Field Theory of Equations with Many Masses

Abstract

This paper develops the field theory of many mass equations with special attention to spin \textonehalf{} and the operator (1) of the author's previous paper on the irreducible volume character of events. The field is assumed to interact with the electromagnetic field which is introduced in a gauge-invariant way. General expressions for the charge-current four-vector and the symmetrical energy-momentum tensor are derived and are shown to satisfy the appropriate conservation theorems. According to a theorem of Leichter, the general solution is shown to be a superposition of nonorthogonal mass states which we designate as the root fields. Nevertheless, the physical quantities, such as the current four-vector, the energy-momentum tensor, etc., are shown to decompose into a sum over those of individual mass states but with an alternation of sign for consecutive roots. The Lagrangian takes the form of an alternating sum over the individual free-field Lagrangians for the mass states, plus the usual term $+(\frac{1}{c}){j}_{\ensuremath{\mu}}{A}_{\ensuremath{\mu}}$ for the interaction with the electromagnetic field. The matter field may be quantized by treating the root fields as independent anticommuting fields. The transformation to the interaction representation is obviously unaltered and the charge and mass renormalization may be treated following Schwinger. To the order of approximation in Schwinger II these renormalizations are not affected. It would seem that these methods of quantization, together with the usual treatment of the electromagnetic field, are at variance with the manifest nonlocal nature of the theory for the irreducible volume character of events.