The scattering of positrons by electrons with exchange on Dirac’s theory of the positron

Abstract

It has been shown by Mott that exchange effects play a considerable part in the collision and consequent scattering of one electron by another. Mott’s original calculation was non-relativistic, and there the exchange effect vanishes when the two electrons have their spins pointing in opposite directions. Møller later developed relativistically invariant expressions for the collision of two charged particles with spin, and it may be seen directly from Møller’s general formula for the collision cross-section that, in the collision of two identical particles, the effect of exchange does not in general vanish even when the two colliding particles initially have their spins pointing in opposite directions. It tends however to zero in this case as the relative velocity of the particles becomes small compared to c , the velocity of light, in agreement with the calculation of Mott. The effect of exchange in the general relativistic case will still be considerable if one of the two electrons be initially (and therefore finally) in a state of negative energy. (If one of the electrons be initially in a negative energy state, then it follows from the conservation of energy and momentum that one of them must be finally in a state of negative energy.) This at once leads to the conclusion that in the collision of an electron with a positron, the calculation of this process on the Dirac theory of the positron, where the positron is considered as an unoccupied state of negative energy, would lead to a result different from that which we should get if we did the calculation considering the positron as an independent positively charged particle in a state of positive energy whose behaviour is descibed by the Dirac equation. The difference would be due to the effect of exchange between the electron we observe initially and the virtual electrons in states of negative energy. As we shall show, the effect of this exchange is considerable. It tends to zero, however, when the relative velocity of the electron and positron becomes small compared to c , as we should expect from what has been said in the last paragraph.