EPR-Bohr and Quantum Trajectories: Entaglement and Nonlocality

Quantum trajectories are used to investigate the EPR-Bohr debate in a modern sense by examining entanglement and nonlocality. We synthesize a single "entanglement molecule" from the two scattered particles of the EPR experiment. We explicitly investigate the behavior of the entanglement molecule rather than the behaviors of the two scattered particles to gain insight into the EPR-Bohr debate. We develop the entanglement molecule's wave function in polar form and its reduced action, both of which manifest entanglement. We next apply Jacobi's theorem to the reduced action to generate the equation of quantum motion for the entanglement molecule to produce its quantum trajectory. The resultant quantum trajectory manifests entanglement and has retrograde segments interspersed between segments of forward motion. This alternating of forward and retrograde segments generates nonlocality and, within the entanglement molecule, action at a distance. Dissection of the equation of quantum motion for the entanglement molecule, while rendering the classical behavior of the two scattered particles, also reveals an emergent "entanglon" that maintains the entanglement between the scattered particles. The characteristics of the entanglon and its relationship to nonlocality are examined.Comment: 15 pages of LaTeX, 2 figures. PACS Nos. 3.65Ta, 3.65Ca, 3.65Ud. Keywords: EPR, entanglement, nonlocality, determinism, quantum trajectories, action at a distanc

Using Rigorous Ray Tracing to Incorporate Reflection into the Parabolic Approximation

We present a parabolic approximation that incorporates reflection. With this approximation, there is no need to solve the parabolic equation for a coupled pair of solutions consisting of the incident and reflected waves. Rather, this approximation uses a synthetic wave whose spectral components manifest the incident and reflected waves.Comment: 4 pages, LaTeX 2.09. No figures. Key words: ocean acoustics, parabolic approximation, parabolic equation, backscatter, propagatio

OPERA Superluminal Neutrinos per Quantum Trajectories

Quantum trajectories are used to study OPERA findings regarding superluminal neutrinos. As the applicable stationary quantum Klein-Gordon equation is real, real quantum reduced actions and subsequent real quantum trajectories follow. The requirements for superluminal neutrinos are examined. A neutrino that is self-entangled by its own backscatter is shown to have a nonlocal quantum trajectory that may generate a superluminal transit time. Various cases are shown to produce theoretical superluminal neutrinos consistent with OPERA neutrinos. Quantum trajectories are also shown to provide insight into neutrino oscillations.Comment: 11 pages of LaTeX2e with 2 figures embedded. Born weighting functions have been applied to the distribution of quantum trajectories of neutrinos that are entangled by backscatter to render superluminal propagation consistent with OPERA observation. This changes findings. Also minor wordsmithin

Action Variable Quantization, Energy Quantization, and Time Parametrization

The additional information within a Hamilton-Jacobi representation of quantum mechanics is extra, in general, to the Schr\"odinger representation. This additional information specifies the microstate of $\psi$ that is incorporated into the quantum reduced action, $W$. Non-physical solutions of the quantum stationary Hamilton-Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue $J$. Eigenvalues $J$ and $E$ mutually imply each other. Jacobi's theorem generates a microstate-dependent time parametrization $t-\tau=\partial_E W$ even where energy, $E$, and action variable, $J$, are quantized eigenvalues. Substantiating examples are examined in a Hamilton-Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of $W$ is shown to ease numerical and analytic computations. Second, a Hamilton-Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics..Comment: Accepted for publication by "Foundations of Physics". Published on-line. Author's final version. Major modifications to improve precision, focus, organization and clarity of exposition. Figures and Tables unchanged. Open universe assume

Extended Version of "The Philosophy of the Trajectory Representation of Quantum Mechanics"

The philosophy of the trajectory representation is contrasted with the Copenhagen and Bohmian philosophies.Comment: 14 pages, LaTeX 2.09. No figures. This is an extended version of "The Philosophy of the Trajectory Representation" which is to appear in the proceedings for Vigier 2000 Symposium, Berkeley, California, USA, 21-25 August 200

Comments on Bouda and Djama's "Quantum Newton's law"

Discussion of the differences between the trajectory representation of Floyd and that of Bouda and Djama [Phys. Lett. A 285 (2001) 27, quant-ph/0103071] renders insight: while Floyd's trajectories are related to group velocities, Bouda and Djama's are not. Bouda and Djama's reasons for these differences are also addressed.Comment: 6 pages LaTeX 2e. No figures. Bouda and Djama's "Quantum Newton's law" has been published in Phys. Lett. A 285 (2001) 27, quant-ph/0103071. Bouda and Djama are submitting to quant-ph a rebuttal, which also has been published in Phys. Lett. A 296 (2002) 312-31

Neutrino Oscillations with Nil Mass

An alternative neutrino oscillation process is presented as a counterexample for which the neutrino may have nil mass consistent with the standard model. The process is developed in a quantum trajectories representation of quantum mechanics, which has a Hamilton-Jacobi foundation. This process has no need for mass differences between mass eigenstates. Flavor oscillations and $\bar{\nu},\nu$ oscillations are examined.Comment: Author's version. In press. Accepted by "Foundations of Physics

Differences between the trajectory representation and Copenhagen regarding the past and present in quantum theory

We examine certain pasts and presents in the classically forbidden region. We show that for a given past the trajectory representation does not permit some presents while the Copenhagen predicts a finite probability for these presents to exist. This suggests another gedanken experiment to invalidate either Copenhagen or the trajectory representation.Comment: 4 pages REVTEX4. No figures. Submitted to "Proceedings for the Eighth International Wigner Symposium" (WYGSYM 8), 27-30 May 2003, CUNY, NY, N

Classical Limit of the Trajectory Representation of Quantum Mechanics, Loss of Information and Residual Indeterminacy

The trajectory representation in the classical limit (\hbar \to 0) manifests a residual indeterminacy. We show that the trajectory representation in the classical limit goes to neither classical mechanics (Planck's correspondence principle) nor statistical mechanics. This residual indeterminacy is contrasted to Heisenberg uncertainty. We discuss the relationship between indeterminacy and 't Hooft's information loss and equivalence classes.Comment: 12 pages LaTeX 2.09. No figures. Accepted by Int. J. Mod. Phys. A. Minor revisions to conform with galley proofs. Acknowledgements expanded. References updated. Key words: classical limits, trajectory interpretation, Planck's correspondence principle, residual indeterminacy, 't Hooft's information loss and equivalence classes, Heisenberg uncertainty principle. Subj-clas: Quantum Physics; Mathematical Physic

Where and why the generalized Hamilton-Jacobi representation describes microstates of the Schr\"odinger wave function

A generalized Hamilton-Jacobi representation describes microstates of the Schr\"odinger wave function for bound states. At the very points that boundary values are applied to the bound state Schr\"odinger wave function, the generalized Hamilton-Jacobi equation for quantum mechanics exhibits a nodal singularity. For initial value problems, the two representations are equivalent.Comment: 6 pages, LaTeX 2.0