Abstract: Quantum mechanics and chaotic fields quantum chaos, complex spectral representation and time-symmetry breaking, T. Petrosky and I. Prigogine scale relativity, fractal space-time and quantum mechanics, L. Nottale introduction to wave phenomena and uncertainty in a fractal space - I, A. Le Mehaute et al young double-slit experiment, Heisenberg uncertainty principle and cantoria space-time, M.S. El Naschie the quantum dimension of space-time, E. Alvarez et al intra-observer chaos - hidden root of quantum mechanics?, O.E. Rossler amplification of superpositional effects through electronic - conformational interactions, M. Conrad quantum cellular automation in 1-D, D.W. Belousek et al an energy-barrier model of biased transport in disordered systems, M. Giona chaos in the Einstein equations, S.E. Rugh.

Abstract: Introduction 1. Nonrelativistic quantum mechanics 2. Relativity and symmetries 3. Quantum field theory 4. Objects of experiences: Quantum states-observables-statistics 5. The event-structure and the spatio-temporal order: local fields 6. Explicit relations and the causal order: Interacting fields 7. Epilogue Appendix A: Measurement and probability: Quantity, quality, modality Appendix B: Fiber bundles Appendix C: The cosmic and the microscopic: An application Notes Bibliography

Abstract: I review the decoherent (or consistent) histories approach to quantum mechanics, due to Griffiths, to Gell-Mann and Hartle, and to Omnes. This is an approach to standard quantum theory specifically designed to apply to genuinely closed systems, up to and including the entire universe. It does not depend on an assumed separation of classical and quantum domains, on notions of measurement, or on collapse of the wave function. Its primary aim is to find sets of histories for closed systems exhibiting negligble interference, and therefore, to which probabilities may be assigned. Such sets of histories are called consistent or decoherent, and may be manipulated according to the rules of ordinary (Boolean) logic. The approach provides a framework from which one may predict the emergence of an approximately classical domain for macroscopic systems, together with the conventional Copenhagen quantum mechanics for microscropic subsystems. In the special case in which the total closed system naturally separates into a distinguished subsystem coupled to an environment, the decoherent histories approach is closed related to the quantum state diffusion approach of Gisin and Percival.

Abstract: Non-differentiable fluctuations in spacetime on a Planck scale introduce stochastic terms into the equations for quantum states, resulting in a proposed new foundation for an existing alternative quantum theory: primary state diffusion (PSD). Planckscale stochastic spacetime structure results in quantum fluctuations, whilst larger-scale curvature is responsible for gravitational forces. The gravitational field and the quantum fluctuation field are the same, differing only in scale. The quantum mechanics of small systems, classical mechanics of large systems and the physics of quantum experiments are all derived dynamically, without any prior division into classical and quantum domains, and without any measurement hypothesis. Unlike the earlier derivation of PSD, the new derivation, based on a stochastic spacetime differential geometry, has essentially no free parameters. However, many features of this structure remain to be determined. The theory is falsifiable in the laboratory, and critical matter interferometry experiments, to distinguish it from ordinary quantum mechanics, might be feasible within the next decade.

Abstract: We discuss quantum-field-theoretic interpretation of the family of observables (the so-called C-algebra) introduced in [ F.V.Tkachov: Preprint FERMILAB-PUB-95/191-T] for a systematic description of multijet structure of multiparticle final states at high energies. We argue that from the point of view of general quantum field theory, all information about the multijet structure is contained in the values of a family of multiparticle quantum correlators that can be expressed in terms of the energy-momentum tensor.

Abstract: The standard formalism of quantum theory is enhanced and definite meaning is given to the concepts of experiment, measurement and event. Within this approach one obtains a uniquely defined piecewise deterministic algorithm generating quantum jumps, classical events and histories of single quantum objects. The wave-function Monte Carlo method of Quantum Optics is generalized and promoted to the level of a fundamental process generating all the real events in Nature. The already worked out applications include SQUID-tank model and generalized cloud chamber model with GRW spontaneous localization as a particular case. Differences between the present approach and quantum measurement theories based on environment-induced master equations are stressed. Questions: what is classical, what is time, and what observers are addressed. Possible applications of the new approach are suggested, among them connection between the stochastic commutative geometry and Connes' noncommutative formulation of the Standard Model, as well as potential applications to the theory and practice of quantum computers.

