On generally covariant mathematical formulation of Feynman integral in Lorentz signature

TL;DR

This paper develops a rigorous mathematical formulation of the Dyson--Schwinger equations as a substitute for Feynman integrals in Lorentz signature quantum field theories, enabling covariant nonperturbative analysis without fixed causal backgrounds.

Contribution

It introduces a mathematically rigorous form of the master Dyson--Schwinger equation applicable in arbitrary signatures and provides conditions for solutions in conformally invariant theories.

Findings

Rigorous formulation of the MDS equation in arbitrary signature

A canonical Wilsonian regularization method for the MDS equation

A convergent iterative algorithm for solving the regularized MDS in conformal theories

Abstract

It is widely accepted that the Feynman integral is one of the most promising methodologies for defining a generally covariant formulation of nonperturbative interacting quantum field theories (QFTs) without a fixed prearranged causal background. Recent literature suggests that if the spacetime metric is not fixed, e.g. because it is to be quantized along with the other fields, one may not be able to avoid considering the Feynman integral in the original Lorentz signature, without Wick rotation. Several mathematical phenomena are known, however, which are at some point showstoppers to a mathematically sound definition of Feynman integral in Lorentz signature. The Feynman integral formulation, however, is known to have a differential reformulation, called to be the master Dyson--Schwinger (MDS) equation for the field correlators. In this paper it is shown that a particular presentation of the MDS equation can be cast into a mathematically rigorously defined form: the involved function spaces and operators can be strictly defined and their properties can be established. Therefore, MDS equation can serve as a substitute for the Feynman integral, in a mathematically sound formulation of constructive QFT, in arbitrary signature, without a fixed background causal structure. It is also shown that even in such a generally covariant setting, there is a canonical way to define the Wilsonian regularization of the MDS equation. The main result of the paper is a necessary and sufficient condition for the regularized MDS solution space to be nonempty, for conformally invariant Lagrangians. This theorem also provides an iterative approximation algorithm for obtaining regularized MDS solutions, and is guaranteed to be convergent whenever the solution space is nonempty. The algorithm could eventually serve as a method for putting Lorentz signature QFTs onto lattice, in the original metric signature.