General quantum-mechanical setting for field-antifield formalism as a hyper-gauge theory

TL;DR

This paper develops a quantum-mechanical framework for the field-antifield formalism as a hyper-gauge theory, formulating a Schrödinger-type equation and expressing its solution as a path integral with a symmetric Weyl symbol.

Contribution

It introduces a novel quantum evolution equation for the field-antifield formalism and provides a closed-form path integral solution, expanding the theoretical understanding of hyper-gauge theories.

Findings

Formulated a Schrödinger-type equation for the formal quantum evolution.

Expressed the solution as a path integral with Weyl's symbol.

Explicitly computed the path integral for a general hyper-gauge Fermion.

Abstract

A general quantum-mechanical setting is proposed for the field-antifield formalism as a unique hyper-gauge theory in the field-antifield space. We formulate a Schr\"odinger-type equation to describe the quantum evolution in a "current time" purely formal in its nature. The corresponding Hamiltonian is defined in the form of a supercommutator of the delta-operator with a hyper-gauge Fermion. The initial wave function is restricted to be annihilated with the delta-operator. The Schr\"odinger's equation is resolved in a closed form of the path integral, whose action contains the symmetric Weyl's symbol of the Hamiltonian. We take the path integral explicitly in the case of being a hyper-gauge Fermion an arbitrary function rather than an operator.