Dirac Hamiltonian

This is a compact Hamiltonian card. The covariant derivation, current, first-order initial-value problem, free operator domain, and QFT boundary are owned by the Covariant Dirac Equation; covariant background-field coupling is owned by Minimal Coupling.

Hamiltonian

The free Dirac Hamiltonian is

H = c\,\boldsymbol\alpha\cdot\hat{\mathbf p} + \beta mc^2,

where

\alpha^i=\gamma^0\gamma^i, \qquad \beta=\gamma^0.

With minimal electromagnetic coupling,

H = c\,\boldsymbol\alpha\cdot \left( \hat{\mathbf p}-q\mathbf A \right) + \beta mc^2 + q\Phi.

Operator setting

The free differential expression defines an operator on $\mathcal H_D=L^2(\mathbb R^3,\mathbb C^4)$ with dense domain $H^1(\mathbb R^3,\mathbb C^4)$ and codomain $\mathcal H_D$ ; on that domain it is self-adjoint. The displayed background-field expression is not, by itself, a complete operator declaration. For bounded real $\Phi$ and bounded real $\mathbf A$ , multiplication by the potentials is a bounded symmetric perturbation, so the $H^1$ domain remains a standard realization. Singular or unbounded backgrounds require a separate relative-boundedness or self-adjoint-extension analysis.

On a region with boundary, one must prescribe a self-adjoint boundary condition that controls the boundary form, equivalently the normal probability flux. Arbitrary componentwise boundary data need not make this first-order operator self-adjoint and can change its spectrum.

If the external potentials depend on time, an instantaneous self-adjoint expression is not enough: a unitary propagator also requires appropriate time regularity and compatible domains for the family $H(t)$ .

Assumptions

Gamma-matrix and metric conventions are fixed.
The Hamiltonian acts on four-component spinors in the operator setting stated above.
External electromagnetic fields are classical backgrounds.
Fixed-particle use is a bridge; interacting relativistic theory requires field quantization.

Reliability checks

With Hermitian $\alpha^i$ and $\beta$ , the free expression is symmetric on its stated domain.
The Clifford relations give $H_0^2=c^2\hat{\mathbf p}^{2}+m^2c^4$ .
Under a gauge transformation, the minimally coupled operator is related by the same unitary phase that transforms the spinor.
Setting $\Phi=\mathbf A=0$ recovers the free Hamiltonian without changing the rest-energy convention.

Common Mistakes

Mixing gamma-matrix conventions from different metric signatures.
Forgetting the scalar potential term in electromagnetic coupling.
Treating negative-energy solutions as ordinary nonrelativistic levels.
Using one-particle Dirac language where pair creation or field degrees of freedom are essential.

Canonical owner and released companions

References

P. A. M. Dirac, “The quantum theory of the electron”, Proceedings of the Royal Society A 117, 610-624, 1928.
J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, 1964.
S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.