Problems in classical electrodynamics: Only for point-like particles?

  • #1
greypilgrim
548
38
Hi.

I was surprised when I first read that there's quite a couple of unsolved problems in classical electrodynamics, such as the Abraham–Lorentz force. I have a couple of questions about that:
  1. Do those difficulties only appear for exact point-like particles? Do they all vanish with continuous charge densities (even if they might be localized around a very small, yet finite, region in space)?
  2. If yes: Isn't the assumption of point-like particles or also quantized charge already quantum, so why would we even expect classical electrodynamics to hold?
 
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  • #2
greypilgrim said:
Do those difficulties only appear for exact point-like particles? Do they all vanish with continuous charge densities (even if they might be localized around a very small, yet finite, region in space)?
As far as I know, yes. All of the mathematical inconsistencies stem from classical point charges.

greypilgrim said:
Isn't the assumption of point-like particles or also quantized charge already quantum, so why would we even expect classical electrodynamics to hold?
I agree. To me these issues speak more to the non-existence of classical point particles than anything else.
 
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FAQ: Problems in classical electrodynamics: Only for point-like particles?

What are the main challenges in classical electrodynamics when dealing with point-like particles?

One of the main challenges is the issue of infinite self-energy, which arises because the electric field of a point charge diverges as you approach the charge. This leads to mathematical difficulties in calculating the energy and momentum of the system. Additionally, point-like particles can lead to singularities in the equations of motion, making it difficult to apply standard analytical methods.

How does the concept of radiation reaction affect point-like particles in classical electrodynamics?

Radiation reaction refers to the self-interaction of a charged particle with its own emitted radiation. For point-like particles, this introduces terms in the equations of motion that account for the energy lost due to radiation. The most well-known formulation is the Lorentz-Abraham-Dirac equation, which includes a term for radiation reaction. However, this equation is notorious for its complex and sometimes non-physical solutions, such as pre-acceleration and runaway solutions.

Can classical electrodynamics provide a complete description of point-like particles?

No, classical electrodynamics cannot provide a complete and consistent description of point-like particles. The theory breaks down at very small scales due to issues like infinite self-energy and radiation reaction. These problems suggest that a more fundamental theory, such as quantum electrodynamics (QED), is needed to accurately describe the behavior of point-like particles.

What are some methods to regularize the infinities associated with point-like particles in classical electrodynamics?

Several methods have been proposed to regularize the infinities, including renormalization techniques borrowed from quantum field theory. Another approach is to model the charge distribution as a small, but finite, extended object rather than a true point-like particle. This can help to smooth out the singularities and provide finite results for quantities like self-energy and radiation reaction forces.

How does the concept of a point-like particle differ between classical electrodynamics and quantum electrodynamics?

In classical electrodynamics, a point-like particle is considered to have no spatial extent, leading to the aforementioned problems with infinities. In quantum electrodynamics (QED), particles are treated as excitations of underlying quantum fields, and the concept of a point-like particle is replaced by the idea of a particle with a probability distribution. This allows QED to handle interactions in a way that avoids the singularities and infinities that plague classical theories.

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