Bell's Theorem and the Demise of Local Reality

1. INTRODUCTION. The advent of quantum physics led to the insight that the act of observation may affect that which is observed. In fact, it can be argued that the act of observation sometimes actually creates that which is observed. One can consider the possibility that the momentum of a particle does not exist until something happens to cause its momentum to exist. That something might well be a human experiment designed to measure the particle's momentum. This possibility was anathema to Einstein, who believed in objective reality; the particle has a momentum regardless of whether or not we observe it. He felt that quantum mechanics, while admittedly right in what it predicted, nonetheless failed to give a complete view of reality. Einstein also believed in the principle of separability. Consider two experiments, say X and Y, performed simultaneously but so far apart that, even traveling at the speed of light, information from the location of experiment X cannot reach experiment Y until after experiment Y is completed. The principle of separability says that under those circumstances the outcome of experiment Y will not depend on how experiment X was performed or even on whether or not X was performed at all. (However, the outcomes of experiments X and Y might be correlated. For instance, if identical twins living in different cities are tested for gender, the results will be completely correlated. Yet the actual testing of the first twin in no way influences the gender of the second twin.) We combine these two beliefs of Einstein and say that he believed in "local hidden variables" or, for short, "local reality." That is, he would have believed experiment Y was measuring something that intrinsically existed as part of reality (the outcome possibly depending on variables that might be hidden from us) but did not depend on whether or not some other experiment was simultaneously performed far away. In 1964, John S. Bell devised a clever argument employing elementary mathematics showing that in certain circumstances the concept of local reality is untenable. Although Bell's theorem is fairly well known to physicists and mathematical physicists, it appears to have escaped the notice of mathematicians. There are now many variants of Bell's theorem and numerous ways of presenting the mathematics, some of them quite slick (for example, see [6]). We choose a presentation that emphasizes a careful translation of the concept of local reality into mathematical terms. We take particular care to explain how the "local" assumption gets translated into mathematics, since it reveals an important and charming insight (the second remark of section 9). 2. ENTANGLED PHOTONS. We will be concerned with an experiment first conceived by Bell and subsequently performed by various people. The experiment involves pairs of entangled photons and certain detectors that interact with those photons. Fortunately, we do not need to explain the physics behind entangled photons: it will suffice to explain in simple terms how the photons and detectors interact. We explain that interaction by first considering an experiment simpler than Bell's. Our preliminary experiment consists of a central source of entangled photons. This source produces a pair of entangled photons, one speeding off to the left (photon L),