^{1}

^{2}

^{3}

^{*}

We present a detailed analysis of the set theoretical proof of Wigner for Bell type inequalities with the following result. Wigner introduced a crucial assumption that is not related to Einstein’s local realism, but instead, without justification, to the existence of certain joint probability measures for possible and actual measurement outcomes of Einstein-Podolsky-Rosen (EPR) experiments. His conclusions about Einstein’s local realism are, therefore, not applicable to EPR experiments and the contradiction of the experimental outcomes to Wigner’s results has no bearing on the validity of Einstein’s local realism.

Einstein challenged the Copenhagen interpretation of quantum mechanics by pro- posing with Podolsky and Rosen [

About 30 years after the EPR paper, Bell [

The work of Bell and his followers had very important consequences for the views on the foundations of physics. A majority of physicists believe that either (i) any physical theory using counterfactually definite functions and obeying Einstein’s local realism (see II) must obey Bell-type inequalities and/or (ii) Wigner’s derivation of Bell-type inequalities is only based on set theory (not involving counterfactual reasoning) and Einstein’s local realism. There is a considerable body of serious mathematical-physical work that has raised objections against both (i) and (ii) (see particularly [

The belief in (ii) arose mainly from the work of Wigner [

There exists, however, the following problem with this latter assessment. Set theory represents a mathematical framework based on axioms that may be regarded as definitions. As such, set theory is not related to any measurements and experiments and, therefore, also not to EPR experiments. It is thus necessary to find a connection between the sets of elements of physical reality that the experiments provide and the sets of mathematical abstractions that Wigner uses in his proof. We analyze this connection in great detail and show that Wigner’s sets of mathematical abstractions contain a topological-combinatorial ordering that is incompatible with the actual ordering of measurement-events in space and time (or space-time). We suggest that it is this incompatibility that leads to the contradictions of Wigner’s work with actual EPR measurements and not any failure of Einstein’s local realism. An analogous situation was discussed by Poincaré and Einstein, who resolved the experimental contradictions to Euclidean geometry by pointing to the connection of Euclidean geometry with the physical reality, a connection achieved by the introduction of the concept of rigid bodies [

As we will show, there exists an analogy to the Einstein-Poincaré discussions, because Wigner connected his mathematical sets to the actual measurements in a way that is inconsistent with the space-time physics of the actual EPR experiments. Key to our reasoning is the fact that the set of data of the actual measurements is necessarily involving space- and time- (or space-time-) coordinates to designate the correlated pairs. These sets of space and time coordinates are inconsistent with Wigner’s sets and subsets that are ordered according to location in three-dimensional space and equip- ment settings in that space, but do not involve measurement times.

We show in detail that Wigner’s derivation of Bell type inequalities (that restrict the possible correlations exhibited by EPR measurements) does actually not use Einstein’s local realism at all. Instead Wigner assumes in a hidden and unjustified way certain topological combinatorial properties of his mathematical sets that are inconsistent with the macroscopic properties of the measurement equipment in space and time as de- termined by clocks, protractors and meter-measures. The contradictions between Wigner’s (Bell-type) inequalities and well known experiments (see II B) must, therefore be blamed on Wigner’s topological-combinatorial assumptions and not on Einstein’s local realism.

To prove this finding, we first analyze the elements of physical reality used in EPR experiments and form the sets of physical elements that must be dealt with in Wigner’s mathematical proof (see Section 2). We then review the essential portion of Wigner’s mathematical proof and point to his unwarranted assumptions (see Section 3). Finally, in Section 4 we deduce from these latter assumptions the topological-combinatorial problems of Wigner’s work and conclude that the well known line “death by experi- ment” applies to Wigner’s theory not to Einstein’s. Our conclusions are summarized in Section 5.

We consider in the following only a variation of EPR experiments as suggested by Bohm and, therefore called EPRB experiments [

It is, therefore, of utmost importance for the discussions of Bell and Wigner to exclude in the actual experiments influences from the other location that propagate with speed slower or equal to that light. How can one make sure that there exist no such other influences and in addition that one indeed measures correlated pairs? This goal has been achieved in [

Key to Bell’s and Wigner’s reasoning is the rapid switching of instruments on both Tenerife and La Palma, just before a measurement of the spin of an incoming particle is performed. In this way one ensures that the instrument settings on the other island cannot influence a given measurement with the speed of light or lower speeds. What Bell and Wigner wished to find out was whether Einstein’s local realism is correct or incorrect. The importance of this fast switching is emphasized over and over by Bell [

However, it is precisely this pairing and timing problem, where Einstein embedded the necessity of using a space-time system, his space-time system, as the basis for his Gedanken-Experiments and the theoretical approach with Podolsky and Rosen, [^{1}

The timing of the measurements is, thus, a most important element of the physical reality, of the data, and we denote the clock-time of the

From what follows in this paper, it appears imperative to perform measurements that involve dependencies on the timing and on the instrument settings in relation to that timing.

