Gupta-Bleuler formalism with scalar fermions in massless Yang-Mills theory

Abstract

In the Yang-Mills theory a formalism corresponding to the conventional Gupta-Bleuler formalism in quantum electrodynamics leads to difficulties with regard to the probability interpretation. It is shown that the tree diagrams of theS-matrix are gauge invariant and that nevertheless theS-matrix leads out of the physical subspace Open image in new window with positive semi-definite metric. TheS-matrix projected onto the physical subspace hence is no longer unitary. In order to arrive at a unitary theory the original Gupta-Bleuler formalism has to be extended by introducing a complex scalar isovector field which obeys anticommutation rules. As a consequence, the usual connection between spin and statistics does not hold in the large state space. This does not pose any difficulty because an indefinite metric is used in the state space. The scalar fermions contribute in such a peculiar way as to cancel all bad effects of the Gupta-Bleuler ghosts in the original formulation, yielding, therefore, the correct Feynman diagrams with the usual analyticity properties. It is possible to give a consistent probability interpretation for the asymptotic particles within this formalism without any additional unitarization procedure. In comparison with quantum electrodynamics the ghost structure is much more complicated and the indefinite metric nontrivial. Nevertheless, our notation of good and bad ghosts with some modification remains the most suitable language also in this extended Gupta-Bleuler formalism.