Some Consequences of Gauge Invariance

Abstract

A simple theorem is .,rated to expand the matrix element Jor a process concerning any number of photons or external elecrrpmagnetic fields into the independent gauge invariant terms. It turns out 'hat the number of such terms is uniquely determined by the initial and final states of the process, regardless of the intermediate states and the type of iqteraction. We have, for example, 4, 14, and 10 independent gauge invariant terms (or the production Qf spin 0 and 1 mesons bv photons . and the Compton scattering, respectively. Some of the well-known absolute selection {ulesfor meson decays can be also derived from this gauge invariant expansion theorem. In their recent works on the production of spinless mesons by photons,')'** Kobaet al. have shown that the requirement of gauge invariance greatly simplified the evaluation and made facile the analysis of the results. In fact they developed. the enormous number of matrix elements into only four independent gauge invariant terms irrespective of the order of approximation. Their conclusion was a direct consequence of i) Lorentz invariance, ii) gaugeinvariance, iii) energy-momentum conservation, iv) spinless field for the meson, and v) Dirac equation for the nucleon, and thus it mattered only about the initial and final states of the process but not about the method of calculation. Such a procedure can therefore be applied 11tlttatis 1nutandis to other cases involving one or more pltotons or external electromagnetic fields. However, since their method itself is very special and seems inconvenient to apply to other processes, its reexamination is desirable. It is the aim of this paper to observe the gauge invariance from the most general points of view and present a method to expand complex matrix elements into their fundamental gauge invariant terms. We shall show in § 2 that a simple consideration indeed a:ffords us such a prescription which is far more concise than theirs· and in addition widely applicable. In the next section, the gauge invariant expansion theorem of the matrix elements is given, according to which it turns out that the number of gauge invariant terms is unambiguously determined by the process, not depending on the method of calculation. We have, for example, 4, 14, and 10 such terms for the production of spin 0 and 1 mesons by photons and the Compton scattering, respectively. In § 4 the possibilities are discussed for utilizing this theorem to derive the absolute selection rules for meson decay. However, as might be expected, the selection rules are obtained only for certain simple cases.