Direct observation of electron zitterbewgung The zitterbewegung (ZBW) frequency of the electron, the Compton frequency, is about 10^21 Hz. This is very high, but we see gamma rays with much higher energies. What's stopping us from probing the electron at this scale to see if ZBW is directly "real" or just an artifact of mathematical modeling? Could we even do so theoretically? Does it even make sense to probe an electron at resolutions this high?
 A: Our current understanding is based on quantum field theory (QFT), which doesn't predict zitterbewegung. Zitterbewegung was a feature of an ancient attempt to interpret a Dirac spinor as a state-vector in the Hilbert space on which observables act, but that interpretation famously doesn't work.
But this question isn't asking whether or not zitterbewegung is predicted. The question is asking whether or not it could be measured. That's a tricky question, because in order to answer it, we need to use established physical principles to predict whether or not electron oscillations with that frequency could be measured! The challenge, then, is to find some way of predicting what is measurable, without presuming whether or not that thing actually exists — which means we can't use our most firmly established physical principles (QFT), at least not for the electron itself.
I don't know of any good way to do that, but if we're willing to delete the word "good," then maybe there's a way. Here's one good way:
Instead of using quantum electrodynamics, which doesn't predict zitterbewegung, let's use a semiclassical model in which the electromagnetic field is still a quantum field but the electron is just a prescribed classical source. The lagrangian looks like this:
$$
 L = -\frac{1}{4}F_{ab}F^{ab} + A_a J^a
$$
where $J^a(x)$ is a prescribed current. The timelike component $J^0(x)$ is the charge density. Gauge invariance requires that the current is conserved: $\partial_a J^a = 0$, but otherwise it can be whatever we want. We can even choose it to be a pointlike source whose location oscillates in space with a frequency of $10^{21}$ Hz.
The question is, does this model predict a different behavior for the quantum EM field, compared to the case of a non-moving pointlike source with the same average location? The answer is yes, it does. However, this model also predicts that energy and momentum are not conserved. Insofar as it allows zitterbewegug to be observed, it also allows non-conservation of energy and momentum to be observed. That's why I crossed out the word "good."
Here's another good way: we can use a fully-quantum but not-fully-relativistic model in which a non-relativistic electron with a well-defined position observable is coupled to the quantum electromagnetic field. This model conserves energy and momentum, but now we face an even bigger problem: we can only prescribe the initial state, not the electron's whole history, and no matter what initial state we prescribe, zitterbewegung does not occur, so of course we can't measure it. Foiled again.
Maybe I'm just not creative enough, or maybe I'm just not motivated enough, but I don't know of any halfway-interesting models that are consistent with the general principles of quantum theory and conserve energy and momentum and accommodate zitterbewegung either as a prediction or as a manual input. Without that, we're stuck — stuck without a way to measure it, and stuck without a motivation for wanting to.
