What only relativistic QED supplies: where QFT begins

Units are SI, with x^μ = (ct, x) and the metric (+, −, −, −), as throughout the course. Bethe's 2S shift, C₂ ln(Λ/K₀) with C₂/h = 135.64 MHz, rests on approximations that fail inside the range it integrates over. The dipole approximation fails at photon energies of order ħc/a₀ = αmc² = 3.73 keV, where the reduced wavelength shrinks to the Bohr radius; non-relativistic kinematics fails as the photon energy approaches mc²; and Λ = mc² is a judgement, not a result. Of the logarithm's 7.72 units, ln(1/α) = 4.92 come from photons above 3.73 keV. The coefficient C₂ is safe: the sum rule fixes it from photon energies far above the binding, where the electron is nearly free and a relativistic treatment agrees. The constant beside the logarithm is not safe: a factor of two in Λ moves 2S by C₂ ln 2 = 94.0 MHz, and nothing in a one-electron theory says which factor is right.

The relativistic theory is assembled rather than derived. Take Unit 9's Dirac field ψ, whose expansion carries electron annihilators b and positron creators d†, and Unit 11's transverse field A, and couple them minimally: H = ∫ψ†(cα⋅(−iħ∇) + βmc²)ψ d³x + Σ ħω a†a + H_C − ∫ j⋅A d³x, with j = qcψ†αψ, q = −e, and H_C the instantaneous Coulomb energy the Coulomb gauge leaves behind. The novelty sits in j⋅A. Because ψ†αψ contains b†d† and db, one photon vertex can create or destroy an electron-positron pair, which topic 13.01's A⋅p Hamiltonian cannot. In the one-electron sector, Unit 6's reduction to order (v/c)² returns the A⋅p coupling with coefficient e/m, the A² term and the spin term, so Bethe's theory is the low-energy edge of this one. Each vertex carries e, so one photon emitted and reabsorbed is order e², that is α, the order Bethe worked at.

Perturbation theory in this H is organised by Dyson's S = T exp(−(i/ħ)∫H_I(t) dt), taken here as a definition. At order e² it holds three processes: the electron emitting and reabsorbing a photon, the self-energy; a photon emitted before the electron scatters off the nucleus and reabsorbed after, the vertex correction; and the nucleus's Coulomb field briefly making a pair, vacuum polarisation. The self-energy is topic 13.03's object with new intermediate states containing a pair, and Weisskopf found in 1939 that these cancel the linear growth, leaving δm c² = (3α/2π)mc² ln(Λ/mc²), quoted. With Λ at the Planck energy, the non-relativistic (4α/3π)Λ is 7.4 × 10¹⁹ mc², while the relativistic δm is 0.18 m. Treated covariantly, δm is a Lorentz scalar, a shift of the rest mass alone, so subtracting it is the same in every frame, and the vacuum loop's divergence is likewise a constant times the charge. Both are absorbed into the measured m and e: renormalisation, at its first order.

Choose a split Λ with K₀ ≪ Λ ≪ mc². Below it the electron is non-relativistic and Bethe's integral is right: C₂ ln(Λ/K₀), derived in topics 13.04 to 13.06. Above αmc² the dipole approximation fails, but the integrand does not, because e^(ik⋅r) commutes with V and leaves the sum rule's ⟨∇²V⟩ unchanged. Above Λ the photon dwarfs the binding, so the electron is effectively free and takes a single kick from the proton's field; the needed quantity is the slope of a free electron's charge form factor, which Kroll and Lamb and, independently, French and Weisskopf computed relativistically in 1949. Their C₂[ln(mc²/2Λ) + 11/24] is quoted, but one part of it is derived here: the coefficient of ln Λ must be −C₂, because Λ marks where we changed methods and the atom cannot know it. The sum, C₂[ln(mc²/2K₀) + 11/24] = 135.644 × 7.4871 = 1015.6 MHz, contains no cutoff. Bethe's formula at Λ = mc² gives 1047.4 MHz with today's K₀ = 16.64 Ry, about 1040 MHz with his 17.8 Ry; for his formula to give 1015.6 MHz, Λ would have to be mc²e¹¹⁄²⁴/2 = 0.791 mc².

The vertex correction also changes the electron's magnetic coupling, by Schwinger's a = α/2π = 0.0011614, quoted. Unit 6's Pauli term multiplies spin-orbit and Darwin by 1 + 2a, so 2S gains 2a times its Darwin energy, aα⁴mc²/8 = 50.87 MHz, and 2P₁/₂ gains 2a times its spin-orbit energy, −aα⁴mc²/24 = −16.96 MHz, computed. In the s-state bracket aα⁴mc²/n³ is (3/8)Cₙ, so self-energy and moment together carry 11/24 + 3/8 = 5/6. Vacuum polarisation is the photon's own loop. Uehling's result, quoted, is a contact potential −(4α/15)(ħ/mc)²(e²/4πε₀)δ³(r): inside ħ/mc = 386 fm the electron is within the cloud of screening pairs, meets more of the proton's charge, and an s level falls by Cₙ/5, 27.13 MHz for 2S, computed with |ψ(0)|². The contact form holds because the Bohr radius is 274 times the 193 fm range; in muonic hydrogen it is not, and polarisation becomes the largest term. Set a = 0 and drop vacuum polarisation, and 2S still lies 1011.5 MHz above 2P₁/₂: the self-energy alone breaks Dirac's degeneracy, which returns only as e → 0.

For 2S: 1015.57 + 50.87 − 27.13 = 1039.31 MHz. For 2P₁/₂: −16.96 MHz from the moment and +4.07 MHz from its own Bethe logarithm, ln(K₀/Ry) = −0.0300, quoted, so −12.89 MHz. The interval is 1052.2 MHz, the 1949 result evaluated with today's constants, against 1057.8 MHz measured: 0.53% short with every piece fixed. Dimensions check, since α is pure and each term is a multiple of α⁵mc²; the free-field limit e → 0 removes all of it. The remaining 5.6 MHz is higher order, quoted: the next power of Zα adds 7.1 MHz, the reduced mass removes 3 parts in 1837 of the total, 1.7 MHz, and recoil, two-loop and proton-size terms close the rest. Those need the S-matrix beyond order e², or the bound electron's propagator to all orders in Zα, and at uranium's Zα = 0.67 nothing less works. Doing that systematically, drawing every term as a Feynman diagram and proving that every divergence is absorbed into m and e at all orders, is quantum field theory, and this course stops there.