A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics

Gsponer, Andre

doi:10.1088/0143-0807/30/1/011

Cited by 30 publications

(38 citation statements)

References 21 publications

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“…Similarly, as explained in [20], it is not necessary to know much about the mathematical details of the Colombeau theory to use it in a context such as the present paper.…”

Section: Definition 2 the Differential Ideal Of Negligible Functions Ismentioning

confidence: 93%

The classical point electron in Colombeau’s theory of nonlinear generalized functions

Gsponer¹

2008

Journal of Mathematical Physics

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The electric and magnetic fields of a pole-dipole singularity attributed to a point-electron-singularity in the Maxwell field are expressed in a Colombeau algebra of generalized functions. This enables one to calculate dynamical quantities quadratic in the fields which are otherwise mathematically illdefined: The self-energy (i.e., 'mass'), the self-angular momentum (i.e., 'spin'), the self-momentum (i.e., 'hidden momentum'), and the self-force. While the total self-force and self-momentum are zero, therefore insuring that the electron-singularity is stable, the mass and the spin are diverging integrals of δ 2 -functions. Yet, after renormalization according to standard prescriptions, the expressions for mass and spin are consistent with quantum theory, including the requirement of a gyromagnetic ratio greater than one. The most striking result, however, is that the electric and magnetic fields differ from the classical monopolar and dipolar fields by δ-function terms which are usually considered as insignificant, while in a Colombeau algebra these terms are precisely the sources of the mechanical mass and spin of the electron-singularity.

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“…Similarly, as explained in [20], it is not necessary to know much about the mathematical details of the Colombeau theory to use it in a context such as the present paper.…”

Section: Definition 2 the Differential Ideal Of Negligible Functions Ismentioning

confidence: 93%

The classical point electron in Colombeau’s theory of nonlinear generalized functions

Gsponer¹

2008

Journal of Mathematical Physics

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“…the multiplication among distributions [43][44][45][46][47][48][49][50][51][52], the division of distributions [53], the random square root of the Dirac delta [54] and the square of the Dirac delta [55] as already discussed above. A brief summary of the problem of the Dirac delta squared is given in [56].…”

Section: Open Problemsmentioning

confidence: 99%

A Paradox of Unity

Fischer¹

2018

Preprint

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Abstract:In previous studies we found that generalized functions can be smooth, discrete, periodic or discrete periodic and they can either be local or global and they are regular or generalized functions. We also saw that these properties were related to Poisson's summation formula on one hand and to Heisenberg's uncertainty principle on the other. In this paper, we interlink these studies and show that scalars (real or complex numbers) considered as trivial functions are discrete and periodic, local and global as well as regular and generalized, simultaneously. However, this is also a paradox because it means that Dirac's δ and 1 (its Fourier transform) coincide. They both are unity. We show that δ and 1 coincide in the sense of scalars (real or complex numbers) but they differ in the sense of (generalized) functions. This result can moreover be related to Max Born's principle of reciprocity. It also answers an open question in present-day quantum mechanics because it means that the Dirac delta squared is simply delta.

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“…While new methods exists to evaluate these products [12], degenerate or self-interacting systems will not be considered in the present paper.…”

Section: Inclusion Of Singularitiesmentioning

confidence: 99%

“…As is well known, there is no other cure than ignoring these divergences, for example by introducing a "covariant" form-factor, and possibly "renormalising" the result afterwards. In the present case, a consistent method is to represent the Υ and δ functions by the limits ε → 0 of a pair of functions such as, for example, 12) and to keep ε very small but finite so that Υ ε (r)/r = 2/πε at r = 0. In that case the radial integration is finite while angular integration gives zero.…”

Section: Electric Dipole In a Locally Uniform Magnetic Fieldmentioning

confidence: 99%

On the electromagnetic momentum of static charge and steady current distributions

Gsponer¹

2007

Eur. J. Phys.

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Abstract.Faraday's and Furry's formulae for the electromagnetic momentum of static charge distributions combined with steady electric current distributions are generalised in order to obtain full agreement with Poynting's formula in the case where all fields are of class C 1 , i.e., continuous and continuously differentiable, and the integration volume is of finite or infinite extent.These three formulae are further generalised to the case where singularities are allowed to exist at isolated points in the fields, and at surfaces separating domains in which the distributions are of class C 1 . Applications are made to electric and magnetic, point-like and finite dipolar systems, with an emphasis on the impact of singularities on the magnitude and location of the electromagnetic momentum.

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A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics

Cited by 30 publications

References 21 publications

The classical point electron in Colombeau’s theory of nonlinear generalized functions

The classical point electron in Colombeau’s theory of nonlinear generalized functions

A Paradox of Unity

On the electromagnetic momentum of static charge and steady current distributions

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