Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory

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

Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory (Extended Abstract)

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.Comment: In Proceedings QPL 2014, arXiv:1412.810

Quantum Histories

There are good motivations for considering some type of quantum histories formalism. Several possible formalisms are known, defined by different definitions of event and by different selection criteria for sets of histories. These formalisms have a natural interpretation, according to which nature somehow chooses one set of histories from among those allowed, and then randomly chooses to realise one history from that set; other interpretations are possible, but their scientific implications are essentially the same. The selection criteria proposed to date are reasonably natural, and certainly raise new questions. For example, the validity of ordering inferences which we normally take for granted --- such as that a particle in one region is necessarily in a larger region containing it --- depends on whether or not our history respects the criterion of ordered consistency, or merely consistency. However, the known selection criteria, including consistency and medium decoherence, are very weak. It is not possible to derive the predictions of classical mechanics or Copenhagen quantum mechanics from the theories they define, even given observational data in an extended time interval. Attempts to refine the consistent histories approach so as to solve this problem by finding a definition of quasiclassicality have so far not succeeded. On the other hand, it is shown that dynamical collapse models, of the type originally proposed by Ghirardi-Rimini-Weber, can be re-interpreted as set selection criteria within a quantum histories framework, in which context they appear as candidate solutions to the set selection problem. This suggests a new route to relativistic generalisation of these models, since covariant definitions of a quantum event are known.Comment: 19 pages, TeX with harvmac. Contribution to Proceedings of the 104th Nobel Symposium, ``Modern Studies of Basic Quantum Concepts and Phenomena'', Gimo, June 1997. To appear in Physica Script

Time and Spacetime: The Crystallizing Block Universe

The nature of the future is completely different from the nature of the past. When quantum effects are significant, the future shows all the signs of quantum weirdness, including duality, uncertainty, and entanglement. With the passage of time, after the time-irreversible process of state-vector reduction has taken place, the past emerges, with the previous quantum uncertainty replaced by the classical certainty of definite particle identities and states. The present time is where this transition largely takes place, but the process does not take place uniformly: Evidence from delayed choice and related experiments shows that isolated patches of quantum indeterminacy remain, and that their transition from probability to certainty only takes place later. Thus, when quantum effects are significant, the picture of a classical Evolving Block Universe (`EBU') cedes place to one of a Crystallizing Block Universe (`CBU'), which reflects this quantum transition from indeterminacy to certainty, while nevertheless resembling the EBU on large enough scales.Comment: 25 Pages. 3 figure

Quasiclassical Dynamics in a Closed Quantum System

We consider Gell-Mann and Hartle's consistent histories formulation of quantum cosmology in the interpretation in which one history, chosen randomly according to the decoherence functional probabilities, is realised from each consistent set. We show that in this interpretation, if one assumes that an observed quasiclassical structure will continue to be quasiclassical, one cannot infer that it will obey the predictions of classical or Copenhagen quantum mechanics.Comment: Published version, to appear in Phys. Rev. A. Clarificatory remarks added on interpretations outside the scope of the paper. (TeX with harvmac, 13 pages.

Discord and non-classicality in probabilistic theories

Quantum discord quantifies non-classical correlations in quantum states. We introduce discord for states in causal probabilistic theories, inspired by the original definition proposed in Ref. [17]. We show that the only probabilistic theory in which all states have null discord is classical probability theory. Non-null discord is then not just a quantum feature, but a generic signature of non-classicality.Comment: 5 pages, revtex styl