Abstract: Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of space–time, in particular, amounts to a postulated new force or physical effect called cogravity.

Abstract: We show that there is a duality exchanging noncommutativity and non-trivial statistics for quantum field theory on R^d. Employing methods of quantum groups, we observe that ordinary and noncommutative R^d are related by twisting. We extend the twist to an equivalence for quantum field theory using the framework of braided quantum field theory. The twist exchanges both commutativity with noncommutativity and ordinary with non-trivial statistics. The same holds for the noncommutative torus.

Abstract: We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs.

Abstract: Quantum causal histories are defined to be causal sets with Hilbert spaces attached to each event and local unitary evolution operators. The reflexivity, antisymmetry and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators. A quantum causal history in which transitivity holds can be treated as `directed' topological quantum field theory. Two examples of such histories are described.

Abstract: Quantum theory is 100 years old and still going strong. Combininggeneral relativity with quantum mechanics is the last hurdle to be overcomein the 'quantum revolution'.

Abstract: Over the last 50 years quantum mechanics has come to be applied in very sophisticated ways. Some of these applications involve the control and observation of quantum systems using subtle noncommutative effects. However only recently has there been any attempt to look at these from a control theory perspective. In this paper we cast some of the main ideas from Nuclear Magnetic Resonance (NMR) as applied to imaging and spectroscopy in a system theoretic framework. For example, NMR spectroscopy is taken to be a system identification problem. Many key aspects of high resolution NMR spectroscopy involve manipulating and controlling nuclear spins in such a way as to generate a suitable signal for the identification problem. This active control of nuclear spin is presented as a problem in the control of nonlinear systems.

Abstract: The pioneering work of early non-relativistic quantum theory led to the understanding that quantum dynamics on Hilbert space is a comprehensive predictive framework for microscopic phenomena. From the Bohr atom, through the nonrelativistic quantum theory of Schrödinger and Heisenberg, and the relativistic Dirac equation for hydrogen, agreement between calculation and experiment improved rapidly over time. The incorporation of special relativity and field theory into quantum theory extended the scope of perturbative calculations, and these were tested through precision measurements of spectra and magnetic moments. Beginning in the 1940’s, experimental tests of the Lamb shift and the anomalous magnetic moment of the electron detected effects that one can ascribe to fluctuations in quantum electrodynamics. These effects deviated numerically from the predictions arising from equations that describe a fixed number of particles, so they were accurate tests of the quantum field hypothesis. Today these experiments have evolved to yield quantitative agreement with the most precise observations and calculations achieved in physics. For example, the anomalous magnetic moment of the electron is known theoretically and experimentally to amazing precision: (g − 2)/2 = 0.001159652200(±40). The success of this work, as well as the success of other less accurate, but compelling, predictions for weak and strong interactions, convince us to accept quantum field theory as the correct physical arena to describe particle physics down to the Planck scale. But the success of relativistic field theory calculations and of perturbative renormalization also led to a logical puzzle: is there any physically-relevant, relativistic quantum field theory that is also mathematically consistent? Put differently, can one give a mathematically complete example of any non-linear theory, relevant for the description of interacting particles, whose solutions incorporate relativistic covariance, positive energy, and causality? One must understand perturbative renormalization in order to resolve this problem, and have control over renormalization from a non-perturbative (or “exact”) point of view. In fact, one needs to overcome sophisticated problems, such as whether a field theory may appear correct on a perturbative level, while it may have no meaning at a non-perturbative level. Doubts about quantum electrodynamics or scalar meson theory were raised early by Dyson and Landau. They recur from the point of view of the renormalization group in the work of Kadanoff and Wilson, as well as in the analysis of “asymptotic

Abstract: A density matrix ρ may be represented in many different ways as a mixture of pure states, ρ=∑ipi|ψi〉〈ψi|. This paper characterizes the class of probability distributions (pi) that may appear in such a decomposition, for a fixed density matrix ρ. Several illustrative applications of this result to quantum mechanics and quantum information theory are given.

Abstract: I show how quantum mechanics, like the theory of relativity, can be understood as a ‘principle theory’ in Einstein's sense, and I use this notion to explore the approach to the problem of interpretation developed in my book Interpreting the Quantum World .

Abstract: The concept of controlling stochastic resonance has been recently introduced [L. Gammaitoni et al., Phys. Rev. Lett. 82, 4574 (1999)] to enhance or suppress the spectral response to threshold-crossing events triggered by a time-periodic signal in background noise. Here, we develop a general theoretical framework, based on a rate equation approach. This generic two-state theory captures the essential features observed in our experiments and numerical simulations.

Abstract: We study properties of a scalar quantum field theory on the two-dimensional noncommutative plane with Eq(2) quantum symmetry. We start from the consideration of a firstly quantized quantum particle on the noncommutative plane. Then we define quantum fields depending on noncommutative coordinates and construct a field theoretical action using the Eq(2)-invariant measure on the noncommutative plane. With the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quantization of the particle theory on the noncommutative plane and that this field theory has (contrary to the common belief) even more severe ultraviolet divergences than its counterpart on the usual commutative plane. Finally we introduce the symmetry transformations of physical states on noncommutative spaces and discuss them in detail for the case of the Eq(2) quantum group.

