Electromagnetic Coupling

Electromagnetic coupling is where convention mistakes become physical sign errors. This chapter uses one signed charge $q$, one SI-normalized four-potential, and one gauge phase from the covariant scalar and spinor equations through the Pauli limit. It then develops systematic relativistic corrections and two exact external-field spectra. The central distinction is between a change of representation, an approximation with a declared order, and an exact solution of a specified background problem.

Enter this chapter

Use Metric and Units, Four-Vectors, and Energy–Momentum Relation for the relativistic conventions. The Klein–Gordon Equation supplies the scalar comparison. For the spinor branch, complete Gamma Matrices, Gamma-Matrix Conventions, and the Covariant Dirac Equation.

The first three pages establish the interacting problem:

1. Minimal Coupling constructs the covariant derivative and the kinetic-momentum algebra.
2. Gauge Covariance applies finite gauge transformations to Hamiltonians, propagators, and a uniform electric field in two gauges.
3. Dirac Dynamics in Electromagnetic Fields derives exact force and work identities and explains the extra commutator required when squaring an electric-field eigenproblem.

From Pauli theory to controlled corrections

Begin with Dirac to Pauli, which derives the leading small component and magnetic term. Minimal Coupling in Wave Mechanics owns the independent Schrödinger construction and its probability current. The Pauli Equation then supplies the low-energy dynamics, Zeeman splitting, spin precession, and signed magnetic-moment convention.

Continue through the following distinctions:

Page	Question it answers
Foldy–Wouthuysen Transformation	How does a unitary representation change separate free energy sectors and transform observables?
Foldy–Wouthuysen Expansion	Which ordered static-field operators survive through order $1/c^2$?
Magnetic Moment	How does a covariant Pauli interaction match to intrinsic spin response and form factors?
Spin–Orbit Coupling	How does the Thomas factor enter, and how does a central potential split angular multiplets?
Darwin Term	Why does the electric-gradient correction become a Coulomb contact, including for S states?

The transformation and expansion have distinct jobs. The former changes the representation of states and observables; the latter truncates a field-dependent calculation under explicit scale assumptions. A correct free transformation alone does not justify an external-field approximation.

Exact spectra as independent checks

Relativistic Landau Levels constructs full Dirac modes in a uniform magnetic field. It explains the single internal state of the lowest level, the two states at higher levels, and the distinction between seed spin labels and full-spinor observables. Its low-energy expansion checks the complete magnetic square in the FW Hamiltonian.

For the electric problem, first evaluate the three shifts on Relativistic Corrections to Hydrogen. Then use Hydrogen Fine Structure Revisited to solve the coupled Dirac–Coulomb radial equations. The agreement tests coefficients and state counting. The exact point-source model still has a specified origin boundary and does not contain recoil or the Lamb shift.

Sign ledger

Throughout this chapter,

$$\begin{aligned} \boldsymbol\pi&=-i\hbar\nabla-q\mathbf A,\\ H_D&=c\boldsymbol\alpha\cdot\boldsymbol\pi+\beta mc^2+q\Phi. \end{aligned}$$

The electron has $q=-e$ with $e>0$. The Pauli page uses a positive magnetic-factor magnitude $g$ and retains the charge sign in $q$; importing a separately signed electron $g$ without translation would double count the sign.

Checks to carry between pages

• A gauge transformation preserves kinetic observables while a time-dependent phase can shift the Hamiltonian’s energy reference.
• The electric spin–orbit term includes the Thomas correction already; adding it again doubles the frame correction.
• A Coulomb Laplacian includes the contact at the origin, even though it vanishes at every ordinary point away from the source.
• A squared Dirac equation supplies candidate energies; the first-order equation and boundary conditions still determine physical spinors.
• Agreement with an exact spectrum checks a specified model, not every correction measured in an experiment.

The potentials remain prescribed classical backgrounds. Quantizing the matter field is necessary to define particle number and pair-production observables; a strong classical electric background can already produce pairs in that field theory. Dynamical photons and radiative corrections require the further QED description. Appending a matched anomalous moment to a one-particle Hamiltonian does not perform its QED calculation.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964.
• W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd ed., Springer, 2000.
• J. J. Sakurai, Advanced Quantum Mechanics, Addison–Wesley, 1967.