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arXiv:1006.1986

Superluminal velocities and nonlocality in relativistic mechanics with scalar potential

Hrvoje Nikolić

Published Jun 10, 2010• 3 citations
arXivHTMLPDF

Abstract

Even though the usual form of relativistic mechanics does not allow superluminal particle velocities and nonlocal interactions, these features are not forbidden by relativity itself. To understand this on a deeper level, we study a generalized form of relativistic mechanics in which the particle is influenced not only by the usual tensor (gravitational) and vector (electromagnetic) potentials, but also by the scalar potential. The scalar potential promotes the mass squared M^2 to a dynamical quantity. Negative values of M^2, which lead to superluminal velocities, are allowed. The generalization to the many-particle case allows a nonlocal scalar potential, which makes nonlocal interactions compatible with relativity. Particle trajectories are parameterized by a scalar parameter analogous to the Newton absolute time. An example in which all these general features are explicitly realized is provided by relativistic Bohmian mechanics.

DOI
10.48550/arxiv.1006.1986

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Even though the usual form of relativistic mechanics does not allow superluminal particle velocities and nonlocal interactions, these features are not forbidden by relativity itself. To understand this on a deeper level, we study a generalized form of relativistic mechanics in which the particle is influenced not only by the usual tensor (gravitational) and vector (electromagnetic) potentials, but also by the scalar potential. The scalar potential promotes the mass squared M^2 to a dynamical quantity. Negative values of M^2, which lead to superluminal velocities, are allowed. The generalization to the many-particle case allows a nonlocal scalar potential, which makes nonlocal interactions compatible with relativity. Particle trajectories are parameterized by a scalar parameter analogous to the Newton absolute time. An example in which all these general features are explicitly realized is provided by relativistic Bohmian mechanics.
We discuss a relativistic particle affected simultaneously by two static potentials, a Lorentz scalar and the time component of a four-vector. We show that the scalar potential has effects on the motion of a particle that are distinctively different from those of a four-vector potential. With the aid of an example we show that, in both classical and quantum mechanics, there are two inequivalent but compatible criteria of nonrelativistic behavior: (mean) kinetic energy much smaller than rest energy and nonrelativistic potential energy independent of total energy. The second of these is the correct condition for use of nonrelativistic methods, but it is the first that is usually invoked to justify their use. The two criteria are equivalent only when the scalar potential has a (mean) value much less in magnitude than the particle’s rest energy. If this condition is not satisfied, nonrelativistic formalisms such as ordinary Newtonian mechanics and the Schrödinger equation may be inadequate even if v/c is small. On the other hand, it is possible that a large S may permit relativistic energy relations to be written in a nonrelativistic form even if v/c is not small. Such an influence of the size of the scalar potential on what is considered nonrelativistic behavior is unfamiliar and is rarely mentioned in textbooks and other publications. In our opinion, it should be better known.
Bohmian mechanics can be generalized to a relativistic theory without preferred foliation, with a price of introducing a puzzling concept of spacetime probability conserved in a scalar time. We explain how analogous concept appears naturally in classical statistical mechanics of relativistic particles, with scalar time being identified with the proper time along particle trajectories. The conceptual understanding of relativistic Bohmian mechanics is significantly enriched by this classical insight. In particular, the analogy between classical and Bohmian mechanics suggests the interpretation of Bohmian scalar time as a quantum proper time different from the classical one, the two being related by a nonlocal scale factor calculated from the wave function. In many cases of practical interest, including the macroscopic measuring apparatus, the fundamental spacetime probability explains the more familiar space probability as an emergent approximate description. Requiring that the quantum proper time in the classical limit should reduce to the classical proper time, we propose that only massive particles have Bohmian trajectories. An analysis of the macroscopic measuring apparatus made up of massive particles restores agreement with the predictions of standard quantum theory.
