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Geometry of quantum states

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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.

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Authors and Affiliations

  1. Institute of Theoretical Physics, Warsaw University, Poland

    Bogdan Mielnik

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  1. Bogdan Mielnik
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Mielnik, B. Geometry of quantum states. Commun.Math. Phys. 9, 55–80 (1968). https://doi.org/10.1007/BF01654032

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  • DOI: https://doi.org/10.1007/BF01654032

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