Feynman-path analysis of Hardy's paradox: Measurements and the uncertainty principle

Abstract

Hardy's paradox is analysed within Feynman's formulation of quantum mechanics. A transition amplitude is represented as a sum over virtual paths which different intermediate measurements convert into different sets of real pathways. Contradictions arise if conflicting statements are applied to the same statistical ensemble. Usefulness of "strange" weak values for resolving the paradox is disputed.

Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”

“…One simple example is the so-called "three-box case" [5,20,29] (for a path analysis of the so-called Hardy's paradox [4] see, for example, [30]). Here, the final state can be reached via three virtual pathways with amplitudes:…”

Asking photons where they have been in plain language

“…This led to a number of intriguing claims, all allegedly supported by the weak measurements argument. The claims included the evidence of "negative kinetic energy" [16], "negative number of particles" [17], "having one particle in several places simultaneously" [18], "photon disembodied from its polarisation" [19], "electron with disembodied charge and mass" [19], and "an atom with the internal energy disembodied from the mass" [19], Earlier we argued [13], [20]- [23] that most of these notions are easily dismissed once the discussion is returned to the framework of standard quantum mechanics. We ask whether the same could be said about the strange behaviour attributed to the photons in [1] and [9].…”

Path integrals, the ABL rule and the three-box paradox

“…Feynman path analysis has been applied to the Hardy's paradox in [14] and in this paper we will use it to analyse measurements performed on pre-and post-selected systems and in particular to the three box paradox mentioned above. The rest of the paper is organised as follows.…”