Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 090, 45 pages      arXiv:0802.2198      https://doi.org/10.3842/SIGMA.2009.090
Contribution to the Special Issue on Deformation Quantization

Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology

Stefan Hollands
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK

Received September 19, 2008, in final form September 01, 2009; Published online September 18, 2009

Abstract
In this paper, we propose a new framework for quantum field theory in terms of consistency conditions. The consistency conditions that we consider are ''associativity'' or ''factorization'' conditions on the operator product expansion (OPE) of the theory, and are proposed to be the defining property of any quantum field theory. Our framework is presented in the Euclidean setting, and is applicable in principle to any quantum field theory, including non-conformal ones. In our framework, we obtain a characterization of perturbations of a given quantum field theory in terms of a certain cohomology ring of Hochschild-type. We illustrate our framework by the free field, but our constructions are general and apply also to interacting quantum field theories. For such theories, we propose a new scheme to construct the OPE which is based on the use of non-linear quantized field equations.

Key words: quantum field theory; operator product expansion; quantum algebra; Hochschild cohomology.

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