Axiomatic foundations of nonrelativistic quantum mechanics: A realistic approach

A realistic axiomatic formulation of nonrelativistic quantum mechanics for a single microsystem with spin is presented, from which the most important theorems of the theory can be deduced. In comparison with previous formulations, the formal aspect has been improved by the use of certain mathematical theories, such as the theory of rigged spaces, and group theory. The standard formalism is naturally obtained from the latter, starting from a central primitive concept: the Galilei group.     

The foundations of quantum mechanics

Starting from a set of assumptions mainly of an operational or experimentally based nature, a derivation of quantum mechanics is presented, with the aim of clarifying the essential features of the theory and their interpretation. Various properties of quantum mechanics such as the addition of amplitudes, the calculation of probabilities, de Broglie's equations, and energy-momentum conservation are derived from first principles. It is investigated whether quantum amplitudes may be constructed from quantities of higher order than complex numbers. Measurable physical quantitics, as traditionally understood, are seen to play a role distinct from and supplementary to the behavior of the quantum amplitudes themselves. This is related to two distinct aspects of the nature of time in the context of quantum mechanics.     

A statistical approach to quantum mechanics

A Monte Carlo method is used to evaluate the Euclidean version of Feynman's sum over particle histories. Following Feynman's treatment, individual paths are defined on a discrete (imaginary) time lattice with periodic boundary conditions. On each lattice site, a continuous position variable x_i specifies the spacial location of the particle. Using a modified Metropolis algorithm, the low-lying energy eigenvalues, |ψ₀(x)|², the propagator, and the effective potential for the anharmonic oscillator are computed, in good agreement with theory. For a deep double-well potential, instantons were found in our computer simulations appearing as multi-kink configurations on the lattice.     

On the quantum logic approach to quantum mechanics

A quantum logic structure for quantum mechanics which contains the concepts of a physical space, localizability, and symmetry groups is formulated. It is shown that there is an underlying Hilbert space which mirrors much of this axiomatic structure. Quantum fields are defined and shown to arise naturally from the quantum logic structure. The fields ofHaag andWightman are generalized to this theory and an attempt is made to find a local equivalence for these fields.     

Epistemological and mathematical foundations of quantum mechanics

The concepts of measurement and measurable quantity are discussed. A probabilistic interpretation independent of the arrow of time is recommended and a definition of quantizable physical systems is given. The space of states of information about the physical system is Schwarz space rather than Hilbert space.     

Particle interactions and quantum mechanics: Mathematical foundations

This review contains chapters from the first volume of S. S. Sannikov-Proskuryakov’s collected papers. The first and the second chapters are compilations of his early publications. The four papers included in Chapters 3–7 were written during the last years of his life. The most valuable results obtained by S. S. Sannikov-Proskuryakov can be useful for deep understanding of the current status and contemporary developments of high-energy physics.     

On the consistent histories approach to quantum mechanics

We review the consistent histories formulations of quantum mechanics developed by Griffiths, Omnès, and Gell-Mann and Hartle, and describe the classification of consistent sets. We illustrate some general features of consistent sets by a few simple lemmas and examples. We consider various interpretations of the formalism, and examine the new problems which arise in reconstructing the past and predicting the future. It is shown that Omnès' characterization of true statements—statements which can be deduced unconditionally in his interpretation—is incorrect. We examine critically Gell-Mann and Hartle's interpretation of the formalism, and in particular their discussions of communication, prediction, and retrodiction, and conclude that their explanation of the apparent persistence of quasiclassicality relies on assumptions about an as-yetunknown theory of experience. Our overall conclusion is that the consistent histories approach illustrates the need to supplement quantum mechanics by some selection principle in order to produce a fundamental theory capable of unconditional predictions.     

Functional approach to scattering in quantum mechanics

A functional approach to scattering theory in quantum mechanics is developed by deriving an explicit functional expression fortransition amplitudes. In applications, the formalism avoids dealing with noncommutativity problems of operators, solving the Schrödinger equation (or the integral equation of the Green's function), or dealing with the often quite complicated continual (path) integrals and, most importantly, applies to short- and long-range interactions. The basic idea is the use of the quantum action principle followed by a systematic analysis of the concept of an intervening source developed earlier in the study of stimulated emission. A comparison with the standard approach is also made.     

