Towards a Gauge Theory for Evolution Equations on Vector-valued Spaces

We investigate symmetry properties of vector-valued diffusion and Schrödinger equations. For a separable Hilbert space H we characterize the subspaces of L(R , 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. 1. The abstract setting: global symmetries In mathematical physics, one is often interested in the formulation of gauge theories. These are field theories where solutions of the relevant equations are symmetric – i.e., invariant under some transformation group of the functional values. A prototypical example is given by quantum electrodynamics, which is a gauge theory with respect to the symmetry group U(1) (the unitary group) and leads to the introduction of the electromagnetic field. The usual framework to deal with gauge theories in a mathematically rigorous way is that of differential geometry. Aim of this note is to propose a possible approach based on operator theoretic methods, instead, borrowing some ideas from the theory of vector bundles. Let H be a separable complex Hilbert space and consider the Bochner space H := L(R ;H), which is a Hilbert space with respect to the canonical inner product