Quantum field theory and Hopf algebra cohomology

TL;DR

This paper introduces a Hopf superalgebra framework for quantum field theory that simplifies calculations of operator products and expectation values, and extends to interacting theories using cohomology.

Contribution

It constructs a Hopf superalgebra structure for QFT operators and develops a twist deformation approach for products, enabling more efficient computations and broader applicability.

Findings

Formulas for perturbative products and expectation values are more efficient.

Extension of the framework to non-perturbative interacting QFT.

Reconstruction theorem for time-ordered products and vacua description.

Abstract

We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.