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On classical models of Spin
Authors:Marek Czachor
Institution:(1) Centre for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland;(2) Present address: Laboratory of Dielectrics and Organic Semiconductors, Technical University of Gdanacutesk, Majakowskiego 11/12, Gdanacutesk, Poland
Abstract:We discuss two classical situations that lead to probabilities characteristic for systems with spin-1/2. (a) Pitowsky model: It is demonstrated that the definition of spin functions does not imply which circle (a parallel or a great circle) on the sphere should be taken as a probability space in calculation of conditional probabilities. Pitowsky's choice of parallels must be formulated as an assumption about the model. It is shown that the model explicitly avoiding this difficulty is possible and no contradiction with the Bell Theorem is found. The modification is based on a new pathological decomposition of the sphere and belongs to a class of hidden variable theories with undetected signals. (b) Aerts model: We show the importance of the ldquopolarization effectrdquo of the measurements for the sake of obtaining a non-Kolmogorovian probability model. It is also shown that the conditioning by a change of state leads in general to the non-Kolmogorovian probability calculus.2. For example, let forallCw,z existxisinCw,zµc ({y isinC w,z capS +|delta(x,y) lepgr/2})=µ c(C w,z capS +)= 1/2µ c(C w,z). Pitowsky spheres with white points distributed onC w,z in this way exist. The proof is exactly analogous to this of Theorem 1 in 1]. Then (**) = 1 -theta/pgr, if one takesC w,z as the probability space. This example seems very instructive.3. If the chargeq falls down on some point then it clearly has not fallen down on another one. Having given a result of a measurement we cannot, within the model with polarization, talk in a sensible way about its alternative: We can think either aboutsuccessive measurements (then the Bell inequality is not derivable) or ask ldquoWhat would have happened if...,rdquo but then we deal with a different problem (in Aerts' terminology 2b] this is anobservation) and obtain again Eq. (8) (there is no complementarity but there is no model of spin either).
Keywords:quantum probability  nonmeasurable sets  Bell inequality  hidden variables
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