- Also, the Pauli equation will be written as:
$$\hat{H}\psi \equiv \hat{H}\psi^{B} \equiv H^{A}_{B}\psi^{B}. \tag{6}$$
Then,
$$H^{A}_{B}\psi^{B} = \frac{1}{2m}\Bigg[ \sigma^{jA}_{B}\Big(p_{j}-eA_{j}\Big)\Bigg]^{2}\psi^{B}+q\phi\delta^{A}_{B}\psi^{B}. \tag{7}$$
Now, the $\sigma^{jA}_{B}$ are the Pauli matrices and $\delta^{A}_{B} = \mathbb{I}_{2\times2} $ is the "$SU(2)$ identity matrix". The $j ={1,2,3}$ and $A,B={1,2}$, are, respectively, the spatial index and the spinor indices.
- Note that in the relativistic case (Dirac equation), the identity matrix and the gamma matrices are $4 \times 4$. This fact is a consequence of the Lie group structure of $SU(2) \times SU(2)$, which is the underlying structure of Weyl spinors $[2]$. A Pauli spinor transforms with a $SU(2)$ matrix, and therefore is a "two-colunm complex vector" $[3]$. Moreover, we have the spin up and down for Pauli spinors, which descrive only matter: $A,B= \{1,2\} \equiv \{\uparrow_{\mathrm{matter}}, \downarrow_{\mathrm{matter}} \} $; in the Dirac equation we thus have Weyl spinors, which describe both matter and antimatter in both spin up and down configurations: $A,B= \{1,2,3,4\} \equiv \{\uparrow_{\mathrm{matter}}, \downarrow_{\mathrm{matter}},\uparrow_{\mathrm{antimatter}}, \downarrow_{\mathrm{antimatter}} \} $.
$[1]$ https://www2.physics.ox.ac.uk/sites/default/files/2014-03-31/qcdgrad_rojo_oxford_tt14_2_basics_pdf_40958.pdf - Page 7
$[2]$ https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group (The unitarian trick section)
$[3]$https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Thvedt.pdf Page 5, above Theorem 4.4
