Hidden BRS invariance in classical mechanics. II

In this paper we develop a path-integral formulation of classical Hamiltonian dynamics, that means we give a functional-integral representation of classical transition probabilities. This is done by giving weight "one" to the classical paths and weight "zero" to all the others. With the help of anticommuting ghosts this measure can be rewritten as the exponential of a certain action $\tilde{S}$. Associated with this path integral there is an operatorial formalism that turns out to be an extension of the well-known operatorial approach of Liouville, Koopman, and von Neumann. The new formalism describes the evolution of scalar probability densities and of $p$-form densities on phase space in a unified framework. In this work we provide an interpretation for the ghost fields as being the well-known Jacobi fields of classical mechanics. With this interpretation the Hamiltonian $\tilde{H}$, derived from the action $\tilde{S}$, turns out to be the Lie derivative associated with the Hamiltonian flow. We also find that the action $\tilde{S}$ presents a set of Becchi-Rouet-Stora- (BRS-)type invariances mixing the original phase-space variables with the ghosts. Together with a Sp(2) symmetry of the pure ghosts sector, they form a universal invariance group ISp(2) which is present in any Hamiltonian system. The physical and geometrical meaning of the ISp(2) generators is discussed in detail: in particular the conservation of one of the generators is shown to be equivalent to the Liouville theorem. The ISp(2) algebra is then used to give a modern operatorial reformulation of the old Cartan calculus on symplectic manifolds.