Noncommutative Bloch analysis of Bochner Laplacians with nonvanishing gauge fields

TL;DR

This paper develops a noncommutative Bloch analysis framework for Bochner Laplacians with nonvanishing gauge fields on manifolds with discrete symmetries, generalizing known physics results to a broader mathematical setting.

Contribution

It introduces a construction of the Bloch decomposition for noncommutative gauge fields and symmetry groups, extending classical results to more general geometric and algebraic contexts.

Findings

Decomposition of Hamiltonians into direct integrals over dual group representations

Expressing propagators and Green functions in terms of group sums

Establishing the inverse relationship between the constructions

Abstract

Given an invariant gauge potential and a periodic scalar potential \tilde{V} on a Riemannian manifold \tilde{M} with a discrete symmetry group \Gamma, consider a \Gamma-periodic quantum Hamiltonian \tilde{H}=-\tilde{\Delta}_{B}+\tilde{V} where \tilde{\Delta}_{B} is the Bochner Laplacian. Both the gauge group and the symmetry group \Gamma can be noncommutative, and the gauge field need not vanish. On the other hand, \Gamma is supposed to be of type I. To any unitary representation \Lambda of \Gamma one relates a Hamiltonian H^{\Lambda}=-\Delta_{B}^{\Lambda}+V on M=\tilde{M}/\Gamma where V is the projection of \tilde{V} to M. We describe a construction of the Bloch decomposition of \tilde{H} into a direct integral whose components are H^{\Lambda}, with \Lambda running over the dual space \hat{\Gamma}. The evolution operator and the resolvent decompose correspondingly. Conversely, given \Lambda\in\hat{\Gamma}, one can express the propagator \mathcal{K}_{t}^{\Lambda}(y_{1},y_{2}) (the kernel of \exp(-itH^{\Lambda})) in terms of the propagator \tilde{\mathcal{K}}_{t}(y_{1},y_{2}) (the kernel of \exp(-it\tilde{H})) as a weighted sum over \Gamma. Such a formula is known in theoretical physics for the case when the gauge field vanishes and \tilde{M} is a universal covering of a multiply connected space M. We show that these constructions are mutually inverse. Analogous formulas exist for resolvents and their kernels (Green functions) as well.