Batalin-Vilkovisky formalism in locally covariant field theory

TL;DR

This work develops a comprehensive, covariant Batalin-Vilkovisky formalism for classical and quantum field theories within a locally covariant framework, emphasizing infinite-dimensional geometry and algebraic quantization methods.

Contribution

It introduces a fully covariant BV complex construction using functors, extends it to classical gravity, and formulates a path integral-independent quantum BV framework within perturbative algebraic QFT.

Findings

Constructed the BV complex for classical gravity.

Established a covariant, functorial BV formalism.

Defined the renormalized quantum master equation.

Abstract

The present work contains a complete formulation of the Batalin-Vilkovisky (BV) formalism in the framework of locally covariant field theory. In the first part of the thesis the classical theory is investigated with a particular focus on the infinite dimensional character of the underlying structures. It is shown that the use of infinite dimensional differential geometry allows for a conceptually clear and elegant formulation. The construction of the BV complex is performed in a fully covariant way and we also generalize the BV framework to a more abstract level, using functors and natural transformations. In this setting we construct the BV complex for classical gravity. This allows us to give a homological interpretation to the notion of diffeomorphism invariant physical quantities in general relativity. The second part of the thesis concerns the quantum theory. We provide a framework for the BV quantization that doesn't rely on the path integral formalism, but is completely formulated within perturbative algebraic quantum field theory. To make such a formulation possible we first prove that the renormalized time-ordered product can be understood as a binary operation on a suitable domain. Using this result we prove the associativity of this product and provide a consistent framework for the renormalized BV structures. In particular the renormalized quantum master equation and the renormalized quantum BV operator are defined. To give a precise meaning to theses objects we make a use of the master Ward identity, which is an important structure in causal perturbation theory.