An attempt is made to show that it is possible to constnuct a theory of particle systems in accord with the basic ideas of the causal interpretation of wave mechanics by the theory of the double solution. After a discussion of the theory of the individual particle, the de Broglie and Vigier theories on the causal interpretation of the theory of systems are examined. It is then shown that it is possible to reframe the analytical mechanics of systems and, in particular, the Hamilton-Jacobi theory in a space with three dimensions. The question of the form of the basic equations in physical space is discussed in detail. The statistical theory of systems of particles of different nature is explained. (J.S.R.)
An attempt is made to show that it is possible to constnuct a theory of particle systems in accord with the basic ideas of the causal interpretation of wave mechanics by the theory of the double solution. After a discussion of the theory of the individual particle, the de Broglie and Vigier theories on the causal interpretation of the theory of systems are examined. It is then shown that it is possible to reframe the analytical mechanics of systems and, in particular, the Hamilton-Jacobi theory in a space with three dimensions. The question of the form of the basic equations in physical space is discussed in detail. The statistical theory of systems of particles of different nature is explained. (J.S.R.)
First, we show that there exists in classical mechanics three actions corresponding to different boundary conditions: two well-known actions, the Euler-Lagrange classical action S_cl(x,t;x_0), which links the initial position x_0 and its position x at time t, the Hamilton-Jacobi action S(x,t), which links a family of particles of initial action S_0(x) to their various positions x at time t, and a new action, the deterministic action S(x,t;x_0,v_0), which links a particle in initial position x_0 and initial velocity v_0 to its position x at time t. We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density. In the first case, where the initial quantum density is a classical density, the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the initial density. In the second case where initial density converges to a Dirac density, the density converges to the Dirac function and the quantum action converges to a deterministic action. Therefore we introduce into classical mechanics non-discerned particles, which satisfy the statistical Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned particles, which satisfy the deterministic Hamilton-Jacobi equations. Finally, we propose an interpretation of the Schrodinger wave function that depends on the initial conditions (preparation). This double interpretation seems to be the interpretation of Louis de Broglie's "double solution" idea.
Pursuing the Hamiltonian formulation of the de Broglie-Bohm (deBB) theory presented in the preceding paper, the Hamilton-Jacobi (HJ) theory of the wave-particle system is developed. It is shown how to derive a HJ equationfor the particle, which enables trajectories to be computed algebraically using Jacobi's method. Using Liouville's equation in the HJ representation we find the restriction on the Jacobi solutions which implies the quantal distribution. This gives a first method for interpreting the deBB theory in HJ terms. A second method proceeds via an explicit solution of the field+particle HJ equation. Both methods imply that the quantum phase may be interpreted as an incomplete integral. Using these results and those of the first paper it is shown how Schrodinger's equation can be represented in Liouvillian terms, and vice versa. The general theory of canonical transformations that represent quantum unitary transformations is given, and it is shown in principle how the trajectory theory may be expressed in other quantum representations. Using the solution found for the total HJ equation, an explicit solution for the additional field containing a term representing the particle back-reaction is found. The conservation of energy and momentum in the model is established, and a weak form of the action-reaction principle is shown to hold. Alternative forms for the Hamiltonian are explored and it is shown that, within this theoretical context, the deBB theory is not unique. The theory potentially provides an alternative way of obtaining the classical limit.