Abstract: We remark that the often ignored quantum probability current is fundamental for a genuine understanding of scattering phenomena and, in particular, for the statistics of the time and position of the first exit of a quantum particle from a given region, which may be simply expressed in terms of the current. This simple formula for these statistics does not appear as such in the literature. It is proposed that the formula, which is very different from the usual quantum mechanical measurement formulas, be verified experimentally. A full understanding of the quantum current and the associated formula is provided by Bohmian mechanics.

Abstract: In the last chapter of this volume we are going to pave the way for the quantum statistics of open systems. So far we have been concerned with the one-component rarefied gas of structureless particles (the Boltzmann gas). In the next volume our expansion will proceed in two main directions, and we shall consider two more basic models.

Abstract: Classical Statistical Dynamics: Introduction, Probability Theory Linear Dynamics Isolated Dynamics Isothermal Dynamics Driven Systems Fluid Dynamics Quantum Statistical Dynamics: Introduction, Quantum Probability Linear Quantum Dynamics Isolated Quantum Dynamics Isothermal and Driven Systems Infinite Systems Information Geometry.

Abstract: As is highlighted in the excellent books by Meyer [21] and by Parthasarathy [26] quantum stochastic calculus on (Boson) Fock space has developed into a new field of mathematics keeping a profound contact with physical applications. Since Hudson and Parthasarathy [12] first formulated quantum stochastic integrals of Ito type in 1984 a crucial role has been played by three basic quantum stochastic processes:

Abstract: In continuation of our study of the existence of solutions of quantum stochastic differential inclusions, we first introduce and develop some aspects of the theory of maximal [resp. hypermaximal] monotone multifunctions, including the description of a number of properties of their resolvents and Yosida approximations, in the present noncommutative setting. Then, it is proved that, under a certain continuity assumption, a quantum stochastic differential inclusion of hypermaximal monotone type has a unique adapted solution which is obtained as the limit of the unique adapted solutions of a one-parameter family of Lipschitzian quantum stochastic differential equations. As examples, we show that a large class of quantum stochastic differential inclusions which satisfy the assumptions and conclusion of our main result arises as perturbations of certain quantum stochastic differential equations by some multivalued stochastic processes.

Abstract: Conventional discrete approximations of a manifold do not preserve its nontrivial topological features. In this article we describe an approximation scheme due to Sorkin which reproduces physically important aspects of manifold topology with striking fidelity. The approximating topological spaces in this scheme are partially ordered sets (posets). Now, in ordinary quantum physics on a manifold M, continuous probability densities generate the commutative C*-algebra C(M) of continuous functions on M. It has a fundamental physical significance, containing the information to reconstruct the topology of M, and serving to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C*-algebra A. As noncommutative geometries are based on noncommutative C*-algebras, we therefore have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. Various methods for doing quantum physics using A are explored. Particular attention is paid to developing numerically viable approximation schemes which at the same time preserve important topological features of continuum physics.

Abstract: It is generally believed that Bohm's version of quantum mechanics is observationally equivalent to standard quantum mechanics. A more careful statement is that the two theories will always make the same predictions for any question or problem that is well posed in both interpretations. The transit time of a “particle” between two points in space is not necessarily well defined in standard quantum mechanics, whereas it is in Bohm's theory since there is always a particle following a definite trajectory. For this reason tunneling times (in a scattering configuration through a potential barrier may be a situation in which Bohm's theory can make a definite prediction when standard quantum mechanics can make none at all. I summarize some of the theoretical and experimental prospects for an unambiguous comparison in the hope that this question will engage the attention of more physicists, especially those experimentalists who now routinely actually do gedanken experiments.

