Generalized Schroedinger equation in Euclidean field theory

TL;DR

This paper introduces a generalized evolution equation for Euclidean quantum fields based on the general boundary formulation, extending the traditional Schrödinger equation to arbitrary spacetime regions.

Contribution

It provides a precise definition of an evolution kernel for Euclidean fields and derives a generalized Schrödinger and Hamilton-Jacobi equation for arbitrary boundary surfaces.

Findings

The evolution kernel satisfies a generalized evolution equation.

The classical counterpart is a Hamilton-Jacobi equation.

The framework applies to arbitrary spacetime regions.

Abstract

We investigate the idea of a "general boundary" formulation of quantum field theory in the context of the Euclidean free scalar field. We propose a precise definition for an evolution kernel that propagates the field through arbitrary spacetime regions. We show that this kernel satisfies an evolution equation which governs its dependence on deformations of the boundary surface and generalizes the ordinary (Euclidean) Schroedinger equation. We also derive the classical counterpart of this equation, which is a Hamilton-Jacobi equation for general boundary surfaces.