirac wave.

t    

^   Fourier component of 1 / r

|           |        |        |

|  ______________________  e-  p= 0

|           |        |        |

|  ______________________

|           |        |        |

  -------------------------------> x

We can assume that our electron has a small momentum p. Its Dirac wave is almost horizontal crests in our standard diagram above.

But the diagram above cannot really explain the Schwinger effect. The "photon" (= the sine wave potential) only transfers momentum to the electron, no energy.

We need to check on what grounds Schwinger himself argued that there is pair production.