For Peter G. Bergmann at seventy

N G Basov, F V Bunkin, Виктор Георгиевич Веселаго +11 more

G. S. Agarwal, Z. Y. Ou

Lev Vaidman A

It is argued that the proof of Cohen [Phys. Rev. A 51, 4373 (1995)] which shows that an application of the Aharonov-Bergmann-Lebowitz (ABL) rule leads to contradiction with predictions of quantum theory is erroneous. A generalization of the ABL rule for the case of an incomplete final measurement (which is needed for the analysis of Cohen's proof) is presented.

R. E. Kastner

An apparent paradox proposed by Aharonov and Vaidman in which a single particle can be found with certainty in two (or more) boxes is analyzed. It is found that the paradox arises from a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule. The paradox is resolved by observing that the counterfactual usage of the rule is, in general, not valid. 1. Background. The Aharonov-Bergmann-Lebowitz (ABL) rule is a well-known formula for calculating the probabilities of the possible outcomes of observables measured at an intermediate time t between pre- and post-selection measurements at times t1 and t2, respectively [Aharonov, Bergmann, and Lebowitz, 1964]. If an intermediate measurement of nondegenerate observable C with eigenvalues {ci} is performed at time t, the ABL rule states that the probability of outcome cj in the case of a preselection for the state |ψ1 〉 and a post-selection for the state |ψ2 〉 is given by: 1 PABL(cj|ψ1, ψ2) = |〈ψ2|cj〉 | 2 |〈cj||ψ1〉 | 2 i |〈ψ2|ci〉 | 2 |〈ci||ψ1〉 | 2 (1) It has recently been shown 2 that the ABL rule cannot be used in a counterfactual sense: i.e., in general, it cannot be used to calculate the probabilities of possible outcomes of observables that have not actually been measured at time t. 2. The three-box example.

Max Jammer

Abner Shimony

Many of the pioneers of quantum mechanics – notably Planck, Einstein, Bohr, de Broglie, Heisenberg, Schrodinger, Born, Jordan, Landé, Wigner, and London – were seriously concerned with philosophical problems. In each case one can ask a question of psychological and historical interest: was it a philosophical penchant which drew the investigator towards a kind of physics research which is linked to philosophy, or was it rather that the conceptual difficulties of fundamental physics pulled him willynilly into the labyrinth of philosophy? I shall not undertake to discuss this question, but shall cite an opinion of Peter Bergmann, which I find congenial: he learned from Einstein that “the theoretical physicist is … a philosopher in workingman's clothes” ([1], q. v.). The problems with which I am preoccupied concern the philosophical implications of quantum mechanics – either epistemological, bearing on the extent, validity, and character of human knowledge; or metaphysical, bearing on the character of reality. Although quantum mechanics is not a system of philosophy, one can wonder whether it is susceptible to coherent incorporation in a philosophical system. I propose to examine the thought of three masters of quantum mechanics – Bohr, Heisenberg, and Schrödinger – not with a critical or historical intention, but in hope of finding some enlightenment concerning the problems posed by contemporary physics. I can say in advance that enlightenment will continue to elude us; nevertheless, the ideas of Bohr, Heisenberg and Schrödinger are rich and evocative for new studies.