Quantum Dynamical Reduction

Time-reparameterization-invariance of General Relativity and cosmological model systems derived from it manifests itself through the Hamiltonian constraint: the Hamiltonian of the system is constrained to vanish. Dynamical evolution follows the orbits generated by the Hamiltonian constraint, however their time-parameterization is physically irrelevant. As a result, on the constraint surface (where the Hamiltonian constraint vanishes), its Hamiltonian flow generates both the dynamical time evolution and the physically irrelevant, “gauge”, time-reparameterization. Taking the quotient of the constraint surface by the flow of the Hamiltonian constraint leaves one with a “frozen” symplectic phase space of constants of motion. In this talk we first review dynamical symplectic reduction, where a suitable clock function is used to provide a preferred parameterization of Hamiltonian orbits, and a suitable set of coordinates on constant-clock surfaces is used to separate these orbits. The rest of the talk will focus on the quantum version of this problem, where we assume that the non-reduced system has been quantized, with the Hamiltonian constraint to be implemented after quantization a la Dirac. We present the quantum version of the dynamical symplectic reduction, which parallels Dirac constraint quantization, performed by identifying a clock within the quantized system. Our recent result derives sufficient conditions that the quantum clock must satisfy in order to generate a consistent reduction. This talk will complement (but not depend on) the recent presentation by Martin Bojowald.