Quantum variational calculus on a lattice

TL;DR

This paper develops a quantum variational calculus framework on noncommutative lattices, deriving equations of motion, conservation laws, and energy relations for scalar fields in discrete noncommutative geometries.

Contribution

It introduces a quantum version of the Anderson variational double complex for noncommutative spaces, solving variational calculus problems on lattices with new theoretical tools.

Findings

Derived Euler-Lagrange equations for scalar fields on a lattice.

Obtained conserved stress-energy tensor and Noether charges.

Established modified energy-momentum relations on the lattice.

Abstract

We solve the long-standing problem of variational calculus on a noncommutative space or spacetime for a significant class of models with trivial jet bundle. Our approach entails a quantum version of the Anderson variational double complex $\Omega(J^\infty)$ and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a $\Bbb Z^m$ lattice regarded as a discrete noncommutative geometry, obtaining the Klein-Gordon equation for a scalar field, including with a general metric and gauge field background, as the Euler-Lagrange equations of motion for an action. In the case of a flat metric we also obtain an exactly on-shell conserved stress-energy tensor and Noether charges for a scalar field on the lattice and modified energy-momentum relations.