We propose stochastic hidden variables model in which hidden variables have a p-adic probability distribution ρ(λ) and at the same time conditional probabilistic distributions P(U,λ), U=A,A′,B,B′, are ordinary probabilities defined on the basis of the Kolmogorov measure-theoretical axiomatics. A frequency definition of p-adic probability is quite similar to the ordinary frequency definition of probability. p-adic frequency probability is defined as the limit of relative frequencies νn but in the p-adic metric. We study a model with p-adic stochastics on the level of the hidden variables description. But, of course, responses of macroapparatuses have to be described by ordinary stochastics. Thus our model describes a mixture of p-adic stochastics of the microworld and ordinary stochastics of macroapparatuses. In this model probabilities for physical observables are the ordinary probabilities. At the same time Bell’s inequality is violated.
I define a notion of local causality LOC closely modelled on the Bell principle, construed as the condition that single case probabilities cannot be modified by actions at spacelike separation. The new principle, like that of Bell, forces Bell inequalities, but with two loopholes: one is retrocausation, known to Bell, but the other is nonuniqueness of remote outcomes, a loophole only for LOC, not for the Bell principle. I also set out a theory of physical probability, applicable to the Everett interpretation, in agreement with the Born rule, and therefore violating Bell inequalities. I show it is consistent with LOC. Surprisingly, both loopholes are exploited. I conclude not only that probability in the Everett interpretation involves no action at a distance, but that the observed violations of Bell inequalities is powerful evidence for many worlds.
A finite non-classical framework for qubit physics is described that challenges the conclusion that the Bell Inequality has been shown to have been violated experimentally, even approximately. This framework postulates the primacy of a fractal-like ‘invariant set’ geometry I U in cosmological state space, on which the universe evolves deterministically and causally, and from which space-time and the laws of physics in space-time are emergent. Consistent with the assumed primacy of I U , a non-Euclidean (and hence non-classical) metric g p is defined in cosmological state space. Here, p is a large but finite integer (whose inverse may reflect the weakness of gravity). Points that do not lie on I U are necessarily g p -distant from points that do. g p is related to the p-adic metric of number theory. Using number-theoretic properties of spherical triangles, the Clauser-Horne-Shimony-Holt (CHSH) inequality, whose violation would rule out local realism, is shown to be undefined in this framework. Moreover, the CHSH-like inequalities violated experimentally are shown to be g p -distant from the CHSH inequality. This result fails in the singular limit p = ∞ , at which g p is Euclidean and the corresponding model classical. Although Invariant Set Theory is deterministic and locally causal, it is not conspiratorial and does not compromise experimenter free will. The relationship between Invariant Set Theory, Bohmian Theory, The Cellular Automaton Interpretation of Quantum Theory and p-adic Quantum Theory is discussed.
I define a notion of locality LOC, closely modelled on Bell's principle of local causality, construed as the condition that single-case probabilities cannot be modified by actions at spacelike separation. The new principle, like Bell's, forces Bell inequalities, but with two loopholes: one is violation of measurement independence, known to Bell, but the other is non-uniqueness of remote outcomes, a loophole only for LOC, not for Bell's principle. I also set out a theory of physical probability, applicable to the Everett interpretation, in which the Born rule is derived, and which therefore violates Bell inequalities. I show it is consistent with LOC. Surprisingly, both loopholes are exploited. I conclude not only that physical probability in the Everett interpretation involves no action at a distance, in the sense of LOC, but that the observed violation of Bell inequalities is powerful evidence for many worlds.