Gauge-covariant extension of the finite-range FIR operator

Develop a gauge-covariant extension of the finite-range finite impulse response (FIR) lattice momentum operator using finite-range parallel transporters for gauge interactions.

Background

The paper constructs and numerically validates the FIR momentum operator only for free fermion propagation in one spatial dimension. Introducing gauge interactions requires replacing each finite lattice displacement by an appropriately ordered product of gauge links, thereby producing finite-range parallel transporters. Although the one-dimensional straight-path construction is relatively direct, higher-dimensional theories introduce path dependence and may require an explicit path choice or path averaging. Establishing this extension and determining whether the operator retains its favorable spectral and propagation properties in interacting gauge theories are left unresolved.