Hidden BRS invariance in classical mechanics. II

Journal Article · Wed Nov 15 00:00:00 EST 1989 · Physical Review (Section) D: Particles and Fields; (USA)

DOI:https://doi.org/10.1103/PhysRevD.40.3363· OSTI ID:5216078

Gozzi, E; Reuter, M ^[1]; Thacker, W D ^[2]

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1. CERN, Geneva (Switzerland)
2. Fachbereich Physik, Universitaet Kaiserlautern, Kaiserlautern, (Federal Republic of Germany)

In this paper we develop a path-integral formulation of {ital classical} Hamiltonian dynamics, that means we give a functional-integral representation of {ital 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 {ital {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 {ital 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 {ital {tilde H}}, derived from the action {ital {tilde S}}, turns out to be the Lie derivative associated with the Hamiltonian flow. We also find that the action {ital {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 {ital 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.

• OSTI ID: 5216078
• Journal Information: Physical Review (Section) D: Particles and Fields; (USA), Vol. 40:10; ISSN 0556-2821
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CLASSICAL MECHANICS
INVARIANCE PRINCIPLES
EQUATIONS OF MOTION
FEYNMAN PATH INTEGRAL
HAMILTONIANS
PHASE SPACE
PROBABILITY
QUANTUM MECHANICS
DIFFERENTIAL EQUATIONS
EQUATIONS
INTEGRALS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
SPACE
657002* - Theoretical & Mathematical Physics- Classical & Quantum Mechanics