A Structurally Relativistic Quantum Theory. Part 1: Foundations

2002

The present article is the first in a program that aims at generalizing quantum mechanics by keeping its structure essentially intact, but constructing the Hilbert space over a new number system much richer than the field of complex numbers. We call this number system ``the Quantionic Algebra''. It is eight dimensional like the algebra of octonions, but, unlike the latter, it is associative. It is not a division algebra, but "almost" one (in a sense that will be evident when we come to it). It enjoys the minimum of properties needed to construct a Hilbert space that admits quantum-mechanical interpretations (like transition probabilities), and, moreover, it contains the local Minkowski structure of space-time. Hence, a quantum theory built over the quantions is inherently relativistic. The algebra of quantions has been discovered in two steps. The first is a careful analysis of the abstract structure of quantum mechanics (the first part of the present work), the ...

This paper is concerned with the possibility and nature of relativistic hidden-variable formulations of quantum mechanics. Both ad hoc teleological constructions and framedependent constructions of spacetime maps are considered. While frame-dependent constructions are clearly preferable, a many-maps theory based on such constructions fails to provide dynamical explanations for local quantum events. Here the hiddenvariable dynamics used in the frame-dependent constructions is just a rule that serves to characterize the set of all possible spacetime maps. While not having dynamical explanations of the values of quantum-mechanical measurement records is a significant cost, it may prove too much to ask for dynamical explanations in relativistic quantum mechanics.

From Nanoscale Systems to Cosmology, 2012

A general formulation of classical relativistic particle mechanics is presented, with an emphasis on the fact that superluminal velocities and nonlocal interactions are compatible with relativity. Then a manifestly relativistic-covariant formulation of relativistic quantum mechanics (QM) of fixed number of particles (with or without spin) is presented, based on many-time wave functions and the spacetime probabilistic interpretation. These results are used to formulate the Bohmian interpretation of relativistic QM in a manifestly relativistic-covariant form. The results are also generalized to quantum field theory (QFT), where quantum states are represented by wave functions depending on an infinite number of spacetime coordinates. The corresponding Bohmian interpretation of QFT describes an infinite number of particle trajectories. Even though the particle trajectories are continuous, the appearance of creation and destruction of a finite number of particles results from quantum theory of measurements describing entanglement with particle detectors.

1999

The objective of this paper is to axiomatically derive quantum mechanics from three basic axioms. In this paper, the Schr odinger equation for a characteristic function is ÿrst obtained and from it the Schr odinger equation for the probability amplitudes is also derived. The momentum and position operators acting upon the characteristic function are deÿned and it is then demonstrated that they do commute, while those acting upon the probability amplitudes obey the usual commutation relation. We also show that, for dispersion free ensembles, the Schr odinger equation for the characteristic function is equivalent to Newton's equations, thus providing us with a correspondence between both theories. As an application of the method, we show how it can be used to make quantization in generalized coordinates.

2012

The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schroendiger’s equation for the density matrix is fist obtained and from it Schroedinger’s equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli’s equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm’s hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free ensembles, Schroedinger’s equation for the density matrix is equivalent to Newton’s equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which a...

Physics Essays, 2011

When determining the coefficients a i and b of the Dirac equation (which is a relativistic wave equation), Dirac assumed that the equation satisfies the Klein-Gordon equation. The Klein-Gordon equation is an equation that quantizes Einstein's relationship E 2 ¼ c 2 p 2 þ E 2 0. Therefore, this paper derives an equation similar to the Klein-Gordon equation by quantizing the relationship E 2 re;n þ c 2 p 2 n ¼ E 2 0 between energy and momentum of the electron in a hydrogen atom derived by the author. By looking into the Dirac equation, it is predicted that there is a relativistic wave equation, which satisfies that equation, and its coefficients are determined. With the Dirac equation, it was necessary to insert a term for potential energy into the equation when describing the state of the electron in a hydrogen atom. However, in this paper, a potential energy term was not introduced into the relativistic wave equation. Instead, potential energy was incorporated into the equation by changing the coefficient a i of the Dirac equation. It may be natural to regard the equation derived in this paper and the Dirac equation as physically equivalent. However, if one of the two equations is superior, this paper predicts it will be the relativistic wave equation derived by the author. V

2008

In this article, the axioms presented in the first one are reformulated according to the special theory of relativity. Using these axioms, quantum mechanic’s relativistic equations are obtained in the presence of electromagnetic fields for both the density function and the probability amplitude. It is shown that, within the present theory’s scope, Dirac’s second order equation should be considered the fundamental one in spite of the first order equation. A relativistic expression is obtained for the statistical potential. Axioms are again altered and made compatible with the general theory of relativity. These postulates, together with the idea of the statistical potential, allow us to obtain a general relativistic quantum theory for ensembles composed of single particle systems. 1

Physical Review Letters, 2002

This talk examines Hamiltonians H that are not Hermitian but do exhibit space-time reflection (PT ) symmetry. If the (PT ) symmetry of H is not spontaneously broken, then the spectrum of H is entirely real and positive. Examples of PT -symmetric non-Hermitian Hamiltonians are H = p 2 + ix 3 and H = p 2 − x 4 . The apparent shortcoming of quantum theories arising from PT -symmetric Hamiltonians is that the PT norm is not positive definite. This suggests that it may be difficult to develop a quantum theory based on such Hamiltonians. In this talk it is shown that these difficulties can be overcome by introducing a previously unnoticed underlying physical symmetry C of Hamiltonians having an unbroken PT symmetry. Using C, it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables in these theories exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.

Applied Mathematics, 2014

The Special Relativity Theory cannot recognize speed faster than light. New assumption will be imposed that matter has two intrinsic components, 1) mass, and 2) charge, that is M = m + iq. The mass will be measured by real number system and charged by an imaginary unit. This article presents a Complex Matter Space in Relativistic Quantum Mechanics. We are hoping that this approach will help us to present a general view of energy and momentum in Complex Matter Space. The conclusion of this article on Complex Matter Space (CMS) theory will lead help to a better understanding toward the conversion of mass and energy equation, unifying the forces, and unifying relativity and quantum mechanics.

We argue that special relativity, instead of quantum theory, should be radically reformulated to resolve inconsistencies between those two theories. A new relativistic transformation recently-proposed renders physical laws form-invariant via transformation of physical quantities, instead of space-time coordinates. This new perspective on relativistic transformation provides an insight into the very meaning of the principle of relativity. The principle of relativity means that the same physical laws hold in all inertial frames, rather than their mathematical formulas are Lorentz-covariant under the Lorentz transformation of space-time coordinates. The space-time concept underlying this new relativistic transformation is Newtonian absolute space and absolute time. With this new perspective, quantum theory becomes compatible with the principle of relativity. A new theory of relativistic quantum mechanics is formulated. The new relativistic quantum mechanics thus obtained maintains the ...