The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality relation for a pair of arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the probabilistic interpretation of quantum theory to define the probability of equality relation for a pair of arbitrary observables. Applications of this new interpretation to measurement theory are discussed briefly.
The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.
One of the most challenging problems in modern physics is the application of quantum theory to the universe as a whole. Progress in this area has been plagued by two types of problem: conceptual and technical. The conceptual problems arise from the old difficulties of interpreting quantum theory. The standard interpretations require that the measuring instruments and observers are outside the quantum system described by the wavefunction. In the late 1950s, however, Hugh Everett proposed an interpretation of quantum theory that might apply to systems that include the observers and measuring instruments, but the adequacy of such interpretations has remained controversial to this day.