Traditionally for a free electron, we presume the expectation of its location (place of the center of mass) and the center of charge at the same place. Although this seemed to be reasonable for a classical approximation (see: Why isn't there a centre of charge? by Lagerbaer), I wasn't sure if it's appropriate for quantum models, and especially for some extreme cases, such as high energy and quark models.

My questions are:

  1. Is there any experimental evidence to support or suspect that the center of mass and charge of an electron must coincide?

  2. Is there any mathematical proof that says the center of mass and charge of an electron must coincide? Or are they permitted to be separated? (By electric field equation from EM, it didn't give enough evidence to separate $E$ field with $G$ field. But I don't think it's the same case in quantum or standard model, i.e. although electrons are leptons, consider $uud$ with $2/3,2/3,-1/3$ charges.)

  3. What's the implication for dynamics if the expectation of centers does not coincide?


Can the center of charge and center of mass of an electron differ in quantum mechanics?

They can. Particle physics does allow for electrons (and other point particles) to have their centers of mass and charge in different locations, which would give them an intrinsic electric dipole moment. For the electron, this is unsurprisingly known as the electron electric dipole moment (eEDM), and it is an important parameter in various theories.

The basic picture to keep in mind is something like this:

Image source

Now, because of complicated reasons caused by quantum mechanics, this dipole moment (the vector between the center of mass and the center of charge) needs to be aligned with the spin, though the question of which point the dipole moment uses as a reference isn't all that trivial. (Apologies for how technical that second answer is - I raised a bounty to attract more accessible responses but none came.) Still, complications aside, it is a perfectly standard concept.

That said, the presence of a nonzero electron electric dipole moment does have some important consequences, because this eEDM marks a violation of both parity and time-reversal symmetries. This is because the dipole moment $\mathbf d_e$ must be parallel to the spin $\mathbf S$, but the two behave differently under the two symmetries (i.e. $\mathbf d_e$ is a vector while $\mathbf S$ is a pseudovector; $\mathbf d_e$ is time-even while $\mathbf S$ is time-odd) which means that their projection $\mathbf d_e\cdot\mathbf S$ changes sign under both $P$ and $T$ symmetries, and that is only possible if the theory contains those symmetry violations from the outset.

As luck would have it, the Standard Model of particle physics does contain violations of both of those symmetries, coming from the weak interaction, and this means that the SM does predict a nonzero value for the eEDM, which falls in at about $d_e \sim 10^{-40} e\cdot\mathrm m$. For comparison, the proton sizes in at about $10^{-15}\:\mathrm m$, a full 25 orders of magnitude bigger than that separation, which should be a hint at just how small the SM's prediction for the eEDM is (i.e. it is absolutely tiny). Because of this small size, this SM prediction has yet to be measured.

On the other hand, there's multiple theories that extend the Standard Model in various directions, particularly to deal with things like baryogenesis where we observe the universe to have much more asymmetry (say, having much more matter than antimatter) than what the Standard Model predicts. And because things have consequences, those theories $-$ the various variants of supersymmetry, and their competitors $-$ generally predict much larger values for the eEDM than what the SM does: more on the order of $d_e \sim 10^{-30} e\cdot\mathrm m$, which do fall within the range that we can measure.

How do you actually measure them? Basically, by forgetting about high-energy particle colliders (which would need much higher collision energies than they can currently achieve to detect those dipole moments), and turning instead to the precision spectroscopy of atoms and molecules, and how they respond to external electric fields. The main physics at play here is that an electric dipole $\mathbf d$ in the presence of an external electric field $\mathbf E$ acquires an energy $$ U = -\mathbf d\cdot \mathbf E, $$ and this produces a (minuscule) shift in the energies of the various quantum states of the electrons in atoms and molecules, which can then be detected using spectroscopy. (For a basic introduction, see this video; for more technical material see e.g. this talk or this one.)

The bottom line, though, as regards this,

Is there any experimental evidence to support or suspect the center of mass and charge of an electron must coincide?

is that the current experimental results provide bounds for the eEDM, which has been demonstrated to be no larger than $|d_e|<8.7\times 10^{−31}\: e \cdot\mathrm{m}$ (i.e. the current experimental results are consistent with $d_e=0$), but the experimental search continues. We know that there must be some spatial separation between the electron's centers of mass and charge, and there are several huge experimental campaigns currently running to try and measure it, but (as is often the case) the only results so far are constraints on the values that it doesn't have.

