Systems of classical particles in the grand canonical ensemble, scaling limits and quantum field theory

TL;DR

This paper explores the connection between Euclidean quantum fields, classical particle systems, and their scaling limits, providing insights into ultraviolet finite interactions and continuum limits in quantum field theory.

Contribution

It introduces a class of ultra-violet finite local interactions for Euclidean quantum fields via stochastic PDEs driven by Poisson noise, linking them to classical particle systems in the grand canonical ensemble.

Findings

Finite volume limits for models with trigonometric interactions

Representation of models as Widom-Rowlinson systems at imaginary temperature

Analysis of continuum limits and their triviality in simple cases

Abstract

Euclidean quantum fields obtained as solutions of stochastic partial pseudo differential equations driven by a Poisson white noise have paths given by locally integrable functions. This makes it possible to define a class of ultra-violet finite local interactions for these models (in any space-time dimension). The corresponding interacting Euclidean quantum fields can be identified with systems of classical "charged" particles in the grand canonical ensemble with an interaction given by a nonlinear energy density of the "static field" generated by the particles' charges via a "generalized Poisson equation". The infinite volume limit of such systems is discussed for models with trigonometric interactions using a representation of such models as Widom-Rowlinson models associated with a (formal) Potts models at imaginary temperature. The continuum limit of the particle systems under consideration is also investigated and the formal analogy with the scaling limit of renormalization group theory is pointed out. In some simple cases the question of (non-) triviality of the continuum limits is clarified.