BELL THEOREM WITHOUT INEQUALITIES - A SINGLE-PARTICLE FORMULATION

It is demonstrated that the following two postulates are inconsistent: (1) the locality postulate, stating that if two physical systems are noninteracting then no real change can take place in one system in consequence of anything that may be done to the other one; and (2) the hidden-variable postulate, stating that a physical quantity has a pre-existing definite value before a measurement. The proof is based on a particular Mach-Zehuder interferometer in which the first beam splitter is an analyzer of circular polarizations. A left-handed beam of light propagates in this interferometer through only one optical path and does not interact with the interferometer's phase shifter. Still, it is shown that, having detected the photon in one of the output ports of the interferometer, an acceptance of the above hidden-variable postulate implies that we know what would have been the result of the measurement had we chosen a phase shift alpha + pi instead of alpha: The photon would be with certainty detected in the other port. It follows that the phase shifter deterministically influences the behavior of photons at the second beam splitter. The proof is formulated for a single-photon state. Another difference with respect to the ordinary proofs of the Bell theorem is the fact that the alternative settings of the phase shifter correspond to measurements of the same observable (in all other proofs one simultaneously considers the values of complementary observables). The proof is formulated without any use of inequalities and no additional assumption such as ''the emptiness of the path not taken'' is made.