We present a new interpretation of quantum mechanics, called the double-scale theory, which expends on the de Broglie-Bohm (dBB) theory. It is based, for any quantum system, on the simultaneous existence of two wave functions in the laboratory reference frame : an external wavefunction and an internal one. The external wave function corresponds to a field that pilots the center-of-mass of the quantum system. The external wave spreads out in space over time. Mathematically, the Schrödinger equation converges to the Hamilton-Jacobi statistical equations when the Planck constant tends towards zero and the Newton trajectories are therefore approximations of the dBB trajectories. The internal wave function corresponds to the interpretation proposed by Edwin Schrödinger for whom the particle is extended. Then, the internal wave remains confined in space. Its converges, when h -> 0, to a Dirac distribution. Furthermore, we show that non-stationary solutions can exist such that the 3N-dimensional configuration space of the internal wave function can be rewritten as the product of N individual 3-dimensional internal wave functions.
We present a semiclassical gravity in the framework of the double scale theory, a new interpretation of quantum mechanics which expands on the de Broglie-Bohm (dBB) theory. In this interpretation, any quantum system is associated with two wave functions, one external and one internal. The external wave function drives the center-of-mass, as in the dBB theory, and spreads with time. The internal wave function corresponds to the density of an extended particle. It remains confined in space and its average position corresponds to the center of mass of the particle. We define, for an N-body system, new semi-classical gravity equations called Einstein-de Broglie equations. The external wave function of each body depends only on the internal wave functions of the N--1 others. They are deduced from Einstein's semi-classical equation by using the internal wave functions instead of the usual wave function. If we restrict to the dBB interpretation, we find the Newton-de Broglie equations that Struyve and Lalo{ë} have recently proposed independently. We show that the Newton-de Broglie and Einstein-de Broglie equations converge to Newton's gravity when h
→ 0, which gives a theoretical validation.
We propose an interpretative framework for quantum mechanics inspired by 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 along with other macroscopic degrees of freedom, and an internal wave function corresponding to the evolution of its internal variables in the center-of-mass system. 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 de Broglie pilot wave. For the internal wave function, we argue in favor of the interpretation proposed by Erwin Schrödinger at the Solvay Congress in 1927: 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 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.
We define the de Broglie-Bohm (dBB) weak interpretation as the dBB interpretation restricted to particles in unbound states whose wave function is defined in the three-dimensional physical space, and the dBB strong interpretation as the usual dBB interpretation applied to all wave functions, in particular to particles in bound states whose wave function is defined in a 3N-dimensional configuration space in which N is the number of particules. We show that the current criticisms of the dBB interpretation do not apply to this weak interpretation and that, furthermore, there are theoritical and experimental reasons to justify the weak dBB interpretation. Theoretically, the main reason concern the continuity existing for such particles between quantum mechanics and classical mechanics: we demonstrate in fact that the density and the phase of the wave function of a single-particle (or a set of identical particles without interaction), when the Planck constant tends to 0, converges to the density and the action of a set of unrecognizable prepared classical particles that satisfy the statistical Hamilton-Jacobi equations. As the Hamilton-Jacobi action pilots the particle in classical mechanics, this continuity naturally concurs with the weak dBB interpretation. Experimentally, we show that the measurement results of the main quantum experiments (Young's slits experiment, Stern and Gerlach, EPR-B) are compatible with the de Broglie-Bohm weak interpretation and everything takes place as if these unbounded particles had trajectories. In addition, we propose two potential solutions to complete the dBB weak interpretation.
In the 1927 Solvay conference, three apparently irreconcilable interpretations of the quantum mechanics wave function were presented: the pilot-wave interpretation by de Broglie, the soliton wave interpretation by Schrödinger and the Born statistical rule by Born and Heisenberg. In this paper, we demonstrate the complementarity of these interpretations corresponding to quantum systems that are prepared differently and we deduce a synthetic interpretation: the double-preparation theory. We first introduce in quantum mechanics the concept of semi-classical statistically prepared particles, and we show that in the Schrödinger equation these particles converge, when , to the equations of a statistical set of classical particles. These classical particles are undiscerned, and if we assume continuity between classical mechanics and quantum mechanics, we conclude the necessity of the de Broglie–Bohm interpretation for the semi-classical statistically prepared particles (statistical wave). We then introduce in quantum mechanics the concept of a semi-classical deterministically prepared particle, and we show that in the Schrödinger equation this particle converges, when , to the equations of a single classical particle. This classical particle is discerned and assuming continuity between classical mechanics and quantum mechanics, we conclude the necessity of the Schrödinger interpretation for the semi-classical deterministically prepared particle (the soliton wave). Finally we propose, in the semi-classical approximation, a new interpretation of quantum mechanics, the 'theory of the double preparation', which depends on the preparation of the particles.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function
ψ. Given a Schrodinger wave function
ψ, we use its value
ψ0 at some fixed time (say,
t=0) to define the probability measure
∣ψ0∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process
ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the
ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.