On retroactive occurrence and twin events in quantum mechanics

Bohr's well-known claim that only a registered phenomenon is a true phenomenon is further elaborated into occurrence in the past: If ideal occurrence of an eventP ((1–P)) is a state at a timet_i makes another eventQ ((1–Q))certain at a later timet_f, and, finallyU is the evolution operator fromt_i tot_f, then, it is proved that the final collapsed stateQ(U U⁺)Q/TrQU U⁺, which comes about in ideal occurrence ofQ att_f,equals the initial collapsed stateU(P P/TrP)U⁺, which evolves from the state resulting from the ideal occurrence ofP in att_i. Utilizing the latter state is called theretroactive apparent ideal occurrence (RAIO) ofP in. A number of consequences, including the general notion of twin events (the case whent_f=t_i, andU=1) is derived. It is pointed out that RAIO is relevant in second-kind quantum measurement, in Wheeler's delayed-choice experiments in second-kind (or conditional) quantum preparators.