Relativistic QM Formulas

These cards are a compact lookup layer for the reviewed relativistic quantum-mechanics spine. They state usable formulas, assumptions, and failure conditions; the derivations remain on their visibly linked canonical owners.

All cards use

$$\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1), \qquad x^\mu=(ct,\mathbf x), \qquad e^{-ip\cdot x/\hbar},$$

with signed charge $q$, $e>0$, and electron charge $q=-e$. Electromagnetic coupling uses

$$A^\mu=(\Phi/c,\mathbf A), \qquad D_\mu=\partial_\mu+\frac{iq}{\hbar}A_\mu, \qquad \boldsymbol\pi=-i\hbar\nabla-q\mathbf A.$$

Released cards

Card	Compact result	Canonical owner
Klein–Gordon Equation	$(\Box+m^2c^2/\hbar^2)\phi=0$	Klein–Gordon Equation
Dirac Equation	$(i\hbar c\gamma^\mu\partial_\mu-mc^2)\psi=0$	Covariant Dirac Equation
Pauli Equation	$H=(\boldsymbol\sigma\cdot\boldsymbol\pi)^2/(2m)+q\Phi$ for minimal $g=2$	Pauli Equation
Gamma-Matrix Identities	$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I_4$	Gamma-Matrix Conventions

The gamma convention owner fixes Dirac and chiral bases, $\gamma^5$, $\sigma^{\mu\nu}$, epsilon signs, traces, and source translations. The representation-independent Clifford derivation remains on Gamma Matrices.

Choose by task

• To test a scalar plane wave or current caveat, use the Klein–Gordon card.
• To translate between covariant and Hamiltonian spinor forms, use the Dirac card.
• To check the signed magnetic term or minimal $g=2$ factorization, use the Pauli card.
• To reduce slash products or verify a trace, use the gamma card after checking its four-dimensional convention boundary.
• To understand why a formula is true, leave the card and follow its canonical owner link.

Translation checklist

Before importing a result from another source, compare:

1. metric signature and d’Alembertian sign;
2. $x^0=t$ versus $x^0=ct$;
3. plane-wave and Fourier phase;
4. SI, Gaussian, or natural-unit normalization;
5. signed charge $q$ versus positive magnitude $e$;
6. the linked signs of $D_\mu$, the matter phase, and the gauge-potential transformation;
7. $\epsilon^{0123}$, $\gamma^5$, $\sigma^{\mu\nu}$, and slash definitions.

Changing one line of a convention package without the dependent lines is a common source of formulas that are dimensionally plausible but physically wrong.

Interpretation boundary

The Klein–Gordon current is not a positive Born density over the full solution space. The Dirac density is positive, but fixed-particle Dirac theory is still incomplete when pair creation or dynamical photons matter. The Pauli equation is a low-energy two-component approximation, not a Lorentz-covariant replacement for Dirac or QED.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.