Towards a gauge theory for evolution equations on vector-valued spaces

TL;DR

This paper explores symmetry properties of vector-valued evolution equations, characterizing local subspaces in Hilbert spaces and examining their invariance under evolution, with potential links to gauge symmetries in physics.

Contribution

It provides a characterization of local subspaces in vector-valued $L^2$ spaces and investigates their invariance under evolution equations, connecting to gauge symmetry concepts.

Findings

Characterization of local subspaces in $L^2( abla, H)$

Analysis of invariance under evolution operators

Discussion of gauge symmetry connections

Abstract

We investigate symmetry properties of vector-valued diffusion and Schr\"odinger equations. For a separable Hilbert space $H$ we characterize the subspaces of $L^2(\Omega, H)$ that are local (i.e., defined pointwise) and discuss the issue of their invariance under the time evolution of the differential equation. In this context, the possibility of a connection between our results and the theory of gauge symmetries in mathematical physics is explored.