Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions

TL;DR

This paper rigorously constructs scale-invariant non-Gaussian stochastic processes over three-dimensional p-adic space, confirming the existence of anomalous dimensions and universality, using advanced renormalization group techniques with space-dependent couplings.

Contribution

It provides a complete proof of the construction of these processes, including the squared field with anomalous dimension, and introduces a space-dependent coupling renormalization group formalism.

Findings

Confirmed the existence of anomalous dimensions in p-adic quantum field models.

Established a form of universality for the constructed models.

Developed a renormalization group approach with space-dependent couplings.

Abstract

In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional p-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. G. Wilson. We also prove a mild form of universality for the model under consideration. Our main innovation is that our rigourous renormalization group formalism allows for space dependent couplings. We derive the relationship between mixed correlations and the dynamical systems features of our extended renormalization group transformation at a nontrivial fixed point. The key to our control of the composite field is a partial linearization theorem which is an infinite-dimensional version of the Koenigs Theorem in holomorphic dynamics. This is akin to a nonperturbative construction of a nonlinear scaling field in the sense of F. J. Wegner infinitesimally near the critical surface. Our presentation is essentially self-contained and geared towards a wider audience. While primarily concerning the areas of probability and mathematical physics we believe this article will be of interest to researchers in dynamical systems theory, harmonic analysis and number theory. It can also be profitably read by graduate students in theoretical physics with a craving for mathematical precision while struggling to learn the renormalization group.