  • $\begingroup$ Nice answer, perhaps you could mention at the end an example or two of experiments where this is being measured? (I personally don't know) $\endgroup$ – Kai Sep 18 '18 at 15:58
  • $\begingroup$ @Kai See the talks already linked to. $\endgroup$ – Emilio Pisanty Sep 18 '18 at 16:00
  • $\begingroup$ Can the center of electric charge, and color charge differ in a quark? $\endgroup$ – Anders Gustafson Sep 18 '18 at 16:56
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    $\begingroup$ @AndersGustafson My knowledge of SU(3) theory is nowhere near up to scratch to answer that. It's an interesting question; you should ask it separately. $\endgroup$ – Emilio Pisanty Sep 18 '18 at 16:58
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    $\begingroup$ I'm not sure that a fundamental EDM of a point particle should be described as if that particle had size. I get the analogy you're going for, but the reason we don't typically use it is that the analogous analogies for spin and magnetic dipole moment don't work at all. $\endgroup$ – knzhou Sep 18 '18 at 22:09

For the simple reason that the current standard model of particle physics has the electron as a point particle, i.e. volume =0 , and the charge,spin and mass all reside at that zero point. The same is true for all particles in the table. The model is validated continually by experiments, and is the mainstream one at this time.

1)Is there any experimental evidence to support or suspect the center of mass and charge of an electron must coincide ?

When one defines a point particle mathematically, as in the standard model, it is a mathematical constraint.

Please note that the statement " the center of mass and the center of charge must be the same", can arise by mathematical symmetry arguments. Even if charge distributions vary in composite objects, depending on the geometry, the symmetry also for free particles will mathematically give a coincidence.

This answers 2) also

What's the implication for dynamics if the expectation of centers does not coincide

It might be connected with the hypothesis of compositeness, i.e. that the elementary particles are not elementary, but have an internal structure. One would need an experiment that would distort the mathematical symmetry of the charge distribution, possibly with a strong electric field, and a model where the point particle and the composite hypothetical charge interactions are studied.

There are limits given by experiments on the compositeness scale. See my answer here to a related question.

Even if it turns out that elementary particles at some high scale display compositness, I would be surpised if they would display asymmetry in the charge distribution versus the mass distribution.

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    $\begingroup$ This answer is rather wide of the mark, I think. You do not need to assume any compositeness hypothesis to have a nonzero electron dipole model (which is the only serious way to understand "the centers of mass and charge do not coincide"), and indeed the Standard Model already predicts a nonzero separation. (It's some ten orders of magnitude beyond the current molecular-spectroscopy experiments, but still.) $\endgroup$ – Emilio Pisanty Sep 18 '18 at 11:02
  • $\begingroup$ As for the claim that "symmetry arguments" entail that the center of mass and the center of charge must be the same -- I don't doubt that one can come up with such arguments, but they'd be wrong. The electron isn't symmetric enough (it has spin) and the SM isn't symmetric enough (it contains both P, T, and C violations) to provide enough symmetry grounds for such an argument to hold. This is clearly seen in the fact that the SM predicts a nonzero eEDM. $\endgroup$ – Emilio Pisanty Sep 18 '18 at 11:05
  • $\begingroup$ @EmilioPisanty please enlighten me , isn't the quantum mechanical charge distribution of a dipole symmetric? The center would be at the center which would coincide with the center of mass distribution as seen in the link I gave for the molecules. Maybe we have a different definition? The center of mass of a rod is at the center. A charged rod + on one side - on another would be a dipole, but the center of charge would be at the center of the rod. $\endgroup$ – anna v Sep 18 '18 at 11:33
  • $\begingroup$ In my definition one would need more charge on one side in order to have a different center of charge to center of mass. $\endgroup$ – anna v Sep 18 '18 at 11:39
  • $\begingroup$ @annav To the extent that classical pictures make sense (which is obviously extremely limited), the symmetric dipolar charge distribution is entirely the wrong model; the correct mental picture is an offset between the charge distribution and the mass distribution, as shown at the 2:30-2:45 mark in this video. Perhaps you're thinking that only neutral systems can have dipole moments? That's a common misconception, but it's just not true. $\endgroup$ – Emilio Pisanty Sep 18 '18 at 11:42

you'd see that in the bubble/cloud chamber, if the center of mass and charge were different, then you'd have some torque effect that would completely change the electrons trajectories(when they are under electric/magnetic field).

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    $\begingroup$ A cloud chamber would be "too coarse" to be able to detect the eEDM! $\endgroup$ – Guill Sep 19 '18 at 22:42
  • $\begingroup$ yes I learned that reading the other answers. Apparently everything is too coarse by 10 orders of magnitudes at least. $\endgroup$ – Manu de Hanoi Sep 20 '18 at 0:15
  • $\begingroup$ But anyways , OP didnt specify the distance between center of mass and center of charge $\endgroup$ – Manu de Hanoi Sep 20 '18 at 1:03

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