In his book "The Road to Reality" chapter 25.2 Roger Penrose describes the motion of an electron as a permanent change of helicity states in form of a zig-zag. He takes the Dirac-equation in Weyl-representation (I guess, Penrose changed the dot on the $\beta$-spinor to a prime in order to not to clutter up the notation in a book that is more orientated to the interested layman than to real physicists):
$$ \nabla^A_{\,B'}\alpha_A = 2^{-1/2} M \beta_{B'}, \quad \nabla^{B'}_{\,A} \beta_{B'} = 2^{-1/2} M \alpha_A$$
He considers the Dirac electron being composed of two ingredients $\alpha$ and $\beta'$ and gives a physical interpretation of these ingredients. It consists of two particles, one described by $\alpha_A$ and the other described by $\beta_B'$, both massless. Each one is continually converting itself into the other one. He calls one of these the 'zig' particle (described by $\alpha_A$) and the other the 'zag' particle (described by $\beta_{B'}$). Being massless both particles travel at speed of light. A picture (that I imitated) illustrates that well:
The picture is supposed to describe the electron in its rest-frame. If the electron is observed moving from the laboratory's rest-frame, the 'zig's and 'zag's no longer appear to contribute equally to the overall motion.
He even calls the jiggling motion of the electron as "Zitterbewegung" which was first described by Schroedinger. However, it is rather a jiggling between to the two helicity states (instead of positive and negative energy states), the zig-particle is supposed to be left-handed whereas the zag-particle is supposed to be right-handed. Both "particles" are supposed to be coupled by the mass term.
The Penrose's picture is very compelling since it also includes an intuitive description for the chirality of the electron in weak interaction. While the 'zig' particle couples to $W^{\pm}$, the 'zag' particle does not. So it looks that the 'zig' particle actually corresponds to $e_L$ (left-handed electron) and the 'zap' particle to the $e_R$ (right-handed electron). A couple of pages later (p.644) he mentions that the coupling between the 'zig' and the 'zag' particle can be considered as caused by the Higgs field. This means that his description seems to be perfect. Another detail in that context is that according to Penrose the velocities of the 'zig' and 'zag' particle are $\pm c$ --- as both are considered as massless --- which corresponds to the value of the velocity operator $\mathbf{\alpha} = \gamma^0 \mathbf{\gamma}$ of the Dirac theory:
$$-i[\mathbf{x}, H ]=\mathbf{\alpha}$$
whose eigenvalues are -- in units of $c$ -- $\pm 1$.
Today "Zitterbewegung" is considered as a failure of relativistic Quantum mechanics, the Dirac equation should be better considered as an equation for a field operator than for an one-particle wave-function. But actually what is so wrong about the "Zitterbewegung" ? The jiggling computed by Schroedinger is extremely high-frequent, would it be ever possible to observe such a high-frequent effect ? And if it cannot be observed, why should it be then a problem ?
My question might appear not very clear, in particular in this post I refer to Penrose's "Zitterbewegung" and not the one found by Schroedinger. Therefore I pose another question: Is there any relationship imaginable between Schroedinger's Zitterbewegung and Penrose's one ?
It is tempting to describe Schroedingers "Zitterbewegung" in a similar way but a 'zig' particle having positive energy and a 'zag' particle having negative energy and the coupling between both would be generated by a virtual photon (instead of a Higgs), but that would require that the mass-term in the Dirac-equation would be replaced by something proportional to the electron charge, so it does not seem to be possible.
It therefore seems that Penrose's approach using instead of (positive & negative) energy states helicity states fits much better. This leads to my final question: As Penrose's picture of "Zitterbewegung" fits so well, would it not be a second opportunity to give the so "miscarried" "Zitterbewegung" a new sense? I am aware that this is a rather controversial topic, I would appreciate a fair discussion.
