Manifestly Covariant and Manifestly Unitary Formulations of Massive Vector-Boson Theory in Indefinite-Metric Spaces

Abstract

A theory is formulated in which massive neutral vector bosons are coupled to a conserved current. The scalar bosons are excluded by using a Gupta-Bleuler-type subsidiary condition which requires the use of an indefinite-metric space. There are two different versions of the theory. One version exhibits manifest covariance and simple Feynman rules that are readily renormalized, but is burdened by unitarity problems. The other version is not manifestly covariant, involves reference to special Lorentz frames, and requires nonlocal operators. It is however manifestly unitary and probabilistically interpretable, since the sum of all transition probabilities = 1, and all probabilities are separately positive semidefinite. The two versions are related by a theorem which demonstrates that $S$-matrix elements are identical in the two versions, except for singular cases such as occur in wave-function renormalization terms. This identity also extends to the Proca theory, although, unlike the Proca theory, the theory treated in this work permits vector bosons to be massive, without requiring it; therefore no singularities or consistency problems arise in its $M\ensuremath{\rightarrow}0$ limit. The allowed nontransverse mode in this theory physically behaves like longitudinally polarized Proca bosons. The gauge invariance of this theory is discussed and compared with the gauge invariance of quantum electrodynamics.