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arXiv:gr-qc/0307014 · Lecture notes in physics

Quantum Mechanics and Discrete Time from “Timeless” Classical Dynamics

Hans-Thomas Elze

Published Dec 9, 2003• 17 citations
arXivHTMLPDF
DOI
10.1007/978-3-540-40968-7_15

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We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. In this way, a physical clock with discrete states is introduced, which presently is still treated as decoupled from the system. This is motivated by the recent discussion of ``timeless'' reparametrization invariant models, where discrete physical time has been constructed based on quasi-local observables. Employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. We derive the emergent quantum Hamiltonian in terms of the underlying classical one. Such Hamiltonians typically need a regularization - here performed by discretization - in order to arrive at models with a stable groundstate in the continuum limit. This is demonstrated in several examples, recovering and generalizing a model advanced by 't Hooft.
The basic tenet of the present work is the assumption of the lack of external and fixed time in the Universe. This assumption is best embodied by general relativity, which replaces the fixed space-time structure with the gravitational field, which is subject to dynamics. The lack of time does not imply the lack of evolution but rather brings to the forefront the role of internal clocks which are some largely arbitrary internal degrees of freedom with respect to which the evolution of timeless systems can be described. We take this idea seriously and try to understand what it implies for quantum mechanics when the fixed external time is replaced by an arbitrary internal clock. We put the issue in a solid, mathematically rigorous framework. We find that the dynamical interpretation of a quantum state of a timeless system depends on the employed internal clock. In particular, we find that the continuous spectra of well-known dynamical observables like the position of a free particle on the real line may turn discrete if measured in unusual clocks. We discuss the meaning of our result for attempts at quantization of global gravitational degrees of freedom.