Abstract: We reconsider the crucial 1927 Solvay conference in the context of current research in the foundations of quantum theory Contrary to folklore, the interpretation question was not settled at this conference and no consensus was reached; instead, a range of sharply conflicting views were presented and extensively discussed Today, there is no longer an established or dominant interpretation of quantum theory, so it is important to re-evaluate the historical sources and keep the interpretation debate open In this spirit, we provide a complete English translation of the original proceedings (lectures and discussions), and give background essays on the three main interpretations presented: de Broglie's pilot-wave theory, Born and Heisenberg's quantum mechanics, and Schroedinger's wave mechanics We provide an extensive analysis of the lectures and discussions that took place, in the light of current debates about the meaning of quantum theory The proceedings contain much unexpected material, including extensive discussions of de Broglie's pilot-wave theory (which de Broglie presented for a many-body system), and a "quantum mechanics" apparently lacking in wave function collapse or fundamental time evolution We hope that the book will contribute to the ongoing revival of research in quantum foundations, as well as stimulate a reconsideration of the historical development of quantum physics A more detailed description of the book may be found in the Preface (Copyright by Cambridge University Press (ISBN: 9780521814218))

Abstract: Physical background.- Mathematical detour: operator theory.- Dynamics.- Mathematical detour: the Fourier transform.- Observables.- The uncertainty principle.- Spectral theory.- Scattering states.- Special cases.- Many-particle systems.- Density matrices.- The Feynman path integral.- Mathematical detour: the calculus of variations.- Mathematical detours: the stationary phase method and operator determinants.- Quasi-classical analysis.-Resonances.- Introduction to quantum field theory.- Quantum electrodynamics of non-relativistic particles: the theory of radiation.- Supplement: renormalization group.-Comments on missing topics, literature, and further reading.

Abstract: The current understanding of the quantum origin of cosmic structure is discussed critically. We point out that in the existing treatments a transition from a symmetric quantum state to an (essentially classical) non-symmetric state is implicitly assumed, but not specified or analysed in any detail. In facing this issue, we are led to conclude that new physics is required to explain the apparent predictive power of the usual schemes. Furthermore, we show that the novel way of looking at the relevant issues opens new windows from where relevant information might be extracted regarding cosmological issues and perhaps even clues about aspects of quantum gravity.

Abstract: The dynamics of a particle in a gravitational quantum well is studied in the context of nonrelativistic quantum mechanics with a particular deformation of a two-dimensional Heisenberg algebra. This deformation yields a new short-distance structure characterized by a finite minimal uncertainty in position measurements, a feature it shares with noncommutative theories. We show that an analytical solution can be found in perturbation and we compare our results to those published recently, where noncommutative geometry at the quantum mechanical level was considered. We find that the perturbations of the gravitational quantum well spectrum in these two approaches have different signatures. We also compare our modified energy spectrum to the results obtained with the GRANIT experiment, where the effects of the Earth's gravitational field on quantum states of ultracold neutrons moving above a mirror are studied. This comparison leads to an upper bound on the minimal length scale induced by the deformed algebra we use. This upper bound is weaker than the one obtained in the context of the hydrogen atom but could still be useful if the deformation parameter of the Heisenberg algebra is not a universal constant but a quantity that depends on the energetic content of the system.

Abstract: An experimental demonstration of quantum contextuality with neutrons is presented, which intended to exhibit a Kochen-Specker-like phenomenon. Since no perfect correlation is expected in practical experiments, inequalities are derived to distinguish quantitatively the obtained results from predictions by a noncontextual hidden variable theory. Experiments were accomplished with the use of a neutron interferometer combined with spinor manipulation devices. The results clearly violate the prediction of noncontextual theories.

Abstract: I analyse the conceptual and mathematical foundations of Lagrangian quantum field theory (QFT) (that is, the ‘naive’ (QFT) used in mainstream physics, as opposed to algebraic quantum field theory). The objective is to see whether Lagrangian (QFT) has a sufficiently firm conceptual and mathematical basis to be a legitimate object of foundational study, or whether it is too ill-defined. The analysis covers renormalisation and infinities, inequivalent representations, and the concept of localised states; the conclusion is that Lagrangian QFT (at least as described here) is a perfectly respectable physical theory, albeit somewhat different in certain respects from most of those studied in foundational work.

Abstract: As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable.

