Consistency Conditions for Probabilities of Quantum Histories

In the framework of the histories approach to quantum mechanics developed by Griffiths and Omnès, we consider the question of the uniqueness of the probability assigned to the histories; the question was solved by Omnès only in special cases. We find conditions which ensure uniqueness of such probability in the general case.     

On the existence of inequivalent quasideterministic domains

In the framework of the history approach to quantum mechanics and, in particular, of the formulation of Gell-Mann and Hartle, the question of the existence of inequivalent decoherent sets of histories is reconsidered. A simple but acceptably realistic model of the dynamics of the universe is proposed and a particular set of histories is shown to be decoherent. By suitable tranformations of this set, a family of sets of histories is then generated, such that the sets, first, are decoherent on the basis of the assumed dynamics of the universe and, secondly, arc certainly inequivalent, apart from trivial special cases. Finally, the original set of histories is refined to get a model of the usual quasiclassical domain and it is shown that, applying to it the previously considered transformations, a family of sets of histories is obtained which share typical properties of the usual quasiclassical domain.Supported in part by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica and by the Istituto Nazionale di Fisica Nucleare.     

Are there realistically interpretable local theories?

Although it rests on strongly established proofs, the statement that no realistically interpretable local theory is compatible with some experimentally testable predictions of quantum mechanics seems at first sight to be incompatible with a few general ideas and clear-cut statements occurring in recent theoretical work by Griffiths, Omnès, and Ballentine and Jarrett. It is shown here that in fact none of the developments due to these authors can be considered as a realistically interpretable local theory, so that there is no valid reason for suspecting that the existing proofs of the statement in question are all flawed.     

Preferred Consistent History Sets

The theme of this paper is the multiplicity ofthe consistent sets of histories emerging in theconsistent-histories approach to quantum mechanics. Wepropose one criterion for choosing preferred families among them: that the physically realizablequasiclassical domain ought to be one corresponding toclassical histories. We examine the way classicalmechanics arises as a particular window, and theimportant role played by the canonical group and theHamiltonian. We finally discuss a possible implicationof our selection criterion: that only a class of Hilbertspace operators corresponds to physical quantities, and hence the full nondistributivity of thelattice of quantum propositions is redundant.     

Consistent histories and operational quantum theory

In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach, nonrelativistic quantum mechanics is most generally formulated in terms of effects, states, and operations. We formulate a generalized consistent histories theory using the concepts and the terminology which have proven useful in the operational formulation of quantum mechanics. The logical rule of the logical interpretation of quantum mechanics is generalized to the present context. The algebraic structure of the generalized theory is studied in detail.     

Decoherent Histories and Realism

We reconsider the decoherent histories approach to quantum mechanics and analyze some problems related to its interpretation which we believe have not been adequately clarified by its proponents. We put forward some assumptions which, in our opinion, are necessary for a realistic interpretation of the probabilities that the formalism attaches to decoherent histories. We prove that such assumptions, unless one limits the set of the decoherent families which can be taken into account, lead to a logical contradiction. The line of reasoning we follow is conceptually different from other arguments which have been presented and which have been rejected by the supporters of the decoherent histories approach. The conclusion is that the decoherent histories approach, to be considered as an interesting realistic alternative to the orthodox interpretation of quantum mechanics, requires the identification of a mathematically precise criterion to characterize an appropriate set of decoherent families which does not give rise to any problem.     

A global equilibrium as the foundation of quantum randomness

We analyze the origin of quantum randomness within the framework of a completely deterministic theory of particle motion—Bohmian mechanics. We show that a universe governed by this mechanics evolves in such a way as to give rise to the appearance of randomness, with empirical distributions in agreement with the predictions of the quantum formalism. Crucial ingredients in our analysis are the concept of the effective wave function of a subsystem and that of a random system. The latter is a notion of interest in its own right and is relevant to any discussion of the role of probability in a deterministic universe.Research supported in part by NSF Grant DMS-9105661.     

