The free electron's self-energy as an electromagnetic mass

Units are SI throughout, with energies quoted in eV; where a four-vector appears the metric is (+, −, −, −). Topic 13.01's result is quoted, not re-derived: δEₙ = (2α/(3πm²c²)) ∫₀^K dE E Σₘ |pₘₙ|²/(Eₙ − Eₘ − E), with E the photon energy. Put the electron in a periodic box with no potential. Its states are plane waves |p⟩, and the momentum operator returns p|p⟩, so pₘₙ = p when m = n and vanishes otherwise. The sum over m collapses to one term, the denominator becomes Eₙ − Eₙ − E = −E, and the E in the measure cancels it: δE(p) = −(2α/(3πm²c²)) p² ∫₀^K dE = −(p²/2m)(4αK/(3πmc²)), derived. Every photon energy below K contributes the same amount, which is why the result is linear in K. Dimensions: α and K/mc² are pure numbers, so δE carries the units of p²/2m. Limits: δE vanishes as α → 0, as K → 0, and for an electron at rest.

A correction that grows as p² is not an offset of the energy zero. It changes how the energy depends on momentum, and that dependence is what inertia means. Added to the bare kinetic energy, it gives p²/2m₀ − (δm/m₀)p²/2m₀ = p²/(2(m₀ + δm)) to first order in α, with δm/m = 4αK/(3πmc²), derived. Lowering E at fixed p flattens E(p), so the velocity ∂E/∂p = p/(m₀ + δm) falls: the dressed electron is heavier. With K = mc², δm/m = 4α/(3π) = 3.10 × 10⁻³ and δm c² = 1.58 keV. The A² term, first order in e², adds (e²/2m)⟨0|A²|0⟩ = αK²/(2πmc²), derived from the same mode sum, ⟨0|A²|0⟩ = Σₖₛ ħ/(2ε₀ωV) = K²/(4π²ε₀ħc³), 0.59 keV at K = mc², equally to every momentum state: it moves the zero of energy, changes no velocity, and contributes nothing to the mass. The picture is that the electron carries its transverse field along, so pushing the electron means pushing the field's momentum too.

ħ cancels from δm: with K = ħckₘₐₓ, δm = e²kₘₐₓ/(3π²ε₀c²), a classical quantity, and classical electrodynamics has it. A charge moving slowly with velocity v carries B = v × E/c², so its field holds momentum P = ε₀∫E × B d³r = (ε₀/c²)∫[vE² − E(E⋅v)] d³r. Averaging over directions replaces E(E⋅v) by vE²/3, giving P = (4U_C/(3c²))v, derived, with U_C = (ε₀/2)∫E² d³r the Coulomb self-energy. If the field has no Fourier components above kₘₐₓ, U_C = e²kₘₐₓ/(4π²ε₀), and 4U_C/(3c²) equals δm exactly. U_C itself, 1.19 keV for kₘₐₓ = mc/ħ, does not depend on v; in Coulomb gauge it sits in the instantaneous Coulomb term as a constant, set apart from the transverse photons, and never enters E(p). The factor 4/3 between mₑₘ c² and U_C is the old puzzle of Abraham's and Lorentz's charged sphere, named here: the field's energy and momentum alone do not form a four-vector, and the non-electromagnetic stresses that hold the charge together must be added.

A cutoff K corresponds to a length ħc/K: K = mc² means 386 fm, the reduced Compton wavelength. Setting δm = m gives K = 3πmc²/(4α) = 323 mc² = 165 MeV, or ħc/K = 1.20 fm, which is (4/(3π)) times the classical electron radius rₑ = 2.82 fm. An electron whose whole mass is electromagnetic would be a femtometre or two across. The model has no right to that conclusion. The dipole step set e^(ik⋅r) = 1 and so dropped the recoil: the intermediate electron really has momentum p − ħk and an energy higher by about E²/(2mc²), which equals the photon energy itself at E = 2mc². Keeping it turns the constant integrand into 1/(1 + E/(2mc²)) and the result into (8α/(3π)) ln(1 + K/(2mc²)), derived in the Hard example: linear below 2mc², logarithmic above. Measurement rules the classical picture out on its own: high-energy e⁺e⁻ scattering finds no electron structure down to below 10⁻¹⁸ m, quoted, more than a thousand times smaller than 1.20 fm.

Near K = mc² photons carry momentum comparable to mc, and the electron's relativistic structure can no longer be left out. Weisskopf's 1939 calculation in Dirac's hole theory, quoted here and confirmed by one-loop quantum electrodynamics, gives δm/m ≈ (3α/2π) ln(K/mc²) for K ≫ mc². Two features matter. The growth is logarithmic: in Weisskopf's picture the filled negative-energy sea, through the exclusion principle, smears the electron's charge over about ħ/mc and cancels the linear term. And δm is proportional to m itself, where the dipole model's δm c² = 4αK/(3π) did not depend on m. The numbers are modest: δm/m = 0.016 at K = 100 mc², 0.024 at 1000 mc², and only 0.18 with K at the Planck energy, 1.22 × 10¹⁹ GeV. A divergence this slow keeps δm a small correction even at absurd cutoffs, so treating it at first order and absorbing it into the measured mass is self-consistent. This lesson quotes the relativistic result and derives nothing from it.

The Hamiltonian's m₀ is the mass the electron would have with its field switched off, and the field cannot be switched off. Every way of weighing an electron, a cyclotron frequency in a Penning trap, the bending of a beam in a magnet, the Rydberg constant of hydrogen, sees m = m₀ + δm, with mc² = 0.51099895 MeV, measured. In the dipole model m₀c² is 509.42 keV if K = mc², 507.83 keV if K = 2mc², and negative above 165 MeV: a number that moves with an arbitrary cutoff is not a property of the electron. Hydrogen shows the bookkeeping is consistent. Topic 13.02's linear term is −(δm/m)⟨n|p²/2m|n⟩, and the virial theorem gives ⟨p²/2m⟩ = Ry/n² for every l, so it shifts each level by −(δm/m)Ry/n², exactly what replacing m₀ by m₀ + δm does to Ry = α²mc²/2. At K = mc² that is 2.55 THz for every n = 2 level, 2400 times the measured 1057.8 MHz Lamb shift but identical for 2s and 2p. The measured Rydberg constant already contains it; topic 13.04 makes the subtraction.