The physical interpretation of wave mechanics. I

Summarizing and extending the ideas of many authors and also of our own work, we try to show that the wave equation of the one-body problem can be transformed into a system of equations describing the motion of a deformable medium carrying charge and having permanent magnetic polarization. The wave equation and the system of transformed equations are connected by a strict one-to-one correspondence. The transformation which is not uniquely determined from a mathematical point of view can be chosen so that the new variables have directphysical significance. The latter requirement permits consequently a unique choice of the mathematical possibilities.     

The physical interpretation of wave mechanics. II

Continuing the considerations given in the first part of this series (I), we use the analysis of the Aharonov-Bohm effect to show that the hydrodynamical variables by which the quantum mechanical one-body problem can be represented are of direct physical significance. It is shown in a particular case that the final state of a system can be obtained from its initial state in a unique manner if both states are characterized by hydrodynamical variables.     

The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics

Letters in Mathematical Physics - Recently Ohsawa (Lett Math Phys 105:1301–1320, 2015) has studied the Marsden–Weinstein–Meyer quotient of the manifold $$T^*\mathbb {R}^n\times...     

Cosmology and the pilot wave interpretation of quantum mechanics

Bell has recently revived the pilot wave interpretation of de Broglie and Bohm as a possible scheme for interpreting wave functions in quantum cosmology. I argue that the pilot wave interpretation cannot be applied consistently to systems whose wave functions split into macroscopically distinguishable states. At some stage the pilot wave interpretation must either tacitly invoke wave function reduction in the same manner as the Copenhagen interpretation, or else abandon locality by requiring physical particles to move faster than light. Consequently, the many-worlds interpretation is the only known realist interpretation of the quantum mechanical formalism which can be extended to quantum cosmology.     

On the semiclassical interpretation of heavy ion reactions

The semiclassical interpretation of heavy ion reactions is discussed. It is found that deflection to negative angles is inconsistent with optical model calculations. Due to the uncertainty principle trajectories are not well defined if the reaction is localized at the nuclear surface. It is, therefore, suggested that quantum diffraction is the dominating physical process for oscillating heavy ion transfer reactions.     

De Broglie's wave particle duality in the stochastic interpretation of quantum mechanics: A testable physical assumption

If one starts from de Broglie's basic relativistic assumptions, i.e., that all particles have an intrinsic real internal vibration in their rest frame, i.e., hv ₀ =m ₀ c ² ; that when they are at any one point in space-time the phase of this vibration cannot depend on the choice of the reference frame, then, one can show (following Mackinnon ⁽¹⁾ ) that there exists a nondispersive wave packet of de Broglie's waves which can be assimilated to the nonlinear soliton wave U ₀ introduced by him in his double solution model of wave mechanics. ⁽²⁾ Since de Broglie's linear pilot waves can be considered to be real waves propagating as collective motions on a covariant subquantum chaotic aether, ⁽³⁾ these new solition waves can be considered as describing the particle's immediate neighborhood, i.e., the aether's reaction to the particle's motion in the stochastic interpretation of quantum mechanics. The existence of such a physical aether (which provides a perfectly causal interpretation of the action-a-distance implied by the Einstein-Podolsky-Rosen experiments) can now be proved by establishing the reality of de Broglie's waves in realizable experiments.     

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     

Spreading of atomic wave packets and semiclassical chaos

The correspondence between the statistical properties of the evolution of a quantum system and Lyapunov instability and the chaos of its semiclassical analog has been demonstrated. The results of the analyses of atomic motion in a laser field in the semiclassical approximation (dynamics is described by several nonlinear equations) and without this approximation (dynamics is described by an infinite system of linear equations) are compared. In the ranges of the parameters for which the semiclassical dynamics of point-like atoms is unstable, the fast “spreading” of quantized wave packets in the momentum space is observed. Thus, deterministic chaos “imitates” the statistics of the quantum nondeterministic effects, although the semiclassical and quantum solutions are fundamentally different.     

The evolution of the ideas of Louis de Broglie on the interpretation of wave mechanics

This paper is devoted to an analysis of the intellectual itinerary of Louis de Broglie, from the discovery of wave mechanics, until today. Essential attention is paid to the fact that this itinerary is far from being linear, since after a first attempt to develop his own views on wave mechanics through the theory of singular waves, Louis de Broglie abandoned it for twenty five years, under the influence of the Copenhagen School (even embracing the conceptions of the latter), until the beginning of the fifties, when he definitively came back to his primary theory. This evolution of the Louis de Broglie's views on wave mechanics is told here and explained through an analysis of the evolution of all of quantum mechanics and, more generally, the dominating conceptions of theoretical physics in our century.This paper is written in a quite personal form, which is not exactly one to which the readers of scientific journals are accustomed, because it reproduces, in fact, the preface of a book (to be published) of Louis de Broglie, which is precisely devoted to the fundamental problems of quantum mechanics and closely linked to the second turnabout of the author.     

Spreading of atomic wave packets and semiclassical chaos

JETP Letters - The correspondence between the statistical properties of the evolution of a quantum system and Lyapunov instability and the chaos of its semiclassical analog has been demonstrated....     

Nonlinear wave mechanics

Nonlinear wave mechanics is constructed, based on Schrödinger-type equation with nonlinearity −bψ ln | ψ |². This nonlinearity is selected by assuming the factorization of wavefunctions for composed systems. Its most attractive features are: existence of the lower energy bound and validity of Planck's relation . In any number of dimensions, soliton-like solutions (gaussons) of our equation exist and move in slowly varying fields like classical particles. The Born interpretation of the wavefunction is consistent with logarithmic nonlinearity and we tentatively estimate the order of magnitude of the universal constant b.     

The zitterbewegung interpretation of quantum mechanics

Thezitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac theory shows that thezitterbewegung need not be attributed to interference between positive and negative energy states as originally proposed by Schroedinger. Rather, it provides a physical interpretation for the complex phase factor in the Dirac wave function generally. Moreover, it extends to a coherent physical interpretation of the entire Dirac theory, and it implies azitterbewegung interpretation for the Schroedinger theory as well.     

The Ithaca interpretation of quantum mechanics

I list several strong requirements for what I would consider a sensible interpretation of quantum mechanics and discuss two simple recent theorems which have important implications for such an interpretation. My talk will not clear everything up; indeed, you may conclude that it has not cleared anything up. But I hope it will provide a different perspective from which to view some old and vexing puzzles (or, if you believe nothing needs to be cleared up, some ancient verities.)