Covariant Jacobi brackets for test particles

Abstract

We show that the space of observables of test particles carries a natural Jacobi structure which is manifestly invariant under the action of the Poincaré group. Poisson algebras may be obtained by imposing further requirements. A generalization of Peierls procedure is used to extend this Jacobi bracket on the space of time-like geodesics on Minkowski spacetime.

Jacobi sigma models

“…The model exhibits first class constraints, which generate gauge transformations. On using a consistent definition of Hamiltonian vector fields for Jacobi manifolds (see for example [20,21]), we show that the latter can be associated with gauge transformations and verify that they close under Lie bracket, generating space-time diffeomorphisms. The model results to be topological, with a finite number of degrees of freedom, on the boundary.…”

“…Contact forms are defined up to multiplication by a non-vanishing function. It is possible to endow the algebra of functions on a contact manifold with a Lie algebra structure [21], which reads…”

Contact manifolds and dissipation, classical and quantum

“…Remark 3 (Contact manifolds and Jacobi brackets). Given a manifold with contact structure (η, ω, ξ) it is possible define a Lie algebra structure of the space of functions by means of [3] [F , G ] Ω = (n − 1) (dF ∧ dG ∧ η) ∧ ω n−1 + (F dG − G dF ) ω n (102)…”

Topological and Dynamical Aspects of Jacobi Sigma Models

“…Non-trivial examples of LCS manifolds may be easily constructed by considering the product M × S 1 , with M a contact manifold [31]. In Section 5.1.1, we shall consider in some detail the case of the Lie group SU(2), which has been been widely studied in the literature in relation with Poisson sigma models (see, for example, [33,34]) and we shall only sketch its LCS counterpart…”