Classical particle dynamics in quantum space

It is suggested that if space-time is quantized at small distances, then even at the classical level particle motion in space is complicated and described by a nonlinear equation. In the quantum space the Lagrangian function or energy of the particle consists of two parts: the usual kinetic terms, and a rotation term determined by the square of the inner angular momentum-a torsion torque caused by the quantum nature of space. Rotational energy and rotational motion of the particle disappear in the limitl0, wherel the value of the fundamental length. In the free particle case, in addition to the rectilinear motion, the particle undergoes a rotation given by the inner angular momentum. Different possible types of particle motion are discussed. Thus, the scheme may shed light on the appearance of rotating or twisting, stochastic, and turbulent types of motion in classical physics and, perhaps, on the notion of spin in quantum physics within the framework of the quantum character of space-time at small distances.     

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inR ^v is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.     

Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the “Minkowski vacuum” for a box at rest in an inertial frame and a “Rindler vacuum” for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. The classical analysis gives no grounds for the “heating effects of acceleration through the vacuum” which appear in the literature of quantum field theory. Thus this distinction provides (in principle) an experimental test to distinguish the two theories.     

A new approach to the Efinger model for a nonlinear quantum theory for gravitating particles

A general solution is obtained for a model of a nonlinear quantum theory for gravitating particles proposed by H. Efinger. The solution procedure is easily generalized to space-time or stochastic formulations.     

Quantum conformal fluctuations near the classical space-time singularity

This paper investigates the behavior of conformal fluctuations of space-time geometry that are admissible under the quantized version of Einstein's general relativity. The approach to quantum gravity is via path integrals. It is shown that considerable simplification results when only the conformal degrees of freedom are considered under this scheme, so much so that it is possible to write down a formal kernel in the most general case where the space-time contains arbitrary distributions of particles with no other interaction except gravity. The behavior of this kernel near the classical space-time singularity then shows that quantum fluctuations inevitably diverge near the singularity. It is shown further that the root cause of this divergence lies in the fact that the Green's function for the conformally invariant scalar wave equation diverges at the singularity. The limitations on the validity of classical general relativity near the space-time singularity are discussed and it is argued that the notion of singularity itself needs to be radically modified once the quantum effects are taken into account.On leave of absence from the Tata Institute of Fundamental Research, Bombay, India     

Quantum relativity theory and quantum space-time

A quantum relativity theory formulated in terms of Davis' quantum relativity principle is outlined. The first task in this theory as in classical relativity theory is to model space-time, the arena of natural processes. It is shown that the quantum space-time models of Banai introduced in another paper is formulated in terms of Davis' quantum relativity. The recently proposed classical relativistic quantum theory of Prugoveki and his corresponding classical relativistic quantum model of space-time open the way to introduce, in a consistent way, the quantum space-time model (the quantum substitute of Minkowski space) of Banai proposed in the paper mentioned. The goal of quantum mechanics of quantum relativistic particles living in this model of space-time is to predict the rest mass system properties of classically relativistic (massive) quantum particles (elementary particles). The main new aspect of this quantum mechanics is that provides a true mass eigenvalue problem, and that the excited mass states of quantum relativistic particles can be interpreted as elementary particles. The question of field theory over quantum relativistic model of space-time is also discussed. Finally it is suggested that quarks should be considered as quantum relativistic particles.Supported by the Hungarian Academy of Sciences.     

Geometric phases and the informational completeness of quantum frames

The basic results on geometric phases are rederived by using infinite dimensional coordinate charts in line bundles, in Hopf bundles, and in projective Hilbert spaces. The determination of a quantum state can be then geometrically described as the measurement of Fubini-Study distances from that state to the elements of informationally complete quantum frames. The basic geometric features of such quantum frames are formulated, and their relationships to corresponding classical frames are analyzed.     

On the Feynman proof of the exact relation between a class of path sums and the general nonlinear Fokker-Planck equation

A simple sum over paths will be considered for the general nonlinear diffusion process described by complex coordinates. In order to derive the corresponding stochastic differential equation the Feynman method used in quantum mechanics will be generalised for the present case of a coordinate dependent variance or diffusion function by means of a nonlinear coordinate transformation. The resulting equation will be seen to be the general nonlinear Fokker-Planck equation. The relevance of the present formulation for nonequilibrium phenomena, such as for example those occuring in nonlinear quantum optics, will be discussed.     

