Yes / Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley's methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the 'probabilistic quadrant' , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables. / The full-text of this article will be released for public view at the end of the publisher embargo on 15 Oct 2020. / Research Development Fund Publication Prize Award winner, October 2019.
| Identifer | oai:union.ndltd.org:BRADFORD/oai:bradscholars.brad.ac.uk:10454/17437 |
| Date | 22 October 2019 |
| Creators | Vourdas, Apostolos |
| Source Sets | Bradford Scholars |
| Language | English |
| Detected Language | English |
| Type | Article, Accepted manuscript |
| Rights | (c) 2019 IoP Publishing. Full-text reproduced in accordance with the publisher's self-archiving policy. |
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