= Solution
In two spatial dimensions there are only two kinetic matrices and one <mass> matrix. The three <Pauli matrices> square to identity and anticommute pairwise, so
$$
\boxed{H_{2+1}=c\sigma_xp_x+c\sigma_yp_y+mc^2\sigma_z}
$$
is a two-component Dirac <Hamiltonian> with $H_{2+1}^2=c^2(p_x^2+p_y^2)+m^2c^4$. It therefore reproduces the stated first-order equation and its Klein-Gordon square. Requiring a fourth anticommuting matrix in three spatial dimensions is precisely what prevented this two-component construction above.
\b[Printed angular-momentum qualification.] Define the dimensionless <spin> $s_i=-i\varepsilon_{ijk}\alpha_j\alpha_k/4$ and the physical orbital <angular momentum> $\ell_i=\varepsilon_{ijk}x_jp_k$, with $[x_i,p_j]=i\hbar\delta_{ij}$. Then
$$
[s_i,s_j]=i\varepsilon_{ijk}s_k,\qquad
[\ell_i,\ell_j]=i\hbar\varepsilon_{ijk}\ell_k,\qquad [\ell_i,s_j]=0.
$$
The source prints $L_i=i\ell_i$ instead. Taken literally, it is anti-Hermitian, has incompatible units with the dimensionless <spin> when they are added, and gives
$$
\boxed{[L_i,L_j]=-\hbar\varepsilon_{ijk}L_k,\qquad
[J_i,J_j]=\varepsilon_{ijk}(-\hbar L_k+i s_k).}
$$
It cannot yield the requested usual angular-momentum algebra or conservation law. For a concrete counterexample set $\hbar=1$ and take a wave with <momentum> along $z$. Since
$$
[H_D,\ell_i]=-ic(\boldsymbol\alpha\times\mathbf p)_i,
\qquad [H_D,s_i]=ic(\boldsymbol\alpha\times\mathbf p)_i,
$$
the printed $J_i=i\ell_i+s_i$ obeys $[H_D,J_x]=(1+i)c\alpha_y p_z\ne0$.
The physically consistent operator is $\mathbf J_{\rm phys}=\boldsymbol\ell+\hbar\mathbf s$. It satisfies
$$
\boxed{[J_{{\rm phys},i},J_{{\rm phys},j}]=i\hbar\varepsilon_{ijk}J_{{\rm phys},k},\qquad
[H_D,J_{{\rm phys},i}]=0.}
$$
The cancellation follows directly from the two mixed <commutators> above. In a spherical basis the free spinor can be labelled by <energy>, $j=1/2,3/2,\ldots$, and $m_j=-j,\ldots,j$, together with a <parity> or equivalent Dirac angular label distinguishing the two orbital couplings $\ell=j\pm1/2$. Alternatively one can choose a <momentum> and <helicity> basis. One cannot simultaneously assign all components of <momentum> and all angular labels as independent commuting observables.
Back to article page
