Introduction

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Einstein, Podolsky, and Rosen (EPR) (1) challenged Bohr (2) and the completeness of quantum mechanics by designing a gedanken experiment that suggested the existence of "hidden parameters" and of a theory that was more complete than quantum mechanics. The EPR design was later realized in various implementations (3), with experimental results close to the quantum mechanical prediction. These experimental results by themselves have no bearing on the EPR claim that quantum mechanics was incomplete nor on the existence of hidden parameters. However, inequalities derived by Bell (4) that are based on the assumption that local hidden parameters exist, taken together with the experimental results that happen to be inconsistent with the result of the Bell inequalities, do appear to prove that local hidden parameters cannot exist. This issue has been discussed in great detail in refs. 5 and 6.

The Bell theorem is based on a mathematical model of the EPR experiments. It has, by itself, no experimental confirmation, because its conclusion contradicts the results of the EPR experiments. The standing of the Bell theorem therefore has unique features in the history of modern physics: the mathematical model and the theorem of Bell are taken to be correct and are seen by many as being as valid as the second law of thermodynamics, whereas there exists no experimental confirmation. However, instead of discarding altogether a mathematical model that contradicts experiment, the contradiction to the experiment is used to prove that the basic assumption of the theorem, the existence of local hidden parameters, is incorrect. The framework of research that has developed around the Bell theorem claims the necessity of "gross nonlocalities." In simple words, the correlated spins of the EPR experiment are in some contact over arbitrary space-like distances of our space-time continuum, and if one spin is measured in one station, the correlated spin in another station is instantaneously influenced. This fact contradicts the locality conditions of Einstein and Einstein's very argument, for lack of completeness of quantum mechanics. Einstein called the instantaneous interaction of the spatially separated spins "spukhafte Fernwirkungen" (spooky action at a distance). He did not accept the possibility of such spooky action and, because quantum mechanics appeared to demand it, it had to be at least incomplete. The Bell theorem and its standard interpretation have turned the logic around. Its supporters now claim that local hidden parameters do not exist and cannot explain the EPR experiments. Quantum mechanics does agree with these experiments, and spooky action at a distance must be accepted as a fact of nature. However, it has been shown in a series of papers, of which we cite only two of the more recent (7, 8), that Bell's theorem does contain more than self-evident locality assumptions. These additional assumptions are related to the role of time in the experiments and the admissibility of more general probability measures.

We show in this paper that the assumption of the existence of local hidden variables is not the only assumption in the proof of the Bell inequalities. We show that the mathematical model of Bell excludes a large set of hidden variables and a large variety of probability densities of these variables, all of which fulfill Einstein's locality conditions perfectly. This exclusion has neither a physical nor a mathematical basis but is based on Bell's mathematical interpretation of what Einstein locality means in terms of probability theory. Our additional set of hidden variables or, as we will call them, parameter random variables, includes time-like correlated parameters and a generalized probability density that is a sum of what we later define as setting-dependent subspace product measures (SDSPMs). We demonstrate that Bell-type proofs cannot go forward by using our extended space of hidden variables.

The paper is organized as follows. We first review the theorem of Bell. We then analyze the restrictions that Bell's proof puts on the parameter space and probability measure and show that a much larger space and a more general probability measure can be constructed without violation of Einstein locality conditions. We demonstrate that a variety of proofs of theorems similar to that of Bell cannot be performed in this larger parameter space and with the more general probability measure, and that these theorems and inequalities are therefore not valid in this space. We finally point toward a mathematical model that uses this larger space and permits the construction of a hidden parameter theory that does agree with EPR experiments.

	The Theorem of Bell

Top Abstract Introduction The Theorem of Bell Extension of Bell's Parameter... Conclusion References

We first present a short summary of the work of Bell. In EPR experiments, two particles having their spins in a singlet state are emitted from a source and are sent to spin analyzers (instruments) at two spatially separated stations, S₁ and S₂. The spin analyzers are described by Bell by using unit vectors a, b, etc., of three-dimensional Euclidean space and functions A = ±1 (operating at station S₁) and B = ±1 (operating at station S₂): furthermore, A does not depend on the settings b of station S₂, nor B on the settings a of station S₁ (Einstein separability or locality). Bell permits particles emitted from the source to carry arbitrary hidden parameters  of a set  that fully characterize the spins and are "attached" to the particles with a probability density  (we denote the corresponding probability measure by µ). Neither the parameters  nor the probability measure µ are permitted to depend on the settings at the stations. Einstein separability is again cited as the reason for this restriction. The analyzer settings are changed rapidly in the experiments and do change after emission from the source. Therefore, the source parameters and their probability measure must not depend on the settings at the time of measurement. Bell further assumes that the values of the functions A and B are determined by the spin analyzer settings and by the parameters such that:

[ 1 ]

Thus A_a() and B_b() can be considered as stochastic processes on , indexed by the unit vectors a and b, respectively. Quantum theory and experiments show that, for a given time of measurement for which the settings are equal in both stations, we have for singlet-state spins

[ 2 ]

with probability one. Bell further defines the spin pair expectation value P(a, b) by

[ 3 ]

From Eqs. 1-3, Bell derives his celebrated inequality (4)

[ 4 ]

and observes that this inequality is in contradiction with the result of quantum mechanics.

