A Cohomological Perspective on Algebraic Quantum Field Theory

TL;DR

This paper explores algebraic quantum field theory through Hochschild cohomology, providing a new framework for understanding deformations and symmetries, which may reveal previously unrecognized symmetries and facilitate model construction.

Contribution

It introduces a cohomological perspective to AQFT, generalizing the framework and defining symmetries via Hochschild cohomology, independent of Lagrangian formulations.

Findings

Hochschild cohomology classes describe first-order deformations.

Computed cohomology classes for interaction terms.

Provides a more concrete approach to perturbative AQFT.

Abstract

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model. This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited. To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.