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Operator product expansion algebra. (English) Zbl 1287.81085
The operator product expansion (OPE) in quantum field theory has been introduced and first studied by K. G. Wilson and W. Zimmermann in 1970–71. It seems helpful for practical calculations as well as getting a deeper understanding of the mathematical structure. Originally, it was suggested that the OPE should be an asymptotic expansion. However, in 2012, S. Hollands and C. Kopper showed, within the framework of the massive Euclidean $$\phi^4$$-quantum field theory in four dimensions, that the Wilson 2-point operator product expansion is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. This result is now extended to the N-point OPE, and so the first result of this paper is the convergence proof in perturbative QFT to more than two points, though in order to simplify the discussion, Holland and Hollands restrict to $$N=3$$. For concreteness they restrict their attention to the case of massive 4-dimensional $$\phi^4$$-theory. The second aim of this paper is to clarify the nature of the algebraic relations of the OPE coefficients. Much of the proof is based on the Wilson-Wegner-Polchinki renormalization group flow equation.

##### MSC:
 81T17 Renormalization group methods applied to problems in quantum field theory 81T08 Constructive quantum field theory 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T18 Feynman diagrams
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