Abstract: The dynamics of a particle in a gravitational quantum well is studied in the context of nonrelativistic quantum mechanics with a particular deformation of a two-dimensional Heisenberg algebra. This deformation yields a new short-distance structure characterized by a finite minimal uncertainty in position measurements, a feature it shares with noncommutative theories. We show that an analytical solution can be found in perturbation and we compare our results to those published recently, where noncommutative geometry at the quantum mechanical level was considered. We find that the perturbations of the gravitational quantum well spectrum in these two approaches have different signatures. We also compare our modified energy spectrum to the results obtained with the GRANIT experiment, where the effects of the Earth’s gravitational field on quantum states of ultracold neutrons moving above a mirror are studied. This comparison leads to an upper bound on the minimal length scale induced by the deformed algebra we use. This upper bound is weaker than the one obtained in the context of the hydrogen atom but could still be useful if the deformation parameter of the Heisenberg algebra is not a universal constant but a quantity that depends on the energetic content of the system.

Abstract: We consider noncommutative quantum mechanics with phase space noncommutativity. In particular, we show that a scaling of variables leaves the noncommutative algebra invariant, so that only the self-consistent effective parameters of the model are physically relevant. We also discuss the recently proposed relation of direct proportionality between the noncommutative parameters, showing that it has a limited applicability.

Abstract: Preface by the Editors Preface 1985 On Weizsacker's Philosophy of Physics (by H. Lyre) Chapter 1: Introduction. 1.1. The Question. 1.2. Outline Part I: The Unity of Physics Chapter 2: The System of theories. 2.1. Preliminary. 2.2. Classical point mechanics. 2.3. Mathematical forms of the Laws of Nature. 2.4. Chemistry. 2.5. Thermodynamics. 2.6. Field theories. 2.7. Non-Euclidan geometry and semantical consistency. 2.8. The relativity problem. 2.9. Special theory of relativity. 2.10. General theory of relativity. 2.11. Quantum theory, historical. 2.12. Quantum theory, plan of reconstruction. Chapter 3: Probability and abstract quantum theory. 3.1. Probability and experience. 3.2. The classical concept of probability. 3.3. Empirical determination of probabilities. 3.4. Second quantization. 3.5. Methodical: reconstruction of abstract quantum theory. 3.6. Reconstruction via probabilities and the lattice of propositions. Chapter 4: Quantum theory and space-time. 4.1. Concrete quantum theory. 4.2. Reconstruction of quantum theory via variable alternatives. 4.3. Space and time. Chapter 5: Models of particles and interaction. 5.1. Open questions. 5.2. Representations in tensor space. 5.3. Quasi-particles in rigid coordinate spaces. 5.4. Model of quantum electrodynamics. 5.5. Elementary particles. 5.6. General theory of relativity. Chapter 6: Cosmology and particle physics (by Th. Gornitz). 6.1. Quantum theory of abstract binary alternatives and cosmology. 6.2. Ur-theoretical vacuum and particle states. 6.4. Outlook. Part II: Time and Information Chapter 7: Irreversibility and entropy. 7.1. Irreversibility as problem. 7.2. A model of irreversible processes. 7.3. Documents. 7.4. Cosmology and the theory of relativity. Chapter 8: Information and evolution. 8.1. The systematic place of the chapter. 8.2. What is information? 8.3. What is evolution? 8.4. Information and probability. 8.5. Evolution as growth of potential information. 8.6. Pragmaticinformation: novelty and confirmation. 8.7. Biological preliminaries to logic. Part III: On the Interpretation of Physics Chapter 9: The problem of the interpretation of quantum theory. 9.1. About the history of the interpretation. 9.2. The semantical consistency of quantum theory. 9.3. Paradoxa and alternatives. Chapter 10: The stream of information. 10.1. The quest for substance. 10.2. The stream of information in quantum theory. 10.3. Mind and form. Chapter 11: Beyond quantum theory. 11.1. Crossing the frontier. 11.2. Facticity of the future. 11.3. Possibility of the past. 11.4. Comprehensive present. 11.5. Beyond physics. Chapter 12: In the language of philosophers. 12.1. Exposition. 12.2. Philosophy of science. 12.3. Physics. 12.4. Metaphysics. References Index

Abstract: The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

Abstract: A remarkable theorem by Clifton et al [Found Phys. 33(11), 1561–1591 (2003)] (CBH) characterizes quantum theory in terms of information-theoretic principles. According to Bub [Stud. Hist. Phil. Mod. Phys. 35 B, 241–266 (2004); Found. Phys. 35(4), 541–560 (2005)] the philosophical significance of the theorem is that quantum theory should be regarded as a “principle” theory about (quantum) information rather than a “constructive” theory about the dynamics of quantum systems. Here we criticize Bub’s principle approach arguing that if the mathematical formalism of quantum mechanics remains intact then there is no escape route from solving the measurement problem by constructive theories. We further propose a (Wigner-type) thought experiment that we argue demonstrates that quantum mechanics on the information-theoretic approach is incomplete.

