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Register a new userFrame Indifferent Formulation of Maxwells Elastic Fluid and the Rational Continuum Mechanics of the Electromagnetic Field
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Authors
Christo I. Christov
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We show that the linearized equations of the incompressible elastic medium admit a `Maxwell form in which the shear component of the stress vector plays the role of the electric field, and the vorticity plays the role of the magnetic field. Conversely, the set of dynamic Maxwell equations are strict mathematical corollaries from the governing equations of the incompressible elastic medium. This suggests that the nature of `electromagnetic field may actually be related to an elastic continuous medium. The analogy is complete if the medium is assumed to behave as fluid in shear motions, while it may still behave as elastic solid under compressional motions. Then the governing equations of the elastic fluid are re-derived in the Eulerian frame by replacing the partial time derivatives by the properly invariant (frame indifferent) time rates. The `Maxwell from of the frame indifferent formulation gives the frame indifferent system that is to replace the Maxwell system. This new system comprises terms already present in the classical Maxwell equations, alongside terms that are the progenitors of the Biot--Savart, Oersted--Amperes, and Lorentz--force laws. Thus a frame indifferent (truly covariant) formulation of electromagnetism is achieved from a single postulate that the electromagnetic field is a kind of elastic (partly liquid partly solid) continuum.
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The dynamics of any classical-mechanics system can be formulated in the reparametrization-invariant (RI) form (that is we use the parametric representation for trajectories, ${bf x}={bf x}(tau)$, $t=t(tau)$ instead of ${bf x}={bf x}(t)$). In this pedagogical note we discuss what the quantization rules look like for the RI formulation of mechanics. We point out that in this case some of the rules acquire an intuitively clearer form. Hence the formulation could be an alternative starting point for teaching the basic principles of quantum mechanics. The advantages can be resumed as follows. a) In RI formulation both the temporal and the spatial coordinates are subject to quantization. b) The canonical Hamiltonian of RI formulation is proportional to the quantity $tilde H=p_t+H$, where $H$ is the Hamiltonian of the initial formulation. Due to the reparametrization invariance, the quantity $tilde H$ vanishes for any solution, $tilde H=0$. So the corresponding quantum-mechanical operator annihilates the wave function, $hat{tilde H}Psi=0$, which is precisely the Schrodinger equation $ihbarpartial_tPsi=hat HPsi$. As an illustration, we discuss quantum mechanics of the relativistic particle.
A matrix basis formulation is introduced to represent the 3 x 3 dyadic Greens functions in the frequency domain for the Maxwells equations and the elastic wave equation in layered media. The formulation can be used to decompose the Maxwells Greens functions into independent TE and TM components, each satisfying a Helmholtz equation, and decompose the elastic wave Greens function into the S-wave and the P-wave components. In addition, a derived vector basis formulation is applied to the case for acoustic wave sources from a non-viscous fluid layer.
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V. A. Fateev (Laboratoire de Physique Mathematique
, Universite den Montpellier II
, Montpellier
2004
A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an exact eigenfunction expansion; these can be associated to Riemannian symmetric spaces of rank one with positive, negative or vanishing curvature. For all other cases an accurate semiclassical approximation is derived, based on the identification of the operators with a peculiar Schroedinger-like operator.
The contraction of the Poincare group with respect to the space trans- lations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincare group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIR) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non- relativistic limit of Maxwells equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.
Conditions for the appearance of topological charges are studied in the framework of the universal C*-algebra of the electromagnetic field, which is represented in any theory describing electromagnetism. It is shown that non-trivial topological charges, described by pairs of fields localised in certain topologically non-trivial spacelike separated regions, can appear in regular representations of the algebra only if the fields depend non-linearly on the mollifying test functions. On the other hand, examples of regular vacuum representations with non-trivial topological charges are constructed, where the underlying field still satisfies a weakened form of spacelike linearity. Such representations also appear in the presence of electric currents. The status of topological charges in theories with several types of electromagnetic fields, which appear in the short distance (scaling) limit of asymptotically free non-abelian gauge theories, is also briefly discussed.
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