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Type: Preprint
2. Observable currents and a covariant Poisson algebra of physical observables
Abstract:
Observable currents are conserved gauge invariant 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. Hamiltonian observable currents are those satisfying d_v F=-\iota_V \Omega_L+ d_h \sigma^F. The presence of the boundary term allows for rich families of Hamiltonian observable currents. We show that Hamiltonian observable currents are capable of distinguishing solutions which are gauge inequivalent. Hamiltonian observable currents are endowed with a bracket, and the resulting algebraic structure is a generalization of a Lie algebra in which the Jacobi relation has been modified by the presence of a boundary term. When integrating over a hypersurface with no boundary, the bracket induced in the algebra of observables makes it a Poisson algebra. With the aim of modelling spacetime local physics, we work on spacetime domains which may have boundaries and corners. In the resulting framework algebras of observable currents are associated to bounded domains, and the local algebras obey interesting glueing properties. These results are due to a revision of the concept of gauge invariance. 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 glueing field equation, and (ii) the perturbation vanishes at the boundary of the domain. A current is gauge invariant if the variation induced by any gauge perturbation vanishes up to boundary terms.
Observable currents are conserved gauge invariant 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. Hamiltonian observable currents are those satisfying d_v F=-\iota_V \Omega_L+ d_h \sigma^F. The presence of the boundary term allows for rich families of Hamiltonian observable currents. We show that Hamiltonian observable currents are capable of distinguishing solutions which are gauge inequivalent. Hamiltonian observable currents are endowed with a bracket, and the resulting algebraic structure is a generalization of a Lie algebra in which the Jacobi relation has been modified by the presence of a boundary term. When integrating over a hypersurface with no boundary, the bracket induced in the algebra of observables makes it a Poisson algebra. With the aim of modelling spacetime local physics, we work on spacetime domains which may have boundaries and corners. In the resulting framework algebras of observable currents are associated to bounded domains, and the local algebras obey interesting glueing properties. These results are due to a revision of the concept of gauge invariance. 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 glueing field equation, and (ii) the perturbation vanishes at the boundary of the domain. A current is gauge invariant if the variation induced by any gauge perturbation vanishes up to boundary terms.
MSC: 70S05, 81T13, 70S15, 53C80
Year Preprint: 2017
Report Number: UNAM-CCM-2017-2
arXiv: arXiv:1704.07955 [gr-qc]



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