The main purpose of this paper is to provide a new proof of the impossibility of local theories of hidden variables based on simpler and more general assumptions than those used by Bell (1964, 1966), Wigner (1970), and Suppes and Zanotti (1976). In particular, no assumptions requiring specific quantum mechanical calculations are required. They are replaced by the principle of exchangeability and the principle of identical conditional distributions given the hidden variable. The results may be most easily discussed in terms of a system of two spin-½ particles initially in the singlet state, but generalizations to other quantum mechanical systems of a similar nature are apparent. In essential terms, the analysis of Bell, stated in a particularly clear form by Wigner, depends upon the following assumptions, which we state in intuitive form. (Mathematically explicit axioms are formulated in Appendix A.) 1. Axial symmetry . For any direction of the measuring apparatus the expected spin is 0, where spin is measured by +1 and -1 for spin +½ and spin -½, respectively. Further, the expected product of the spin measurements is the same for different orientations of the measuring apparatuses, as long as the angle between the measuring apparatuses remains the same. 2. Opposite measurement for same orientation . The correlation between the spin measurements is – 1 if the two measuring apparatuses have the same orientation. 3. Independence of λ . The expectation of any function of λ is independent of the orientation of the measuring apparatus.
The main purpose of this paper is to provide a new proof of the impossibility of local theories of hidden variables based on simpler and more general assumptions than those used by Bell (1964, 1966), Wigner (1970), and Suppes and Zanotti (1976). In particular, no assumptions requiring specific quantum mechanical calculations are required. They are replaced by the principle of exchangeability and the principle of identical conditional distributions given the hidden variable. The results may be most easily discussed in terms of a system of two spin-½ particles initially in the singlet state, but generalizations to other quantum mechanical systems of a similar nature are apparent. In essential terms, the analysis of Bell, stated in a particularly clear form by Wigner, depends upon the following assumptions, which we state in intuitive form. (Mathematically explicit axioms are formulated in Appendix A.) 1. Axial symmetry . For any direction of the measuring apparatus the expected spin is 0, where spin is measured by +1 and -1 for spin +½ and spin -½, respectively. Further, the expected product of the spin measurements is the same for different orientations of the measuring apparatuses, as long as the angle between the measuring apparatuses remains the same. 2. Opposite measurement for same orientation . The correlation between the spin measurements is – 1 if the two measuring apparatuses have the same orientation. 3. Independence of λ . The expectation of any function of λ is independent of the orientation of the measuring apparatus.
INTRODUCTION The main purpose of this chapter is to prove a lemma about random variables, and then to apply this lemma to the characterization of local theories of hidden variables by Bell (1964, 1966) and Wigner (1970), which are focused around Bell's inequality. We use the results of the lemma in two different ways. The first is to show that the assumptions of Bell and Wigner can be weakened to conditional statistical independence rather than conditional determinism because determinism follows from conditional independence and the other assumptions that are made about systems of two spin-½ particles. The second direction is to question the attempt of Clauser and Home (1974) to derive a Bell-type inequality for local stochastic theories of hidden variables which use an assumption of conditional statistical independence for observables. The main thrust of our analysis obviously arises from the probabilistic lemma we prove. Roughly speaking, this lemma asserts that if two random variables have strict correlation, that is, the absolute value of the correlation is one, and it is in addition assumed that their expectations are conditionally independent when a third random variable λ is given, then the conditional variance of X and Y given λ is zero. In other words, given the hidden variable λ the observables X and Y are strictly determined. The lemma itself, of course, depends on no assumptions about quantum mechanics. It may be regarded as a limitation on any theories that assume both strict correlation between observables and their conditional independence on the basis of some prior or hidden variable.
Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.
The study of hidden-variable interpretations of quantum mechanics was transformed by two papers of Bell. In his 1966 paper in Reviews of Modern Physics [1] he analysed the argument of von Neumann [2] and the more powerful arguments of Jauch and Piron [3] and of Gleason [4], all leading to similar conclusions about the nonassignability of simultaneous values to all the observables of a nontrivial quantum system; and he showed that these arguments do not exclude what may be called “contextualistic” hidden-variable theories, in which the value of an observable O is allowed to depend not only upon the hidden state λ, but also upon the set C of compatible observables measured along with O . In Bell's 1964 paper in Physics [5] (written later than the other, in spite of the earlier publication date) he gave a new kind of argument against a hidden-variable interpretation of quantum mechanics, by showing that no hidden-variables theory which satisfies a certain condition of locality can agree with all the statistical predictions of quantum mechanics. The main purpose of this lecture is to discuss an experiment, designed by Clauser, Horne, Holt and myself [6], which is based on Bell's proof of the incompatibility of quantum mechanics and local hidden-variable theories. Before doing this, however, I wish to make several comments.
Abstract. In conformity with our previous work on the measurement process in quantum mechanics in terms of first passage random walks in Hilbert space through interactions with the measuring devices, we here consider the problem of measurement of entangled states at spatially separated measuring devices. We find that Bell’s inequality need not be an obstacle if the set of hidden variables include local ones, and we present a simple model that includes specific probability distributions of the source and local hidden variables, giving the quantum mechanical result for the expectation value of the spin-spin correlation for arbitrary orientations of the frames of measurement, which is known to violate Bell’s relation for chosen angles of orientation. Local measurements and entangled states 2 1.
The Bell's theorem stands as an insuperable roadblock in the path to a very desired intuitive solution of the EPR paradox and, hence, it lies at the core of the current lack of a clear interpretation of the quantum formalism. The theorem states through an experimentally testable inequality that the predictions of quantum mechanics for the Bell's polarization states of two entangled particles cannot be reproduced by any statistical model of hidden variables that shares certain intuitive features. In this paper, we show, however, that the proof of the Bell's theorem involves a subtle, though crucial, assumption that is not required by fundamental physical principles and, hence, it is not necessarily fulfilled in the experimental setup that tests the inequality. Indeed, this assumption can neither be properly implemented within the standard framework of quantum mechanics. Namely, the proof of the theorem assumes that there exists a preferred absolute frame of reference, supposedly provided by the lab, which enables to compare the orientation of the polarization measurement devices for successive realizations of the experiment and, hence, to define jointly their response functions over the space of hypothetical hidden configurations for all their possible alternative settings. We notice, however, that only the relative orientation between the two measurement devices in every single realization of the experiment is a properly defined physical degree of freedom, while their global rigid orientation is a spurious gauge degree of freedom. Hence, the preferred frame of reference required by the proof of the Bell's theorem does not necessarily exist. In fact, it cannot exist in models in which the gauge symmetry of the experimental setup under global rigid rotations of the two detectors is spontaneously broken by the hidden configurations of the pair of entangled particles and a non-zero geometric phase appears under some cyclic gauge symmetry transformations. Following this observation, we build an explicitly local model of hidden variables that reproduces the predictions of quantum mechanics for the Bell's states.
An argument, due originally to J. S. Bell, is somewhat simplified and made more specific. It deals primarily with a quantum mechanical system consisting of the spins of two spin-12 particles. It shows that a description of the quantum mechanical measurement of the spin components of these two particles by means of hidden parameters is impossible if we assume that the parameters determining the outcome of the measurement of the spin of each particle are independent of the direction in which we decide to measure the spin of the other particle. The mathematical reason for the impossibility is analyzed.