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Perturbative quantum field theory via vertex algebras

Hollands, Stefan and Olbermann, Heiner 2009. Perturbative quantum field theory via vertex algebras. Journal of Mathematical Physics 50 (11) , 112304. 10.1063/1.3255442

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In this paper, we explain how perturbative quantum field theory can be formulated in terms of (a version of) vertex algebras. Our starting point is the Wilson–Zimmermann operator product expansion (OPE). Following ideas of a previous paper ( S. Hollands, e-print arXiv:0802.2198 ), we consider a consistency (essentially associativity) condition satisfied by the coefficients in this expansion. We observe that the information in the OPE coefficients can be repackaged straightforwardly into “vertex operators” and that the consistency condition then has essentially the same form as the key condition in the theory of vertex algebras. We develop a general theory of perturbations of the algebras that we encounter, similar in nature to the Hochschild cohomology describing the deformation theory of ordinary algebras. The main part of the paper is devoted to the question how one can calculate the perturbations corresponding to a given interaction Lagrangian (such as λφ4) in practice, using the consistency condition and the corresponding nonlinear field equation. We derive graphical rules, which display the vertex operators (i.e., OPE coefficients) in terms of certain multiple series of hypergeometric type.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Uncontrolled Keywords: algebra, Lagrangian field theory, nonlinear field theory, perturbation theory, series (mathematics)
Additional Information: 42 pp.
Publisher: American Institute of Physics
ISSN: 0022-2488
Last Modified: 04 Feb 2018 01:24

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