= Dirac Hamiltonian
{c}
The Dirac Hamiltonian is the first-order time-evolution operator obtained from the <Dirac equation>. For a free massive particle,
$$
H=c\boldsymbol\alpha\cdot\mathbf P+\beta mc^2,
$$
where the <Dirac matrices> satisfy $\{\alpha_i,\alpha_j\}=2\delta_{ij}I$, $\{\alpha_i,\beta\}=0$ and $\beta^2=I$. These identities give $H^2=c^2P^2+m^2c^4$, recovering the <relativistic energy-momentum relation>.
Minimal electromagnetic coupling replaces the <canonical momentum> by $\mathbf P-q\mathbf A$ and adds $q\phi$, in SI units. The wavefunction has spinor components, and the free operator has positive- and negative-energy branches. The <Dirac field> interprets particle and antiparticle excitations with a stable vacuum; it is more complete than treating the negative branch as an ordinary one-particle spectrum with no lower bound.
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