A class of covariant commutators for fields of integer spinJ andm ≠ 0, and form = 0 without the necessity of introducing an indefinite metric

Abstract

We show that it is possible to give covariant commutators for a field of integer spinJ, for nonzero and for zero mass, as particular cases of a more general commutator. We calculate this explicitly for spin 1 and 2. We generate the decompositions of the momentum amplitude for a field of integer spinJ, by decomposing the momentum amplitude of a vector fieldAμ in three directions in such a way that the condition of zero divergence is satisfied. In the nonzero-mass case, as is well known, quantum conditions are given on the (2J + 1) independent components; in this way the usual commutators are obtained. In the massless case only two of the components appear as gauge invariants, and for this reason quantum conditions are given only on these two components, since the others are physically irrelevant. Moreover, the commutators thus obtained are compatible with the field equations and with the subsidiary conditions, and the energy is automatically positive definite, without it being necessary to introduce an indefinite metric.