Physics and the Measurement of Continuous Variables

Abstract

This paper addresses the doubts voiced by Wigner about the physical rele- vance of the concept of geometrical points by exploiting some facts known to all but honored by none: Almost all real numbers are transcendental; the explicit represen- tation of any one will require an infinite amount of physical resources. An instrument devised to measure a continuous real variable will need a continuum of internal states to achieve perfect resolution. Consequently, a laboratory instrument for measuring a continuous variable in a finite time can report only a finite number of values, each of which is constrained to be a rational number. It does not matter whether the variable is classical or quantum-mechanical. Now, in von Neumann's measurement theory (von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Uni- versity Press, Princeton, 1955), an operator A with a continuous spectrum—which has no eigenvectors—cannot be measured, but it can be approximated by operators with discrete spectra which are measurable. The measurable approximant F( A) is not canonically determined; it has to be chosen by the experimentalist. It is argued that this operator can always be chosen in such a way that Sewell's results (Sewell in Rep. Math. Phys. 56: 271, 2005; Sewell, Lecture given at the J.T. Lewis Memorial Conference, Dublin, 2005) on the measurement of a hermitian operator on a finite- dimensional vector space (described in Sect. 3.2) constitute an adequate resolution of the measurement problem in this theory. From this follows our major conclusion, which is that the notion of a geometrical point is as meaningful in nonrelativistic quantum mechanics as it is in classical physics. It is necessary to be sensitive to the fact that there is a gap between theoretical and experimental physics, which reveals itself tellingly as an error inherent in the measurement of a continuous variable.