Common Relativistic Hamiltonians

Hamiltonians that share a low-energy limit can act on different state spaces and retain different energy sectors. This comparison identifies those choices before listing formulas. Each entry links to its derivation; the general Hamiltonian index covers a wider range of quantum models.

Required background. The Dirac Hamiltonian defines the free operator and sectors. Helpful background. Minimal Coupling fixes charge signs; the Foldy–Wouthuysen Expansion states the approximation behind the correction terms.

State space and energy sector determine the generator

Keep $\hbar,c$ explicit and use SI electromagnetic potentials:

$$\begin{aligned} \mathbf p&=-i\hbar\nabla,\\ \boldsymbol\pi&=\mathbf p-q\mathbf A,\\ V&=q\Phi,\qquad M=mc^2. \end{aligned}$$

The charge $q$ is signed; an electron has $q=-e$ with $e>0$. Write $\mathcal E=\sqrt{M^2+c^2\mathbf p^2}$ for the positive free energy operator.

Description and owner	Generator	State space and sector
Free Dirac	$H_D=c\boldsymbol\alpha\cdot\mathbf p+\beta M$	Four components; both energy signs
Dirac in prescribed EM fields	$H_D=c\boldsymbol\alpha\cdot\boldsymbol\pi+\beta M+V$	Four components; sector separation depends on background
Exact free FW representation	$H_{\rm FW}=\beta\mathcal E$	Four components; two diagonal energy blocks
Positive square-root theory	$H_+=\mathcal E$	Positive sector; scalar or a specified spin degeneracy
Pauli approximation	$H_P=\boldsymbol\pi^2/(2m)+V-q\hbar\boldsymbol\sigma\cdot\mathbf B/(2m)$	Two components; leading positive-sector reduction, $m>0$

The first four rows retain the full relativistic mass term; their free positive-energy branch includes $mc^2$. The Pauli row uses a wavefunction with the rest phase $e^{-iMt/\hbar}$ removed, so its generator is measured relative to $M$. One may also use $H_+-M$ for the square-root theory, but that subtraction must be stated.

On all of $\mathbb R^3$, the free Dirac realization has domain $H^1(\mathbb R^3,\mathbb C^4)$ and Hilbert space $L^2(\mathbb R^3,\mathbb C^4)$. A bounded real scalar potential and bounded real vector potential give a bounded Hermitian perturbation of that operator. Singular Coulomb fields, unbounded vector potentials, boundaries and time-dependent evolution require their owners’ additional assumptions. Hermitian matrices alone do not specify a self-adjoint boundary problem.

The exact free FW transformation changes states and observables. Selecting its positive block is a further restriction, not the basis change itself. Likewise the position-space nonlocality of $\mathcal E$ does not, by itself, establish superluminal signalling by local field observables; see Localization Problems.

Static low-energy corrections need a stated counting

For $m>0$ and a smooth, static electrostatic potential with $\mathbf A=0$, the positive FW block through the stated $c^{-2}$ order is

$$\begin{aligned} H_{\rm FW,+}={}&M+V+\frac{\mathbf p^2}{2m} -\frac{\mathbf p^4}{8m^3c^2}\\ &+\frac{\hbar^2\nabla^2V}{8m^2c^2}\\ &+\frac{\hbar}{4m^2c^2} \boldsymbol\sigma\cdot(\nabla V\times\mathbf p) +\cdots . \end{aligned}$$

The terms are respectively rest energy, electrostatic energy, leading kinetic energy, kinetic correction, Darwin term and spin–orbit term. Remove $M$ when working with the rest phase already factored out.

This is the static asymptotic expansion with $m,q,\hbar,V$ and spatial scales held fixed as $c$ grows, applied to low-momentum states and sufficiently regular fields. It is not a uniform operator-norm expansion over arbitrarily large momentum. For example, the truncated free kinetic expression becomes negative at large enough momentum; that is outside its low-energy use, not a new negative-energy particle sector.

For a general static magnetic background, the corresponding kinetic correction contains the full ordered square

$$F^2,\qquad F=\boldsymbol\pi^2-q\hbar\boldsymbol\Sigma\cdot\mathbf B,$$

not merely $\boldsymbol\pi^4$. The FW expansion owner supplies the full expression and electric commutator. Dropping magnetic terms requires an additional weak-field counting. The Pauli row has the minimal Dirac value $g=2$; radiative and structure-dependent coefficients belong to Nonrelativistic QED.

Klein–Gordon evolution is a second-order Cauchy problem

The charged scalar equation is

$$(i\hbar\partial_t-V)^2\phi =\left(c^2\boldsymbol\pi^2+m^2c^4\right)\phi.$$

The left side is an operator composition: for a time-dependent potential it also differentiates $V$. The problem needs two Cauchy data, such as $\phi$ and its covariant time derivative. It is not a one-component Schrödinger equation with a positive ordinary $L^2$ norm for arbitrary frequency data.

Even in a static background, replacing this equation by $i\hbar\partial_t\phi =(V+\sqrt{c^2\boldsymbol\pi^2+m^2c^4})\phi$ generally loses commutator terms when $V$ does not commute with the square root. A doubled first-order formulation must state its own pairing and sector choice. See External Potentials and the Klein–Gordon Inner Product.

Energy dependence and changing representations

Eliminating a Dirac component can produce an energy-dependent Schur operator. Its reduced component generally needs a nontrivial normalization metric; it is not automatically a normalized Pauli wavefunction. Use Effective Hamiltonians for that construction.

For a time-dependent unitary change $\psi'=U(t)\psi$, the generator is

$$H'=UHU^\dagger+i\hbar\dot U U^\dagger.$$

The connection term can mix instantaneous energy sectors even when $UHU^\dagger$ is block diagonal. The conditions for suppressing that mixing belong to Antiparticle Decoupling. None of these one-particle generators alone supplies multiparticle creation probabilities in a quantized field theory.

References

• Foldy, Leslie L., and Siegfried A. Wouthuysen. “On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit.” Physical Review 78, 29–36 (1950). doi:10.1103/PhysRev.78.29.
• Greiner, Walter. Relativistic Quantum Mechanics: Wave Equations. Third edition. Springer, 2000. doi:10.1007/978-3-662-04275-5. Scalar and spinor wave equations in external fields.
• Thaller, Bernd. The Dirac Equation. Springer, 1992. doi:10.1007/978-3-662-02753-0. Operator domains, spectral sectors and representation changes.