Editing Dirac equation (section)

=== Coupled Weyl Spinors ===

As mentioned [[Dirac equation#Axial symmetry|above]], the ''massless'' Dirac equation immediately reduces to the homogeneous [[Weyl equation]]. By using the [[Gamma matrices#Weyl (chiral) basis|chiral representation of the gamma matrices]], the nonzero-mass equation can also be decomposed into a pair of ''coupled'' inhomogeneous Weyl equations acting on the first and last pairs of indices of the original four-component spinor, i.e. <math>\psi = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}</math>, where <math>\psi_L</math> and <math>\psi_R</math> are each two-component [[Weyl spinor]]s. This is because the skew block form of the chiral gamma matrices means that they swap the <math>\psi_L</math> and <math>\psi_R</math> and apply the two-by-two Pauli matrices to each:
<math display="block">\gamma^\mu \begin{pmatrix}\psi_L \\ \psi_R \end{pmatrix} = \begin{pmatrix}\sigma^\mu \psi_R \\ \overline{\sigma}^\mu \psi_L \end{pmatrix} .</math>

So the Dirac equation
<math display="block">
(i\gamma^\mu \partial_\mu - m)\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} = 0
</math>
becomes
<math display="block">
i\begin{pmatrix} \sigma^\mu \partial_\mu \psi_R \\ \overline{\sigma}^\mu \partial_\mu \psi_L \end{pmatrix} = m\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}
, </math>
which in turn is equivalent to a pair of inhomogeneous Weyl equations for massless left- and right-[[Helicity (particle physics)|helicity]] spinors, where the coupling strength is proportional to the mass:
<math display="block">
i\sigma^\mu \partial_\mu \psi_R = m \psi_L
</math>
<math display="block">
i\overline{\sigma}^\mu \partial_\mu \psi_L = m \psi_R
.</math>{{clarify|reason=In the Penrose source the RHS is divided by \sqrt{2} and there is no imaginary unit on the LHS, but he does not go into the derivation. Other sources -- and the Axial symmetry section above -- seem to agree with the form given here.|date=June 2023}}

This has been proposed as an intuitive explanation of [[Zitterbewegung]], as these massless components would propagate at the speed of light and move in opposite directions, since the helicity is the projection of the spin onto the direction of motion.<ref name="PenroseZigzag">{{cite book |last1=Penrose |first1=Roger |title=The Road to Reality |date=2004 |publisher=Alfred A. Knopf |isbn=0-224-04447-8 |pages=628–632 |edition=Sixth Printing}}</ref> Here the role of the "mass" <math>m</math> is not to make the velocity less than the speed of light, but instead controls the average rate at which these reversals occur; specifically, the reversals can be modeled as a [[Poisson process]].<ref name="PRL_1984_07_30">{{cite journal |last1=Gaveau |first1=B. |last2=Jacobson |first2=T. |last3=Kac |first3=M. |last4=Schulman |first4=L. S. |title=Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion |journal=Physical Review Letters |date=30 July 1984 |volume=53 |issue=5 |pages=419–422|doi=10.1103/PhysRevLett.53.419 |bibcode=1984PhRvL..53..419G }}</ref>