Nonrelativistic QED Preview

Nonrelativistic quantum electrodynamics, or NRQED, is a quantum field theory of slowly moving charged matter and low-energy electromagnetic fields. It organizes interactions in powers of momentum, energy, and field scales relative to the mass. Its coefficients are fixed by matching physical amplitudes to an underlying relativistic theory. This page develops a bounded example: the magnetic, Darwin, and spin–orbit coefficients through order $1/m^2$. It does not derive a complete higher-order operator basis, a loop correction, or the Lamb shift.

Required background. Dirac to Pauli fixes the leading magnetic coupling; Foldy–Wouthuysen Expansion derives the static Darwin and spin–orbit terms.

Helpful background. Relativistic Normalization converts external states; Antiparticle Decoupling identifies retained sectors; EFT and Effective Hamiltonians provides the general matching framework.

A low-energy fermion field and its operators

Use $\hbar=c=1$, $m>0$, and a nonzero signed charge $q$. The rest-energy phase has been removed from the two-component fermionic field $\psi$. Define

$$D_t=\partial_t+iq\Phi,$$ $$\mathbf D=\boldsymbol\nabla-iq\mathbf A,\qquad \boldsymbol\pi=-i\mathbf D.$$

Take parity- and time-reversal-even interactions and a canonical operator basis in which higher time derivatives on the fermion have been removed by field redefinitions. The one-fermion terms through $1/m^2$ can be written

$$\mathcal L=\psi^\dagger\mathcal K\psi +\mathcal L_\gamma+\mathcal L_{\rm other},$$ $$\begin{aligned} \mathcal K={}&iD_t+\frac{\mathbf D^2}{2m} +c_F\frac{q\,\boldsymbol\sigma\cdot\mathbf B}{2m}\\ &+c_D\frac{q\,\boldsymbol\nabla\cdot\mathbf E}{8m^2}\\ &+ic_S\frac{q\,\boldsymbol\sigma\cdot (\mathbf D\times\mathbf E-\mathbf E\times\mathbf D)} {8m^2}\\ &+O(m^{-3}). \end{aligned}$$

The divergence in the Darwin term acts only on $\mathbf E$. In the spin–orbit term the covariant derivatives act as ordered operators on everything to their right. $\mathcal L_\gamma$ contains the dynamical photon sector, and $\mathcal L_{\rm other}$ stands for additional matter and contact sectors required by the process. This is not merely a one-particle wave equation with three adjustable constants.

For a prescribed background, the corresponding one-particle Hamiltonian is

$$\begin{aligned} H_{\rm NR}={}&q\Phi+\frac{\boldsymbol\pi^2}{2m} -c_F\frac{q\,\boldsymbol\sigma\cdot\mathbf B}{2m}\\ &-c_D\frac{q\,\boldsymbol\nabla\cdot\mathbf E}{8m^2}\\ &-c_S\frac{q\,\boldsymbol\sigma\cdot (\mathbf E\times\boldsymbol\pi-\boldsymbol\pi\times\mathbf E)} {8m^2}+\cdots. \end{aligned}$$

The ordered spin–orbit combination is Hermitian even when $\boldsymbol\nabla\times\mathbf E$ does not vanish. Setting $c_F=c_D=c_S=1$ reproduces the minimally coupled point-Dirac result at this order.

The expansion assumes soft external momenta and frequencies, and controlled field insertions, for example $|q\mathbf E|/m^2$ and $|q\mathbf B|/m^2$ small in a weak-background application. A statement of inverse-mass order does not by itself specify every bound-state or background power counting. At $1/m^3$, the free kinetic correction $-\boldsymbol\pi^4/(8m^3)$ is only one of several operators. Magnetic, electric, and two-photon terms must also be included at the claimed order; the static FW owner already shows why dropping the full magnetic square can give an incomplete expansion.

Match amplitudes with the same state normalization

Factor $q$ out of the relativistic electromagnetic vertex and write

$$\Gamma^\mu =F_1(t)\gamma^\mu +\frac{iF_2(t)}{2m}\sigma^{\mu\nu}k_\nu,$$ $$k=p'-p,\qquad t=k^2,\qquad F_1(0)=1.$$

Here $\sigma^{\mu\nu}=i[\gamma^\mu,\gamma^\nu]/2$. For the algebraic matching below assume form factors regular at the expansion point; loop-level infrared qualifications are stated below. Let $a=F_2(0)$ and $s_1=F_1'(0)$, where the derivative is with respect to the dimensional invariant $t$.

Take static elastic kinematics $E'=E$ and put $\mathbf K=\mathbf p'-\mathbf p$. With covariant spinors normalized by $u^\dagger u=2E$, the charge vertex in nonrelativistic external-state normalization is

$$j_{\rm NR}^0 =\frac{\bar u(p')\Gamma^0u(p)}{\sqrt{2E'\,2E}}.$$

This denominator matters for the Darwin coefficient. Expanding the normalized spinors and using $(\boldsymbol\sigma\cdot\mathbf p') (\boldsymbol\sigma\cdot\mathbf p) =\mathbf p'\cdot\mathbf p +i\boldsymbol\sigma\cdot(\mathbf p'\times\mathbf p)$ gives, between two-component spinors, the three contributions

$$\begin{aligned} \mathcal J_{\rm Dirac} &=1-\frac{\mathbf K^2}{8m^2} +\frac{i\boldsymbol\sigma\cdot(\mathbf p'\times\mathbf p)} {4m^2},\\ \mathcal J_{\rm Pauli} &=a\left[-\frac{\mathbf K^2}{4m^2} +\frac{i\boldsymbol\sigma\cdot(\mathbf p'\times\mathbf p)} {2m^2}\right],\\ \mathcal J_{\rm slope}&=-s_1\mathbf K^2. \end{aligned}$$

