Functionals and the Quantum Master Equation

TL;DR

This paper develops an invariant formulation of the quantum master equation using graded tensor algebras, extending the anti-bracket and Laplacian to more general functional spaces.

Contribution

It introduces a new invariant definition of the anti-bracket and Laplacian for functionals valued in graded tensor algebras, broadening the mathematical framework of the quantum master equation.

Findings

Invariant definitions of anti-bracket and Laplacian for graded tensor algebra-valued functionals

New anti-bracket for ordinary functionals

Extension of quantum master equation formulation

Abstract

The quantum master equation is usually formulated in terms of functionals of the components of mappings from a space-time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the anti-bracket (odd Poisson bracket) of functionals and the other is the Laplacian of a functional. Both of these terms seem to depend on the fact that the mappings on which the functionals act are vector-valued. It turns out that neither this Laplacian nor the anti-bracket is well-defined for sections of an arbitrary vector bundle. We show that if the functionals are permitted to have their values in an appropriate graded tensor algebra whose factors are the dual of the space of smooth functions on M, then both the anti-bracket and the Laplace operator can be invariantly defined. Additionally, one obtains a new anti-bracket for ordinary functionals.