Symmetries of the classical path integral on a generalized phase-space manifold

In this paper we generalize previous work done on the path-integral approach to classical mechanics and its symmetries. We study in particular the case that the components of the symplectic two-form ${\omega }_{\mathrm{ab}}$, expressed in arbitrary coordinates, are allowed to depend on the phase-space coordinates. This lifts the restriction that the path integral and its symmetry generators be expressed only in terms of canonical coordinates. We show, in particular, that an extra term must be added to the anti-Becchi-Rouet-Stora (anti-BRS) charge in order to preserve the ISp(2) symmetry which reflects the geometry of phase space. The cohomology of this new anti-BRS operator is found to be isomorphic to the de Rham cohomology of phase space. The modification of the anti-BRS charge leads to a modification of one of the supersymmetry generators associated with the classical Hamiltonian. Despite this change in the form of the generators, the classical Kubo-Martin-Schwinger conditions can still be derived from this supersymmetry. We also prove that the requirement of supersymmetric invariance of the states results in a new set of equations that, despite their new form, are still satisfied by the Gibbs states on a general phase-space manifold.