Abstract: Certain concrete "ontological models" for quantum mechanics (models in which measurement outcomes are deterministic and quantum states are equivalent to classical probability distributions over some space of `hidden variables') are examined. The models are generalizations of Kochen and Specker's such model for a single 2-dimensional system - in particular a model for a three dimensional quantum system is considered in detail. Unfortunately, it appears the models do not quite reproduce the quantum mechanical statistics. They do, however, come close to doing so, and in as much as they simply involve probability distributions over the complex projective space they do reproduce pretty much everything else in quantum mechanics. The Kochen-Specker theorem is examined in the light of these models, and the rather mild nature of the manifested contextuality is discussed.

Abstract: In standard nonrelativistic quantum mechanics the expectation of the energy is a conserved quantity. It is possible to extend the dynamical law associated with the evolution of a quantum state consistently to include a nonlinear stochastic component, while respecting the conservation law. According to the dynamics thus obtained, referred to as the energy-based stochastic Schrodinger equation, an arbitrary initial state collapses spontaneously to one of the energy eigenstates, thus describing the phenomenon of quantum state reduction. In this paper, two such models are investigated: one that achieves state reduction in infinite time and the other in finite time. The properties of the associated energy expectation process and the energy variance process are worked out in detail. By use of a novel application of a nonlinear filtering method, closed-form solutions—algebraic in character and involving no integration—are obtained of both these models. In each case, the solution is expressed in terms of a random variable representing the terminal energy of the system and an independent noise process. With these solutions at hand it is possible to simulate explicitly the dynamics of the quantum states of complicated physical systems.

Abstract: We show that the quantum central-force problems can be modeled and solved exactly by quantum Hamilton–Jacobi formulation, from which the quantum operators z, 2, and can be derived without using the quantization principle p (/i)∂/∂x. Quantum conservation laws expressed by the Poisson bracket show that the eigenvalues of these quantum operators are just equal to the constants of motion along the eigen-trajectories defined in a complex domain. The shell structure observed in bound systems, such as the hydrogen atom, is found to stem from the structure of the quantum potential, by which the quantum forces acting on the electron can be uniquely determined, the stability of atomic configuration can be justified, and the quantum trajectories of the electron can be obtained by integrating the related quantum Lagrange equations. On solving the quantum equations of motion, the solution of the Schrodinger equation serves as the first integration of the second-order quantum Lagrange equations. The stable equilibrium points of the derived first-order nonlinear quantum dynamics are shown to be identical to the positions with maximum probability predicted by standard quantum mechanics. The internal mechanism of how the quantum dynamics evolve continuously to classical dynamics and of how the quantum conservation laws transit continuously to the classical conservation laws as n ∞ are analyzed in detail. The construction of the quantum scattering trajectory by searching for an unbound solution for the Schrodinger equation is investigated. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Abstract: A key concept in the modern theory of open quantum systems is the notion of indirect measurement as introduced by Kraus [5]. An indirect measurement on a quantum system is a (direct) measurement of some quantity in its environment, made after some interaction with the system has taken place. When we make such a measurement, our description of the quantum system changes in two ways: we account for the flow of time by a unitary transformation (following Schrodinger), and we update our knowledge of the system by conditioning on the measurement outcome (following von Neumann). If we then repeat the indirect measurement indefinitely, we obtain a chain of random outcomes. In the course of time we may keep record of the updated density matrix Qt, which at time t reflects our best estimate of all observable quantities of the quantum system, given the observations made up to that time. This information can in its turn be used to predict later measurement outcomes. The stochastic process St of updated states, is the quantum trajectory associated to the repeated measurement process. By taking the limit of continuous time, we arrive at the modern models of continuous observation: quantum trajectories in continuous time satisfying stochastic Schrodinger equations [3], [4], [2], [1]. These models are employed with great success for calculations and computer simulations of laboratory experiments such as photon counting and homodyne field detection. In this paper we consider the question, what happens to the quantum trajectory at large times. We do so only for the case of discrete time, not a serious restriction indeed, since asymptotic behaviour remains basically unaltered in the continuous time limit.

Abstract: We review quantum causal histories starting with their interpretations as a quantum field theory on a causal set and a quantum geometry. We discuss the difficulties that background independent theories based on quantum geometry encounter in deriving general relativity as the low energy limit. We then suggest that general relativity should be viewed as a strictly effective theory coming from a fundamental theory with no geometric degrees of freedom. The basic idea is that an effective theory is characterized by effective coherent degrees of freedom and their interactions. Having formulated the pre-geometric background independent theory as a quantum information theoretic processor, we are able to use the method of noiseless subsystems to extract such coherent (protected) excitations. We follow the consequences, in particular, the implications of effective locality and time.

