Singularity in Classical E&M

[Alan Grayson]

The Coulomb field of a point charge diverages as the distance to the charge decreases to zero. Is this singularity resolved in the classical or quantum theory of E&M? AG

[John Clark]

On Wed, May 14, 2025 at 4:39 PM Alan Grayson <agrays...@gmail.com> wrote:

> The Coulomb field of a point charge diverages as the distance to the charge decreases to zero. Is this singularity resolved in the classical or quantum theory of E&M?

In classical physics the amount of energy in a point electrical charge such as an electron is infinite, quantum electrodynamics avoids infinity in a process called "renormalization". The point charge interacts with a cloud a virtual particles that pop in and out of existence with each having their own Feynman diagram;  the infinity from one part of the calculation is canceled out by another infinity in another part of the calculation, so you're left with a finite charge that agrees with experimental results better than one part in a billion. It has been called the most accurate prediction in the entire history of science.

Richard Feynman had more to do with developing renormalization than anyone and received the Nobel prize for it, but he was never satisfied with it because, although it worked wonderfully well,  this canceling out inconsistencies business is not mathematically rigorous and so it cannot be proven to contain no inconsistencies. Feynman said this:

"The shell game that we play is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! It's a way of sweeping the problems under the rug."

A few years later during his Nobel Prize acceptance speech he said: 

"It has not yet become obvious to me that there's no real problem. I cannot define the real problem; therefore, I suspect there's no real problem, but I'm not sure there's no real problem."

John K Clark    See what's on my new list at  Extropolis

w3v7f2

[Alan Grayson]

Do point charges exist in quantum field theory? Is it the electric field which is quantized? If not that, then what?  TY, AG 

w3v7f2

[Alan Grayson]

What I am asking is whether Feynman's renormalization procedure is specifically applied to a single point charge in quantum E&M? AG 

w3v7f2

[John Clark]

On Fri, May 16, 2025 at 11:34 PM Alan Grayson <agrays...@gmail.com> wrote:

 > Is it the electric field which is quantized?

Electrical charge is quantized in the sense that, unless you have access to a particle accelerator, everything you see in your daily life is made up of electrons with a -1 charge, top quarks with +2/3 charge, and bottom quarks with -1/3 charge. But that alone can't prevent a singularity or a point charge that contains an infinite amount of energy.

> Do point charges exist in quantum field theory? 

Currently the most accurate possible answer to that question is "sort of". From a distance it looks like a point charge but the closer you look the fuzzier the picture gets because the point charge is always surrounded by a sea of virtual particles, and the closer you look the denser that sea becomes. 

If you find that explanation unsatisfactory and you are unable to form a mental picture of what's going on then welcome to the club. Even the man who invented renormalization, Richard Feynman, was not shy about saying something didn't smell right about it despite it being able to make experimental predictions with fantastic accuracy, in fact the most accurate predictions in all of science. 

John K Clark    See what's on my new list at  Extropolis

tnw

[Brent Meeker]

You mean, "Is the electron assumed a point particle?".   Yes, but you exchange photons not charge.  At each photon-electron vertex you get a coupling constant of g=sqrt(4pi alpha) where alpha is the fine-structure constant    e^2/(hbar*c^2)=1/137  So that's the only way that the electron charge, e, enters and it's the experimental charge value.  No renormalization is needed since g<<1 and the terms don't blow up. 

For the strong force, even the vacuum propagator term blows up, so you subtract it off from all the higher order terms to get a finite remainder.

Brent

[Alan Grayson]

Is the classical singularity in the Coulomb force caused by the assumption of point sized particles, that have zero volume? If particles were modeled as very smal but continuous regions of charge, would the singularity go away when the distance to the center of the charge is less than its radius? AG 

[Brent Meeker]

Yes.

If particles were modeled as very smal but continuous regions of charge,would the singularity go away

Yes.

when the distance to the center of the charge is less than its radius? AG

??
Brent

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[Alan Grayson]

I was referring to the situation where the test charge is within the boundary of the source charge, assuming the latter has a finite radius. If the singularity is resolved by modifying the model of the source charge, can we conclude that a quantum theory of the EM field is NOT necessary to resolve this particular singularity? If so, what's the motivation of developing a quantum theory of EM? AG

[Brent Meeker]

The notional singularity isn't the only reason for wanting a quantum theory of gravity.  It's a also a matter of consistency in interactions and what goes on the right-hand side of Einstein's equation.

Brent

[Alan Grayson]

I was referring to a quantum theory of EM, not GR. Are all the singularities in the classical theory of EM resolved by changing the model of particles? AG 

[Brent Meeker]

They're resolved by quantum field theory, which I suppose could the thought of as a drastic change in the model of particles.

Brent

[Alan Grayson]

Modeling particles as having finite volume resolves the singularity of the Coulomb force at the center of a point charge. Are there other singularites in classical EM which remain despite this change in modeling? If so, can you describe some of them? TY, AG