Mass renormalisation: Bethe's subtraction

SI units throughout, with energies in eV and shifts quoted as frequencies ΔE/h; the electron stays non-relativistic, so the course metric (+, −, −, −) never enters. The Hamiltonian of topic 13.01 carries the bare mass m₀, and topic 13.03 derived that the field adds δm/m = 4αK/(3πmc²), quoted here. Experiments weigh m = m₀ + δm, and the Rydberg constant that spectroscopy measures is built on m. Kramers' proposal was to say so in the Hamiltonian. Expanding 1/m₀ = 1/(m − δm) gives p²/2m₀ = p²/2m + (δm/m)p²/2m + O(α²), derived. Take H₀ = p²/2m + V and treat the counterterm (δm/m)p²/2m as a perturbation beside the coupling to the field: both are of order α, the self-energy at second order in a coupling proportional to e, the counterterm at first order. Check it on a free electron. Its self-energy, −(δm/m)p²/2m from topic 13.03, and the counterterm cancel, so E(p) = p²/2m with the measured m to order α. That is the renormalisation condition, and it fixes δm once, before any atom is considered.

Topic 13.02's linear term came from closure, Σₘ |pₘₙ|² = ⟨n|p²|n⟩. Run it backwards on the counterterm: ⟨n|(δm/m)p²/2m|n⟩ = (2αK/(3πm²c²))⟨n|p²|n⟩ = (2α/(3πm²c²)) Σₘ |pₘₙ|² ∫₀^K dE, a sum over the same intermediate states with integrand 1. Add it term by term, using the identity E/(Eₙ − Eₘ − E) + 1 = (Eₘ − Eₙ)/(E + Eₘ − Eₙ), and ΔEₙ = (2α/(3πm²c²)) Σₘ |pₘₙ|² (Eₘ − Eₙ) ∫₀^K dE/(E + Eₘ − Eₙ), derived. At K = mc² the counterterm is +10.19 THz for 1s and +2.547 THz for every n = 2 level, each cancelling that level's linear term. Dimensions: |pₘₙ|²/(m²c²) is a pure number and the rest an energy. Free limit: p is diagonal, the only term has Eₘ = Eₙ, and ΔE = 0, as the renormalisation condition demands. The cancellation needs both sums over the same states. Keep only 2s's bound states, 31.4 per cent of ⟨p²⟩ (topic 13.02), against the full counterterm, and 1.75 THz of linear divergence survives.

Write Δ = Eₘ − Eₙ. The subtracted integrand Δ/(E + Δ) is 1 for photons far below Δ, where the bound term is negligible and the counterterm stands alone, and falls as Δ/E above it. For a state above the level, ∫₀^K dE Δ/(E + Δ) = Δ ln(1 + K/Δ), derived: positive, so such terms raise the level, and every state with weight lies above 2s, the direction Lamb and Retherford found. Count by decades of photon energy. Before the subtraction each decade of the −1 plateau contributed ten times the decade below it; after it, each decade above Δ contributes the same Δ ln 10, which is what a logarithm is. For K far above the intervals that carry the weight, ΔEₙ → (2α/(3πm²c²)) Σₘ |pₘₙ|² (Eₘ − Eₙ) ln(K/|Eₘ − Eₙ|): its coefficient of ln K is topic 13.05's sum rule, and the weighted mean of ln|Eₘ − Eₙ| is topic 13.06's Bethe logarithm. The subtraction moved the weight down but not out of reach: for an interval of 10 eV, photons above αmc² = 3.73 keV, where e^(ik⋅r) ≈ 1 fails, still supply 45 per cent of the logarithm.

If m lies below n, Δ is negative and Δ/(E + Δ) blows up at E = |Δ| = ħωₙₘ. There |m, 1ₖₛ⟩ has exactly the energy of |n, 0⟩: the photon is real, and the integral as written has no value. Unit 12 supplied the prescription, quoted: for a state switched on adiabatically the energy denominators carry +iη with η → 0⁺, so the subtracted integrand becomes Δ/(E + Δ − iη), and 1/(E + Δ − iη) = P[1/(E + Δ)] + iπδ(E + Δ). The principal value, the integral with a vanishing symmetric gap around the pole, gives Δ ln((K − |Δ|)/|Δ|), derived: negative once K exceeds 2|Δ|, so a lower state pushes the level down, and for K ≫ |Δ| it takes the same form Δ ln(K/|Δ|) as a state above. That is why one formula in ln|Eₘ − Eₙ| covers both. For hydrogen's 2p, the 1s term at K = mc² is −172 MHz and moves by −11.0 MHz each time K doubles. The states above 2p push the other way, and topic 13.05 shows that the ln K pieces cancel for p states.

The δ-function adds iπΔ to each lower term, so Im ΔEₙ = −(2α/3) Σₘ |pₘₙ|² (Eₙ − Eₘ)/(m²c²) over states below n. Write the complex energy as Eₙ − iħΓₙ/2 and use pₘₙ = imωₘₙrₘₙ, quoted from Quantum Mechanics II: Γₙ = (4α/(3c²)) Σₘ ωₙₘ³ |rₘₙ|², derived, which is Unit 12's Weisskopf-Wigner rate, now the other half of one complex self-energy. Dimensions: ω³r²/c² is s⁻³ m² ÷ m² s⁻² = s⁻¹. The counterterm is real, so the subtraction never touches a width, and Γ needs no cutoff beyond K above ħωₙₘ, because a real photon has a single energy. Against measurement: for 2p, ħω = 10.20 eV and |⟨1s|r|2p⟩|² = (2¹⁵/3¹⁰)a₀² give Γ = 6.27 × 10⁸ s⁻¹, a 1.60 ns lifetime as measured, and a natural width Γ/2π = 99.8 MHz, a tenth of the Lamb shift, which is why Lamb and Retherford's resonances were so broad. 2s has no lower state with a dipole matrix element, 1s being excluded by parity, so its shift is real at this order; its 0.12 s two-photon lifetime is fourth order in e, quoted.

Kramers urged at the Shelter Island conference of June 1947 that the electron's electromagnetic mass be separated out and the theory written in the observed mass; Bethe worked out the non-relativistic calculation within days, and topic 13.07 recomputes his 1040 MHz. Nothing in the subtraction is adjustable. δm is fixed by the free electron, the same counterterm acts on every level, and it must use the same regulator as the bound self-energy: cut the free electron at K and the bound one at 1.01K and 2s keeps 1 per cent of its linear term, 25.5 GHz, twenty-four times the Lamb shift. What the subtraction leaves is a logarithm of K, and K stays; this course sets it at mc² by argument, where the dipole and non-relativistic treatment must fail. Removing the cutoff needs a covariant calculation, quoted in topic 13.09. Quantum electrodynamics goes further, renormalising mass and charge at every order in α, the programme of Tomonaga, Schwinger, Feynman and Dyson set out by 1949. This course names it and stops at this one subtraction.