Generalized Tomonaga-Schwinger equation from the Hadamard formula

TL;DR

This paper derives a generalized Tomonaga-Schwinger equation within continuous Euclidean field theory directly from the functional integral formalism, utilizing Hadamard's formula, advancing the theoretical framework for background-independent quantum field theory.

Contribution

The paper provides a direct derivation of the generalized Tomonaga-Schwinger equation from the functional integral approach, bypassing lattice assumptions.

Findings

Equation derived using Hadamard's formula

Applicable to continuous Euclidean field theory

Enhances understanding of background-independent QFT

Abstract

A generalized Tomonaga--Schwinger equation, holding on the entire boundary of a {\em finite} spacetime region, has recently been considered as a tool for studying particle scattering amplitudes in background-independent quantum field theory. The equation has been derived using lattice techniques under assumptions on the existence of the continuum limit. Here I show that in the context of continuous euclidean field theory the equation can be directly derived from the functional integral formalism, using a technique based on Hadamard's formula for the variation of the propagator.