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Series on knots and everything

The Complex Form of Relativistic Maxwell's Equations

Published Mar 1, 2011• 0 citations
Publisher

Abstract

We outline the relativistic formalism which gives a more comprehensive explanation of the complexification scheme. Such issues as considering the Higgs Boson as a soliton depends on Lorentz invariance and relativistic causality constraints. We relate the complexification of Maxwell's equations to models of nonlocal micro and macro phenomena. In this chapter we relate the electromagnetic fields, F and A , the potentials to the gravitational field, G . We examine, for example, the manner in which advanced potentials may explain the remote connectedness which is indicated by the Clauser test of Bell's theorem. Similar arguments apply to Young's double slit experiment. The collective coherent phenomena of superconductivity is also explainable by considering the relativistic field theoretic approach in which wave equations are solved in the complex Minkowski space.

DOI
10.1142/9789814324250_0007

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We outline the relativistic formalism which gives a more comprehensive explanation of the complexification scheme. Such issues as considering the Higgs Boson as a soliton depends on Lorentz invariance and relativistic causality constraints. We relate the complexification of Maxwell's equations to models of nonlocal micro and macro phenomena. In this chapter we relate the electromagnetic fields, F and A , the potentials to the gravitational field, G . We examine, for example, the manner in which advanced potentials may explain the remote connectedness which is indicated by the Clauser test of Bell's theorem. Similar arguments apply to Young's double slit experiment. The collective coherent phenomena of superconductivity is also explainable by considering the relativistic field theoretic approach in which wave equations are solved in the complex Minkowski space.
We develop a formalism for the Schrodinger equation in an eight dimensional (8D) complex Minkowski space and discuss its relation to the Dirac equation, properties of nonlocality, remote connectedness, Young’s double slit experiment, Bell’s Theorem, the EPR paradox and anticipatory parameters of spacetime; and also identify an imaginary temporal component as a small nonlinear term and find soliton or solitary wave solutions. These coherent solutions can carry information over long distances, are consistent with Lorentz invariance and appear to provide a fundamental methodology for describing the issue of quantum measurement and a new context for the basis of quantum theory. In the Copenhagen view models of reality are not desirable. However, our new approach may enable the redefinition of concepts of reality from a new nonlocal and anticipatory quantum theory. Certainly, the most desirable consequence of scientific discovery is the ability to refine our concepts of reality.
Abstract. We develop a formalism for the Schrödinger equation in an eight dimensional complex Minkowski space and discuss its relation to the Dirac equation, properties of nonlocality, remote connectedness, Young’s double slit experiment, Bell’s Theorem, the EPR paradox and anticipatory parameters of spacetime; and also identify an imaginary temporal component as a small nonlinear term and find soliton or solitary wave solutions. These coherent solutions can carry information over long distances, are consistent with Lorentz invariance and appear to provide a fundamental methodology for describing the issue of quantum measurement and a new context for the basis of quantum theory. In the Copenhagen view models of reality are not desirable. However our new approach may enable the redefinition of concepts of reality from a new nonlocal anticipatory quantum theory. Certainly the most desirable consequence of scientific discovery is the ability to redefine our concepts of reality.
The issue of whether Bell's theorem and other remote connectedness phenomena, such as Young's double slit experiment, demands superluminal or space-like signals or prior generated luminal signals is an area of hot debate. This also relates to the existence of advanced vs. retarded potentials and annihilation creation operators which are of interest in this regard. Using the complex model of we will examine the issue of the nonlocality of Bell's theorem as quantum mechanical ‘transactions’ providing a microscopic communication path between detectors across space-like intervals, which violate the EPR locality postulate [1]. See Chap. 4. This picture appears consistent with the remote connectedness properties of complex Minkowski space. Also, there are implications for macroscopic communications channels; another area of debate. Detailed discussions of Bell's theorem are given in [2].
The present study explores a theoretical extension of relativistic quantum wave equations through the introduction of complex space-time coordinates. The motivation for this approach arises from persistent conceptual challenges in relativistic quantum mechanics, particularly those related to probability interpretation and particle localization. By reformulating the Klein?Gordon and Dirac equations within a complex coordinate framework, the paper examines how the mathematical structure of these equations can be broadened without compromising fundamental physical principles. The analysis shows that treating the wave function as an analytic entity in complex space-time provides additional flexibility in handling boundary conditions and singularities. It also offers a refined perspective on the interpretation of solutions, especially in cases where traditional real-valued formulations lead to ambiguities. The study further demonstrates that Lorentz invariance can be preserved through analytic continuation, ensuring consistency with relativistic requirements. Although the framework remains theoretical, it suggests that complexification of space-time may serve as a useful tool in addressing certain limitations of relativistic quantum theory. The findings indicate that this approach has the potential to contribute to ongoing developments in quantum field theory and may provide a conceptual bridge toward more generalized formulations of quantum physics.
In this chapter, we consider a quantum particle wave function with a bound spectrum of velocity c, and obtain the relativistic momentum based on the group velocity of this wave function. With a space-time isometry condition, the Lorentz transformation and the relativistic dynamics were obtained. Considering a field interacting with a quantum particle as a four-vector conjugated to the space-time vector in the time-dependent phase, we obtain the Lorenz force and the Maxwell equations. It is interesting that only the Ampère-Maxwell equation of a magnetic circuit is specific to the electromagnetic field, while the other equations are general for a field interacting with a charged quantum particle. Considering the time-dependent phase of a quantum particle interacting with an electromagnetic field with a space-time homogeneity condition, we obtain Lorentz transformations for this field. For a quantum particle at a non-relativistic velocity, we obtain a wave function with a very rapidly-varying factor, of a frequency proportional to the rest energy of this particle. From the Schrödinger equation of a particle with a relativistic Hamiltonian, we obtain a split of the wave function into four components, describing a proper rotation of this particle with an angular momentum called spin (Dirac’s relativistic electron theory). Moreover, we also calculate electron potential in the magnetic field, and two-electron interaction potential.