Wick theorem for analytic functions of Gaussian fields

TL;DR

This paper develops a comprehensive framework for analyzing correlations of analytic functions of Gaussian fields, connecting discrete lattice models to continuum limits and exploring dualities with fermionic fields.

Contribution

It introduces a novel method to compute correlations using multigraphs and Feynman diagrams, and links these to tensor functionals and fermionic Gaussian states.

Findings

Correlation functions expressed via multigraphs and Feynman diagrams.

Scaling limits connect to Fock space tensor functionals.

Duality between bosonic and fermionic Gaussian fields reformulated as a matrix minors problem.

Abstract

We compute the correlation of analytic functions of general Gaussian fields in terms of multigraphs and Feynman diagrams on the lattice Z^d. Then, we connect its scaling limit to tensors of the correlation functionals of Fock space fields. Afterwards, we investigate the relation with fermionic Gaussian field states for even functions. For instance, we characterize the correlation functionals of the exponential of a continuous Gaussian Free Field or general analytic functions of fractional Gaussian fields as limits of quantities constructed via a sequences of discrete fields. Finally, we show that the duality between even powers of bosonic Gaussian fields and "complex" fermionic Gaussian fields can be reformulated in terms of a principal minors assignment problem of the corresponding covariance matrices.