By P. A. M. DIRAC
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Extra info for Classical Theory of Radiating Electrons
This hints at the future identification of γ with the electric charge of the particle emitting a photon. Indeed, let us consider an antiparticle which propagates from x2 , emits a photon at x and then propagates to x1 . For the corresponding amplitude we would write ↔ ∂ G(x − x2 ) , γa · G(x1 − x) ∂xµ where we used Ga = G and the subscript a refers to antiparticle. The same process can be described as photon emission by a particle by taking the reversed time sequence, t2 < t < t1 , ↔ γ · G(x2 − x) ∂ G(x − x1 ) .
Henceforth, we shall write Ψ+ = Ψ. In the case of the stationary state Ψ(x) = Ψ(r)e−iEt , the function ρ(r) plays the rˆole of the probability density ρ(r) = 2E |Ψ(r)|2 ρ(x) = Ψ∗+ (x) i 38 1 Particles and their interactions in the same way as for the photon. 96) for a free particle. 96) for G(x). ) As before, G(x) = G(−x) . e. positive frequencies propagate forward in time. 7 Interactions of spinless particles How can an interaction be described in relativistic quantum theory? There is no potential, there are no forces – all these are entirely nonrelativistic notions.
1) Both poles are in the lower half-plane (marked as ◦). In this case, if t < 0, the contour has to be closed in the upper half-plane, and GR = 0 , t < 0. If t > 0, the contour has to be closed in the lower half-plane, and we have GR = d3 k e−i|k|t+ik·r − (2π)3 2|k| d3 k ei|k|t+ik·r , (2π)3 2|k| t > 0. The retarded Green function GR contains negative frequencies and therefore does not suit us: we cannot use it to describe propagation of relativistic particles. (2) The pole on the negative axis is shifted upwards, while that on the positive axis is shifted downwards (marked as ×).