Observable currents in lattice field theories

TL;DR

This paper introduces a framework for observable currents in lattice field theories, linking local spacetime objects to physical observables, and explores their algebraic structures and applications to various models, including gauge theories.

Contribution

It develops a multisymplectic approach to observable currents in discrete spacetime, including local, nonlocal, and improved local currents, with applications to multiple field theories.

Findings

Observable currents induce physical observables via surface integration.

A Lie algebra structure is established on observable currents.

Weak observable currents can be localized and used to distinguish solutions.

Abstract

Observable currents are spacetime local objects that induce physical observables when integrated on an auxiliary codimension one surface. Since the resulting observables are independent of local deformations of the integration surface, the currents themselves carry most of the information about the induced physical observables. I introduce observable currents in a multisymplectic framework for Lagrangian field theory over discrete spacetime. One family of examples is composed by Noether currents. A much larger family of examples is composed by currents, spacetime local objects, that encode the symplectic product between two arbitrary vectors tangent to the space of solutions. A weak version of observable currents, which in general are nonlocal, is also introduced. Weak observable currents can be used to separate points in the space of physically distinct solutions. It is shown that a large class of weak observable currents can be "improved" to become local. A Poisson bracket gives the space of observable currents the structure of a Lie algebra. Peierls bracket for bulk observables gives an algebra homomorphism mapping equivalence classes of bulk observables to weak observable currents. The study covers scalar fields, nonlinear sigma models and gauge theories (including gauge theory formulations of general relativity) on the lattice. Even when this paper is entirely classical, this study is relevant for quantum field theory because a quantization of the framework leads to a spin foam model formulation of lattice field theory.