# Why isn't it obvious that a particle doesn't interact with its own field, classically?

The Wheeler-Feynman absorber theory or any other theory that tries to avoid the notion of field as an independent degree of freedom has always been concerned about infinite self energy of a charged particle. I don't feel confident that I appreciate the gravity of this problem.

According to the theory of fields every charged particle has a field of its own whose strength varies with distance from the particle and this field is what which the particle uses to exert force on other charged particles. So, if the field is supposed to be an inalienable part attached to the particle that it uses to interact with other particles why would the field be used to exert a force on the particle itself and as such why an infinite self energy at the points very close to it? Am I ignoring any implication of this theory that leads to a logical debacle? I hope with your elaborate answers I can see why this isn't something to be deemed obvious and can appreciate the seriousness of the Abraham force.

• You might consider reading Feynman's Nobel lecture on "The Development of the Space-Time View of Quantum Electrodynamics". It has some interesting insight on this problem of self-energy. Sure, it is older and dated but clear and understandable. – K7PEH Apr 16 '15 at 14:44
• @K7PEH : Hope you can provide some links to that. – Sheldon Kripke Apr 16 '15 at 14:48
• Sorry, I assumed you could easily find it via Google. It is actually on the Internet in at least a dozen different locations, some printed PDFs, others are formatted web pages such as the Nobel location I am including here: nobelprize.org/nobel_prizes/physics/laureates/1965/… – K7PEH Apr 16 '15 at 14:53
• @K7PEH : Could you walk me through the infinite degrees of freedom problem mentioned here ? – Sheldon Kripke Apr 23 '15 at 3:48
• which problem are you referring to? I think you should post another question and specifically include sufficient quotes from the paper to give context to the question. If you mean infinite number of H.O. states then it is just that. There are an infinite number of states for each H.O. from ground state all the way up. – K7PEH Apr 24 '15 at 0:16

## 3 Answers

The main problem is radiation reaction. It's a fact that if I take a charge and jiggle it about, it will somehow "inject" energy into the electromagnetic field around it; we know this because we can detect this energy in the form of electromagnetic radiation. This means that when I grabbed the charge and jiggled it, I must have performed extra work on it (above the work needed to accelerate its mass) which then ends up as energy in the far field. This extra work can only have been performed against a force, which is electrical in nature since it depends on the charge. Since the only electric field present is that of the charge, we conclude that the charge must have interacted with its own field in some way.

However, it's also obvious that a point charge cannot interact with its own electric field in the same way that it does with the field of other charges. At $\mathbf r=0$, the electric field $$\mathbf E=q\frac{\mathbf r}{r^3}$$ is singular and ill-defined; it doesn't even have a direction. So the self-interaction needs to be something different.

Everything that follows is an attempt to patch together these two disparate facts. How do you account exactly for that radiated energy, and how can you formulate consistent laws that have energy conservation built in intrinsically? In the far field, it is easy to account for the radiated energy via an electromagnetic energy density proportional to $|\mathbf E|^2$, but if you take this at face value then it blows up if you have point charges. If you try to take this directly but remove the point charge self energies by hand, the resulting theory is clunky, hard to use, and its results are not always Lorentz invariant. If you don't ascribe an energy content to the field (which means you cannot be considering it as a dynamical variable, as done by e.g. Wheeler-Feynman), how do you account for the fact that one can transfer energy via electromagnetic radiation?

More fundamentally, though: do the electromagnetic fields at a given point depend on how they were made? Is it enough to say "the electric field at $\mathbf r$ is 5V/m pointing in the $\hat{\mathbf z}$ direction"? Or do we need to specify which point charges created it? The latter stands very much against the concept of field, and how we use it in practice to calculate and prove things, so we'd need to have a very close look at all of electrodynamics (i.e. where is the 'which-charge' information in Maxwell's equations?). If the former is true, however, with what justification can we just yank out certain specific charge-dependent components from the EM field energy?

So you see, these are tough questions, and they are nowhere near solved, so it's an interesting area (but also it's not clear whether it's even possible to solve these questions, so keep that in mind).

• I am extremely thankful for your effort . I feel I need to give a bit more thought to it ,so will come back to you soon – Sheldon Kripke Apr 18 '15 at 8:50
• Take your time - these are hard matters to digest. Do read Ribarič and Šušteršič's work (e.g. arXiv:1005.3943) - it can be a bit hard to take in but they are very upfront about the stuff that's nowhere near solved. – Emilio Pisanty Apr 18 '15 at 11:19
• Ok, I marked the phrase "inject energy into the electromagnetic field "..what's wrong with omitting the term electromagnetic field and replacing it by "space" or "vacuum" ? Basically , in the spirit of my first question on this website ,I would like to ask why do we assume the notion of a physical entity field ? – Sheldon Kripke Apr 22 '15 at 6:30
• :As I have said in this question , why can't we say that a particle uses it's field only to exert force on others or why can't we define that a particle's field is an entity that is attached with a particle and is used to exert force on other particles ? – Sheldon Kripke Apr 22 '15 at 6:41
• Just to confirm , could you walk me through the reason as to why a charged particle when accelerated should radiate energy ? – Sheldon Kripke Apr 22 '15 at 6:41

I don't really know any high energy QFT, so I don't think I can explain a QFT concept in terms of classical physics and relate the two, but if I've understood the question I can give a pretty classical guess. Coulomb interaction is mediated by the electromagnetic field, and this interaction travels at the speed of light in vaccuum. So the electron "broadcasts" this signal, but it can't interact with the signal because the signal is flying away at the speed of light. As a naive side note: I think that if you tried to construct a theory of superluminal charged particles, you would need to consider the interaction of the particle with itself.

It does interact with its own field! This is the only way to get an accelerating charge to radiate away energy -- it has to lose energy itself, so it has to be repulsed by its own field.

• could you elaborate this as the only way to get an accelerating charge to radiate ? – Sheldon Kripke Apr 16 '15 at 14:42
• Emilio's answer is very good. – Scott Lawrence Apr 16 '15 at 19:36