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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Time scales and the problem of measurement in quantum mechanics

L. Lanz, Giovanni Maria Prosperi, A. G. Sabbadini

Published Apr 1, 1971• 14 citations
Publisher
DOI
10.1007/bf02723082

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Abstract Many theories of quantum gravity suggest that there should exist a minimal scale with which it is possible to measure distances and time. We present a formalism of quantum mechanics exchibiting a minimal length and time scale. The work is based on the Page-Wootters formalism which gives a possible solution to the problem of time in quantum mechanics. This formalism is an example of a relational approach to quantum dynamics. It is based on the idea that reference frames with respect to which motion is described are themselves physical systems interacting with the degrees of freedom they wish to describe. Quantum dynamics is described in terms of quantum correlations between a clock and a system. In our work we modify the commutation relations between the time and frequency operators leading to a minimal uncertainty of time measurement. This results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We further investigate the e ects of such modi cation on a couple simple quantum systems.
Recent years have shown a revival of interest in the conceptual problem raised by the incorporation of quantum mechanics into physics. The problem is an old one, but the use of both waves and particles as visualizations of electrons has presented it in a new and acute form. Several observations based on the methods of quantum mechanics can be made. (1) The time development of an isolated system is completely determinate, as in classical mechanics. (2) A precisely defined quantum mechanical state is an eigenstate of some operator described in classical terms. The production of such a state requires a suitable large scale apparatus. (3) If the state is not precisely defined as an eigenstate, quantum statistical mechanics must be used. (4) The idea of measurement is of doubtful applicability in the analysis of idealized experiments, since the process of measurement destroys the identity of the object (an electron) whose properties are measured. Quantum mechanics may be viewed strictly as a method of describing, in classical terms, the behavior of large scale apparatus, although the use of visualizations with a properly conditioned intuition is probably essential to progress in the analysis of new phenomena.