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NON LINEAR ELECTROMAGNETISM AND PHOTONS IN THE FUNCTIONAL THEORY OF PARTICLWS

Florence A. Aeschlimann, Jean-Louis Destouches

Published Nov 1, 1957• 0 citations
OpenAlex

Abstract

>It is shown that Mie's or Born's equations of non linear electromagnetism are obtained from the non linear equations of the photon (functional theory), by means of linear combinations of these equations-defined by the fundamental operators of the Louis de Broglie photon. The magnetic and electric fields are defined exactly as in Louis de Broglie's theory of photon whereas the inductions are defined by other linear terms according to the second definitions of the fields by Leuis de Broglie. Only the polarizations and the expressions deriving from them are expressed by means of non linear terms. In this way a microphysical theory can be constructed, which unifies Louis de Broglie's theory of the photon and Mie's or Barn's non linear electromagnetism. (auth)

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It is shown that Mie's or Born's equations of non linear electromagnetism are obtained from the non linear equations of the photon (functional theory), by means of linear combinations of these equations-defined by the fundamental operators of the Louis de Broglie photon. The magnetic and electric fields are defined exactly as in Louis de Broglie's theory of photon whereas the inductions are defined by other linear terms according to the second definitions of the fields by Leuis de Broglie. Only the polarizations and the expressions deriving from them are expressed by means of non linear terms. In this way a microphysical theory can be constructed, which unifies Louis de Broglie's theory of the photon and Mie's or Barn's non linear electromagnetism. (auth)
The present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.
I. La Nouvelle théorie de la lumière de M. L. de Broglie est interprétée d'une façon géométrique qui est suggérée par le principe de Relativité. — II. On discute et l'on classe différents types d'équations complémentaires généralisant le type de M. L. de Broglie et susceptibles de servir de base à une théorie du photon. - III. Parmi ces représentations, on précise, en particulier, celle qui conduit à un Électromagnétisme purement maxwellien (avec spin 1) et celle qui conduit à un Électromagnétisme purement non maxwellien (avec spin 0). - IV. On indique comment se modifie la fonction d'annihilation lorsque l'on modifie la représentation utilisée par M. L. de Broglie.
According to modern quantum field theory all elementary particles are quantum excitations of an underlying quantum field. In the case of the photon the underlying field is the electromagnetic field. In photon wave mechanics the photon concept was reached by a reinterpretation of the physical meaning of the free classical electromagnetic field. Such a reinterpretation does not give us a quantum theory of light quanta because the elementary quantum of action is foreign to classical physics. Although Planck’s constant does occur in the various Schro¨dinger-like wave equations we have presented in Part III it just plays the role of a multiplicative factor, see e.g., Eq. (10.36). The “justification” for bringing ~ into the formalism is the wish to make contact to the Einstein-de Broglie relations [Eqs. (10.40) and (10.41)]. In a sense one may call the light particle appearing from classical electrodynamics the classical photon.
The classical source-free electric field is a polar transverse-vector field, but the classical magnetic field is an axial transverse-vector field. A derivative-related linear transformation (which is its own inverse) of the classical axial magnetic field in fact produces an alternate polar transverse-vector representation of the classical magnetic field. The classical source-free complex-valued electromagnetic polar transverse-vector field whose real part is the classical source-free polar electric field, and whose imaginary part is the alternate polar representation of the classical source-free magnetic field, turns out to satisfy the time-dependent Schroedinger equation whose Hamiltonian operator is that of the free photon. That classical source-free complex-valued electromagnetic polar transverse-vector field can, moreover, be slightly linearly modified to become the normalized wave function of the free photon.
Dans le but de faire rentrer la théorie du photon de M. Louis de Broglie dans le cadre de la théorie structurale des corpuscules élémentaires, on établit les bases d'une théorie corpusculaire générale appliquant les principes de la mécanique ondulatoire tout en suivant un formalisme voisin de celui de la théorie électromagnétique classique. Le cas de l'équation d'ondes de l'électron de Dirac étant écarté, l'étude du cas le plus simple permet de retrouver d'une façon très générale la théorie du photon de M. Louis de Broglie sous ses deux aspects corpusculaire et électromagnétique sans nécessiter de spécialisation des matrices servant à représenter le système des équations d'ondes.
We derive a non-linear wave equation directly from de Broglie’s fundamental relation λ= h/p without invoking the Hamiltonian formalism or probability interpretations. By treating ψ = 1/λ as an ontological spatial compaction field, we obtain an exact, general non-linear field equation. The nonlinear term is written as |Ψ|^2 Ψ to maintain phase coherence between the linear propagation and the self-interaction. Both isotropic and anisotropic forms are presented. Planck’s constant cancels out completely, revealing the geometric nature of the propagation.