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Quantic Superpositions and the Geometry of Complex Hilbert Spaces

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  • Daniel Lehmann
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    Abstract

    The concept of a superposition is a revolutionary novelty introduced by Quantum Mechanics. If a system may be in any one of two pure states x and y, we must consider that it may also be in any one of many superpositions of x and y. This paper proposes an in-depth analysis of superpositions. It claims that superpositions must be considered when one cannot distinguish between possible paths, i.e., histories, leading to the current state of the system. In such a case the resulting state is some compound of the states that result from each of the possible paths. It claims that states can be compounded, i.e., superposed in such a way only if they are not orthogonal. Since different classical states are orthogonal, the claim implies no non-trivial superpositions can be observed in classical systems. It studies the parameters that define such compounds and finds two: a proportion defining the mix of the different states entering the compound and a phase difference describing the interference between the different paths. Both quantities are geometrical in nature: relating one-dimensional subspaces in complex Hilbert spaces. It proposes a formal definition of superpositions in geometrical terms. It studies the properties of superpositions.

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    File URL: http://ratio.huji.ac.il/sites/default/files/publications/dp447.pdf
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    Bibliographic Info

    Paper provided by The Center for the Study of Rationality, Hebrew University, Jerusalem in its series Discussion Paper Series with number dp447.

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    Length: 32 pages
    Date of creation: Feb 2007
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    Handle: RePEc:huj:dispap:dp447

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    Keywords: Superpositions in Quantum Mechanics; Geometry of Hilbert Spaces; Quantum measurements; Measurement algebras; Quantum Logic;

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