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Correlation functions for the Gross-Neveu model

J. Dimock

TL;DR

The paper develops a non-perturbative, finite-volume analysis of correlation functions in the two-dimensional massless Gross-Neveu model using a Grassmann-algebra and renormalization-group framework. By constructing a generating-functional formalism and performing a controlled RG flow, it proves that all correlation functions, treated as distributions, are uniformly bounded in the ultraviolet cutoff, with explicit expressions and bounds for the two-point and higher-point functions. The authors derive first-order coupling corrections and present a pathway to higher-order computations, highlighting the emergence of a renormalized propagator and the applicability of the method to bosonic-field models where perturbation theory may fail. This work provides a robust, non-perturbative toolkit for controlling quantum field-theoretic correlation functions in finite volume, with implications for broader classes of interacting fermionic systems.

Abstract

This is a non-perturbative treatment of correlation functions for the weakly coupled massless Gross-Neveu model in a finite volume. The main result is that all correlation functions, treated as distributions, are uniformly bounded in the ultraviolet cutoff.

Correlation functions for the Gross-Neveu model

TL;DR

The paper develops a non-perturbative, finite-volume analysis of correlation functions in the two-dimensional massless Gross-Neveu model using a Grassmann-algebra and renormalization-group framework. By constructing a generating-functional formalism and performing a controlled RG flow, it proves that all correlation functions, treated as distributions, are uniformly bounded in the ultraviolet cutoff, with explicit expressions and bounds for the two-point and higher-point functions. The authors derive first-order coupling corrections and present a pathway to higher-order computations, highlighting the emergence of a renormalized propagator and the applicability of the method to bosonic-field models where perturbation theory may fail. This work provides a robust, non-perturbative toolkit for controlling quantum field-theoretic correlation functions in finite volume, with implications for broader classes of interacting fermionic systems.

Abstract

This is a non-perturbative treatment of correlation functions for the weakly coupled massless Gross-Neveu model in a finite volume. The main result is that all correlation functions, treated as distributions, are uniformly bounded in the ultraviolet cutoff.

Paper Structure

This paper contains 13 sections, 13 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of the Grassmann algebra
  4. derivatives
  5. integrals
  6. auxiliary fields
  7. Correlation functions
  8. generating functions
  9. correlation functions
  10. first order corrections
  11. concluding remarks
  12. The weight function $\Gamma(X)$
  13. Smooth approximations to distributions

Key Result

Theorem 1

DiYu24 Let $L, h, C_Z, C_E$ be sufficiently large and chosen in that order, and let $g_{\max}$ be sufficiently small depending on these constants. Suppose $S_k$ has the form (bongo) with Then $S_{k+1}$ has the same form with new parameters

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Lemma 1
  • ...and 3 more