Abstract: Chaos implies unpredictability, fluctuations, and the need for statistical modelling. Quantum optics has developed into one of the most advanced subdisciplines of modern physics in terms of the control of matter on a microscopic scale, and, in particular, of isolated, single quantum objects. Prima facie, both fields therefore appear rather distant in philosophy and outset. However, as we shall discuss in the present review, chaos, and, more specifically, quantum chaos opens up novel perspectives for our understanding of the dynamics of increasingly complex quantum systems, and of ultimate quantum control by tailoring complexity.

Abstract: We give a short overview over recent developments on quantum graphs and outline the connection between general quantum graphs and so-called quantum random walks.

Abstract: Familiar textbook quantum mechanics assumes a fixed background spacetime to define states on spacelike surfaces and their unitary evolution between them. Quantum theory has changed as our conceptions of space and time have evolved. But quantum mechanics needs to be generalized further for quantum gravity where spacetime geometry is fluctuating and without definite value. This paper reviews a fully four-dimensional, sum-over-histories, generalized quantum mechanics of cosmological spacetime geometry. This generalization is constructed within the framework of generalized quantum theory. This is a minimal set of principles for quantum theory abstracted from the modern quantum mechanics of closed systems, most generally the universe. In this generalization, states of fields on spacelike surfaces and their unitary evolution are emergent properties appropriate when spacetime geometry behaves approximately classically. The principles of generalized quantum theory allow for the further generalization that would be necessary were spacetime not fundamental. Emergent spacetime phenomena are discussed in general and illustrated with the example of the classical spacetime geometries with large spacelike surfaces that emerge from the `no-boundary' wave function of the universe. These must be Lorentzian with one, and only one, time direction. The essay concludes by raising the question of whether quantum mechanics itself is emergent.

Abstract: We propose several parametrization-free solutions to the problem of quantum state reduction control by means of continuous measurement and smooth quantum feedback. In particular, we design a feedback law for which almost global stochastic feedback stabilization can be proved analytically by means of Lyapunov techinques. This synthesis arises very naturally from the physics of the problem, as it relies on the variance associated with the quantum filtering process.

Abstract: A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

Abstract: We discuss in an introductory manner structural similarities between the polylogarithm and Green functions in quantum field theory.

Abstract: The debate on the nature of quantum probabilities in relation to Quantum Non Locality has elevated Quantum Mechanics to the level of an Operational Epistemic Theory. In such context the quantum superposition principle has an extraneous non epistemic nature. This leads us to seek purely operational foundations for Quantum Mechanics, from which to derive the current mathematical axiomatization based on Hilbert spaces.In the present work I present a set of axioms of purely operational nature, based on a general definition of “the experiment”, the operational/epistemic archetype of information retrieval from reality. As we will see, this starting point logically entails a series of notions [state, conditional state, local state, pure state, faithful state, instrument, propensity (i.e. “effect”), dynamical and informational equivalence, dynamical and informational compatibility, predictability, discriminability, programmability, locality, a‐causality, rank of the state, maximally chaotic state, maximally entan...

Abstract: Topics in Non-Equilibrium Quantum Statistical Mechanics.- Fermi Golden Rule and Open Quantum Systems.- Decoherence as Irreversible Dynamical Process in Open Quantum Systems.- Notes on the Qualitative Behaviour of Quantum Markov Semigroups.- Continual Measurements in Quantum Mechanics and Quantum Stochastic Calculus.

Abstract: In this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Levy–Laplacian is obtained as the usual Volterra–Gross Laplacian using the Cesaro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise.

Abstract: This paper serves as a preparation of work that focuses on extracting cosmological sectors from Loop Quantum Gravity. We start with studying the extraction of subsystems from classical systems. A classical Hamiltonian system can be reduced to a subsystem of ''relevant observables'' using the pull-back under the Poisson-embedding of the ''relevant part of phase space'' into full phase space. Since a quantum theory can be thought of as a noncommutative phase space, one encounters the problem of embedding noncommutative spaces. We solve this problem for a physically interesting set of quantum systems and embeddings by constructing the noncommutative analogue of the construction of an embedding as the projection to the base space of an embedding of fibre bundles over the involved spaces. This paper focuses on the physical ideas that enter our programme of reduction of quantum theories and tries to explain these on examples rather than abstractly, which will be the focus of a forthcoming paper.

Abstract: The Virasoro--Zamolodchikov Lie algebra $w_{\infty}$ has been widely studied in string theory and in conformal field theory, motivated by the attempts of developing a satisfactory theory of quantization of gravity. The renormalized higher powers of quantum white noise (RHPWN) *-Lie algebra has been recently investigated in quantum probability, motivated by the attempts to develop a nonlinear generalization of stochastic and white noise analysis. We prove that, after introducing a new renormalization technique, the RHPWN Lie algebra includes a second quantization of the $w_{\infty}$ algebra. Arguments discussed at the end of this note suggest the conjecture that this inclusion is in fact an identification