New approach to the semiclassical limit of quantum mechanics

We propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the Schrödinger operator in the presence of tunneling based on the theory of diffusion processes. Our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which, as far as we know, have never been treated rigorously in the literature. In particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1) symmetric double well potentials 2) potentials periodic and symmetric over a finite interval. Theses situations are characterized by a remarkable dependence on such deformations. Our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems.Supported in part by GNSMGNFM     

On the logical foundations of the Jauch-Piron approach to quantum physics

We make a critical analysis of the basic concepts of the Jauch-Piron (JP) approach to quantum physics. Then, we exhibit a formalized presentation of the mathematical structure of the JP theory by introducing it as a completely formalized syntactic system, i.e., we construct a formalized languageL _e and formally state the logical-deductive structure of the JP theory by means ofL _e . Finally, we show that the JP syntactic system can be endowed with an intended interpretation, which yields a physical model of the system. A mathematical model endowed with a physical interpretation is given which establishes (in the usual sense of the model theory) the coherence of the JP syntactic system.     

A new approach to relativistic multi-configuration quantum mechanics in titanium

Multiply-ionized atoms in plasmas and astronomical systems are predominantly of intermediate atomic numbers with open electron shells. The spectra seen in laboratory plasmas and astrophysical plasmas are dominated by characteristic Kα1,2 photoemission lines. Modelling these transitions requires advanced relativistic frameworks to begin to formulate solutions. We present a new approach to relativistic multi-configuration determination of Kα1,2 diagram and satellite energies in titanium to a high level of convergence, allowing accurate fitting of satellite contributions and the first agreement with profile to negligible residuals. These developments also apply to exciting frontiers including temporal variation of fundamental constants, theoretical chemistry and laboratory astrophysics.     

Brouwer-Zadeh logic and the operational approach to quantum mechanics

This paper is concerned with a logical system, called Brouwer-Zadeh logic, arising from the BZ poset of all effects of a Hilbert space. In particular, we prove a representation theorem for Brouwer-Zadeh lattices, and we show that Brouwer-Zadeh logic is not characterized by the MacNeille completions of all BZ posets of effects.     

The stochastic quantum mechanics approach to the unification of relativity and quantum theory

The stochastic phase-space solution of the particle localizability problem in relativistic quantum mechanics is reviewed. It leads to relativistically covariant probability measures that give rise to covariant and conserved probability currents. The resulting particle propagators are used in the formulation of stochastic geometries underlying a concept of quantum spacetime that is operationally based on stochastically extended quantum test particles. The epistemological implications of the intrinsic stochasticity of such quantum spacetime frameworks for microcausality, the EPR paradox, etc., are discussed.Supported in part by NSERC Grant A5208.     

A molecular quantum mechanics approach to evaluate molecular properties in liquid phase using statistical mechanics

A method for the calculation of molecular properties in liquid phase is proposed. The theoretical development links in a natural fashion Molecular Quantum Mechanics and Statistical Mechanics. The method can be applied to molecular liquids in thermodynamical equilibrium without strong interactions between molecules. After the Introduction (in which some fundamental ideas are discussed) the model is developed in its simplest form, giving a numerical application of interest for IR spectroscopy in liquid phase. Immediatly, this preliminar version is generalized for pure liquids taking into account the polarization effects on all the molecules of the sample. A numerical example on carbon disulphide shows the convergence of this method and its sensitivity to the structural changes in the liquid. In the last section, some future improvements are suggested and several topics related with the study of molecular liquids are considered.     

Axiomatic foundations of quantum mechanics revisited: The case for systems

We present an axiomatization of nonrelativistic quantum mechanics for a system with an arbitrary number of components. The interpretation of our system of axioms is realistic and objective. The EPR paradox and its relation to realism is discussed in this framework. It is shown that there is no contradiction between realism and recent experimental results.     

Trajectories and causal phase-space approach to relativistic quantum mechanics

We analyze phase-space approaches to relativistic quantum mechanics from the viewpoint of the causal interpretation. In particular, we discuss the canonical phase space associated with stochastic quantization, its relation to Hilbert space, and the Wigner-Moyal formalism. We then consider the nature of Feynman paths, and the problem of nonlocality, and conclude that a perfectly consistent relativistically covariant interpretation of quantum mechanics which retains the notion of particle trajectory is possible.     

A constructive approach to gene expression dynamics

Recently, experiments on mRNA abundance (gene expression) have revealed that gene expression shows a stationary organization described by a scale-free distribution. Here we propose a constructive approach to gene expression dynamics which restores the scale-free exponent and describes the intermediate state dynamics. This approach requires only one assumption: Markov property.