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Foundations of Physics Letters

Uncertainty principle and uncertainty relations: extension of a result of Uffink and Hilgevoord

A. B. Evans

Published Aug 1, 1990• 2 citations
Publisher
DOI
10.1007/bf00769711

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First, I briefly review the different conceptions of time held by three rival interpretations of quantum theory: the collapse of the wave-packet, the pilot-wave interpretation, and the Everett interpretation (Section 2). Then I turn to a much less controversial task: to expound the recent understanding of the time-energy uncertainty principle, and indeed of uncertainty principles in general, that has been established by such authors as Busch, Hilgevoord and Uffink. Although this may at first seem a narrow topic, I point out connections to other conceptual topics about time in quantum theory: for example, the question under what circumstances there is a time operator (Section 4.3).
In the context of (far-field) multislit interferometry we investigate the utility of two formulations of uncertainty in accounting for the complementarity of spatial localization and fringe width. We begin with a characterization of the relevant observables and general considerations regarding the suitability of different types of measures. The detailed analysis shows that both of the discussed uncertainty formulations yield qualitatively similar results, confirming that they correctly capture the relevant tradeoff. One approach, based on an idea of Aharonov and co-workers, is intuitively appealing and relies on a modification of the Heisenberg uncertainty relation. The other approach, developed by Uffink and Hilgevoord for single- and double-slit experiments, is readily applied to multislits. However, it is found that one of the underlying concepts requires generalization and that the choice of the parameters requires more consideration than was known.
We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of "quantum blobs" introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by Hilgevoord and Uffink.