3

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?

nimish
  • 152

1 Answers1

2

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.

Chiral Anomaly
  • 54,035
  • I don't believe this is correct -- we need a ZBW-like effect to explain certain phenomena and if you want to be pedantic in QED you treat it as virtual electron/positron pairs interacting with the particle: https://arxiv.org/pdf/quant-ph/0612090.pdf

    Now I agree that the position operator as you described doesn't make sense, but the idea here is to probe at the scale where an electron would be said to be in trembling oscillatory motion by Dirac & Schrodinger but interacting with spontaneously created/destroyed virtual particles by Feynman & Schwinger. The name is arbitrary.

     – nimish Jun 29 '21 at 15:36
  • s/correct/relevant/ -- this is an important point when using QFT but I'm more interested in the "physical" behavior of the particle as it is, not as how we choose to model it. The map is not the territory! – nimish Jun 29 '21 at 15:42
  • @nimish Ah, then I think i misunderstood the question. Are you asking what would be observed using measurements with that resolution, according to qed? – Chiral Anomaly Jun 29 '21 at 18:58
  • Even in nonrelativistic qm, doing high resolution position measurements in rapid succession would give results that jump around randomly. That happens in qed, too, with the added complication of particle creation. Pair creation is one way (but not the best way) to understand why an electron position operator doesn't exist in qed. I'll delete the answer after you see this, but I'm still curious to understand what you're asking. – Chiral Anomaly Jun 29 '21 at 19:05
  • More like what is blocking us from doing such high frequency measurement? Is there a theoretical or physical block against us from seeing the matter wave/pair production/whatever somehow? I'd keep the answer, it's reasonable – nimish Jun 30 '21 at 00:00
  • @nimish To directly observe oscillations with freq $2mc^2/\hbar$, the measurements would need to have a time-resolution finer than $\hbar/2mc^2$. That requires using energies $E\sim 2mc^2$, which is enough energy to create an electron-positron pair. And that's just for one measurement. To directly observe oscillations, we'd need to use a rapid sequence of those measurements, which would tend to produce many electron-positron pairs. Electrons are indistinguishable, so I don't know how one could justify attributing the results as being due to oscillations in the position of any one electron. – Chiral Anomaly Jun 30 '21 at 01:01
  • @nimish For perspective: Experiments don't directly measure the lifetime $\tau$ of extremely short-lived particles. They measure the width $\Gamma=\hbar/\tau$ instead, which is the spread of energies over which the particle's signature can be observed in a scattering experiment. A lifetime of $\hbar/2mc^2\sim 10^{-21}$ s corresponds to a width of $\Gamma\sim 2mc^2$, which is $\sim 10\times$ the reported width of the famous $J/\psi$ particle. Its lifetime is $\sim 10\times$ longer than the period of zitterbewegung oscillations. – Chiral Anomaly Jun 30 '21 at 01:06
  • hmm, that makes more sense. Hestenes claims some sort of ZBW resonance could be observed, but part of my question is basically, would that mean anything? (Whether you chalk it up to high frequency oscillation or virtual pair production is i guess a philosophical thing) -- https://www.ime.unicamp.br/~agacse2018/abstracts/InvitedSpeakers/Hestenes-ElectronClock.pdf inspired this question – nimish Jul 05 '21 at 18:16
  • 1
    @nimish Hestenes' geometric perspective on Clifford algebra was helpful to me when I was first learning about Clifford algebra (which is what most mathematicians have always called it), and I'm grateful for that. However, based on some spot-checks, his 2018 paper seems to be describing a personal theory. His theory and QFT both involve the Dirac equation, but that doesn't mean much. Same equations $\neq$ same postulates. Hestenes' postulates seems pretty non-standard, so I'd have to spend more time studying it before I could say anything really helpful about what it predicts. – Chiral Anomaly Jul 05 '21 at 22:57
  • Yes, it's an interpretation shift, though the appearance of a spinor is required in both formalisms. I think his idea is that there's a real, geometric interpretation of spin as rotation (ZBW) vs the abstract rotation that spin in regular QFT uses. Not sure you could get away without some Dirac/$Cl_{1,3}(C)$ algebra though!

    So my question here was based on: if we did take spin to be a literal rotation, Hestenes claims it's basically ZBW, can we even measure such a thing or is it beyond us?

     – nimish Jul 05 '21 at 23:13