Stochastic microcausality in relativistic quantum mechanics

A recently formulated concept of stochastic localizability is shown to be consistent with a concept of stochastic microcausality, which avoids the conclusions of Hegerfeldt's no-go theorem as to the inconsistency of sharp localizability of quantum particles and Einstein causality. The proposed localizability on quantum space-time is shown to lead to strict asymptotic causality. For finite time evolutions, upper bounds on propagation to the exterior of stochastic light cones are derived which show that the resulting probabilities are too small to be actually observable in a realistic context.Supported by an NSERC Fellowship.Suported in part by NSERC research grant No. A5206.     

Symplectic structure of quantum phase and stochastic simulation of qubits

Starting with the projective interpretation of the Hilbert space special stochastic representation of the wave function in Quantum Mechanics (QM), based on soliton realization of extended particles, is considered with the aim to model quantum states via classical computer. Entangled solitons construction having been earlier introduced in the nonlinear spinor field model for the calculation of the Einstein-Podolsky-Rosen (EPR) spin correlation for the spin-1/2 particles in the singlet state, the other example is now studied. The latter concerns the entangled envelope solitons in Kerr dielectric with cubic nonlinearity, where we use two-solitons configurations for modeling the entangled states of photons. Finally, the concept of stochastic qubits is used for quantum computing modeling.     

The vanishing likelihood of space-time singularity in quantum conformal cosmology

A general formalism is developed for studying the behavior of quantized conformal fluctuations near the space-time singularity of classical relativistic cosmology. It is shown that if the material contents of space-time are made of massive particles which obey the principle of asymptotic freedom and interact only gravitationally, then it is possible to estimate the quantum mechanical probability that, of the various possible conformal transforms of the classical Einstein solution, the actual model had a singularity in the past. This probability turns out to be vanishingly small, thus indicating that within the regime of quantum conformal cosmology it is extremely unlikely that the universe originated out of a space-time singularity.     

Quantum mechanics in noninertial reference frames: Relativistic accelerations and fictitious forces

One-particle systems in relativistically accelerating reference frames can be associated with a class of unitary representations of the group of arbitrary coordinate transformations, an extension of the Wigner–Bargmann definition of particles as the physical realization of unitary irreducible representations of the Poincaré group. Representations of the group of arbitrary coordinate transformations become necessary to define unitary operators implementing relativistic acceleration transformations in quantum theory because, unlike in the Galilean case, the relativistic acceleration transformations do not themselves form a group. The momentum operators that follow from these representations show how the fictitious forces in noninertial reference frames are generated in quantum theory.     

Semi-classical quantum mechanics and stochastic calculus of variations

Within the framework of the general theory of stochastic calculus of variations, we examine mainly the notion of second variation in the stochastic mechanics of E. Nelson, a representative of quantum mechanics in which the concept of path for particles keep a sense. We show that the two approaches used in classical calculus of variation to know if a path is not only an extremum but also the minimum of the action, namely, the local one (weak minimum) and the global one (strong minimum), can be generalized to include the quantum-mechanical paths. Thus, we can prove that locally, a solution of the classical equation of motion is really the minimum, even in a large class of quantum paths containing the semi-classical trajectories. By introducing a stochastic version of the excess function of Weierstrass, we show the analogous global property. There, of course, one can speak of the principle of least action in a strict sense. Several explicit examples are discussed.     

Center-of-mass tomographic approach to quantum dynamics

In the tomography representation we propose a new approach, which describes the dynamics of quantum particles by the Kolmogorov equations for non-negative propagators. To solve the Kolmogorov equations we use a diffusive Markovian random processes described by the related nonlinear stochastic Langevin equations. As a result the dynamics of quantum particles is described by the proposed numerical scheme combining both Langevin dynamics and Monte Carlo methods. We test the developed approach by applying it to the wave packet dynamics in harmonic potentials and to particle tunneling through a barrier.     

A Relativistic Version of the Ghirardi–Rimini–Weber Model

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times.     

Electromagnetic Field in Finsler and Associated Spaces

The electromagnetic field and its interaction with the leptons is introduced in Finsler space. This space is also considered as the microlocal space-time of the extended hadrons. The field equations for the Finsler space have been obtained from the classical field equations by quantum generalization of this space-time below a fundamental length-scale. On the other hand, the classical field equations are derived from a property of the fields on the autoparallel curve of the Finsler space. The field equations for the associated spaces of the Finsler space, which are macroscopic spaces, such as the large-scale space-time of the universe and the usual Minkowski space-time, can also be obtained for the case of Finslerian bispinor fields separable as the direct products of fields depending on the position coordinates with those depending on the directional arguments. The equations for the coordinate-dependent fields are the usual field equations with the cosmic time-dependent masses of the leptons. The other equations of the directional variable-dependent fields are solved here. Also, the lepton current and the continuity equation are considered. The form-invariance of the field equations under the general coordinate transformations of the Finsler spaces has been discussed.