Difficulty with a Point Charge Particle in Electrodynamics

[gpran]

Point Particle in Relativity and Electrodynamics:

“The Classical Theory of Fields” – by Landau and Lifshitz, in its discussion about classical size of a particle, concludes that:- Thus we come to the conclusion that in classical (non-quantum) ‘relativistic mechanics’, we cannot ascribe finite dimensions to particles which we regard as elementary. In other words, within the framework of classical theory elementary particles must be treated as ‘points’.

In a paper – “Classical radiation theory and point charges”- by F H J Cornish, Journal-Proceedings of the Physical Society, the ‘abstract’ says that:- The usual theory for a continuous system may not be applied without some alteration to a system of point charges because the orthodox electromagnetic energy-momentum tensor for the field leads to infinite self energies. In this paper, a class of modified energy-momentum tensors are constructed in such a way that the field singularities do not lead to difficulty and the corresponding equations of motion are derived. (Couldn’t read the complete paper as it is not freely available).

In “Classical Electrodynamics (third edition)” by J. D. Jackson, the Poynting’s theorem analysis starts with:- If there exists a ‘continuous distribution’ of charge and current, the total rate of doing work by the fields in a finite volume is given by volume integral of (J.E). For a point charge described by a delta function, this derivation faces difficulties as discussed in- “A relook at radiation by a point charge. I”, Canadian Journal of Physics (available on T- space). However, this Larmor formula based on Poynting’s theorem seems to have been proved ‘experimentally’ for an accelerating charge. Maybe, F H J Cornish in above mentioned paper is suggesting a remedy.

Even in quantum mechanics, the particle is assumed as a point particle which may follow a probabilistic distribution pattern under potential field. My question is: How to reconcile the issue of size of a fundamental particle, under these circumstances of contradictory requirements by these important theories? Also, if a particle has finite size, then how to define volume (Vol= L1. L2 X L3 ) of a particle in a four dimensional space ?

 

[Dale]

How to reconcile the issue of size of a fundamental particle, under these circumstances of contradictory requirements by these important theories?

You just stay away from the issue. Meaning, only treat things as classical point particles at distances much larger than where quantum effects become important. If you need to go to smaller distances then you need to use QED.

 

[gpran]

The Poynting’s vector analysis in Classical Electrodynamics (Jackson J. D.) begins by stating that the charge and current density are continuous distributions. When the source is a point charge (delta function ); volume integral (E.J) is not possible. Field E(r)=1/ r is not Dirac-measurable, hence we can't compute the volume integral ∫E(r)δ(r)dV. Reason is, both δ(r) and E(r) are infinite at r = 0. This can lead to problems to the ‘Larmor radiation formula’ giving point charge radiation, when accelerating. The Larmor radiation formula is based on the Poynting vector.

Even though, the above mentioned papers are concentrating on point charges, in my view, this problem may remain as long as the source ‘charge density’ is not a continuous distribution i.e. when electron has some finite size. We generally assume that the charge sits on the particle and field extends outside the particle. Current J outside the particle will be zero giving E.J=0, outside this particle of finite size. Therefore, as suggested, best solution is to keep away from this issue.

The volume in a three dimensional vector analysis is by definition; Vol= L1. (L2 X L3). With finite particle volume, I also wanted to know definition of volume in a four dimension analysis. I agree that QED becomes important at smaller distances as it is not dependent on the actual size and shape of any particle.

Thanks for your suggestion and presently, I am trying to understand general relativity which is far more complex. Thanks for advice once again.

 

[Dale]

Even though, the above mentioned papers are concentrating on point charges, in my view, this problem may remain as long as the source ‘charge density’ is not a continuous distribution

No, discontinuities are fine. It is only the infinite charge density a a point that causes a problem.

 

[nnunn]

Since Dirac had to use a superposition of 4 (2 pairs of 2) virtual spinors to model his fermions, is this a hint that our renormalized electron may be a very crude model? For example, the fact that the [left hand state] of an electron sees a Higgs field, while the [right hand state] does not, must be telling us something about the real electron... ?

As Einstein intuited, "You know, it would be sufficient to really understand the electron."

 

[Vanadium 50]

For example, the fact that the [left hand state] of an electron sees a Higgs field, while the [right hand state] does not, must be telling us something about the real electron... ?

Well, it might if it were true.

 

[gpran]

Since Dirac had to use a superposition of 4 (2 pairs of 2) virtual spinors to model his fermions, is this a hint that our renormalized electron may be a very crude model? For example, the fact that the [left hand state] of an electron sees a Higgs field, while the [right hand state] does not, must be telling us something about the real electron... ?

As Einstein intuited, "You know, it would be sufficient to really understand the electron."

Values of Scalar potential = q/r and Field E(r)=q / r2 can be obtained from Maxwell equations taking source as a delta function only. If electron is not a point particle at very small distances, then the formulas need modifications when used for very Small distances. But, even in Quantum mechanics, the same formulas are being used. If we try to modify the source, then as suggested in the above mentioned Two Papers (F H J Cornish and Canadian J. Physics), a ‘relook at all aspects is required’; including the radiation.

(I wish to repeat the my earlier note here to highlight the contradiction: “The Classical Theory of Fields” – by Landau and Lifshitz, in its discussion about classical size of a particle, concludes that:- Thus we come to the conclusion that in classical (non-quantum) ‘relativistic mechanics’, we cannot ascribe finite dimensions to particles which we regard as elementary. In other words, within the framework of classical theory elementary particles must be treated as ‘points’.)

We may still give some shape and size to particle. But volume of an object seems to be a tricky concept in four dimensional analyses. In the three dimension vector analysis, Vol= L1. L2 X L3. In a book titled ‘Tensor analysis’ by I. S. Sokolnikoff (Pdf freely available on web), it is stated during discussion on N-Dimensional spaces that ‘ the idea of distance becomes devoid of any physical sense’. It is now widely accepted that the space is actually four dimensional. Under such circumstances how do we define volume of an object in vector analysis? Please guide.

 

[fresh_42]

But volume of an object seems to be a tricky concept in four dimensional analyses. In the three dimension vector analysis, Vol= L1. L2 X L3. In a book titled ‘Tensor analysis’ by I. S. Sokolnikoff (Pdf freely available on web), it is stated during discussion on N-Dimensional spaces that ‘ the idea of distance becomes devoid of any physical sense’. It is now widely accepted that the space is actually four dimensional. Under such circumstances how do we define volume of an object in vector analysis? Please guide.

This is a different topic and should be discussed in a separate thread, although I expect some basic misunderstandings to arise. The topic as stated tends to become a bit of a mixture between numerology and philosophy, neither of which we discuss here.

Thread closed.