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Journal of Mathematical Physics

On the equivalence of nonrelativistic quantum mechanics based upon sharp and fuzzy measurements

S. Twareque Ali, H. D. Doebner

Published Jul 1, 1976• 58 citations
Publisher

Abstract

Starting from the idea that physical measurements may have residual imprecisions, the possibility of replacing the nonrelativistic, three-dimensional configuration space by a so-called fuzzy configuration space, having an isomorphic Borel structure, is discussed. A quantization procedure with respect to such a space is developed, and the invariance of nonrelativistic quantum mechanics under such Borel isomorphisms is exploited to prove the equivalence of this quantization procedure to the usual quantization procedure on a fuzzy-free configuration space. Further, any Galilean invariant dynamics is shown to be insensitive to such imprecisions in the measurements of position and momentum.

DOI
10.1063/1.523034

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Starting from the idea that physical measurements may have residual imprecisions, the possibility of replacing the nonrelativistic, three-dimensional configuration space by a so-called fuzzy configuration space, having an isomorphic Borel structure, is discussed. A quantization procedure with respect to such a space is developed, and the invariance of nonrelativistic quantum mechanics under such Borel isomorphisms is exploited to prove the equivalence of this quantization procedure to the usual quantization procedure on a fuzzy-free configuration space. Further, any Galilean invariant dynamics is shown to be insensitive to such imprecisions in the measurements of position and momentum.
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We investigate the consistency of coherent state (or Berezin-Klauder-Toeplitz, or anti-Wick) quantization in regard to physical observations in the non- relativistic (or Galilean) regime. We compare this procedure with the canonical quantization (on both mathematical and physical levels) and examine whether they are or not equivalent in their predictions: is it possible to dif- ferentiate them on a strictly physical level? As far as only usual dynamical observables (position, momentum, energy, ...) are concerned, the quantization through coherent states is proved to be a perfectly valid alternative. We successfully put to the test the validity of CS quantization in the case of data obtained from vibrational spectroscopy (data that allowed to validate canonical quantization in the early period of Quantum Mechanics).
The quantum spacetime and the phase space with fuzzy structure are investigated as the possible quantization formalism. In this theory, the state of the nonrelativistic particle m corresponds to the element of fuzzy ordered set (Foset), i.e. the fuzzy point. Due to Foset partial (weak) ordering, the m space coordinate x acquires principal uncertainty σ x . It is shown that Schrödinger formalism of quantum mechanics can be completely derived from consideration of m evolution in fuzzy phase space with minimal number of axioms.
This paper defines, on the Galilean space-time, the group of asymptotically Euclidean transformations (AET), which are equivalent to Euclidean transformations at space-time infinity, and proposes a formulation of nonrelativistic quantum mechanics which is invariant under such transformations. This formulation is based on the asymptotic quantum measure, which is shown to be invariant under AET's. This invariance exposes an important connection between AET's and Feynman path integrals, and reveals the nonmetric character of the asymptotic quantum measure. The latter feature becomes even clearer when the theory is formulated in terms of the coordinate-free formalism of asymptotically Euclidean manifold, which do not have a metric structure. This mathematical formalism suggests the following physical interpretation: (i) Particles evolution is represented by trajectories on an asymptotically Euclidean manifold; (ii) The metric and the law of motion are not defined a priori as fundamental entities, but they are properties of a particular class of reference frames; (iii) The universe is considered as a probability space in which the asymptotic quantum measure plays the role of a probability measure. Points (ii) and (iii) are used to build the asymptotic measurement theory, which is shown to be consistent with traditional quantum measurement theory. The most remarkable feature of this measurement theory is the possibility of having a nonchaotic distribution of the initial conditions (NCDIC), an extremely counterintuitive but not paradoxical phenomenon which allows to interpret typical quantum phenomena, such as particle diffraction and tunnel effect, while still providing a description of their motion in terms of classical trajectories.