P. Hajicek
Group Quantization of Parametrized Systems. I. Time Levels
The paper is published: J. Math. Phys. 36 (1995) 4612-4638

MSC:

58F06 Geometric quantization (applications of representation theory), See also {22E45, 81S10}

81S10 Geometric quantization, symplectic methods, See Also {

83A05 Special relativity

83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems)

58F05 Hamiltonian and Lagrangian systems; symplectic geometry, See also {70Hxx, 81S10}

81T13 Yang-Mills and other gauge theories, See also {53C07,

Abstract: Dirac's ideas on forms of relativistic dynamics are generalized to all
finite-dimensional first-class parametrized constrained systems. The
Poincar\'{e} group and the algebra of its generators is replaced by a
canonical group of symmetries and by an algebra of elementary observables.
The quantum theory is determined by two choices: that of the group (or
algebra) and that of its representation. Achoiceof gauge and time is
represented by a transversal surface in the phase space. Projection of
symmetry transformations and of observables to transversal surfaces is
defined and shown to preserve both the group and the algebraicstructures.
Thus, the choice of gauge and time has no influence on the structure of
the quantum mechanics. Any observable moves a given transversal surface
via Poisson bracket around the phase space. This defines a one-dimensional
family of time levels (in the phase space) and time slices (in classical
solutions). A family of observables is called a complete family of partial
Hamiltonians, if each point of each classical solution is contained in
at least one time slice. The corresponding more-dimensional family of time
slices form a subset of the full many-finger time-slice system. Any two of
the corresponding time levels in the phase space are related by a
symmetry. This fact is exploited to define ``the same measurement at
different times'', to develop a complete quantitative theory of ``changes
in time'' on this basis, and, finally, to construct the Schr\"{o}dinger
and Heisenberg picture of quantum dynamics.

Keywords: quantization, gauge invariance, Einstein's general theory of relativity, parametrized systems, relativistic dynamics, Dirac's theory