Two Mathematically Equivalent Versions of Maxwell’s Equations

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell’s equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers.     

Maxwell’s Equations as the One-Photon Quantum Equation

Maxwell's equations (the Faraday and Ampère-Maxwell laws) can be presented as a three-component equation in a way similar to the two-component neutrino equation. However, in this case, the electric and magnetic Gauss laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have also shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable and well understood quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum theories.     

Dynamics of Maxwell’s pendulum

The stability of motion of Maxwell’s pendulum is investigated in a uniform gravity field. By means of several canonic transforms of the equations of pendulum motion and the method of the surfaces of Poincaré sections, the problem is reduced to investigation of the immobile-point stability retaining the area of mapping of the plane into itself. In the space of dimensionless parameters, the stability and instability regions are singled out.     

Solution of Maxwell’s equations on a de Sitter background

The Maxwell equations for the electromagnetic potential, supplemented by the Lorenz gauge condition, are decoupled and solved exactly in de Sitter space–time studied in static spherical coordinates. There is no source besides the background. One component of the vector field is expressed, in its radial part, through the solution of a fourth-order ordinary differential equation obeying given initial conditions. The other components of the vector field are then found by acting with lower-order differential operators on the solution of the fourth-order equation (while the transverse part is decoupled and solved exactly from the beginning). The whole four-vector potential is eventually expressed through hypergeometric functions and spherical harmonics. Its radial part is plotted for given choices of initial conditions. We have thus completely succeeded in solving the homogeneous vector wave equation for Maxwell theory in the Lorenz gauge when a de Sitter space–time is considered, which is relevant both for inflationary cosmology and gravitational wave theory. The decoupling technique and analytic formulae and plots are completely original. This is an important step towards solving exactly the tensor wave equation in de Sitter space–time, which has important applications to the theory of gravitational waves about curved backgrounds.     

About Maxwell’s equations on fractal subsets of ℝ3

In this paper we have generalized $F^{\bar \xi }$ -calculus for fractals embedding in ?³. $F^{\bar \xi }$ -calculus is a fractional local derivative on fractals. It is an algorithm which may be used for computer programs and is more applicable than using measure theory. In this Calculus staircase functions for fractals has important role. $F^{\bar \xi }$ -fractional differential form is introduced such that it can help us to derive the physical equation. Furthermore, using the $F^{\bar \xi }$ -fractional differential form of Maxwell’s equations on fractals has been suggested.     

Octonic Form of Proca-Maxwell’s Equations and Relativistic Derivation of Electromagnetism

We propose to write the Proca-Maxwell’s equations, Klein-Gordon equation and electromagnetic energy conservation with magnetic monopole in terms of a new algebraic structure named as octon. The Lorentz transformations of electromagnetic fields are also taken into consideration.     

Feynman’s Proof of Maxwell Equations: in?the?Context?of Quantum Gravity

Many of us are familiar with Feynman’s “proof” of 1948, as revealed by Dyson, which demonstrates that Maxwell equations of electromagnetism are a consequence of Newton’s laws of motion of classical mechanics and the commutation relations of coordinate and momentum of quantum mechanics. It was Feynman’s purpose to explore the universality of dynamics of particles while making the fewest assumptions. We re-examine this formulation in the context of quantum gravity and show how Feynman’s derivation can be extended to include quantum gravity.     

On the Local Structure of the Klein–Gordon Field on Curved Spacetimes

This paper investigates wave equations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the non-characteristic Cauchy problem to show that a solution to a wave equation vanishing in an open set vanishes in the envelope of this set, which may be considerably larger and in the case of timelike tubes may even coincide with the spacetime itself. We apply this result to the real scalar field on a globally hyperbolic spacetime and show that the field algebra of an open set and its envelope coincide. As an example, there holds an analog of Borchers' timelike tube theorem for such scalar fields and, hence, algebras associated with world lines can be explicitly given. Our result applies to cosmologically relevant spacetimes.     

Maxwell’s demon and the management of ignorance in stochastic thermodynamics

It is nearly 150 years since Maxwell challenged the validity of the second law of thermodynamics by imagining a tiny creature who could sort the molecules of a gas in such a way that would decrease entropy without exerting any work. The demon has been discussed largely using thought experiments, but it has recently become possible to exert control over nanoscale systems, just as Maxwell imagined, and the status of the second law has become a more practical matter, raising the issue of how measurements manage our ignorance in a way that can be exploited. The framework of stochastic thermodynamics extends macroscopic concepts such as heat, work, entropy and irreversibility to small systems and allows us explore the matter. Some arguments against a successful demon imply a second law that can be suspended indefinitely until we dissipate energy in order to remove the records of his operations. In contrast, under stochastic thermodynamics, the demon fails because on average, more work is performed upfront in making a measurement than can be extracted by exploiting the outcome. This requires us to exclude systems and a demon that evolve under what might be termed self-sorting dynamics, and we reflect on the constraints on control that this implies while still working within a thermodynamic framework.     

