We construct, for any finite dimension n, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For n=2 our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions n .≥ 3 We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our hidden variables scheme and their relationship with the results of Fine [J. Math. Phys. 23 1306 (1982)].
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References
D. Aerts (1986) ArticleTitle“A possible explanation for the probabilities of quantum mechanics” J. Math. Phys. 27 202
S. L. Adler D. C. Brody T. A. Brun L. P. Hughston (2001) ArticleTitle“Martingale models for quantum state reduction” J. Phys. A 34 8795
N. Gisin C. Piron (1981) ArticleTitle“Collapse of the wave packet without mixture” Lett. Math. Phys. 5 379
D. Aerts B. Coecke B. D’Hooghe F. Valckenborgh (1997) ArticleTitle“A mechanistic macroscopic physical entity with a three-dimensional Hilbert space description” Helv. Phys. Acta 70 793
M. Czachor (1992) ArticleTitle“On classical models of spin” Found. Phys. Lett. 5 249
K. A. Kirkpatrick (2003) ArticleTitle“Classical three-box “paradox” J. Phys. A 36 4891
K. A. Kirkpatrick (2003) ArticleTitle““Quantal” behavior in classical probability” Found. Phys. Lett. 16 199
I. Pitowsky (1983) ArticleTitle“Deterministic model of spin and statistics” Phys. Rev. D 27 2316
S. Aerts, “The Born rule from a consistency requirements on hidden measurements in complex Hilbert space”, preprint: quant-ph/0212151.
D. Aerts (1998) ArticleTitle“The hidden measurement formalism: what can be explained and where quantum paradoxes remain” Int. J. Theor. Phys. 37 291
B. Coecke (1995) ArticleTitle“Generalization of the proof on the existence of hidden measurements to experiments with an infinite set of outcomes” Found. Phys. Lett. 8 437
B. Coecke (1995) ArticleTitle“Hidden measurement model for pure and mixed states of quantum physics in Euclidean space” Int. J. Theor. Phys. 34 1313
B. Coecke ( 1995) ArticleTitle“A hidden measurement representation for quantum entities described by finite-dimensional complex Hilbert spaces” Found. Phys. 25 1185
B. Coecke (1998) ArticleTitle“A representation for a spin-s entity as a compound system in R3 consisting of 2s individual spin-1/2 entities”, Found. Phys. 28 1347
B. Coecke (1998) ArticleTitle“A representation for compound quantum systems as individual entities: hard acts of creation and hidden correlations” Found. Phys. 28 1109
B. Coecke (1997) ArticleTitle“Classical representations for quantum-like systems through an axiomatics for context dependence” Helv. Phys. Acta 70 442
B. Coecke (1997) ArticleTitle“A classification of classical representations for quantum-like systems” Helv. Phys. Acta 70 462
B. Coecke F. Valckenborgh (1998) ArticleTitle“Hidden measurements, automorphisms, and decompositions in context-dependent components” Int. J. Theor. Phys. 37 311
A. Fine (1982) ArticleTitle“Joint distributions, quantum correlations, and commuting observables” J. Math. Phys. 23 1306
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer- Verlag, 1994.
T. A. Schilling, Geometry of quantum mechanics, Ph.D. thesis, Pennsylvania State University, 1996.
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Oliynyk, T.A. Hidden Measurements, Hidden Variables and the Volume Representation of Transition Probabilities. Found Phys 35, 85–107 (2005). https://doi.org/10.1007/s10701-004-1921-x
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DOI: https://doi.org/10.1007/s10701-004-1921-x