[ 5 ]

Here a·b is the scalar product of a and b.

The proof of Bell's inequality is based on the obvious fact that for x, y, z = ±1, we have

[ 6 ]

Substituting x = A_b(), y = A_c(), z = A_a() and integrating with respect to the measure µ, one obtains Eq. 4 in view of Eq. 3. Thus, from the vantage point of mathematics, the Bell inequality is a straightforward consequence of the set of hypotheses and assumptions that are imposed.

	Extension of Bell's Parameter Space and Probability Measure

Top Abstract Introduction The Theorem of Bell Extension of Bell's Parameter... Conclusion References

We are going to argue below that Bell's parameter space is not general enough and excludes with no necessity a manifold of parameters that has at least the cardinality of the continuum. Bell's probability measure, correspondingly, is not as general as the physics of relativity would permit. To show this, we start with a discussion of the parameter space and corresponding probabilities out of Bell's book (8).

Bell (8) defines the following parameter sets that are in the backward light cone, as defined by relativity and as illustrated in Fig. 1. He lets N denote the specification of all entities that are represented by parameters and belong to the overlap of the backward light cones of both space-like separated stations S₁ and S₂. In addition, he considers sets of parameters L_a (our notation) that are in the remainder of the backward light cone of S₁ and M_b for S₂, respectively. Bell (see p. 56 of ref. 8) denotes the conditional probability that the function A_a assumes a certain value with A_a = ±1 by {A_a |L_a, N} and similarly for B_b = ±1. Then he claims that in a local causal theory, we have:

[ 7 ]

Eq. 7 appears entirely plausible as a consequence of the finite speed of light: whatever happens at station S₁ to the result of A_a cannot be causally connected to the result of B_b in station S₂ within a local theory. When the switching of the settings is fast enough, the probability that A_a assumes a certain value must be the same no matter what value B_b might assume. Although this conclusion is undoubtedly correct at a given instance of measurement, Bell's use of Eq. 7 as identical and valid for all times of measurement with a given setting is fatally flawed. The reason is the possible dependence of A_a and B_b on time-like correlated parameters that may be setting dependent. The mathematics of Bell-type proofs requires complete statistical independence of A_a and B_b for the whole set of measurements and not only at a given time. It also contains the assumption of identical L_a and M_b for all measurements of a run. This assumption, however, cannot be guaranteed, because physical phenomena other than the setting of the polarizer by the experimenter can occur in the stations, and these can be correlated.

View larger version (9K): [in this window] [in a new window]   Fig. 1.   Light cone used by Bell. X denotes the space coordinate and t the time.

Consider, for example, two clocks, one in each station. These clocks may have different settings (e.g., pendulum length and/or starting time, etc.). The time that one clock shows is certainly not the causal reason for the time of the other clock. It is the same physical law that is at work in both stations and that causes a correlation in the periodicity of the processes in the clocks or in some general periodic processes, for that matter. It is, of course well known that two gyroscopes in the two stations could also be used as clocks, as they may indicate the rotation of the earth. As mentioned, there may also be other periodic processes that cause correlations, and these correlations may be influenced by the settings a and b. Although there are clear analogies of gyroscopes and spin properties, we do not wish to push this comparison too far. We do, however, wish to point out the dangers of using of Eq. 7 without proper caution. Bell's argument resulting in Eq. 7 does not include the vital fact that the experiments are made in a time sequence and that the backward light cones change and evolve with time. The situation is illustrated in Fig. 2, which shows that for each instant of measurement there is a different light cone. Fig. 2 illustrates our point for measurements at two different times t₁ and t₂.

View larger version (28K): [in this window] [in a new window]   Fig. 2.   Light cones at randomly chosen times t1 and t2 of the measurements. The clocks indicate time-like and setting-dependent correlations of parameters in stations S1 and S2. (In this example, the difference between the times indicated by the clocks in the two stations stays constant.)

The backward light cones contain sets of parameters L_a,t1, M_b,t1, N_t1, and L_a,t2, M_b,t2, N_t2, respectively. It is clear that the set L_a,t1 and the set L_a,t2 may contain setting-dependent parameters ^*_a with different probability densities. It is also clear from the discussion with clocks that the sets L_a,t1,L_a,t2, M_b,t1,M_b,t2, etc., need not be statistically independent. This fact has several consequences that do not permit Bell-type proofs to go forward. We outline below the most crucial problems.

Bell uses combinations of Eq. 7 for different settings in his proofs as follows (see, e.g., Bell's equation 10 of ref. 9, p. 56).

[ 8 ]

However, because the measurements with setting b and c are necessarily taken at different times, this relation needs to be written in the form

[ 9 ]

which, in general, is clearly incorrect.

