Infinite Self-Energy of a Charged Point Particle

In summary: If you perturb the electron cloud close to the nucleus you can see the perturbation spreading outwards like ripples on a pond.In summary, the problem of a heavy electron in bound atomic orbits is solved by deriving its motion using the Uehling integral.
  • #1
Himanshu
67
0
Recently I became acquainted with the problem of Infinite Self-Energy of a Charged Point Particle as described by both classical and quantum theory. Infinities of this kind certainly hint at the inconsistency and incompleteness of the theory itself. I was speculating that infinity is arising dew to the dimensionless nature of the particle. But in the context of string theory this issue could have been resolved as we now do no longer have to deal with point particles. I was wondering if this issue has been resolved by other theories.
 
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  • #2
How are the singularities get resolved, I thought singularities are inherent in every mathematical theory of physical phenomena, is it not?
 
  • #3
Himanshu, what references were you reading? I understood that particular classical infinity becomes zero in quantum mechanics.
 
  • #4
You can "renormalize" in classical physics just like you do in quantum physics. Of course, there are no problems at all with classical continuum mechanics.
 
  • #5
cesiumfrog said:
Himanshu, what references were you reading? I understood that particular classical infinity becomes zero in quantum mechanics.

I am referring to Griffiths- Introduction to Electrodynamics. The book says that the infinite energy of a point charge is a reccuring source of embarassment for electromagnetic theory afflicting quantum version as well as the classical. I have no Idea about how the self-energy becomes infinite in the context of QM, but in classical EM theory its a straightforward calculation.
 
  • #6
Stingray said:
You can "renormalize" in classical physics just like you do in quantum physics. Of course, there are no problems at all with classical continuum mechanics.

Do you have any example of a problem that can be solved classically by renormalisation?
 
  • #7
cesiumfrog said:
Do you have any example of a problem that can be solved classically by renormalisation?

The typical place where the self-energy of a point charge is considered is in deriving the motion of such an object. Roughly, you get an equation that looks like

[tex]
m a = q E - (q^2 / r) a + \ldots
[/tex]

for a body with charge q and radius r. The first term is the ordinary force on a charged particle. The second is due to the inertia of its self-field. Rearranging,

[tex]
(m + q^2/r) a = q E + \ldots
[/tex]

The left-hand side diverges if r is shrunk while keeping q and m fixed. So you say that only [itex]m + q^2/r[/itex] is observable. That's the effective or renormalized mass. You imagine that m is infinitely negative in order to get a finite sum.

This is closely analogous to things done in QFT. The difference is that the real classical theory has objects with finite r. The mass still shifts (so it may be said to "renormalize"), but there are no infinities that get swept away.
 
  • #8
Many years ago I had to solve the problem of a heavy electron (like the negative muon) in bound atomic orbits around nuclei. This is a two body problem, because the muon is inside (nearly) all the electron cloud. When the muon is within a few Fermi of the nucleus, the electric field is so strong that it can create virtual electron-positron pairs (vacuum polarization) (within the limits of the uncertainty principal), and modify the one-over-r^2 dependence of the electric field. This was solved using the Uehling integral (Phys Rev, about 1937). The Uehling integral has been validated with both muonic and pionic atoms, and (I think) electron scattering. The cloud of virtual electrons and positrons in vacuum polarization shield the bare (un-normalized) charge.
 
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Related to Infinite Self-Energy of a Charged Point Particle

1. What is the concept of infinite self-energy of a charged point particle?

The concept of infinite self-energy of a charged point particle refers to the idea that an infinitely small particle, or point, with a non-zero charge would theoretically have an infinite amount of energy. This concept arises from the classical theory of electromagnetism and is a result of the mathematical divergence when trying to calculate the energy of a point particle.

2. Why is the concept of infinite self-energy important in physics?

The concept of infinite self-energy is important in physics as it highlights the limitations of classical theories and the need for more advanced theories, such as quantum mechanics, to accurately describe and predict the behavior of particles at a subatomic level. It also has implications in the study of gravity and the creation of a unified theory of physics.

3. Can the infinite self-energy of a charged point particle be physically observed?

No, the concept of infinite self-energy is a theoretical concept and cannot be physically observed or measured. It is a mathematical result that arises when trying to calculate the energy of a point particle and is not a physical quantity that can be measured.

4. How does the concept of infinite self-energy relate to the concept of singularity?

The concept of infinite self-energy is often linked to the concept of a singularity, which is a point in space where the laws of physics break down and become infinite. This is because the infinite self-energy of a point particle is a mathematical singularity, meaning it is undefined and has no physical meaning.

5. Are there any proposed solutions to the problem of infinite self-energy?

Various solutions have been proposed to address the issue of infinite self-energy, such as renormalization in quantum field theory and the introduction of cutoffs in calculations. However, these solutions are still subject to debate and further research is needed to fully understand and resolve the problem of infinite self-energy.

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