I was reading through Penrose's *Road to Reality* when I saw his interesting description of the Dirac electron (Chapter 25, Section 2). He points out that in the two-spinor formalism, Dirac's one equation for a massive particle can be rewritten as two equations for two interacting massless particles, where the coupling constant of the interaction is the mass of the electron. In the Dirac formalism, we can write the electron field as $\psi = \psi_L + \psi_R$ where $\psi_L = \frac{1}{2}(1-\gamma_5)\psi$ and $\psi_R = \frac{1}{2}(1+\gamma_5)\psi$. Then the Lagrangian is:

$$
\mathcal{L}=i\bar{\psi}\gamma^\mu\partial_\mu\psi - m\bar{\psi}\psi
$$
$$
\mathcal{L}=i\bar{\psi}_L\gamma^\mu\partial_\mu\psi_L+i\bar{\psi}_R\gamma^\mu\partial_\mu\psi_R - m(\bar{\psi}_L\psi_R + \bar{\psi}_R\psi_L)
$$

This is the Lagrangian of two massless fields, one left-handed and one right-handed, which interact with coupling constant $m$.

He then pictorially explains his interpretation by drawing this interaction Feynman-diagram-style. The initial particle (L or R) travels at speed $c$ (with luminal momentum) until it "interacts" and transforms into the other particle (R or L) which also travels at speed $c$ **but in the opposite direction**. This "zig-zagging" of the particles causes the two-particle system to travel at a net velocity which is less than $c$, thus giving the electron a subluminal momentum, thereby granting it a mass of $m$. He later states that this interaction is mediated by the Higgs boson, which we get if we replace the coupling constant $m$ with the Higgs field.

I've tried to put this argument into a mathematical setting, deriving the massive propagator from the two massless propagators via perturbation methods, but what I can't seem to get around is the **conservation of momentum**. When a "zig" particle changes into a "zag", the direction changes and thus the new particle gains momentum out of "nowhere." If we involve Higgs, then the Higgs boson could carry away/grant the necessary momentum, but I want to know if the model can work without involving the Higgs. Is this possible?

This post imported from StackExchange Physics at 2024-08-15 20:45 (UTC), posted by SE-user FrancisFlute