Observable currents and a covariant Poisson algebra of physical observables

TL;DR

This paper develops a covariant algebra of gauge-invariant observable currents in bounded spacetime domains, generalizing Lie algebra structures and emphasizing local, boundary-sensitive properties for modeling spacetime physics.

Contribution

It introduces a framework for observable currents with a modified Lie algebra structure in bounded domains, incorporating boundary effects and a revised gauge invariance concept.

Findings

Observable currents form a generalized Lie algebra with boundary modifications.

The algebraic structure obeys gluing properties across subdomains.

Gauge invariance is redefined to include boundary conditions.

Abstract

Observable currents are locally defined gauge invariant conserved currents; physical observables may be calculated integrating them on appropriate hypersurfaces. Due to the conservation law the hypersurfaces become irrelevant up to homology, and the main objects of interest become the observable currents them selves. Gauge inequivalent solutions can be distinguished by means of observable currents. With the aim of modeling spacetime local physics, we work on spacetime domains $U\subset M$ which may have boundaries and corners. Hamiltonian observable currents are those satisfying ${\sf d_v}F=-\iota_V\Omega_L+{\sf d_h}\sigma^F$ and a certain boundary condition. The family of Hamiltonian observable currents is endowed with a bracket that gives it a structure which generalizes a Lie algebra in which the Jacobi relation is modified by the presence of a boundary term. If the domain of interest has no boundaries the resulting algebra of observables is a Lie algebra. In the resulting framework algebras of observable currents are associated to bounded domains, and the local algebras obey interesting gluing properties. These results are due to considering currents that defined only locally in field space and to a revision of the concept of gauge invariance in bounded spacetime domains. A perturbation of the field on a bounded spacetime domain is regarded as gauge if: (i) the "first order holographic imprint" that it leaves in any hypersurface locally splitting a spacetime domain into two subdomains is negligible according to the linearized gluing field equation, and (ii) the perturbation vanishes at the boundary of the domain. A current is gauge invariant if the variation in them induced by any gauge perturbation vanishes up to boundary terms.