We introduce into classical mechanics the concept of non-discerned particles for particles that are identical, non-interacting and prepared in the same way. The non-discerned particles correspond to an action and a density which satisfy the statistical Hamilton-Jacobi equations and allow to explain the Gibbs paradox in a simple manner. On the other hand, a discerned particle corresponds to a particular action that satisfies the local Hamilton-Jacobi equations. We then study the convergence of quantum mechanics to classical mechanics when hbar -> 0 by considering the convergence for the two cases. These results provide an argument for a renewed interpretation of quantum mechanics.
We introduce into classical mechanics the concept of non-discerned particles for particles that are identical, non-interacting and prepared in the same way. The non-discerned particles correspond to an action and a density which satisfy the statistical Hamilton-Jacobi equations and allow to explain the Gibbs paradox in a simple manner. On the other hand, a discerned particle corresponds to a particular action that satisfies the local Hamilton-Jacobi equations. We then study the convergence of quantum mechanics to classical mechanics when hbar -> 0 by considering the convergence for the two cases. These results provide an argument for a renewed interpretation of quantum mechanics.
We introduce into classical mechanics the concept of nondiscerned particles for particles that are identical, non-interacting and prepared in the same way. The non-discerned particles correspond to an action and a density which satisfy the statistical Hamilton-Jacobi equations and allow to explain the Gibbs paradox. On the other hand, a discerned particle corresponds to a particular action that satisfies the special Hamilton-Jacobi equations. We then study the convergence of quantum mechanics to classical mechanics when ~ tends to 0 by considering two cases : the convergence to non-discerned classical particles and the convergence to a classical discerned particle. Based on these convergences, we propose an updated interpretation of quantum mechanics.
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.
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 to the initial density . In the first case where the initial quantum density is a classical density rho_0(x), 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 density rho_0(x). In the second case where initial density
The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function ψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.
Identical classical particles are distinguishable. This distinguishability affects the number of ways W a macrostate can be realized on the microlevel, and from the relation S=k ln W leads to a nonextensive expression for the entropy. This result is usually considered incorrect because of its inconsistency with thermodynamics. It is sometimes concluded from this inconsistency that identical particles are fundamentally indistinguishable and that quantum mechanics is indispensable for making sense of this inconsistency. In contrast, we argue that the classical statistics of distinguishable particles and the resulting nonextensive entropy function are perfectly acceptable from both a theoretical and an experimental perspective. The inconsistency with thermodynamics can be removed by taking into account that the entropy concept in statistical mechanics is not completely identical to the thermodynamical one. We observe that even identical quantum particles are in some cases distinguishable, and conclude that quantum mechanics is irrelevant to the Gibbs paradox.