HM01 — Lamb Shift
| Field | Value |
|---|---|
| Domain | Quantum Electrodynamics |
| System | Hydrogen atom, 2S½ and 2P½ levels |
| Group | U(1) gauge symmetry of QED |
| H^k tier | H⁰ → H³ (cross-tier coupling) |
| ISA | Hum (β = it/ℏ) |
| Status | Validated to < 1% |
| Opcodes | ORBIT · EMIT · FLOW · ERASE |
| Papers | Paper 620 |
Physical system
The Lamb shift is the 1058 MHz splitting between the 2S½ and 2P½ levels of hydrogen. Dirac’s equation — which correctly describes all H¹-tier atomic structure — predicts exact degeneracy between these levels. The observed splitting arises entirely from virtual photon emission and reabsorption: the electron couples to the zero-point fluctuations of the electromagnetic field differently in the 2S (spherical, non-zero contact density |ψ(0)|²) and 2P (vanishing contact density) states.
Willis Lamb and Robert Retherford measured the shift in 1947 using microwave spectroscopy. Hans Bethe calculated it within a week using non-relativistic QED. The Lamb shift was the first precision test of quantum field theory and the experimental result that launched renormalisation as a serious programme.
ISA interpretation
The Lamb shift is the minimal experiment that requires the H³ tier.
In the ISA grading, orbital wavefunctions are H⁰ objects (ORBIT opcode: occupancy without phase). The electron-photon vertex is H³ (EMIT opcode: particle-field coupling at the field-theoretic tier). The Lamb shift arises from an H⁰ → H³ cross-tier coupling: the contact density |ψ(0)|² (an H⁰ quantity) feeds into the EMIT vertex amplitude (an H³ process), creating an orbital-dependent shift that is invisible at H¹ (the tier where Dirac works).
Without EMIT, the shift is zero. The Hum ISA makes this explicit: the Lamb shift is the first observable that cannot be computed without the new H³ primitive.
ISA programme
PROGRAM Lamb_shift_1loop
; Input: hydrogen atom in 2S or 2P orbital
ORBIT(electron, n=2, l=0, j=1/2) ; 2S state (l=1 for 2P)
; Virtual photon loop: emit, propagate (self-energy), reabsorb
FOR k IN photon_modes: ; FLOW(beta=it/hbar) — Wick-rotated sum
EMIT(electron, photon, k, e)
PROPAGATE(photon, k, x->x) ; self-energy: same spacetime point
ABSORB(electron, photon, k) ; ABSORB = EMIT-dagger
; The H0->H3 cross-tier coupling: contact density selects S vs P
; 2S: |psi(0)|^2 != 0 -> delta_E != 0
; 2P: |psi(0)|^2 = 0 -> delta_E = 0
RENORM(Sigma_bare) ; subtract free-electron self-energy
OUTPUT delta_E(2S) - delta_E(2P) ; predicted: ~1058 MHz
Computable output
Bethe’s formula (Rydberg units, n=2, 2S state):
δE(2S) = (4α³/3π) · (2/n³) · Ry · B
where B = ln(2/α²) − ⟨ln(ΔE/Ry)⟩ = 10.534 − 2.811 = 7.723 (using Bethe 1947 Table I for the 2S geometrically-averaged excitation energy).
With α = 1/137.036, Ry = 13.606 eV: δE(2S) = 1047.5 MHz.
- vs Bethe original (~1040 MHz): 0.7% agreement
- vs experiment (1057.845 MHz): 1.0% agreement
- Residual (~10 MHz): relativistic corrections (Kroll-Lamb 1949)
Validated by script x620a_bethe_lamb_shift.py (PASS).
Connections
- MO01 (Motive ISA): EMIT is the one new primitive that extends MO01 to QFT. Without EMIT, δE = 0 (Dirac’s result); with EMIT, δE ≠ 0.
- HM07 (Amplituhedron): in the Grassmannian basis, EMIT survives but PROPAGATE and RENORM dissolve. The Lamb shift in the Grassmannian basis would be a single ORBIT on the appropriate positive geometry — an open problem.
- PT01 (PT-symmetric system): the Lamb shift β* (operating frequency 1058 MHz / c) is not the Motive ISA β*, but the coupling hierarchy α → α³ is the same “three IMAGINE steps to reach H³” structure.
Validation
- Lamb & Retherford (1947), Phys. Rev. 72, 241: measurement, 1058 MHz.
- Bethe (1947), Phys. Rev. 72, 339: non-relativistic calculation, ~1040 MHz.
- Kroll & Lamb (1949), Phys. Rev. 75, 388: relativistic correction, closing the 10 MHz gap.
- Modern value: 1057.8446 ± 0.0029 MHz (CODATA 2018).