Higher-Order Kinetic Term for Controlling Photon Mass in Off-Shell Electrodynamics

In relativistic classical and quantum mechanics with Poincaré-invariant parameter, particle worldlines are traced out by the evolution of spacetime events. The formulation of a covariant canonical framework for the evolving events leads to a dynamical theory in which mass conservation is demoted from a priori constraint to the status of conserved Noether current for a certain class of interactions. In pre-Maxwell electrodynamics—the local gauge theory associated with this framework —events induce five local off-shell fields, which mediate interactions between instantaneous events, not between the worldlines which represent entire particle histories. The fifth field, required to compensate for dependence of gauge transformations on the evolution parameter, enables the exchange of mass between particles and fields. In the equilibrium limit, these pre-Maxwell fields are pushed onto the zero-mass shell, but during interactions there is no mechanism regulating the mass that photons may acquire, even when event trajectories evolve far into the spacelike region. This feature of the off-shell formalism requires the application of some ad hoc mechanism for controlling the photon mass in two opposite physical domains: the low energy motion of a charged event in classical Coulomb scattering, and the renormalization of off-shell quantum electrodynamics. In this paper, we discuss a nonlocal, higher derivative correction to the photon kinetic term, which provides regulation of the photon mass in a manner which preserves the gauge invariance and Poincaré covariance of the original theory. We demonstrate that the inclusion of this term is equivalent to an earlier solution to the classical Coulomb problem, and that the resulting quantum field theory is renormalized.     

Spontaneous Collapse Models on a Lattice

We present spontaneous collapse models of field theories on a 1+1 null lattice, in which the causal structure of the lattice plays a central role. Issues such as “locality,” “nonlocality,” and superluminal signaling are addressed in the context of the models which have the virtue of extreme simplicity. The formalism of the models is related to that of the consistent histories approach to quantum mechanics.     

Quantum histories without contrary inferences

In the consistent histories formulation of quantum theory it was shown that it is possible to retrodict contrary properties. We show that this problem do not appear in our formalism of generalized contexts for quantum histories.     

Timeless Configuration Space and the Emergence of Classical Behavior

The inherent difficulty in talking about quantum decoherence in the context of quantum cosmology is that decoherence requires subsystems, and cosmology is the study of the whole Universe. Consistent histories gave a possible answer to this conundrum, by phrasing decoherence as loss of interference between alternative histories of closed systems. When one can apply Boolean logic to a set of histories, it is deemed ‘consistent’. However, the vast majority of the sets of histories that are merely consistent are blatantly nonclassical in other respects, and further constraints than just consistency need to be invoked. In this paper, I attempt to give an alternative answer to the issues faced by consistent histories, by exploring a timeless interpretation of quantum mechanics of closed systems. This is done solely in terms of path integrals in non-relativistic, timeless, configuration space. What prompts a fresh look at such foundational problems in this context is the advent of multiple gravitational models in which Lorentz symmetry is not fundamental, but only emergent. And what allows this approach to overcome previous barriers to a timeless, conditional probabilities interpretation of quantum mechanics is the new notion of records—made possible by an inherent asymmetry of configuration space. I outline and explore consequences of this approach for foundational issues of quantum mechanics, such as the natural emergence of the Born rule, conservation of probabilities, and the Sleeping Beauty paradox.     

The Measurement Process in the Generalized Contexts Formalism for Quantum Histories

In the interpretations of quantum mechanics involving quantum histories there is no collapse postulate and the measurement is considered as a quantum interaction between the measured system and the measured instrument. For two consecutive non ideal measurements on the same system, we prove that both pointer indications at the end of each measurement are compatible properties in our generalized context formalism for quantum histories. Inmediately after the first measurement an effective state for the measured system is deduced from the formalism, generalizing the state that would be obtained by applying the state collapse postulate.     

Positive-operator-valued time observable in quantum mechanics

We examine the longstanding problem of introducing a time observable in quantum mechanics; using the formalism of positive-operator-valued measures, we show how to define such an observable in a natural way and we discuss some consequences.     

Entropy and Wigner Distribution Functions Revisited

Several expressions for quantum entropy proposed in the literature are evaluated within the Weyl—Wigner—Moyal phase-space representation of quantum mechanics, with emphasis on some important subtle points in this approach. It has been found that the Rényi—Süßmann entropy and its linearization are distinguished because of their properties.     

Mathematical Theory of Non-Equilibrium Quantum Statistical Mechanics

We review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics recently proposed in refs. 1–7. In the algebraic formalism of quantum statistical mechanics we introduce notions of non-equilibrium steady states, entropy production and heat fluxes, and study their properties. Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs. We exhibit examples of such systems which have strictly positive entropy production.