On the “Equivalence” of the Maxwell and Dirac Equations

It is shown that Maxwell's equation cannot be put into a spinor form that is equivalent to Dirac's equation. First of all, the spinor in the representation of the electromagnetic field bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac's equation implies a complex structure specific to spin 1/2 particles that has no counterpart in Maxwell's equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell's and Dirac's equations into each other.     

A Possible Reinterpretation of Einstein’s Equations

In this paper, we first review Huei’s formulation in which it is shown that the linearized Einstein equations can be written in the same form as the Maxwell equations. We eliminate some imperfections like the scalar potential which is ill linked to the electric-type field, the Lorentz-type force which is obtained with a time independence restriction and the undesired factor 4 which appears in the magnetic-type part. Second, from these results and in the light of a recent work by C.C. Barros, we propose an extension of the equivalence principle and we suggest a new interpretation for Einstein’s equations by showing that the electromagnetic Maxwell equations can be derived from a new version of Einstein’s ones.     

Electromagnetism,Local Covariance,the Aharonov–Bohm Effect and Gauss’ Law

We quantise the massless vector potential A of electromagnetism in the presence of a classical electromagnetic (background) current, j, in a generally covariant way on arbitrary globally hyperbolic spacetimes M. By carefully following general principles and procedures we clarify a number of topological issues. First we combine the interpretation of A as a connection on a principal U(1)-bundle with the perspective of general covariance to deduce a physical gauge equivalence relation, which is intimately related to the Aharonov–Bohm effect. By Peierls’ method we subsequently find a Poisson bracket on the space of local, affine observables of the theory. This Poisson bracket is in general degenerate, leading to a quantum theory with non-local behaviour. We show that this non-local behaviour can be fully explained in terms of Gauss’ law. Thus our analysis establishes a relationship, via the Poisson bracket, between the Aharonov–Bohm effect and Gauss’ law – a relationship which seems to have gone unnoticed so far. Furthermore, we find a formula for the space of electric monopole charges in terms of the topology of the underlying spacetime. Because it costs little extra effort, we emphasise the cohomological perspective and derive our results for general p-form fields A (p <  dim(M)), modulo exact fields, for the Lagrangian density ${\mathcal{L} = \frac{1}{2} dA\wedge*dA+ A\wedge*j}$ . In conclusion we note that the theory is not locally covariant, in the sense of Brunetti–Fredenhagen–Verch. It is not possible to obtain such a theory by dividing out the centre of the algebras, nor is it physically desirable to do so. Instead we argue that electromagnetism forces us to weaken the axioms of the framework of local covariance, because the failure of locality is physically well-understood and should be accommodated.     

General Dyson–Schwinger Equations and Systems

We classify combinatorial Dyson–Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary (eventually infinite) number of insertion operators. We distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Faà di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.     

Hyperkähler Implosion and Nahm’s Equations

We use Nahm data to describe candidates for the universal hyperkähler implosion with respect to a compact Lie group.     

A non-deteriorating algorithm for computational electromagnetism based on quasi-lacunae of Maxwell’s equations

The performance of many well-known methods used for the treatment of outer boundaries in computational electromagnetism (CEM) may deteriorate over long time intervals. The methods found susceptible to this undesirable phenomenon include some local low order artificial boundary conditions (ABCs), as well as perfectly matched layers (PMLs). We propose a universal algorithm for correcting this problem. It works regardless of either why the deterioration occurs in each particular instance, or how it actually manifests itself (loss of accuracy, loss of stability, etc.). Our algorithm relies on the Huygens’ principle in the generalized form, when a non-zero electrostatic solution can be present behind aft fronts of the propagating waves, i.e., inside the lacunae of Maxwell’s equations. In this case, we refer to quasi-lacunae as opposed to conventional lacunae, for which the solution behind aft fronts is zero. The use of quasi-lacunae allows us to overcome a key constraint of the previously developed version of our algorithm that was based on genuine lacunae. Namely, the currents that drive the solution no longer have to be solenoidal. Another important development is that we apply the methodology to general non-Huygens’ problems.     

Generating Techniques and Analytically Extended Solutions of the Einstein–Maxwell Equations

The correspondence of arbitrary parameters inexact axisymmetric solutions of the Einstein-Maxwellequations constructed with the aid of differentgenerating methods to the analytically extendedparameter sets is discussed and examples of the extendedsolutions are given.