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arXiv:quant-ph/0503171

Can the Salecker-Wigner clock be microscopic?

Andor Frenkel

Published Mar 21, 2005• 1 citation
arXivHTMLPDF

Abstract

Half a century ago H. Salecker and E. P. Wigner examined the functioning of a quantum clock of very simple construction [1]. They raised the question whether such a clock can be microscopic or not, but no clear-cut answer has been reached in [1] (see also [2]). In this note it is shown that their clock can have a microscopic mass and size only if its accuracy is poor, but if the size is macroscopic, a decent accuracy can be achieved even if the mass is microscopic.

DOI
10.48550/arxiv.quant-ph/0503171

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Half a century ago H. Salecker and E. P. Wigner examined the functioning of a quantum clock of very simple construction [1]. They raised the question whether such a clock can be microscopic or not, but no clear-cut answer has been reached in [1] (see also [2]). In this note it is shown that their clock can have a microscopic mass and size only if its accuracy is poor, but if the size is macroscopic, a decent accuracy can be achieved even if the mass is microscopic.
In order to acquire an extended Salecker-Wigner formula from which to derive the optimal accuracy in reading a clock with a massive particle as the signal, von Neumann's classical measurement is employed, by which simultaneously both position and momentum of the signal particle can be measured approximately. By an appropriate selection of wave function for the initial state of the composite system (a clock and a signal particle), the formula is derived accurately. Valid ranges of the running time of a clock with a given optimal accuracy are also given. The extended formula means that contrary to the Salecker-Wigner formula there exists the possibility of a higher accuracy of time measurement, even if the mass of the clock is very small.
This article deals with the limitations which the quantized nature of microscopic systems imposes on the possibility of measuring distances between space-time events. It is proposed to use only clocks for measuring space-time distances and to avoid the use of measuring rods which are essentially macrophysical objects. The accuracy of reading a clock with a given mass is considered and examples for microphysical clocks are given. It is shown that the mass of the clock, and the uncertainty (spread) of this quantity, exceed certain values which depend on the accuracy with which the time interval is to be measured, the magnitude of this time interval (the running time of the clock) and the size of the clock. The minimum mass uncertainty of the clock is given by Heisenberg's relation; the minimum mass itself is higher by the ratio of the running time and the accuracy.If the possibility of constructing states whose wave functions are Gaussian wave packets is admitted, the mass and the mass uncertainty of the clock differs only by logarithmic factors from the uncertainties which follow from general principles of quantum mechanics. The masses are much higher if the possibility of constructing arbitrary wave packets is not admitted.
Current interpretations of quantum mechanics suggest that the classical concepts of space and time are not applicable to microscopic systems. Salecker and Wigner have recently proved that these concepts have no operational meaning for microsystems. Therefore, space-time descriptions may be valid only for macroscopic systems. It is here suggested that space and time themselves arise from, but do not have analogs in, the properties of microscopic particles, in the same way that thermodynamic properties arise as a result of interactions among the many actually existing particles of the universe. Neither the particles nor the interactions need to be described in spatial-temporal terms. This macroscopic interpretation of space-time seems compatible with the known properties of the physical world, suggests a more direct interpretation of the statistical nature of microscopic events, and offers a new approach to some physical problems.