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Why L2 Topology in Quantum Physics

Chris Culellar, University of Texas at El Paso Evan Longpre, University of Texas at El PasoFollow Vladik Kreinovich, University of Texas at El PasoFollow

11-2010

Technical Report: UTEP-CS-10-53

To appear in Journal of Uncertain Systems, 2012, Vol. 6, No. 2.

It is known that in quantum mechanics, the set S of all possible states coincides with the set of all the complex-valued functions f(x) for which the integral of |f(x)|^{2} is 1. From the mathematical viewpoint, this set is a unit sphere in the space L^{2} of all the functions for which this integralis finite. Because of this mathematical fact, usually the set S is considered with the topology induced by L^{2}, i.e., topology in which the basis of open neighborhood of a state f is formed by the open balls. This topology seem to work fine, but since this is a purely mathematical definition, a natural question appears: does this topology have a physical meaning? In this paper, we show that a natural physical definition of closeness indeed leads to the usual L^{2}-topology.

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Technical Report: UTEP-CS-10-53

To appear in

Journal of Uncertain Systems, 2012, Vol. 6, No. 2.