Forward–backward semiclassical and quantum trajectory methods for time correlation functions

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

The known methods, due for instance to G.W. Mackey and T.F. Jordan, which exploit the transformation properties with respect to the Euclidean and Galileian group to determine the formalism of the Quantum Theory of a localizable particle, fail in the case that the considered transformations are not symmetries of the physical system. In the present work we show that the formalism of standard Quantum Mechanics for a particle without spin can be completely recovered by exploiting the covariance properties with respect to the group of Euclidean transformations, without requiring that these transformations are symmetries of the physical system.

Quantum mechanical irrebersibility

The real number line has served classical physics well as a model continuum for both space and time. Quantum mechanics preserves this continuum picture of space and time while extending the range of functions on spacetime to complex numbers. The resulting theory is precise, elegant, experimentally verified and universally accepted. However, since the existence of a ‘realistic’ basis for quantum mechanics remains an open question, there is as yet no consensus on a satisfactory interpretation of the theory. The step in the transition from classical to quantum physics that characteristically impedes attempts to construct realistic models is the special extension of functions from R to C. In quantum mechanics this step is formal and algebraic in nature. This talk discusses an alternative in which the geometry of a spacetime path automatically builds the extension from Z to C, with all spacetime functions simply counting contributions of a single path. The model produces the correct continuum propagator in the continuum limit, but the realistic origin of the propagator is only apparent when the smallest scale in the system (Planck scale) remains finite. This type of model is consequently missed by current formulations of quantum mechanics because the continuum limit is assumed at the outset. This suggests that it may be fruitful to look to discrete formulations of quantum mechanics in the quest for an ontological version of the theory.

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

The spacetime approach to quantum mechanics

We present in a rather pedagogical way a new formulation of quantum mechanics. Our starting point is the path integral representation of the quantum-mechanical propagator analytically continued to imaginary timeW(X″, s″|X′, s′). We view the set of random paths contributing toW(X″, s″|X′, s′) as the manifold of solutions of a Langevin equation with a Gaussian white noise. We thus obtainW(X″, s″|X′, s′) as the noise-average of a suitable functional of the solution of the Langevin equation. The standard quantum-mechanical propagator is finally recovered by analytically continuingW(X″, s″|X′, s′) back to real time. The present approach allows for a straightforward application of standard methods of classical stochastic processes to quantum-mechanical problems and offers a new promising way to perform computer simulations of quantum-dynamical systems.

We present a new full-dimensional diabatic potential energy matrix (DPEM) for electronically nonadiabatic collisions of OH(A 2 Σ+ ) with H2 , and we calculate the probabilities of electronically adiabatic inelastic collisions, nonreactive quenching, and reactive quenching to form H2 O+H. The DPEM was fitted using a many-body expansion with permutationally invariant polynomials in bond-order functions to represent the many-body part. The dynamics calculations were carried out with the fewest-switches with time uncertainty and stochastic decoherence (FSTU/SD) semiclassical trajectory method. We present results both for head-on collisions (impact parameter b equal to zero) and for a full range of impact parameters. The results are compared to experiment and to earlier FSTU/SD and quantum dynamics calculations with a previously published DPEM. The various theoretical results all agree that nonreactive quenching dominates reactive quenching, but there are quantitative differences between the two DPEMs and between the b=0 results and the all-b results, especially for the probability of reactive quenching.

2 - The Formalism of Quantum Mechanics

This paper attempts an interpretation of Everett's relative state formulation of quantum mechanics that avoids the commitment to new metaphysical entities like ‘worlds’ or ‘minds’. Starting from Everett's quantum mechanical model of an observer, it is argued that an observer's belief to be in an eigenstate of the measurement (corresponding to the observation of a well-defined measurement outcome) is consistent with the fact that she objectively is in a superposition of such states. “Subjective states” corresponding to such beliefs are constructed. From an analysis of these subjective states and their dynamics it is argued that Everett's pure wave mechanics is subjectively consistent with von Neumann's classical formulation of quantum mechanics. It follows from the argument that the objective state of a system is in principle unobservable. Nevertheless, an adequate concept of empirical reality can be constructed.

The five laws of relativistic quantum mechanics, according to Feynman's path integral formulation, are concisely stated and applied to experiments. Reflection–diffraction-grating experiments for both photons and electrons are analysed, in particular, the Davisson–Germer experiment in which the wave-like property of electrons was first established. It is shown how classical, purely spatial, effective wave theories for both photons and electrons are predicted by the path integral formulation of quantum mechanics. The standard Copenhagen interpretation of wave mechanics is critically discussed in the light of the described experimental applications of the path integral formulation.

A formulation of quantum mechanics, which begins by postulating assertions for individual physical systems, is given. The statistical predictions of quantum mechanics for infinite ensembles are then derived from its assertions for individual systems. A discussion of the meaning of the “state” of an individual quantum mechanical system is given, and an application is made to the clarification of some of the paradoxical features of the theory.

В статье приводятся аргументы в пользу того, что формализм квантовой механики (КМ) может быть использован для моделирования явлений и систем за пределами КМ и даже физики, в частности в социогуманитарной области. Как известно, это действительно имеет место. Примером является квантовоподобное моделирование (КПМ) А. Хренникова. Мы предлагаем философское обоснование КПМ в рамках контекстуального реализма (КР). Применительно к квантовой физике мы говорим о контекстуальном квантовом реализме (ККР). Принципиальными моментами в КР являются чувствительность онтологии к контексту и понимание контекстуальности как неразрывно связанной с нормативностью. Подробное изложение концепции КР можно найти в других наших работах. Здесь мы лишь вкратце резюмируем ее и применяем для понимания оснований и возможностей КПМ. The article argues that the formalism of quantum mechanics (QM) can be used to model phenomena and systems beyond QM and even physics, in particular in the socio-humanitarian field. As is known, this really takes place; A. Khrennikov’s quantum-like modeling (QLM) is an example. We propose a philosophical justification for QLM within the framework of contextual realism (CR). With regard to quantum physics, we speak of contextual quantum realism (CQR). The fundamental points in CR are the sensitivity of ontology to context and the understanding of contextuality as inseparable from normativity. A detailed account of the concept of CR can be found in our other works. Here we only briefly summarize it and apply it to understand the foundations and possibilities of QLM.

Fluctuations in the Dynamics of Single Quantum Systems

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a ‘limiting case’ of the category Hilb packaging quantum mechanics. Following from this assumption, quantum–classical correspondence can be considered as a mapping of the category Hilb to the category Set, i.e., a functor from Hilb to Set, taking place in the macroscopic limit. As a procedure, which takes us from an object of the category Hilb (i.e., a Hilbert space) in the macroscopic limit to an object of the category Set (i.e., a set of values that describe the configuration of a system), this functor must take a finite number of steps in order to make the equivalence of Hilb and Set verifiable. However, as it is shown in the present paper, such a constructivist requirement cannot be met in at least one case of an Ising model of a spin glass. This could mean that it is impossible to demonstrate the emergence of classicality totally from the formalism of standard quantum mechanics.