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Register a new userMaxwells equations are universal for locally conserved quantities
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Authors
Lucas Burns
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A fundamental result of classical electromagnetism is that Maxwells equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwells equations. This holds true for any conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the Poincare lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.
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In the first sections of this article, we discuss two variations on Maxwells equations that have been introduced in earlier work--a class of nonlinear Maxwell theories with well-defined Galilean limits (and correspondingly generalized Yang-Mills equations), and a linear modification motivated by the coupling of the electromagnetic potential with a certain nonlinear Schroedinger equation. In the final section, revisiting an old idea of Lorentz, we write Maxwells equations for a theory in which the electrostatic force of repulsion between like charges differs fundamentally in magnitude from the electrostatic force of attraction between unlike charges. We elaborate on Lorentz description by means of electric and magnetic field strengths, whose governing equations separate into two fully relativistic Maxwell systems--one describing ordinary electromagnetism, and the other describing a universally attractive or repulsive long-range force. If such a force cannot be ruled out {it a priori} by known physical principles, its magnitude should be determined or bounded experimentally. Were it to exist, interesting possibilities go beyond Lorentz early conjecture of a relation to (Newtonian) gravity.
This paper provides a view of Maxwells equations from the perspective of complex variables. The study is made through complex differential forms and the Hodge star operator in $mathbb{C}^2$ with respect to the Euclidean and the Minkowski metrics. It shows that holomorphic functions give rise to nontrivial solutions, and the inner product between the electric and the magnetic fields is considered in this case. Further, it obtains a simple necessary and sufficient condition regarding harmonic solutions to the equations. In the end, the paper gives an interpretation of the Lorenz gauge condition in terms of the codifferential operator.
Let $V(t) = e^{tG_b},: t geq 0,$ be the semigroup generated by Maxwells equations in an exterior domain $Omega subset {mathbb R}^3$ with dissipative boundary condition $E_{tan}- gamma(x) ( u wedge B_{tan}) = 0, gamma(x) > 0, forall x in Gamma = partial Omega.$ We study the case when $Omega = {x in {mathbb R^3}:: |x| > 1}$ and $gamma eq 1$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $G_b.$
When a measurement is made on a system that is not in an eigenstate of the measured observable, it is often assumed that some conservation law has been violated. Discussions of the effect of measurements on conserved quantities often overlook the possibility of entanglement between the measured system and the preparation apparatus. The preparation of a system in any particular state necessarily involves interaction between the apparatus and the system. Since entanglement is a generic result of interaction, as shown by Gemmer and Mahler[1], and by Durt[2,3] one would expect some nonzero entanglement between apparatus and measured system, even though the amount of such entanglement is extremely small. Because the apparatus has an enormous number of degrees of freedom relative to the measured system, even a very tiny difference between the apparatus states that are correlated with the orthogonal states of the measured system can be sufficient to account for the perceived deviation from strict conservation of the quantity in question. Hence measurements need not violate conservation laws.
Conserved quantities are crucial in quantum physics. Here we discuss a general scenario of Hamiltonians. All the Hamiltonians within this scenario share a common conserved quantity form. For unitary parametrization processes, the characteristic operator of this scenario is analytically provided, as well as the corresponding quantum Fisher information (QFI). As the application of this scenario, we focus on two classes of Hamiltonians: su(2) category and canonical category. Several specific physical systems in these two categories are discussed in detail. Besides, we also calculate an alternative form of QFI in this scenario.
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