In this Letter wetake a diR'erent approach and show, by considering a newgedanken experiment, that it is possible to demonstrateBell's theorem by means of a direct contradiction (i.e.,without the need of inequalities) using a two-particlestate

We findthat, if the "elements of reality" corresponding toLorentz-invariant observables are themselves Lorentz in-variant, then Lorentz-invariant realistic interpretations ofquantum mechanics are not possible.

The framework presented here demonstrates that at least in the two-photon case, the distribution of zero-probability outcomes is not a characteristically quantum phenomenon. This has implications for ongoing discussions over Hardy’s paradox, as discussed in Section 3.3. It is common to interpret Hardy’s result as demonstrating that quantum-mechanical nonlocality can be demonstrated using purely the distribution of zero-probability results, but since we have shown that the zero-probability cases in Hardy’s case can be reproduced within classical electromagnetism, it is now clear that what makes Hardy’s example characteristically quantum is not the assignation of zero probabilities per se, but rather the discrete restriction on the set of possible outcomes. This is an intriguing indication that discreteness may have a closer relationship with quantum no-go theorems than much of the literature would seem to suggest

Hardy Williams