Wigner ignored the measurement times and corresponding connections when considering possible and actual outcomes in his theory and we will see that this negligence has serious consequences for his set theoretic calculations of correlations.

Actual EPRB measurements (e.g. [

As mentioned, the atoms and molecules constituting the measurement equipment have many body interactions with the incoming particles, from both the view of Einstein’s physics and also from the most modern view of quantum physics. Nevertheless, both Bell, Wigner and all their followers describe this equipment, as the Copenhagen school of quantum mechanics does, just by a three-dimensional unit vector of space, indicating the direction of a polarizer or of a Stern-Gerlach magnet. We denote this unit vector for the nth experiment by

where, as mentioned, both

As mentioned, both Bell and Wigner use Einstein locality as a tool of reasoning: For carefully designed EPRB experiments they reason, that the measurement outcome

Wigner discussed in his paper three pairs of measurements with the given settings

Here

For each pair, we are interested with Wigner in only two different possibilities when measuring on the individual particles: either equal outcomes or unequal outcomes. Therefore, for a string of ^{2}

As we have mentioned, Einstein’s local realism has not been used up to now. We show below that it also is not used in Wigner’s further considerations that lead to his variation of Bell’s inequality. Wigner’s theory does thus derive, without reference to Einstein locality, how many different correlations of equal and different outcomes may occur for the

We ask, therefore, the question: What caused the reduction of the possible number of correlations in Wigner’s proof? Our answer is that it was not any assumption regarding Einstein’s local realism. Instead, the reduction of the number of possible correlations arises from Wigner’s unwarranted mathematical assumption that is related to set

e/u | |||
---|---|---|---|

e | e | e | 3/0 |

u | u | e | 2/1 |

u | u | u | 0/3 |

e | u | u | 1/2 |

e | e | u | 2/1 |

u | e | u | 1/2 |

u | u | e | 1/2 |

u | e | e | 2/1 |

theoretic probability theory. The reduction of possible correlations is, therefore, artifi- cial and does not apply to EPR experiments. As mentioned, the celebrated line “death by experiment” is correct but it applies to Wigner’s theory and not, as reported in so many sensationalist articles, to Einstein’s.

Wigner did not consider any dependence of the measurement outcomes on the measurement times and just assumed that the possible or actual outcomes ^{3}

Wigner defines (p. 1007, left column last paragraph of [

all

In the next paragraph Wigner states his most crucial assumption by a definition: “Let

Wigner’s assumption about the joint probability of the domains of variables (parameters) that determine the possible outcomes implies the existence of consistent joint probabilities for the possible and actual outcomes of measurements such as

As an aside, Bernard d’Espagnat [

Wigner himself intended, as did Bell originally, to use exclusively elements of physical reality in the sense of Mach and we discuss his work from this point of view only.

As an illustration of how far reaching Wigner’s assumptions are, consider Bell’s six-tuple (2) which contains three correlated pairs and add to each of the pairs an arbitrary (artificial) third notebook entry for the setting that is not already contained in the pair-outcomes: setting ^{4}

The

However, it is a well known fact of probability theory, particularly the set theoretic probability theory of Kolmogorov, that the existence of joint triple (and higher order) probabilities is not guaranteed [

This is an enormously important point that has been overlooked also by Bell and all his followers [

Several different cases need to be considered to present the full proof that Wigner’s approach must not be applied to EPRB experiments. We just present one example that is typical and list the following six-tuple using Wigner’s subsets related to settings

The symbol “

Wigner never included considerations of both measurement times and topology and used six-tuple (5) to obtain the possible outcomes for all the following 9 possible pairings:

We use again

The

We consider now the possible correlations for the outcomes of the

As seen from the measurement times, only 3 of the pairings of (6) do correspond to originally correlated pairs. Note that

The ordering of the mathematical abstractions that form the subsets that Wigner chose are simply incompatible with the ordering of the original measurements in space and time. The measurement time is the only guarantee for the correct pairing of the original measurements. These facts demonstrate that Wigner oversimplified complex topological-combinatorial factors that are vital for the outcomes of his considerations. Another important fact, for the incorrectness of Wigner’s pairings that arise from his 9 possibilities (6), is that the pairs with equal settings are not derived from originally correlated pairs and therefore may randomly assume all of the value-pairs

Thus, Wigner reduced the number of possible correlations of EPRB measurements by assuming without justification the existence of certain joint probabilities. He did not realize this fact and was, therefore, faced with the problem to explain the contradictions with actual experiments if they occurred (and as indeed were found [

We note in passing that all of our (and Wigner’s) reasoning that is related to the triple of settings

These facts show that Wigner’s procedure to derive Bell’s inequality is set-theoretically neither general nor sound. Wigner’s work, taken in conjunction with experiments such as presented in [

Hess, K., De Raedt, H. and Michielsen, K. (2017) Analysis of Wigner’s Set Theoretical Proof for Bell- Type Inequalities. Journal of Modern Phy- sics, 8, 57-67. http://dx.doi.org/10.4236/jmp.2017.81005