EPR Correlations and EPW Distributions

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It is known that with Bohm’s example of EPR correlations, involving particles with spin, there is an irreducible non-locality. The non-locality cannot be removed by the introduction of hypothetical variables unknown to ordinary quantum mechanics. How is it with the original EPR example involving two particles of zero spin? Here we will see that the Wigner phase space distribution’ illuminates the problem. Of course, if one admits “measurement” of arbitrary “observables” on arbitrary states, it is easy to mimic2 the EPRB situation. Some steps have been made towards realism in that c~nnec t ion .~ Here we will consider a narrower problem, restricted to “measurement” of positions only, on two non-interacting spinless particles in free space. EPR considered “measurement” of momenta as well as positions. But the simplest way to “measure” the momenta of free particles is just to wait a long time and “measure” their positions. Here we will allow position measurements at arbitrary times tl and t2 on the two particles respectively. This corresponds to “measuring’) the combinations t1 + t I f i l / r n l , 42 + t262/m2 (1) at time zero, where ml and m2 are the masses, and the @ and 6 are position and momentum operators. We will be content here with just one space dimension. The times tl and t 2 play the same roles here as do the two poiarizer settings in the EPRB example. One can envisage then some analogue of the CHHS inequality4i5 discriminating between quantum mechanics on the one hand and local causality on the other. The QM probability of finding, at times tl and t 2 respectively, that particles at positions q1 and q2 respectively, is