Abstract: We revive an old proposal of Zeh for the preferred basis in the many-worlds interpretation of quantum mechanics. The algorithm for the basis reduces to the eigenvalue problems for density matrices of subsystems forming the whole system under consideration. We generalize this procedure to the case of degenerate eigenvalues of reduced density matrices. A semiclassical calculational method for these eigenvalues is developed and applied to some model problems. The classical properties of elements of the preferred basis are investigated. It is shown that classicality exists only in some part of many-worlds branches. Moreover, it depends crucially on the initial conditions and Hamiltonians and under some circumstances turns out to be a temporary phenomenon. Applications of the preferred-basis proposal to quantum cosmology are discussed. The relation between the preferred-basis approach and quantum-histories approach is discussed.

Abstract: Two problems will be considered: the question of hidden parameters and the problem of Kolmogorovity of quantum probabilities. Both of them will be analyzed from the point of view of two distinct understandings of quantum mechanical probabilities. Our analysis will be focused, as a particular example, on the Aspect-type EPR experiment. It will be shown that the quantum mechanical probabilities appearing in this experiment can be consistently understood as conditional probabilities without any paradoxical consequences. Therefore, nothing implies in the Aspect experiment that quantum theory is incompatible with a deterministic universe.

Abstract: A new axiomatization for probability theory, including the classical and the quantum case is proposed. It is shown that the quantum formalism can be deduced from this set of (physically meaningful) axioms.

Abstract: The differential-geometric formulation of statistics (the so-called information geometry) concerning the structure of a smooth manifold in the parameter space Θ of classical probabilities, S = {p(·,θ),θ ϵ Θ}, discussed by Amari, is extended to the same manifold but for quantum states (density matrices), S = {ϱ(θ);θ ϵ Θ} in N × N matrix algebras. This is done by introducing an n-tuple of tangent vectors {δ}ni = 1 in analogy to the classical ones {∂i}ni = 1. On this basis, a special problem of quantum information geometry is treated; namely, the analysis of the exponential and the mixture families defined, respectively, as (e) ϱ(θ) = exp(θi Ai − ψ(θ)). θ ϵ Θ = Rn. Ai ϵ Bs(HN) . (m) ϱ(θ) = θiAi + θ0 A0. θ ϵ Θ = (0,1)n + 1. ∑i=0nθi=1. Ai ϵ B+(HN) Tr Ai = 1 (the tensorial summation convention for repeated indices is used). We prove some of the basic theorems known in the classical information geometry by extending the formulation to a non-commutative smooth manifold. We establish the existence of a pair of dual affine coordinate systems in (e) or (m) and a projection theorem in order to ensure the Cramer-Rao inequality and an identification of the efficient estimator.

Abstract: We discuss recent developments in the foundations of quantum theory with a particular emphasis on description of measurement—like couplings between classical and quantum systems. The SQUID-tank coupling is described in some details, both in terms of the Liouville equation describing statistical ensambles and piecewise deterministic random process describing random behaviour of individual systems.

Abstract: An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised temporal analogue of the lattice of propositions of standard quantum logic. Particular emphasis is placed on those cases in which the history propositions can be represented by projection operators in a Hilbert space, and on the associated concept of a `history group'.

Abstract: We develop a rigorous treatment of discontinuous stochastic unitary evolution for a system of quantum particles that interacts singularly with quantum “bubbles" in a cloud chamber at random instants of time This model allows to watch and follow with a quantum particle trajectory like in cloud chamber by sequential unsharp localization of spontaneous scatterings of the bubbles Thus, the continuous reduction and spontaneous localization theory is obtained as the result of quantum …ltering theory, ie, a theory describing the conditioning of the a priori quantum state by the measurement data We show that in the case of indistinguishable particles the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation The latter coincides with the nonstochastic Schrodinger equation only in the mean …eld approximation, whereas the central limit yields Gaussian mixing ‡uctuations described (

Abstract: Quantum information theory is at the confluent of computer science and quantum mechanics. We survey some of the most striking recent developments in the field.