On the formulation of SPDEs leading to local, relativistic QFTs with indefinite metric and nontrivial S-matrix

TL;DR

This paper develops a mathematical framework for constructing relativistic quantum field theories with indefinite metrics using stochastic partial differential equations, providing explicit formulas for scattering amplitudes and conditions for nontrivial scattering.

Contribution

It introduces explicit formulae for Wightman functions from SPDEs and establishes conditions on the mass spectrum for nontrivial scattering in relativistic QFTs.

Findings

Explicit formulas for Fourier transformed truncated Wightman functions

Necessary and sufficient conditions on mass spectrum for nontrivial scattering

Explicit computation of scattering amplitudes

Abstract

We discuss Euclidean covariant vector random fields as the solution of stochastic partial differential equations of the form $DA=\eta$, where $D$ is a covariant (w.r.t. a representation \tau of $SO(d)$) differential operator with "positive mass spectrum" and $\eta$ is a non-Gaussian white noise. We obtain explicit formulae for the Fourier transformed truncated Wightman functions, using the analytic continuation of Schwinger functions discussed by Becker, Gielerak and Lugewicz. Based on these formulae we give necessary and sufficient conditions on the mass spectrum of $D$ which imply nontrivial scattering behaviour of relativistic quantum vector fields associated to the given sequence of Wightman functions. We compute the scattering amplitudes explicitly and we find that the masses of particles in the obtained theory are determined by the mass spectrum of $D$.