We propose an interpretative framework for quantum mechanics corresponding to the specifications of Louis de Broglie's double-solution theory. The principle is to decompose the evolution of a quantum system into two wave functions: an external wave function corresponding to the evolution of its center of mass and an internal wave function corresponding to the evolution of its internal variables in the center-of-mass system. Mathematical decomposition is only possible in certain cases because there are many interactions linking these two parts. In addition, these two wave functions will have different meanings and interpretations. The external wave function "pilots" the center of mass of the quantum system: it corresponds to the Broglie pilot wave. When the Planck constant tends to zero, it results mathematically from the convergence of the square of the module and the phase of the external wave function to a density and a classical action verifying the Hamilton-Jacobi statistical equations. This interpretation explains all the measurement results, namely those yielded by interference, spin measurement (Stern and Gerlach) and non-locality (EPR-B) experiments. For the internal wave function, several interpretations are possible : the one of the pilot wave can be applied in cascade to the internal wave function. However, the interpretation proposed by Erwin Schr{ö}dinger at the Solvay Congress in 1927 and restricted to the internal wave function is also possible. For Schr{ö}dinger, the particles are extended and the square of the module of the (internal) wave function of an electron corresponds to the density of its charge in space. We present many arguments in favour of this interpretation, which like the pilot wave interpretation is realistic and deterministic. Finally, we will see that this double interpretation serves as a frame of reference by which to better understand the debates on the interpretation of quantum mechanics and to review the relationships between gravity and quantum mechanics.
This paper and a following one develop a Hamiltonian formulation of the de Broglie-Bohm (deBB) interpretation of quantum mechanics. Drawing upon a unified Hamiltonian treatment of a classical particle and its associated ensemble, it is shown how the interaction of a particle and a spin-0 quantum wave can be consistently expressed in a canonical framework. A formulation is found in which the particle is acted upon by a force determined by the quantum potential but the particle does not react on the Schrodinger field. This requires the introduction of an additional field which the particle does act upon, and allows the deBB theory to be expressed in Hamiltonian terms. An ensemble theory for the wave-particle system based on Liouville's equation is developed, and it is shown that for a pure quantum state the particle obeys its own Liouville equation. The general particle phase space distribution that is compatible with ‖ψ‖ 2 is derived. A special case of this corresponds to the relation p = ∂S/∂q (and thus the guidance law of the deBB theory) but more general distributions, and hence non-deBB motions, are possible. This relation is interpreted as a constraint on the phase space of the composite system. Using the theory of Hamiltonian constraints it is shown that the first-order guidance law and Hamilton's second-order particle equations are dynamically equivalent.
The Copenhagen interpretation has been the subject of much criticism, notably by De Broglie and Einstein, because it contradicts the principles of causality and realism. The aim of this essay is to study the wave mechanics as an alternative to traditional quantum mechanics, in the continuity of the ideas of Louis de Broglie: the pilot wave theory of De Broglie (where each particle is associated with a wave which guides it), De Broglie-Bohm theory, stochastic electrodynamics (where the stochastic character of particles is caused by the energy field of the fluctuating vacuum), and the analogies between quantum mechanics and hydrodynamics.
The de Broglie–Bohm interpretation (Bohmian mechanics, hidden-variables theory, causal interpretation
or pilot-wave theory) is one of the interpretations of quantum mechanics which opposes
to the claim of standard interpretation on the completeness of description of the state of quantummechanical
system, that description in Bohmian mechanics is completed by defining positions of
the particles of the system i.e. hidden variables and their dynamics. These facts are the basis of motivation
for the introduction of a new theory that can describe dubious axiomatic requirements of
standard interpretation. The presented theory is deterministic and explicitly nonlocal. Originators
of Bohmian mechanics were L. de Broglie (1927.) and D. Bohm (1952.) who presented mathematically
more general and conceptual elaborate formulation. Along with the historical overview,
a note is given about the main properties of the theory, such as the issue of nonlocality and hidden
variables. The aim of this paper is to present and explain the main ideas and properties of the
theory as well as to give a systematic overview of present and historical mathematical construction
of the theory and most of all to discuss implications of de Broglie–Bohm theory on the ontology of
quantum-mechanical picture, showcasing examples of the equivalency of the description of a free
particle as a propagation of the plane wave in space or as the ensemble of the trajectories and of
the spreading of the Gaussian wave packet. The main example of the obvious conceptual clashing
of the interpretations is presented: the double-slit experiment, with a detailed examination of the
Bohmian mechanics’ predictions using presented formalism on the two double-slit experiment.
The graphs of trajectories for given examples are obtained.