Time and observables in quantum gravity via the evolving constants method.

摘要

Rovelli's concept of "evolving constants of motion" offers an attractive unification of the seemingly contradictory notions of evolution and full diffeomorphism invariance in general relativity. In this dissertation Rovelli's unification is used to provide a new angle on the "problem of time," allowing derivation of sufficient criteria for existence of a solution to the "functional evolution problem." The "new angle" is also found to lead to a single particle interpretation of relativistic quantum mechanics which is nonlocal, and hence reminiscent of quantum gravity. The similarity is used to suggest a local observable in quantum gravity.;In Chapter 3 a simple example of the general concept formulated in Chapter 2 is given. The relativistic free particle system in 1 + 1 dimensions is formulated as a "bi-Hamiltonian system." One Hamiltonian generates ordinary time translations, and another generates (essentially) boosts. Any observer, accelerated or not, sees evolution as the continuous unfolding of a canonical transformation which may be described using the two Hamiltonians. When the system is quantized both Hamiltonians become Hermitian operators in the standard positive definite inner product. Hence, each observer sees the evolution of the wave function as the continuous unfolding of a unitary transformation in the standard positive definite inner product. The result appears to be a consistent single particle interpretation of relativistic quantum mechanics. This interpretation has the feature that the wave function is observer dependent, and observables have nonlocal character, similar to what one might expect in quantum gravity. In Chapter 4 a discussion of the roles in physics of experimental apparatus, observers, and background structure is given with an eye toward applying the concepts developed earlier to quantum gravity. The possibility that an observer in quantum gravity may be defined as an "actual collection of matter" is considered. The force on a collection of matter is a local observable. It allows contact to be made with the standard observables of particle physics.;In Chapter 1 the subject is introduced and a general definition of time in a dynamical formulation of general relativity given. In Chapter 2 Rovelli's "quantum mechanics without time" is used to motivate an intrinsically time-slicing-independent picture of reduced phase space quantum gravity, which may be described as "quantization after evolution." Sufficient criteria for carrying out quantization after evolution are developed in terms of a general concept of the classical limit of quantum mechanics. If these criteria are satisfied then it is possible to have consistent unitary evolution of operators, with respect to an infinite parameter family of time-slicings (and probably all time-slicings), with the correct classical limit. The criteria are particularly amenable to study in (2 + 1)-dimensional gravity, where the reduced phase space is finite dimensional.