Dynamics of Totally Constrained Systems II. Quantum Theory

TL;DR

This paper introduces a novel quantum dynamics formulation for totally constrained systems, incorporating time observables as physical quantities, and describes the evolution as linear mappings among acausal subspaces, extending classical statistical dynamics.

Contribution

It develops a new quantum formulation for constrained systems using relative probability amplitudes, differing from Dirac's approach, and demonstrates its consistency and applications.

Findings

Consistent normalization of amplitudes on acausal subspaces for type I von Neumann algebras.

Mappings among acausal subspaces are conformal.

Application to simple models like parametrized quantum mechanics and relativistic particles.

Abstract

In this paper a new formulation of quantum dynamics of totally constrained systems is developed, in which physical quantities representing time are included as observables. In this formulation the hamiltonian constraints are imposed on a relative probability amplitude functional $\Psi$ which determines the relative probability for each state to be observed, instead of on the state vectors as in the conventional Dirac quantization. This leads to a foliation of the state space by linear manifolds on each of which $\Psi$ is constant, and dynamics is described as linear mappings among acausal subspaces which are transversal to these linear manifolds. This is a quantum analogue of the classical statistical dynamics of totally constrained systems developed in the previous paper. It is shown that if the von Neumann algebra $\C$ generated by the constant of motion is of type I, $\Psi$ can be consistently normalizable on the acausal subspaces on which a factor subalgebra of $\C$ is represented irreducibly, and the mappings among these acausal subspaces are conformal. How the formulation works is illustrated by simple totally constrained systems with a single constraint such as the parametrized quantum mechanics, a relativistic free particle in Minkowski and curved spacetimes, and a simple minisuperspace model. It is pointed out that the inner product of the relative probability amplitudes induced from the original Hilbert space picks up a special decomposition of the wave functions to the positive and the negative frequency modes.