P ROOF. Proving this remark is quite simple. In the direction, it follows from the properties of Dirac and Riesz kernels. In the other direction, it is a consequence of the uniqueness of the Hadamard expansion. In this section we shall consider potentials of the form.
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Potentials of the form are called electromagnetic potentials because they correspond to including an electromagnetic field in the physical situation under consideration, except for a multiplication by the complex unit i. If the physical potential is given by the "four-vector" A m the new momentum is given by , so the "physical case", in the expression of v , is given by purely imaginary components.
The second equation is automatically obeyed, as long as F mn is obtained from a potential. In classical physics the field at each point is observable, while the potentials are considered just a technique for calculating such field, having no separate reality This changes dramatically in Quantum Mechanics, as can be seen by the Aharonov-Bohm effect. See, for example Sakurai In terms of its components, this means to change from a m to. As the Huygens' property is a physically sound problem, the validity or not of the Huygens' principle cannot depend on the particular choice of the electromagnetic gauge to understand the idea that Huygens' principle is physically observable we should understand the meaning of , the solution of Dirac equation.
In simpler words, we have a non-zero probability of finding the particle only on the support of the wave-function. The change of the gauge is a particular kind of trivial transformation, as defined in the introduction. We see immediately. Thus, the invariance of Huygens' principle by change of gauge can be concluded by two distinct reasonings, coming from two different points of view, historically unrelated.
We point out that, as far as we know, the fact that the change of electromagnetic gauge is a particular case of trivial transformations as defined by Hadamard was never mentioned in the literature. We shall study compatibility conditions between the last two equations. We start applying Dirac operator to the first:.
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This can be interpreted as the absence of sources. If we relax the hypothesis that the potential is real or purely imaginary, then we can find a large class of Huygens potentials. One family of examples is given by. Huygens' principle has been subject of extensive investigation in recent times.
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For reviews in two different directions, see Berest and Belger et al. One of its most striking characteristics is its relations to integrability see Berest and Veselov In the wave operator case, potentials of Huygens' type are, after a suitable change of variables, solutions of the Korteweg-de Vries equation. For Dirac operators, the relation between Huygens' principle and integrability involves the AKNS after Ablowitz, Kaup, Newell and Segur hierarchy, which in turn is an example of integrable matrix systems.
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This is proved in Chalub and Zubelli b. A natural follow up of the present work is to relate Huygens' principle for Dirac operators associated to the electromagnetic field and integrable systems. I thank Prof. Jorge P.
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Zubelli for introducing me on this subject and for many suggestions. A survey on Huygens' principle. Z Anal Anwendungen, 16 1 : The Huygens principle and integrability. Uspekhi Mat Nauk, 49 6 : Hierarchies of Huygens' operators and Hadamard's conjecture. Acta Appl Math, 53 2 : Pergamon Press, sixth edition. Integrable systems, Huygens' principle, and Dirac operators.
World Sci. J Nonlin Math Phys, 8, Supplement: The Quantum Theory of the Electron. Proc Roy Soc of London A, Ann Chim et Phys, 2 , 1, Fundamental solutions for the wave operator. Exposition Math, 15 1 : Clifford algebras and Dirac operators in harmonic analysis. Cambridge University Press, Cambridge. Lectures on Cauchy's problem in linear partial differential equations. Dover Publications, New York. Reprint of the original, Librarie Marchand. Dirac g -equation, classical gauge fields and Clifford algebra. Adv Appl Clifford Algebras, 8 1 : A treatise of the Reflexion, Inflexion and Colours of Light.
Advanced Quantum Mechanics. Despite the successful closed form solution of many simple geometries, the solutions available were insufficient to analyze many electromagnetic systems. This was the age of approximations, roughly between s and s. The class of solvable problems for which approximate solutions exist, was greatly enlarged. In particular, Harrington was noted for popularizing MOM among the electromagnetics community, while it is know as the boundary element method BEM in other communities.
Finite element has been with the structure and mechanics community, and Silvester was an early worker who brought its use into the electromagnetics community. Other names commonly cited in this field are: Wilton, Mittra, and Taflove. There has always been marriage between electromagnetics and mathematics from the very beginning—a marriage made in heaven perhaps. Actually, quite sophisticated mathematical techniques were used to analyze electromagnetic problems because electromagnetic theory was predated by the theory of fluid and theory of sound.
These fields were richly entwined with mathematics with the work of famous mathematicians such as Euler, Lagrange, Stokes, and Gauss. Moreover, many of the mathematics of low-Reynold number flow in fluid theory and scalar wave theory of sound can be transplanted with embellishment to solve electromagnetic problems. Examples of problems solved during the age of simple shapes are the Mie and Debye scattering by a sphere and Rayleigh scattering by small particles. Rayleigh also solved the circular waveguide problem for electromagnetic waves because he was well versed in the mathematical theory of sound, having written three volumes on the subject while sailing down the Nile River.
Sommerfeld solved the half plane problem as far back as because the advanced mathematical techniques were available then. He also solved the Sommerfeld half space problem in in order to understand the propagation of radio waves over the lossy half-earth. The problem was solved in terms of, what is now known as, the Sommerfeld integrals, an example of which is as follows:. Figure 2. A dipole over a half space. The problem was first solved by Sommerfeld to understand the propagation of radio waves over the lossy earth.
Evaluating the Sommerfeld integrals was an impossibility during his time, but it is a piece of cake now in the modern era. Subsequently, approximation techniques, such as the stationary phase method, the method of steepest descent, and the saddle point methods were used to derive approximations to the Sommerfeld integrals.
Approximate methods generally help to elucidate the physics of the wave interaction with complex geometry. The physical insights offered by approximate solutions spurred the age of approximations, roughly between s and s. A large parameter such as frequency is used to derive asymptotic approximations. Moreover, heuristic ideas were used to derive the physical optics approximation, Kirchhoff approximation, and various geometrical optics approximations.
These approximations eventually lead to the geometrical theory of diffraction and the uniform asymptotic theory of diffraction. The applications of these approximate methods to scattering by complex structures are usually ansatz based. The ansatz assumes that the scattering solution is of the form:. The leading coefficient and the exponent are found from canonical solutions such as the Sommerfeld half plane problem, or scattering by a sphere or a cylinder, followed by the use of Watson transformation.
The use of approximate solution enlarges the class of solvable problems, but the error is usually not controllable. Asymptotic series are semi-convergent series; hence there is not a systematic way to reduce the solution error by including more terms in the ansatz.
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