CXXVII. On field theories with non-localized interaction

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Abstract
It is shown that a method of direct quantization of field equations (completely avoiding the canonical formalism) is applicable to a wide class of fields with a non-local interaction. In § 1 the extended source theory of Peierls and McManus is quantized and discussed. The S-matrix is computed by the method of C. N. Yang. A general rule for an immediate transcription of the traditional S-matrix elements into the new form is given. In §2 another type of a non-local interaction is discussed and it is shown that all the usual covergence difficulties may be easily avoided if we give up the strict localizability of charge, energy, etc., and if we abandon the postulate of gauge invariance of the second kind. This is acceptable since it is possible to guarantee the integral conservation laws without the continuity equations and to secure a vanishing photon rest mass by other devices, not necessarily by the formal gauge invariance. In the limit λ→0 where λ is a fundamental length, the discussed field theory goes over into the traditional quantum electrodynamics.

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