The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

“…We replace h i,r i (y i , z i ), for i ∈ I 23 , by the integral in (16) which introduces |I 23 | new variables of integration x i . We use the forgetful property (3)…”

Section: From Probability To Combinatoricsmentioning

confidence: 99%

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

Abdesselam

2020

Commun. Math. Phys.

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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. Contents 4.3. Graph construction 33 4.4. Two-scale pin and sum argument 34 4.5. Power-counting 36 5. Proof of the main theorem 37 6. A detailed example: the fractional massless free field 38 6.1. The explicit abstract system of pointwise correlations and the BNNFB 38 6.2. The OPE structure 40 6.3. A proof of the ENNFB 44 6.4. A proof of the soft and hard hypotheses on the OPE coefficients 47 6.5. Application of the main theorem to the construction of local Wick monomials as random distributions 48 6.6. Conformal invariance 49 References 50 1.2. Abstract systems of pointwise correlations. Let A be a finite set which we will call an alphabet and will serve to label fields. Define the big diagonal

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“…It was shown in [1] (see [2] for more detail) that the solution of such a reversion problem is given by the perturbation expansion of the following quantum field theory one-point function…”

Section: The "Proof "mentioning

confidence: 99%

“…Aknowledgments : The content of this article is an outgrowth of techniques developed in collaboration with V. Rivasseau and presented in [1]. We thank J. Feldman for his invitation to the Mathematics Department of the University of British Columbia where part of this work was done.…”

Section: Introductionmentioning

confidence: 99%

“…We will use the quantum field theory model introduced in [1], which is related to an earlier formula of G. Gallavotti [6] for the Lindstedt series in KAM theory, in order to express the compositional inverse of a power series in the multivariable setting. Our "proof" of the Lagrange-Good formula will follow from this representation of the formal inverse by straightforward and quite natural field theoretical computations which will, in particular, "explain" the determinantal denominator as a normalization factor for a probability measure.…”

Section: Introductionmentioning

confidence: 99%

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