Hamiltonian Cards

These cards give compact operator formulas only after fixing the space, domain, electromagnetic convention, and validity regime. They are a lookup layer; all derivations remain on the canonical owners in the relativistic bridge.

Released cards

Card	Compact Hamiltonian	Space and regime	Canonical owner
Dirac Hamiltonian	$c\boldsymbol\alpha\cdot\boldsymbol\pi+\beta mc^2+q\Phi$	Four-component spinors; free or prescribed classical background; fixed-particle use below pair-production conditions	Covariant Dirac Equation
Pauli Hamiltonian	$\boldsymbol\pi^2/(2m)+q\Phi-q\hbar\boldsymbol\sigma\cdot\mathbf B/(2m)$	Two-component spinors; leading positive-energy, nonrelativistic, minimal $g=2$ regime	Pauli Equation

Here

$$\boldsymbol\pi=-i\hbar\nabla-q\mathbf A,$$

with signed charge $q$. The electron has $q=-e$, $e>0$.

Comparison

The Dirac operator is first order in space and acts on positive- and negative-energy sectors separated by approximately $2mc^2$ at low momentum. The Pauli operator is second order in spatial derivatives and acts only on the leading positive-energy two-component sector after that separation has been used.

The Pauli magnetic term is not inserted phenomenologically in the minimal case. It follows from

$$(\boldsymbol\sigma\cdot\boldsymbol\pi)^2 = \boldsymbol\pi^2-q\hbar\boldsymbol\sigma\cdot\mathbf B.$$

The derivation is owned by Dirac to Pauli. An anomalous magnetic factor changes the spin coefficient but does not alter the gauge phase or kinetic momentum.

Reading rule

Before using a Hamiltonian card, record:

• the Hilbert space and operator domain;
• boundary conditions or decay behavior;
• whether fields are prescribed or dynamical;
• metric, gamma basis, unit system, and signed-charge convention;
• the energy range and approximation order;
• whether pair creation, radiative corrections, or higher-order relativistic effects are excluded.

A formula can be algebraically correct and still be the wrong operator for a chosen domain or the wrong effective theory for a chosen energy scale.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964.
• L. L. Foldy and S. 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.
• J. J. Sakurai, Advanced Quantum Mechanics, Addison–Wesley, 1967.