The Dirac Equation with a 6x6 Matrix

Question

The Dirac Equation is $$ i\hbar \frac{\partial \Psi}{\partial t}=\left[c\sum_i{\alpha_i p_i}+mc^2\beta\right]\Psi $$ with the constraints $$ \{ \alpha_i,\alpha_j\}=2\delta_{ij} \\ \{ \alpha_i, \beta\}=0 \\ \{ \beta, \beta\}=2 $$ imposed to get the relativistic dispersion $E^2=(mc^2)^2+(pc)^2$. In 3D, the smallest size allowed for the $\alpha$ and $\beta$ matrices is 4x4, and this describes a spin 1/2 particle.

I've heard that using 6x6 matrices describes a spin 1 particle. I've also heard that Maxwell's Equations follow from the Dirac Equation using 6x6 matrices. Is this true? If so, do you have a reference for this? If not, what happens if I try to use a 6x6 or larger matrix to describe a particle?

Also, I know that 2x2 matrices describe a spin 1/2 particle in 2D. Is there an analogous Dirac Equation for 1D? I realize that proper rotations make no sense in 1D, so I don't expect there to be a 1D Dirac Equation, but if there is one, please correct me.

Answer 1

The electromagnetic field tensor expressed in spinor notation:

$F_{A C \dot{B}\dot{D}}=\sigma^{\mu}_{A \dot{B}}\sigma^{\nu}_{C \dot{D}} F_{\mu \nu}$

decomposes into a self dual and an anti-self dual parts:

$F_{A C \dot{B}\dot{D}}=\epsilon_{AC} \phi_{ \dot{B}\dot{D}} + \epsilon_{\dot{B}\dot{D}} \phi_{AC } $

(Where $\phi_{AC}$ is symmetric, thus contains 3 independent components. The components of $\phi_{AC}$ are just $\mathbf{E} + i \mathbf{B}$)

For a sourceless Maxwell theory, The Maxwell equations are equivalent to two homogeneous Dirac equations in the self dual and anti-self dual parts:

$\nabla^{A}_{\dot{B}} \phi_{AC } = 0$

$\nabla^{\dot{B}}_{A} \phi_{\dot{B}\dot{D} } = 0$

Since each equation has 3 independent components, they can be combined to get a signle 6 dimensional matrix equation.

Remark: For an introducton to the two component spinor notation, please see for examples the appendix to the following lecture notes by: Christian Saemann