New relativistic mechanics of two-particle system is set forth, where the two constituent particles are interacting by an arbitrary (central) action-at-a-distance. The fundamental equations are presented in a form covariant under general transformation of parameters parametrizing the world lines of constituent particles. The theory represents the proper relativistic generalization of the usual Newtonian mechanics in the sense that it tends in the non-relativistic (and weak interaction) limit to the usual mechanics of two particles moving under a corresponding non-relativistic potential. For the analysis of theory it is convenient to choose a certain particular gauge (i.e., parametrization) fixed by two gauge relations. This brings the theory to a canonical formalism accompanied by two weak equations, and in this gauge quantization can be performed. The result verifies that the relativistic quantum mechanics for two particles interacting by an action-at-a-distance is just represented by a bilocal wave equation and a subsidiary condition, with the clarification of its correspondence-theoretical fundation and internal dynamics. As an example the case of Hooke-type force is illustrated, where the internal motions are elliptic oscillations in the center-of-mass frame. Its quantum theory just reproduces the original form of bilocal theory giving bound states lying on a straightly rising trajectory and on its daughter trajectories.
The field of scalar and vector potentials in electrodynamics is shown to represent an informational field capable of superluminally transmitting a signal (information) with no energy and momentum transfer. This conclusion strictly follows from Maxwell's equations for electromagnetic field interacting with electric charges and currents in vacuum, without resort to any additional hypotheses. That superluminal communication is possible is seen from the fact that the own field, generated by particles and inseparable from them, transforms the environment into a special physical medium which is capable to instantaneously transfer a signal (information) about any changes, happening to a particle in the region of its basic localization, to arbitrarily large distances. The phenomenon of superluminal communication is caused by the non-local connection of scalar and vector potentials with the electric and magnetic field strengths. The basis for the mechanism of superluminal communication considered in this work is the Aharonov-Bohm effect indicative of the field of electromagnetic potentials as a real physical field, which directly influences the behaviour of electron waves. The conclusion is made that in quantum systems superluminal signals occur constantly, in any quantum processes. The occurrence of superluminal signals is due to the space-time symmetry breaking of a special kind, consisting in that the equations for potentials do not possess relativistic invariance though Maxwell's equations for the field strengths are Lorentz-invariant. The results presented do not contradict the physical principles underlying special relativity and confirm the fundamental conclusion, made for the first time by de Broglie, that gauge invariance is not an absolute law in physics
This work is a revised version (Version 2) of a previous paper by the author, presented in a manner intended to emphasize more clearly the connection between its epistemological postulates and the subsequent development of its formalism. Several modifications have been introduced, and new sections have been added. As in the first version, the study concerns the physical interpretation of the imaginary quantities that arise in the formulation of superluminal relativity (including complex coordinates, velocities, angles, charges, and masses). In this framework, the kinematics of tachyons is analyzed, and the nature of the imaginary velocities appearing therein is examined. Furthermore, it is shown that a physical interpretation of the complex quantities (masses and charges) occurring both in quantum mechanics and in superluminal relativity is indeed possible. In this context, the complex nature of the electric charge is investigated, with particular attention to its divisibility. It is demonstrated that singular fractions of the electric charge exist whose squared values coincide with the observed quark charges. Within the same conceptual framework, the physical nature of a complex mass is also explored. It is shown that the imaginary component of such a mass implies that its gravitational field intensity behaves as a wave, propagating with the velocity of the wave associated with the particle itself. It is further demonstrated that the corresponding wavelength coincides with that given by the De Broglie relation, and that the resulting wave equation—second order in both spatial and temporal coordinates—is compatible with the Klein–Gordon wave equation. This constitutes a significant indication of the physical character of the associated wave and suggests an alternative interpretation to the orthodox probabilistic view of quantum mechanics.
Abstract. I present a relativistic covariant version of the Bohmian interpretation of quantum mechanics and discuss the corresponding measurable predictions. The covariance is incoded in the fact that the nonlocal quantum potential transforms as a scalar, which is a consequence of the fact that the nonlocal wave function transforms as a scalar. The measurable predictions that can be obtained with the deterministic Bohmian interpretation cannot be obtained with the conventional interpretation simply because the conventional probabilistic interpretation does not work in the case of relativistic quantum mechanics.