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Tests of Bell's inequality

At the time Bell's result first became known, the experimental record was reviewed to see if any known results provided evidence against locality. None did. Thus an effort began to develop tests of Bell's inequality. A series of experiments conducted by Aspect ended with one in which polarizer angles were changed while the photons were `in flight'[1]. This was widely regarded at the time as being a reasonably conclusive experiment confirming the predictions of . Aspect's experiment had the arrangement shown in Figure on page .

Pairs of photons in a singlet state are emitted by the source in opposite directions. Each traverses a polarizer and is detected. Quantum mechanics predicts a relationship between the angle of the two polarizers and the probability for detecting both photons from a singlet state pair that no local theory could reproduce provided the delay between when the polarizer angles change and the probability of joint detections changes is less time then it takes light to travel from either polarizer to the detector most distant from it. Three years later Franson showed that Aspect's experiment to test Bell's inequality did not rule out local realistic theories with delayed determinism[17].

Discretized finite difference equation models possess delayed determinism in the sense that a system can begin to converge to a state and then reverse the process as discussed in Section on page . There is no definite time at which a state is determined absolutely. A state is only determined statistically as there is always some nonzero probability that it can be reversed. Discretized finite difference equation models are not hidden variables theories in the sense that they are not theories of particles plus hidden variables. They are theories of `hidden' distributed information stored holographic like throughout a space time region. This information cannot be uniquely associated with individual particles although it determines the results observed in particle interactions. The classical parameters of an interaction are determined as focal points of continuous nonlinear changes in the wave function and not as discrete events. In addition to not violating Bell's inequality this class of theories can in principle be distinguished from standard quantum mechanics by other experiments.

Franson's notion of delayed determinism i.e. that an event may not be determined until some time after it has been completed, may seem strange and unrealistic. However there is no objective definition of event in quantum mechanics. The unobserved microscopic events that Franson discusses (such as the emission of a photon by an excited atom) are hypothetical. It is a mistake to assume that such events occur as macroscopic events do. Quantum mechanics only allows us to compute the probabilities of making observations given certain initial conditions. What happens between the time we set up the initial conditions and make an observation is the terra incognita of quantum mechanics. We cannot base the timing in a test of Bell's inequality on the hypothetical times of hypothetical events.

Franson's objections to Aspect's experiment showed that there is no objective criteria in the formalism of the existing theory for computing the timing in an experimental test of Bell's inequality. One way to understand this is through the thought experiment of Schrödinger's cat[29] as discussed in Section on page . There is nothing in the formalism of that allows us to know when macroscopic events are irreversibly determined. That question is left to interpretations which for the most part are metaphysical and not subject to experimental tests. Thus there is no way to decide among them. This problem applies not only to tests of Bell's inequality but to any experiment that asks questions about the timing of causal sequences of macroscopic events.

If the timing cannot be derived from the formalism of quantum mechanics or from an interpretation of the theory then it must be derived from a competing theory. Developing such alternatives, even if extremely speculative, is a critical element in designing tests of Bell's inequality. The timing constraints I describe in Chapter apply to a broad class of alternative theories and not just the class of models I advocate. These timing constraints are often assumed by experimenters perhaps without fully realizing that they cannot be derived from the formalism of the exiting theory.

A recent analysis which claims to describe how to close all the loopholes in tests of Bell's inequality[26] is incomplete in its analysis of the timing issues. The authors state on page 3210: ``To close this loophole, the analyzer's settings should be changed after the correlated pair has left the source.'' There is no way to know when the pair has left the source unless one detects them at that point which makes the experiment impossible. The speed of the process that generate the photons is only relevant if there is a common trigger for that process and the changing of the polarizer angles. Perhaps this is what the authors are suggesting. The timing can only involve macroscopic events such as setting the polarizers or macroscopic effects from detecting the photons. The basis for determining the times of these events must come from a competing theory. The authors do not discuss this or the need to base timing on purely macroscopic events. In Section we describe what must be done to address the timing issue in practical experiments.

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