Nonrelativistic Limits

A nonrelativistic limit is a scale hierarchy, not the instruction “set $c$ to infinity.” The positive-energy sector must be identified, its rest phase removed, and the relevant residual energies, momenta, background scales, and observation times controlled. This chapter separates the limiting statement, stationary component elimination, dynamical sector control, and quantum-field matching.

Enter this chapter

Start with Nonrelativistic Limit to specify what is held fixed and how states are compared. Its free example proves convergence on each fixed prepared state over bounded times while showing why there is no uniform propagator approximation over all momenta.

1. Complete the common relativistic on-ramp: Metric and Units → Four-Vectors → Energy–Momentum Relation.
2. From that base, complete both free-equation branches: the Klein–Gordon Equation and Gamma Matrices → Gamma-Matrix Conventions.
3. Rejoin them at the Covariant Dirac Equation, then continue through relativistic Minimal Coupling.
4. Continue to Dirac to Pauli for the leading spinor reduction.

The owner performs the leading reduction in the Dirac basis. It derives the exact upper/lower component equations and obtains

$$\chi\simeq \frac{\boldsymbol\sigma\cdot\boldsymbol\pi}{2mc}\varphi,$$

and uses the noncommuting kinetic momenta to prove

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

Substitution gives the Pauli Hamiltonian and the minimal tree-level value $g=2$.

Approximation ledger

The leading particle-sector approximation needs control of:

• momenta small compared with $mc$;
• gauge-covariant residual energies and relevant potential-energy differences small compared with $2mc^2$;
• the strengths and variation scales of prescribed fields;
• preparation and evolution in the intended particle sector;
• the available real pair channels in a field-theory application.

Antiparticle Decoupling turns those last conditions into a dynamical question. A static spectral projection can be exactly invariant; an instantaneous projection for a changing Hamiltonian generally is not. An open gap alone does not control arbitrarily long resonant driving.

The algebraic elimination is not the full Foldy–Wouthuysen transformation. Systematic block diagonalization also transforms states and observables and can generate kinetic, spin–orbit, Darwin, gradient, and time-dependent terms whose form depends on explicit assumptions.

Scalar reduction and systematic corrections

Klein–Gordon to Schrödinger supplies the scalar reduction, current normalization, and a free spectral-cutoff error estimate. The Foldy–Wouthuysen Transformation then distinguishes a unitary representation change from a truncation. Foldy–Wouthuysen Expansion derives the ordered static electromagnetic corrections, including the magnetic terms that a purely electrostatic formula would omit.

Relativistic Corrections to Hydrogen evaluates the kinetic, spin–orbit, and Darwin matrix elements and checks their sum against the exact Dirac–Coulomb spectrum. The approximation is organized by momentum, field strength, and variation scales; a formally short Hamiltonian need not be valid at arbitrary momentum or for a singular source without a prescription.

Choose the effective description

Page	What it determines
Effective Hamiltonians	An ordered, energy-dependent Dirac Schur equation, its reconstructed norm, and the admissible eigenvalue branch
Antiparticle Decoupling	Preparation error and dynamical leakage from a specified spectral sector, with explicit gap and protocol qualifications
Nonrelativistic QED Preview	Magnetic, Darwin, and spin–orbit coefficients matched to consistently normalized relativistic amplitudes

The first applies the general projection method; the second applies adiabatic control. Their outputs answer different questions. A stationary denominator is not automatically a local time-evolution generator, and a one-body leakage norm is not automatically a pair count.

NRQED adds dynamical photons and matching coefficients that can encode radiative and internal-structure effects. Its tree-level point-Dirac coefficients reproduce the Pauli and FW results already derived here. At loop level, an infrared regulator, renormalization convention, and complete operator basis are part of the matching calculation.

An electron–positron system can also be nonrelativistic: both species may be retained as slow degrees of freedom. The relevant decision is which modes and channels are resolved by the effective theory, not whether every antiparticle has been removed by terminology.

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.
• Richard J. Hill, Gabriel Lee, Gil Paz, and Mikhail P. Solon, “The NRQED Lagrangian at Order 1/M⁴,” Physical Review D 87, 053017, 2013, doi:10.1103/PhysRevD.87.053017.
• Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler, “Bounds for the Adiabatic Approximation with Applications to Quantum Computation,” Journal of Mathematical Physics 48, 102111, 2007, doi:10.1063/1.2798382.
• J. J. Sakurai, Advanced Quantum Mechanics, Addison–Wesley, 1967.
• Bernd Thaller, The Dirac Equation, Springer, 1992, doi:10.1007/978-3-662-02753-0.