Abstract
The Operator Product Expansion (OPE) and Renormalization Group (RG) are two of the most advanced and sophisticated structures in Quantum Field Theory (QFT). With this work we aim to show that the complexity in those areas is contained in one and the same universal operad structure. In more detail, this is a symmetric operad (with derivations) and its universality means that it is model independent within a large class of QFT models. The latter operad we call expansion operad. In the context of renormalization theory we find an isomorphic operad, which we call renormalization operad. The applications of the latter are for the description of the so called renormalization group and its action on the space of physical coupling constants via formal diffeomorphisms.
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Notes
- 1.
The analyticity properties of local quantum fields indicate that these expansions have more strong convergence than as asymptotic series. This is clearly seen in the so called Globally Conformal Invariant QFT (see [16, Sect. 8] and Sect. 2). It can be achieved even in the most general case of Wightman QFT [17].
- 2.
In physics literature the minus scaling dimension corresponds to the so called an energy dimension of a field.
- 3.
The series (28) is absolutely convergent on vector states with bounded conformal energy but in this case it is enough to axiomatize the OPE on a level of formal power series.
- 4.
Some authors call them also vertex operator algebras; the adjective “chiral” indicates that the \(D=1\) case is ment.
- 5.
All the sequences below are trivial at \(n=1\), but nevertheless it is convenient to start from \(n=1\).
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Acknowledgements
The author thanks for the useful discussions with Spencer Bloch, Francis Brown, Pierre Cartier and Maxim Kontsevich on various topics related to this work during his visits at Institut des Hautes Études Scientifiques (IHÉS, Bures-sur-Yvette, France). The author is grateful for the support and hospitality by Instituto de Ciencias Matemáticas (ICMAT, Madrid), where this work was presented during the Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory (ICMAT, September 15–December 19, 2014). The author thanks the referees for their careful reading of the manuscript an for suggesting many corrections and improvements. This work was supported in part by the Bulgarian National Science Fund under research grant DN-18/3.
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Nikolov, N.M. (2020). Vertex Algebras and Renormalization. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_12
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