On Ghost Fermions

TL;DR

This paper rigorously derives the path integral for ghost fermions from first principles, clarifies the proper BRST operator extension, and discusses topological issues affecting physical state solutions in constrained quantization.

Contribution

It provides a first-principles derivation of ghost fermion path integrals and proposes a revised nonminimal BRST operator to avoid boundary terms, addressing longstanding issues in the field.

Findings

Derived the path integral for ghost fermions from first principles.

Identified the correct nonminimal BRST operator extension to avoid boundary terms.

Highlighted topological obstructions and anomalies in solving the Dirac condition for physical states.

Abstract

The path integral for ghost fermions, which is heuristically made use of in the Batalin- Fradkin-Vilkovisky approach to quantization of constrained systems, is derived from first principles. The derivation turns out to be rather different from that of physical fermions since the definition of Dirac states for ghost fermions is subtle. With these results at hand, it is then shown that the nonminimal extension of the Becchi-Rouet-Stora-Tyutin operator must be chosen differently from the notorious choice made in the literature in order to avoid the boundary terms that have always plagued earlier treatments. Furthermore it is pointed out that the elimination of states with nonzero ghost number requires the introduction of a thermodynamic potential for ghosts; the reason is that Schwarz's Lefschetz formula for the partition function of the time- evolution operator is not capable, despite claims to the contrary, to get rid of nonzero ghost number states on its own. Finally, we comment on the problems of global topological nature that one faces in the attempt to obtain the solutions of the Dirac condition for physical states in a configuration space of nontrivial geometry; such complications give rise to anomalies that do not obey the Wess-Zumino consistency conditions.