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A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

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Abstract

We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer’s theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson’s operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field.

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Acknowledgements

This work would have been impossible without a semester leave from teaching and administrative duties in the form of a Sesquicentennial Associate Award. Therefore, the support of the Mathematics Department and the College and Graduate School of Arts and Sciences at the University of Virginia is very gratefully acknowledged. For useful discussions or correspondence we thank A. Chandra, J. Dubedat, J. Fageot, M. Furlan, C. Garban, M. Gubinelli, M. Hairer, S. Hollands, C. Hongler, C. Kopper, A. Kupiainen, P. Mitter, J.-C. Mourrat, J. Oikarinen, E. Peltola, V. Rychkov, E. Seiler, D. Simmons-Duffin, V. Vargas and E. Witten. Finally, we thank the anonymous referees for suggestions which helped improve this article.

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    Abdelmalek Abdesselam

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Communicated by M. Hairer

Dedicated to the memory of Roland Sénéor (1938–2016)

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Abdesselam, A. A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion. Commun. Math. Phys. 376, 555–608 (2020). https://doi.org/10.1007/s00220-019-03665-4

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