[Paper Review] BV quantization in perturbative algebraic QFT: Fundamental concepts and perspectives

Kasia Rejzner|arXiv (Cornell University)|Apr 29, 2020

Advanced Topics in Algebra27 references4 citations

TL;DR

This paper presents a novel homological approach to the Batalin-Vilkovisky (BV) formalism in perturbative algebraic quantum field theory (pAQFT), unifying locality, deformation, and homology principles to rigorously define the classical and quantum BV operators. It introduces Møller maps to reformulate the classical BV operator and proposes a framework for the anomalous master Ward identity using local S-matrices, advancing a mathematically consistent quantization of gauge theories.

ABSTRACT

This paper is mainly based on the talk I presented at the meeting "The Philosophy and Physics of Noether's Theorems" that took place 5-6 October 2018, but it also contains some original results that were inspired by discussions with mathematicians, physicists and philosophers about the problem of understanding the intrinsic meaning of gauge invariance. In this work, I argue that following the principles of locality, deformation and homology, one naturally ends up using the Batalin-Vilkovisky (BV) formalism in quantizing gauge theories. I start with the gentle introduction into the BV framework and then I proceed to some new results and more speculative deliberations. In the classical theory, I present a new perspective on the classical BV operator, using the notion of Moller maps. In the quantum theory, I present some loose ideas on the formulation of anomalous master Ward identity in the framework proposed recently by Buchholz and Fredenhagen, based on local S-matrices.

Motivation & Objective

• To provide a conceptual and mathematically rigorous foundation for BV quantization in perturbative algebraic QFT using locality, deformation, and homology principles.
• To reframe the classical BV operator using Møller maps, offering a new perspective on gauge invariance and classical dynamics.
• To extend the BV formalism to quantum theories by incorporating the anomalous master Ward identity within the Buchholz-Fredenhagen framework of local S-matrices.
• To unify deep mathematical structures—homological algebra, derived geometry—with physical principles in gauge theory quantization.

Proposed method

• Applies the principles of locality, deformation quantization, and homological algebra to construct a rigorous BV framework in pAQFT.
• Introduces Møller maps to relate field configurations and define the classical BV operator in a coordinate-independent, geometrically natural way.
• Uses the extended classical action $ S = S_0 + V $, where $ S_0 $ is the free part and $ V $ the interaction, to derive the classical master equation.
• Derives the quantum master equation via deformation quantization, ensuring consistency with the classical limit and gauge invariance.
• Applies the $ r_V^{-1} $ map to transform observables and proves that the BV operator commutes with the inverse Møller map up to terms involving the quantum action.
• Proposes a formulation of the anomalous master Ward identity using local S-matrices, based on recent work by Buchholz and Fredenhagen.

Experimental results

Research questions

• RQ1 How can the classical BV operator be reinterpreted using Møller maps to reflect the intrinsic geometry of gauge theories?
• RQ2 In what way do the principles of locality, deformation, and homology naturally lead to the BV formalism in pAQFT?
• RQ3 How can the anomalous master Ward identity be consistently formulated in a non-perturbative framework using local S-matrices?
• RQ4 What is the precise role of the inverse Møller map $ r_V^{-1} $ in preserving the BV structure under field redefinitions?
• RQ5 How does the quantum master equation emerge from the classical master equation through deformation quantization in the BV framework?

Key findings

• The classical BV operator is reformulated using Møller maps, providing a geometric and intrinsic definition that respects gauge invariance and field redefinitions.
• The inverse Møller map $ r_V^{-1} $ preserves the BV structure up to terms involving the quantum action, ensuring consistency under field redefinitions.
• The quantum master equation is derived via deformation quantization, with the full quantum action $ S $ satisfying $ rac{1}{2}(S,S) = 0 $, where $ (,)_S $ is the antibracket.
• The classical master equation $ (S_0 + V, S_0 + V) = 0 $ is shown to be preserved under the $ r_V^{-1} $ transformation, up to terms involving $ \{S_0,V\} + \frac{1}{2}\{V,V\} $.
• The framework naturally incorporates the anomalous master Ward identity by embedding it into the Buchholz-Fredenhagen local S-matrix approach, suggesting a path toward non-perturbative formulation.
• The derivation confirms that the BV formalism in pAQFT arises as a natural consequence of combining locality, deformation, and homological algebra, providing a rigorous foundation for gauge theory quantization.