a a A ,', the unprimed spin (or "internal") connection, 4~, , B , and a possible cosmological constant A. As in Ref. 8 we restrict the soldering form to be anti-Hermitian from the beginning so that the space-time metric will always be real [with signature ( -+ + + )]. The matter fields consist of minimally coupled, massive Klein-Gordon field 4, massive Dirac field (cA, ?7,.), and Yang-Mills connection one-form 4A, (with an arbitrary internal gauge group). It is easy to extend the framework to $ I 2572

We study the problem of computing the probability for the time-of-arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the "time-of-arrival" operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle, and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schrödinger current. We derive a well defined uncertainty relation between time-of-arrival and energy; this result shows that the well known arguments against the existence of such a relation can be circumvented. Finally, we define a "time-representation" of the quantum mechanics of a free particle, in which the time-of-arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics.

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e. phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through "deparametrization" and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories -based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-are compared and shown to be equivalent.

An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of non-linear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed by one of us using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems.