By F.J. Dyson (lecture notes), Michael J. Moravcsik (editor)
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Extra info for Advanced Quantum Mechanics, 2nd Edition
To avoid this Dirac constructed the many-time theory in which each electron has its own private time coordinate, and satisfies its private Dirac equation. This theory is all right in principle. But it becomes hopelessly complicated when pairs are created and you have equations with new time-coordinates suddenly appearing and disappearing. In fact the whole program of quantizing the electron theory as a theory of discrete particles each with its private time becomes nonsense when you are dealing with an infinite “sea” or an indefinite number of particles.
We do not try to derive or justify the Feynman formula here. We just show that it gives the same results as the usual QM. For a discussion of the difficulties in defining the sum H , and a method of doing it in simple cases, see C. Morette, Phys. Rev. 81 (1951) 848. From formula (174) we derive at once the most general Correspondence Principle giving us back the classical theory in the limit as → 0. For suppose → 0 then the exponential factor in (174) becomes an extremely rapidly oscillating function of H for all histories H except that one for which I(Ω) is stationary.
Roy. Soc. A 63 (1950) 681. K. Bleuler, Helv. Phys. Acta 23 (1950) 567. The older treatment is unnecessary and difficult, so we will not bother about it. By (211), (216) is equivalent to assuming (kµ akµ ) Ψ = 0 (216a) µ for each momentum vector k of a photon. As a result of this work of Gupta and Bleuler, the supplementary conditions do not come into the practical use of the theory at all. We use the theory and get correct results, forgetting about the supplementary conditions. Gauge-Invariance of the Theory The theory is gauge-invariant.