Remarks on a recent article of Ch. Obcemea and E. Brandas on the Theory of subdynamics: M. Courbage, Laboratoire de Probabilités, Tour 56 4, Place Jussieu, 75230 Paris Cedex 05, France

• Publisher: Elsevier Science
• Year: 1985
• Tongue: English
• Weight: 71 KB
• Volume: 163
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(85)90391-4

✦ Synopsis

Poisson brackets between two super-Hamiltonians on canonical coordinates (spatial metrics in geometrodynamics and embedding variables in parametrized theories) is usually regarded as an indication that the Dirac relations cannot be connected with a representation of the complete Lie algebra L Diffd of spacetime diffeomorphisms. It is shown how this difficulty may be overcome and a homomorphic mapping of spacetime vector fields YE L Diff 4 into the Poisson bracket algebra on the phase space of the system constructed. How the technique works in the case of a parametrized field theory is explained in the present paper, and it is generalized to canonical geometrodynamics in the companion paper (Part II). In a parametrized theory, the phase space of the system is the ordinary phase space of the field augmented by the embedding variables X: I --t 4 and their conjugate momenta. The dynamical variable H( I') which represents YE L Diff & generates a deformation of the embedding along the flow lines of Y accompanied by the correct dynamical evolution of the held data and preserves the constraints in the extended phase space of the system. The relation between the representations of Diff z and Diff A is also established.