SL(2,R) model with two Hamiltonian constraints

Details SL(2,R) model with two Hamiltonian constraints SL(2,R) model with two Hamiltonian constraints

1999-01-26

arxiv.org/abs/gr-qc/9901073v2

Phys.Rev.D60:044009,1999

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

9 pages, 1 figure, revised version, to appear in Phys. Rev. D

Scientific paper

10.1103/PhysRevD.60.044009

We describe a simple dynamical model characterized by the presence of two noncommuting Hamiltonian constraints. This feature mimics the constraint structure of general relativity, where there is one Hamiltonian constraint associated with each space point. We solve the classical and quantum dynamics of the model, which turns out to be governed by an SL(2,R) gauge symmetry, local in time. In classical theory, we solve the equations of motion, find a SO(2,2) algebra of Dirac observables, find the gauge transformations for the Lagrangian and canonical variables and for the Lagrange multipliers. In quantum theory, we find the physical states, the quantum observables, and the physical inner product, which is determined by the reality conditions. In addition, we construct the classical and quantum evolving constants of the system. The model illustrates how to describe physical gauge-invariant relative evolution when coordinate time evolution is a gauge.

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