Dirac-Källén Correspondence – The Second Letters

Abstract

In a letter dated 18 October 1966, Dirac answered Källén’s first letter as follows:

Dear Källén,

Thank you for your long letter explaining your method in detail, and also the relativistic cut-off. At one time I worked a good deal with such cut-offs. As far as I know they do not apply to a logarithmic infinity. One can see the difficulty by a simple dimensional argument. One may introduce some procedure which makes $\int _{{0}}^{{\infty}}kdk$ count as zero and also $\int _{{0}}^{{\infty}}dk$ counts as zero. The corresponding result for the l o g ^mic ∞ would be to count $\int _{{a}}^{{\infty}}\frac{dk}{k}$ as zero. Here we must have some lower limit $\underline{a}$, not zero, having the dimensions of frequency or energy. The result cannot be relativistic. It is just the l o g a r ^mic ∞ which comes into renormalization. So I believe one must resign oneself to a non-relativistic cut-off.

There is another difficulty with your treatment. Ingoing fields cannot be defined in general for electrodynamics. Consider the example of two electrons scattering one another with the Coulomb force as the main force between them. Then the waves representing the electrons do not tend to plane waves sufficiently rapidly with increasing distance between the electrons. The ingoing waves for the electrons have definite frequencies and directions, but not definite phases, so they are not completely definable. This is a difficulty that occurs only with the inverse square law of force, but it does stop one from building up a complete electrodynamics in terms of ingoing fields.