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Register a new userA note on conserved quantities for electromagnetic waves
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T.G. Philbin
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Electromagnetic waves carry an infinite number of conserved quantities. We give a simple explanation of this fact, which also shows how to write down conserved quantities at will and calculate their associated symmetry transformations. This framework is then used to discuss decompositions of optical angular momentum, and to prove that magnetic helicity is conserved for beams and pulses. Finally we describe an infinite set of electromagnetic conserved quantities that corresponds to the Virasoro generators of conformal field theories. In the quantum case the Virasoro generators acquire a central charge in their algebra, an example of a quantum anomaly.
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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.
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.
Recent observations of the luminosity-red shift relation of distant type Ia supernovae established the fact that the expansion of the universe is accelerated. This is interpreted by saying that there exists some kind of agent (called dark energy), which exerts an overall repulsive effect on ordinary matter. Dark energy contributes today in the amount of about 73 % to the total energy content of the universe, and its spatial distribution is compatible with perfect uniformity. The simplest possible explanation for dark energy is to assume that it is just a universal constant, the so called cosmological constant. This would mean that the background arena for all natural phenomena, once all physical matter-energy has been ideally removed, is the de Sitter space time. Thus, the Poincare group should be replaced by the de Sitter group, and one is naturally led to a reformulation of the theory of special relativity on these grounds. The absence of a privileged class of equivalent frames (inertial frames) suggests that, in de Sitter relativity it would be desirable, to characterize significant physical quantities in an intrinsic way, namely in a manner independent of the choice of any particular coordinate patch. In this talk we would like to show how this can be accomplished for any set of independent conserved quantities along the geodesic motion of a free de Sitter particle. These quantities allow for a natural discussion of classical pointlike scattering and decay processes.
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.
We provide a unified semiclassical theory for the conserved current of nonconserved quantities, and manifest it in two physical contexts: the spin current of Bloch electrons and the charge current of mean-field Bogoliubov quasiparticles. Several longstanding problems that limit the playground of the conserved spin current of electrons are solved. We reveal that the hitherto overlooked torque quadrupole density and Berry phase correction to the torque dipole density are essential to assure a circulating spin current with vanishing net flow at equilibrium. The band geometric origin of bulk spin transport is ascertained to be the momentum space spin texture and Berry curvature instead of the spin Berry curvature, paving the way for material related studies. In superconductors the attained conserved charge current corresponds to the quasiparticle charge current renormalized by the condensate backflow. Its circulation at equilibrium gives an orbital magnetization, which involves the characteristics of superconductivity, such as the Berry curvature arising from unconventional pairing and an orbital magnetic moment induced by the charge dipole of moving quasiparticles.
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