The high energy limit of the trajectory representation of quantum mechanics

The trajectory representation in the high energy limit (Bohr correspondence principle) manifests a residual indeterminacy. This indeterminacy is compared to the indeterminacy found in the classical limit (Planck's constant to 0) [Int. J. Mod. Phys. A 15, 1363 (2000)] for particles in the classically allowed region, the classically forbiden region, and near the WKB turning point. The differences between Bohr's and Planck's principles for the trajectory representation are compared with the differences between these correspondence principles for the wave representation. The trajectory representation in the high energy limit is shown to go to neither classical nor statistical mechanics. The residual indeterminacy is contrasted to Heisenberg uncertainty. The relationship between indeterminacy and 't Hooft's information loss and equivalence classes is investigated.Comment: 12 pages of LaTeX. No figures. Incorporated into the "Proceedings of the Seventh International Wigner Symposium" (ed. M. E. Noz), 24-29 August 2001, U. of Maryland. Proceedings available at http://www.physics.umd.edu/robo

The Equivalence Postulate of Quantum Mechanics

The Equivalence Principle (EP), stating that all physical systems are connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). Trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transform and of its recently observed relation with second-order linear differential equations. This reflects in an intrinsic psi^D-psi duality between linearly independent solutions of the Schroedinger equation. Unlike Bohm's theory, there is a non-trivial action even for bound states. No use of any axiomatic interpretation of the wave-function is made. Tunnelling is a direct consequence of the quantum potential which differs from the usual one and plays the role of particle's self-energy. The QSHJE is defined only if the ratio psi^D/psi is a local self-homeomorphism of the extended real line. This is an important feature as the L^2 condition, which in the usual formulation is a consequence of the axiomatic interpretation of the wave-function, directly follows as a basic theorem which only uses the geometrical gluing conditions of psi^D/psi at q=\pm\infty as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunnelling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the non-stationary higher dimensional quantum HJ equation and the relativistic extension are derived.Comment: 1+3+140 pages, LaTeX. Invariance of the wave-function under the action of SL(2,R) subgroups acting on the reduced action explicitly reveals that the wave-function describes only equivalence classes of Planck length deterministic physics. New derivation of the Schwarzian derivative from the cocycle condition. "Legendre brackets" introduced to further make "Legendre duality" manifest. Introduction now contains examples and provides a short pedagogical review. Clarifications, conclusions, ackn. and references adde

Interference, reduced action, and trajectories

Instead of investigating the interference between two stationary, rectilinear wave functions in a trajectory representation by examining the two rectilinear wave functions individually, we examine a dichromatic wave function that is synthesized from the two interfering wave functions. The physics of interference is contained in the reduced action for the dichromatic wave function. As this reduced action is a generator of the motion for the dichromatic wave function, it determines the dichromatic wave function's trajectory. The quantum effective mass renders insight into the behavior of the trajectory. The trajectory in turn renders insight into quantum nonlocality.Comment: 12 pages text, 5 figures. Typos corrected. Author's final submission. A companion paper to "Welcher Weg? A trajectory representation of a quantum Young's diffraction experiment", quant-ph/0605121. Keywords: interference, nonlocality, trajectory representation, entanglement, dwell time, determinis

OPERA data and The Equivalence Postulate of Quantum Mechanics

An interpretation of the recent results reported by the OPERA collaboration is that neutrinos propagation in vacuum exceeds the speed of light. It has been further been suggested that this interpretation can be attributed to the variation of the particle speed arising from the Relativistic Quantum Hamilton Jacobi Equation. I show that this is in general not the case. I derive an expression for the quantum correction to the instantaneous relativistic velocity in the framework of the relativistic quantum Hamilton-Jacobi equation, which is derived from the equivalence postulate of quantum mechanics. While the quantum correction does indicate deviations from the classical energy--momentum relation, it does not necessarily lead to superluminal speeds. The quantum correction found herein has a non-trivial dependence on the energy and mass of the particle, as well as on distance travelled. I speculate on other possible observational consequences of the equivalence postulate approach.Comment: 8 pages. Standard LaTex. References adde

From a Mechanical Lagrangian to the Schr\"odinger Equation. A Modified Version of the Quantum Newton's Law

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a non-relativistic spinless system. This Lagrangian is written as a difference between a function $T$, which represents the quantum generalization of the kinetic energy and which depends on the coordinate $x$ and the temporal derivatives of $x$ up the third order, and the classical potential $V(x)$. The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function $T$ is first assumed arbitrary. The development of $T$ in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton-Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton's law. We also analytically establish the famous Bohm's relation % $\mu \dot{x} = \partial S_0 /\partial x$ % outside of the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, it plays really a role of an additional potential as assumed by Bohm.Comment: 20 pages, LateX, no figure, some calculations are reported in appendice

Trajectories in the Context of the Quantum Newton's Law

In this paper, we apply the one dimensional quantum law of motion, that we recently formulated in the context of the trajectory representation of quantum mechanics, to the constant potential, the linear potential and the harmonic oscillator. In the classically allowed regions, we show that to each classical trajectory there is a family of quantum trajectories which all pass through some points constituting nodes and belonging to the classical trajectory. We also discuss the generalization to any potential and give a new definition for de Broglie's wavelength in such a way as to link it with the length separating adjacent nodes. In particular, we show how quantum trajectories have as a limit when $\hbar \to 0$ the classical ones. In the classically forbidden regions, the nodal structure of the trajectories is lost and the particle velocity rapidly diverges.Comment: 17 pages, LateX, 6 eps figures, minor modifications, Title changed, to appear in Physica Script