Quantum variational calculus on a 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 $\hat{I}\text{c◯}(J^{\infty })$ and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a ${\mathbb{Z}}^m$ 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.

42 pages latex, 5 figures, minor improvements since last version