Abstract: A stochastic transformation of quantum similarity matrices is described and its role in the field of quantitative structure–activity relationship (QSAR) analyzed. The possible interest to manipulate in such a way a quantum similarity matrix, computed over a quantum object set, is diverse. First, any quantum similarity matrix column or row can easily become a discrete probability distribution, associable to a corresponding quantum object density function. Second, in order to ease its subsequent use, the resulting quantum stochastic matrix can be easily symmetrized, by means of any usual procedure or, as described here, by an inward matrix product algorithm. Third, the final matrix transform can be considered as a new quantum similarity index and can be used as a new quantum object descriptor in QSAR models. Fourth, such symmetric stochastic transform can acquire an interesting role in the approximate solution of the fundamental quantum QSAR (QQSAR) equation under various assumptions. Finally, a new algorithm, based on inward matrix product algebra, to obtain strictly positive constrained solutions of the fundamental QQSAR equation, is described. Some application examples are provided in order to illustrate the previous theoretical development. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 163–177, 2000 © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 163–177, 2000

Abstract: We treat a relativistically moving particle interacting with a quantum field from an open system viewpoint of quantum field theory by the method of influence functionals or closedtime-path coarse-grained effective actions. The particle trajectory is not prescribed but is determined by the backreaction of the quantum field in a self-consistent way. Coarse-graining the quantum field imparts stochastic behavior in the particle trajectory. The formalism is set up here as a precursor to a first principles derivation of the Abraham-Lorentz-Dirac (ALD) equation from quantum field theory as the correct equation of motion valid in the semiclassical limit. This approach also discerns classical radiation reaction from quantum dissipation in the motion of a charged particle; only the latter is related to vacuum fluctuations in the quantum field by a fluctuation-dissipation relation, which we show to exist for nonequilibrim processes under this type of nonlinear coupling. This formalism leads naturally to a set of Langevin equations associated with a generalized ALD equation. These multiparticle stochastic differential equations feature local dissipation (for massless quantum fields), multiplicative noise, and nonlocal particle-particle correlations, interrelated in ways characteristic of nonlinear theories, through generalized fluctuation-dissipation relations.

Abstract: The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading the research literature in quantum computation and quantum information theory. This paper is a written version of the first of eight one hour lectures given in the American Mathematical Soci- ety (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in Jan- uary 2000, and will be published in the AMS PSAPM volume entitled "Quantum Computation.". Part 1 of the paper is a preamble introducing the reader to the con- cept of the qubit, Part 2 gives an introduction to quantum mechanics covering such top- ics as Dirac notation, quantum measurement, Heisenberg uncertainty, Schrodinger's equation, density operators, partial trace, multipartite quantum systems, the Heisenberg versus the Schrodinger picture, quan- tum entanglement, EPR paradox, quantum entropy. Part 3 gives a brief introduction to quantum computation, cover- ing such topics as elementary quantum computing devices, wiring di- agrams, the no-cloning theorem, quantum teleportation, Shor's algo- rithm, Grover's algorithm. Many examples are given to illustrate underlying principles. A ta- ble of contents as well as an index are provided for readers who wish to "pick and choose." Since this paper is intended for a diverse audi- ence, it is written in an informal style at varying levels of difficulty and sophistication, from the very elementary to the more advanced.

Abstract: The question to be addressed is, In what sense and to what extent do quantum statistics for, and the standard formal quantum-mechanical description of, systems of many identical particles entail that identical quantum particles are indistinguishable? This paper argues that whether or not we consider identical quantum particles as indistinguishable is a matter of theory choice underdetermined by logic and experiment.

Abstract: A nonparametric Bayesian approach is developed to determine quantum potentials from empirical data for quantum systems at finite temperature. The approach combines the likelihood model of quantum mechanics with a priori information on potentials implemented in the form of stochastic processes. Its specific advantages are the possibilities to deal with heterogeneous data and to express a priori information explicitly in terms of the potential of interest. A numerical solution in maximum a posteriori approximation is obtained for one-dimensional problems. As nonparametric estimates, the results depend strongly on the implemented a priori information.

Abstract: It is shown that for a given set of correlations either in a classical or in a quantumprobability space both the classical and the quantum probability spaces areextendable in such a way that the extension contains common causes of thegiven correlations, where common cause is taken in the sense of Reichenbach'sdefinition. These results strongly restrict the possible ways of disprovingReichenbach's common cause principle and indicate that EPR-type quantumcorrelations might very well have a common cause explanation.

Abstract: In the existing expositions of the Karolyhazy model, quantum mechanical uncertainties are mimicked by classical spreads It is shown how to express those uncertainties through entities of the future unified theory of general relativity and quantum theory

Abstract: Without invalidating quantum mechanics as a principle underlying the dynamics of a fundamental theory, it is possible to ask for even more basic dynamical laws that may yield quantum mechanics as the machinery needed for its statistical analysis. In conventional systems such as the Standard Model for quarks and leptons, this would lead to hidden variable theories, which are known to be plagued by problems such as non-locality. But Planck scale physics is so different from field theories in some flat background space-time that here the converse may be the case: we speculate that causality and locality can only be restored by postulating a deterministic underlying theory. A price to be paid may be that the underlying theory exhibits dissipation of information.

Abstract: t is shown that although the de Broglie-Bohm quantum theory of motion is equivalent to standard quantum mechanics when averages of dynamical variables are taken over a Gibbs ensemble of Bohmian trajectories, the equivalence breaks down for ensembles built over clearly separated short intervals of time in special multi-particle systems. This feature is exploited to propose a realistic experiment to distinguish between the two theories.