The above arguments also demonstrate that Bell's use of a single probability density () that is valid for all times of a run of measurements is in contradiction to physical intuition and facts: the parameter space related to light cones changes and evolves, in general, with time. To describe this physical reality (if this word is permitted), one needs at least to admit a time dependence of (), i.e., one needs to replace () by

[ 10 ]

In addition, one needs to include, again in general, setting-dependent parameters denoted, e.g., by ^*_a(t) in station S₁ and by ^*_b^* (t) in station S₂ with

[ 11 ]

if b = a to make it possible to fulfill Eq. 2. Bell has included into his later proofs (after publication of ref. 9) setting-dependent parameters. However, he and everyone else assumed that ^*_a and ^*_b^*  to be statistically independent. He argued this independence from the fact that the parameters are in different stations, and he did not consider time-like correlations as described above. Bell assumes that setting-dependent parameters in the stations analyzers (called instruments by Bell) must be statistically independent or do not exist, as is explicitly stated in his book (ref. 8, p. 38): "... it is necessary that the equality holds in equation 8 (which is equal to our Eq. 1), i.e., for this case the possibility of the results depending on hidden variables in the instruments can be excluded from the beginning... " Of course, in a run with all different settings, that still would be true. However, P(a, b) is evaluated from measurements with fixed a and b. Therefore, a possible time dependence can cause statistical correlations. To visualize such a time dependence, assume that the parameters ^*_a and ^*_b^*  are identified with the hour pointers of two clocks in the two stations. The clock in station S₁ is connected to a plane that is perpendicular to the setting a and the clock in station S₂ to a plane perpendicular to b. Let the direction of the pointers be idealized by unit vector ^*_a in station S₁ and ^*_b^*  in S₂ at each instant of measurement. Clearly, these parameters will exhibit time correlations. Note that it is of no concern that the measurements are taken at random times. It is the time correlation in the two stations at any given time that matters.

In addition to the generalization of Bell's probability densities shown in Eq. 10, one needs a further generalization and replace () by

[ 12 ]

Of course, to obey Einstein locality, ^*_a and ^*_b^*  must be station specific and can be correlated only by time-like correlations, i.e., by some relation to local periodic processes. It is also important to note that the station parameters ^*_a and ^*_b^*  cannot be emulated by the parameter pair a,  or b, , as always implied by Bell by use of his functions A and B. The source parameters  of Bell appear with a given probability density. Because the parameters ^*_a and ^*_b^*  can have different probability distributions for different a and b, which are not related in any way to the parameters , it becomes clear that the joint density _tv(, ^*_a, ^*_b^* ) can depend, a priori, on the setting vectors a and b. It is irrelevant that by lucky coincidence, the triples (a, , ^*_a) and (b, , ^*_b^* ) could perhaps be written as a,  and b,  for some  incorporating  and the station parameters. The probability density that must be considered, in general, for all these parameters is therefore also different from that of Bell and must exhibit a time dependence. This fact implies the necessity of a more general probability measure that includes time-like correlated parameters.

We have shown [K. Hess and W. Philipp, quant-ph, http://xxx.lanl.gov/abs/quant-ph/0103028, March 7 (2000)] that a properly chosen sum of what we call SDSPMs does not violate Einstein separability and does lead to the quantum result of Eq. 5, while still always fulfilling Eq. 2. By this we mean the following. The probability space  is partitioned into a finite number M of subspaces _m

[ 13 ]

For given a and b, a setting-dependent measure (µ_ab)_m is defined on each subspace _m. This measure can be extended to the entire space  by setting

[ 14 ]

The final measure µ is then defined on the entire space  by

[ 15 ]

and the index m indicates the time correlations. In the above notation, we would have m = t_i. We have shown ([K. Hess and W. Philipp, quant-ph, http://xxx.lanl.gov/abs/quant-ph/0103028, March 7 (2000)] that a product measure can be found such that

[ 16 ]

This fact provides assurance that we can avoid any hint of spooky action within our system of SDSPMs.

It is clear that Bell's proof does not go through with such a probability measure, because integrating Eq. 6 to obtain Eq. 4 works only with a single setting-independent probability measure. In addition, one can show in a rather intricate proof [K. Hess and W. Philipp, quant-ph, http://xxx.lanl.gov/abs/quant-ph/0103028, March 7 (2000)] that the quantum result of Eq. 5 can be obtained with a probability measure as in Eq. 15. In other words, hidden parameters are possible if the parameter space is properly extended. We also have shown that the parameters that are considered that way show no trace of spooky action.

	Conclusion

Top Abstract Introduction The Theorem of Bell Extension of Bell's Parameter... Conclusion References

We have presented a mathematical framework that is more extensive than that of Bell and permits the possibility of describing the spin-pair correlation in EPR-type experiments by use of hidden parameters. A key element of our approach is contained in the introduction of time-like correlated parameter random variables that also depend on the setting of the station in which they influence the measurements. This consideration leads in a natural way to a setting-dependent probability measure composed of SDSPMs. Use of such SDSPMs does not permit the proof of Bell to go forward (nor any other proofs of similar theorems known to us as given, e.g., in refs. 8 and 10). We conclude that setting- and time-dependent parameter random variables present a possible loophole in theorems à la Bell.