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Quantic Superpositions and the Geometry of Complex Hilbert Spaces

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  • Daniel Lehmann

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.

Suggested Citation

  • Daniel Lehmann, 2007. "Quantic Superpositions and the Geometry of Complex Hilbert Spaces," Discussion Paper Series dp447, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
  • 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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