Indefinite-Metric Quantum Theory of Genuine and Higgs-Type Massive Vector Fields

On the basis of the indefinite-metric vector field theory proposed previously, Johnson's proposition that the physical mass of a vector field tends to zero as its bare mass goes to zero, is shown to be valid if the vector field couples with a charged scalar field in the minimal interaction. In the case of the theory of spontaneously broken gauge invariance, the reason why the vector field acquires a non-zero mass in spite of the above theorem is clarified. The theory of a vector field which is massive owing to the spontaneous breakdown of gauge invariance is consistently formulated in the framework of the indefinite-metric quantum field theory. In this formalism, both renormalizability and the unitarity of the physical S-rnatrix are self-evident. § I. Introduction Recently, the present author 1 l'*l has proposed an indefinite-metric theroy 2 l of a massive vector field such that as its mass goes to zero the theory smoothly tends to the Landau-gauge quantum electrodynamics.3l' 4 l As one of important consequences of this theory, we can reasonably show the validity of Johnson's proposition 5 l'**l that if the bare mass of the vector field ufo goes to zero, its physical mass must also tend to zero, provided that there are no other massless physical particles, under the assumption that the current j'" is conserved and does not explicitly depend on ufo. On the other hand, in connection with Weinberg's theory of leptons, BJ much attention has been paid to the spontaneous breakdown of gauge invariance in the massless vector field theories. Several years ago, Higgs and others 7 l noted that if gauge invariance of the theory is spontaneously broken, the massless vector field acquires a non-zero mass, but then Goldstone bosons do not appear in the Coulomb gauge because we do not have manifest covariance, which is necessary for the proof of the Goldstone theorem. 8 l If one reconsiders this situation in a covariant gauge, in which we have to introduce indefinite metric, Goldstone bosons appear but they become unphysical. This interesting phenomenon is now called the Higgs phenomenon. Recently, 't Hoofel has applied it to the Yang-Mills *> Unfortunately, the publication of this paper was much delayed. **> Johnson's reasoning was based on the conventional massive vector field theory whose mass-less limit is non-existent.