Quantum Clock in a Model Where the Hamiltonian Is a Difference Between Two Harmonic Oscillators

Yoshiaki Ohkuwa

In order to study the "problem of time", Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a 'clock' for the other oscillator. In this paper we examine a model where the Hamiltonian is a difference between two harmonic oscillators, and we consider one of them which has the minus sign as a 'clock'. Klauder's projection operator approach to generalized coherent states is used to define physical states and operators. The resolution of unity is derived in terms of a gauge invariant coordinate. We investigate the 'quantum clock' and show that the evolution described by it is identical to the classical motion when the energy becomes large.

Dariusz Wciso

Philip L. Bowers

Yoshiaki Ohkuwa

In order to study the ``problem of time in quantum gravity,'' Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a ``clock'' for the other oscillator. In this paper we examine a model where the Hamiltonian is a difference between two harmonic oscillators, and we consider one of them which has the minus sign as a `'clock,'' since the gravitational degree of freedom has a minus sign in the Hamiltonian of quantum cosmology. Klauder's projection operator approach to generalized coherent states is used to define physical states and operators. The resolution of unity is derived in terms of a gauge invariant coordinate. We investigate the ``quantum clock'' and show that the evolution described by it is identical to the classical motion when the energy becomes large.

M. C. Ashworth

The ``problem of time'' has been a pressing issue in quantum gravity for some time. To help understand this problem, Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a ``clock'' for the other oscillator, thus giving a natural time reference frame for the system. Recently, the author has constructed an explicit form for coherent states on the reduced phase space of this system in terms of Klauder's projection operator approach. In this paper, I will discuss the features of this model, and then by using coherent state representations and other tools from coherent state quantization, I investigate the construction of gauge-invariant operators on this reduced phase space and the ability to use a quantum oscillator as a ``clock.''

Carlo Jacoboni