BRST approach to Hamiltonian systems

BRST formulation of cohomological Hamiltonian mechanics is presented. In the path integral approach, we use the BRST gauge fixing procedure for the partition function with trivial underlying Lagrangian to fix symplectic diffeomorphism invariance. Resulting Lagrangian is BRST and anti-BRST exact and the Liouvillian of classical mechanics is reproduced in the ghost-free sector. The theory can be thought of as a topological phase of Hamiltonian mechanics and is considered as one-dimensional cohomological field theory with the target space a symplectic manifold. Twisted (anti-)BRST symmetry is related to global N=2 supersymmetry, which is identified with an exterior algebra. Landau-Ginzburg formulation of the associated $d=1$, N=2 model is presented and Slavnov identity is analyzed. We study deformations and perturbations of the theory. Physical states of the theory and correlation functions of the BRST invariant observables are studied. This approach provides a powerful tool to investigate the properties of Hamiltonian systems.