The Dirac Hamiltonian as an Operator

The formal Dirac Hamiltonian becomes a quantum evolution generator only after its Hilbert space and operator domain are specified. On all of three-dimensional space, the free operator is self-adjoint on the first Sobolev space. Its spectral projectors separate two orthogonal energy sectors while its exact Fourier evolution preserves the full positive spinor norm. These facts do not make its spectrum bounded below.

Required background. The Covariant Dirac Equation derives the Hamiltonian and current; Gamma-Matrix Conventions fixes the Hermitian matrices. Fourier transforms and self-adjoint operator domains are assumed.

The free realization on all of space

Let $m>0$ and use

$$\mathcal H=L^2(\mathbb R^3,d^3x;\mathbb C^4),\qquad H_D=c\boldsymbol\alpha\cdot\mathbf P+\beta mc^2, \quad \mathbf P=-i\hbar\nabla.$$

In momentum space this is multiplication by the Hermitian matrix $h(\mathbf p)=c\boldsymbol\alpha\cdot\mathbf p+\beta mc^2$, with $h(\mathbf p)^2=E_{\mathbf p}^2I$ and $E_{\mathbf p}=\sqrt{c^2\mathbf p^2+m^2c^4}$. The natural domain is

$$D(H_D)=\left\{\psi\in\mathcal H: \int d^3p\,E_{\mathbf p}^2 \|\widetilde\psi(\mathbf p)\|_{\mathbb C^4}^2<\infty\right\} =H^1(\mathbb R^3;\mathbb C^4).$$

The maximal Hermitian matrix multiplier is self-adjoint on this domain. Equivalently, the identity

$$\|H_D\psi\|^2=c^2\|\mathbf P\psi\|^2+m^2c^4\|\psi\|^2$$

shows that its graph norm is equivalent to the first Sobolev norm. Smooth compactly supported spinors form a core for this free realization. The formula specifies a densely defined self-adjoint operator, not just a matrix that happens to be Hermitian at each point.

Spectral projectors and the two continua

At fixed momentum define

$$P_\pm(\mathbf p)=\frac12 \left(I\pm\frac{h(\mathbf p)}{E_{\mathbf p}}\right).$$

Since $(h/E)^2=I$, these are Hermitian orthogonal projectors with $P_++P_-=I$, $P_+P_-=0$, and $hP_\pm=\pm E P_\pm$. Their traces are two, so each energy has two spin states. The operator spectrum is

$$\sigma(H_D)=(-\infty,-mc^2]\cup[mc^2,\infty).$$

For an arbitrary normalized packet, $\|P_+\psi\|^2+\|P_-\psi\|^2=1$. Both terms are nonnegative ordinary Hilbert-space norms. Negative energy is not negative Dirac probability. The interpretation of that second energy sector is developed separately in Negative-Energy Solutions.

Because $P_\pm(\mathbf p)$ depend nonpolynomially on momentum, their position-space action is nonlocal. Discarding one sector is not the same operation as deleting two fixed components of the Dirac-basis spinor. At rest the energy projectors reduce to $(I\pm\beta)/2$, but at nonzero momentum upper and lower components mix.

Exact unitary evolution

The spectral theorem gives

$$\begin{aligned} e^{-iH_Dt/\hbar} &=e^{-iEt/\hbar}P_++e^{+iEt/\hbar}P_-\\ &=\cos(Et/\hbar)I -i\frac{H_D}{E}\sin(Et/\hbar), \end{aligned}$$

where $E=\sqrt{c^2\mathbf P^2+m^2c^4}$ is the scalar positive multiplier and commutes with $H_D$. This expression is useful for exact free spectral stepping. It advances any four-component initial datum and preserves its norm; it does not require a separately chosen initial time derivative.

The free positive and negative sector norms are individually constant. A perturbation that does not preserve these free spectral subspaces can mix them. The resulting c-number evolution is still unitary for a self-adjoint Hamiltonian, but its interpretation as an isolated electron may fail. Unitarity and isolation of a physical particle sector are separate statements.

Potentials and boundaries change the operator problem

A bounded Hermitian matrix-valued multiplication potential $V(\mathbf x)$ is a bounded self-adjoint perturbation, so $H_D+V$ is self-adjoint on the same $H^1$ domain. This simple sufficient criterion does not cover arbitrary singular Coulomb potentials or unbounded vector potentials. Those require their own domain or form analysis. It also does not ensure a gap of exactly $2mc^2$ for the perturbed spectrum.

On a spatial region $\Omega$, integration by parts gives the boundary form

$$\langle\phi,H_D\psi\rangle-\langle H_D\phi,\psi\rangle =-i\hbar c\int_{\partial\Omega} \phi^\dagger(\boldsymbol\alpha\cdot\mathbf n)\psi\,dS.$$

A self-adjoint boundary realization must make this bilinear form vanish on its domain and supply the required maximality condition. Merely checking zero flux for one state is insufficient. Periodic conditions on a box cancel opposite-face contributions and give a standard self-adjoint free example. Imposing every spinor component to vanish at a boundary is not automatically a self-adjoint condition for a first-order operator.

Velocity within a fixed energy sector

The unprojected position operator obeys $\dot X_i=(i/\hbar)[H_D,X_i]=c\alpha_i$ on a suitable common core. Each component of $c\boldsymbol\alpha$ has eigenvalues $\pm c$, but a positive-energy plane wave can have any speed below $c$. There is no contradiction: projecting the observable changes its action. From $\{h,\alpha_i\}=2cp_iI$ one obtains

$$P_\pm c\alpha_iP_\pm =\pm\frac{c^2p_i}{E_{\mathbf p}}P_\pm.$$

This is the group velocity in that energy sector. The off-diagonal parts $P_+\alpha_iP_-$ and its adjoint connect the sectors and are responsible for the rapid interference term studied in zitterbewegung. They should not be confused with a superluminal expectation value.

Exercises

1. Verify $P_\pm^2=P_\pm$ and compute their ranks.

Solution

$P_\pm^2=(I\pm2h/E+h^2/E^2)/4=P_\pm$. The matrices $\alpha_i,\beta$ are traceless, hence $\operatorname{tr}P_\pm=4/2=2$. A finite Hermitian projector has rank equal to its trace.

2. Derive the projected velocity identity by sandwiching $\{h,\alpha_i\}=2cp_iI$ between $P_\pm$.

Solution

Both appearances of $h$ become $\pm E$ on the selected sector, so $\pm2E P_\pm\alpha_iP_\pm=2cp_iP_\pm$. Multiplying by $c$ gives the stated velocity.

3. Does an energy expectation $\langle H_D\rangle=0$ imply a zero spinor or a failure of norm conservation?

Solution

Neither. Equal-norm packets in opposite energy sectors with matching energy distributions can have cancelling energy expectations and nonzero total norm. Each evolves unitarily. Whether such a superposition models an intended physical preparation is a separate question.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964 — free spinor evolution.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, revised and enlarged ed., Academic Press, 1980 — spectral calculus and self-adjoint evolution.
• B. Thaller, The Dirac Equation, Springer, 1992 — domains, spectral projections, and boundary realizations.