The last sign follows from $t=-\mathbf K^2$ in these static kinematics. Combining them yields

$$j_{\rm NR}^0 =\chi'^\dagger\mathcal J_{\rm NR}\chi +O\!\left(\frac{p_{\rm soft}^4}{m^4}\right),$$ $$\mathcal J_{\rm NR} =1-\frac{c_D\mathbf K^2}{8m^2} +\frac{ic_S\boldsymbol\sigma\cdot (\mathbf p'\times\mathbf p)}{4m^2},$$

for regular form factors and uniformly soft momenta, with fixed dimensionless form-factor derivatives. Thus

$$c_D=1+2a+8m^2s_1,\qquad c_S=1+2a.$$

As a Fourier sign check, an electrostatic mode $\Phi(\mathbf K)$ has $\boldsymbol\nabla\cdot\mathbf E\mapsto\mathbf K^2\Phi$ and

$$\langle\mathbf p'| \mathbf E\times\boldsymbol\pi-\boldsymbol\pi\times\mathbf E |\mathbf p\rangle =-2i\Phi(\mathbf K)(\mathbf p'\times\mathbf p)$$

at linear order in the background. Substitution into $H_{\rm NR}$ reproduces the two signs in the matched charge amplitude.

Magnetic response and the Lorentz constraint

The zero-momentum magnetic matching gives

$$c_F=1+F_2(0)=1+a.$$

Its intrinsic moment is $\boldsymbol\mu=q c_F\boldsymbol\sigma/(2m)$, or $g=2c_F$ in the signed-charge convention. The covariant moment calculation is developed in Magnetic Moment. An electron’s negative charge is already in $q$; it is not a reason to make $c_F$ negative.

Together the coefficients obey

$$c_S=2c_F-1.$$

In the chosen canonical basis, this relation is required by the Lorentz symmetry inherited by the low-energy theory. Boosts act on its fields and operators with inverse-mass corrections even though its leading dispersion is nonrelativistic. Heinonen, Hill, and Solon derive this constraint from the effective-theory implementation of Lorentz transformations.

The Darwin coefficient has additional independent charge-distribution information: $c_D=c_S+8m^2F_1'(0)$. An anomalous magnetic moment alone therefore does not determine the complete response to an inhomogeneous electric field.

What matching adds beyond the Dirac equation

For a structureless point particle at tree level, $F_2(0)=F_1'(0)=0$, so all three coefficients equal one. Radiative corrections or internal structure can change them. One-photon form factors still do not determine all two-photon polarizabilities, annihilation operators, or interactions involving several matter fields.

At loop level, the raw on-shell slope $F_1'(0)$ of a charged particle can be infrared divergent. Full theory and effective theory must be matched with the same infrared regulator, subtracting their shared low-energy contributions and specifying renormalization. A coefficient such as $c_D(\mu)$ can depend on the scheme, scale, and operator basis. Reading an unqualified finite “charge radius” directly from an infrared-divergent slope would not define a physical observable.

Field redefinitions and equations of motion can move terms among derivative, Darwin, and contact operators. For example, the photon equation relates $\boldsymbol\nabla\cdot\mathbf E$ to charge density. Consistently matched amplitudes are invariant under a change of basis; individual off-shell coefficients need not be.

Finally, the retained species depend on the process. An electron–positron system with small relative velocity can retain both fields and matched annihilation operators. A single-electron low-energy application may exclude real pair channels while retaining their hard virtual effects in coefficients. Neither choice is equivalent to counting lower Dirac components as real antiparticles.

Exercises

Three independent pieces of information. Suppose a regular model form factor has $F_2(0)=1/10$ and $8m^2F_1'(0)=1/5$. Find $c_F$, $c_S$, and $c_D$. Check the boost constraint.

Solution

$c_F=11/10$, $c_S=6/5$, and $c_D=7/5$. Indeed $2c_F-1=6/5$. The extra $1/5$ in $c_D$ cannot be inferred from the magnetic moment.

A slope convention. A reference expands $F_1(-Q^2)=1-r_1^2Q^2/6+\cdots$. Express $F_1'(0)$ in this page’s convention and the corresponding slope contribution to $c_D$.

Solution

Because $t=-Q^2$, $F_1(t)=1+r_1^2t/6+\cdots$. Thus $F_1'(0)=r_1^2/6$ and the contribution is $4m^2r_1^2/3$. This is an algebraic convention check; it does not remove the infrared qualifications on interpreting such a slope for a charged particle.

Why normalize before matching? In a static spin-independent charge amplitude, what goes wrong if $\bar u'\Gamma^0u$ is compared directly with a unit-normalized Pauli-state matrix element?

Solution

The relativistic external states carry the factors $\sqrt{2E'\,2E}$. Their momentum expansion contributes at the same order as the Darwin term. Leaving those factors in would attribute a normalization effect to an interaction coefficient. Convert both amplitudes to the same external-state convention first.

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. Classical Dirac reduction used as the tree-level checkpoint.
• Heinonen, Johannes, Richard J. Hill, and Mikhail P. Solon. “Lorentz Invariance in Heavy Particle Effective Theories.” Physical Review D 86, 094020 (2012). doi:10.1103/PhysRevD.86.094020; author manuscript. Boost constraints in the low-energy operator basis.
• Hill, Richard J., 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; author manuscript. Canonical operator basis, one-photon matching, infrared subtraction, and field redefinitions; this page retains only its lower-order preview.