Geometry of quantum states View Full Text


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Article Info

DATE

1968-03

AUTHORS

Bogdan Mielnik

ABSTRACT

In the first part of this work, an attempt of a realistic interpretation ofquantum logic is presented. Propositions of quantum logic are interpreted as corresponding to certain macroscopic objects called filters; these objects are used to select beams of particles. The problem of representing the propositions as projectors in a Hilbert space is considered and the classical approach to this question due to Birkhoff and von Neumann is criticized as neglecting certain physically important properties of filters. A new approach to this problem is proposed. The second part of the paper contains a revision of the concept of a state in quantum mechanics. The set of all states of a physical system is considered as an abstract space with a geometry determined by the transition probabilities. The existence of a representation of states by vectors in a Hilbert space is shown to impose strong limitations on the geometric structure of the space of states. Spaces for which this representation does not exist are called non-Hilbertian. Simple examples of non-Hilbertian spaces are given and their possible physical meaning is discussed. The difference between Hilbertian and non-Hilbertian spaces is characterized in terms of measurable quantities. More... »

PAGES

55-80

References to SciGraph publications

  • 1967-12. On the algebraic structure of quantum mechanics in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf01654032

    DOI

    http://dx.doi.org/10.1007/bf01654032

    DIMENSIONS

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