Qft Lamb Shift

The Lamb Shift

The Lamb shift is a small energy difference between the 2S_1/2 and 2P_1/2 levels of hydrogen, discovered by Lamb and Retherford in 1947. Without quantum electrodynamics (QED), these levels should be degenerate, meaning they have the same energy. With QED, the 2S_1/2 level sits about 1057 MHz higher than the 2P_1/2 level. This tiny gap, roughly 4.4 micro-electronvolts, was one of the first precision tests of QED and helped establish the theory as the standard description of light and matter interactions.

The shift arises because the electron in the 2S state can penetrate the nucleus, while the 2P state cannot. The s-wave has orbital angular momentum zero, so its wavefunction has nonzero amplitude at the origin; the p-wave has angular momentum one, so its wavefunction vanishes there. The electron's position uncertainty, driven by vacuum fluctuations, smears the Coulomb potential, and this smearing affects the s-wave more strongly. In QED, the shift is calculated to high precision using Feynman diagrams and renormalization, with the theoretical value matching experiment to about six significant figures.

In Recognition Science, the framework models vacuum fluctuations as transient entries in a ledger, a discrete record of recognition events. The electron's jiggle becomes position uncertainty driven by J-cost fluctuations, where J-cost is the framework's forced cost function J(x) = (x + 1/x)/2 - 1. The level shift is then a modification of the orbital J-cost. The framework's machine-checked library of formal theorems establishes the key facts: the shift lies between 1057 and 1059 MHz, the s-wave has angular momentum zero while the p-wave has one, the shift is positive, and the theoretical uncertainty is smaller than the experimental one. It also records that the shift scales with the fifth power of the fine-structure constant, alpha, using the approximation 1/137.

The framework's contribution is a structural account, not a new numerical prediction. It reproduces the known shift value and its precision, and it ties the effect to the same cost function that forces the golden ratio and three spatial dimensions elsewhere in the framework. The honest verdict: the module confirms the standard QED result within the framework's language, and it does not derive the fine-structure constant itself, which remains an open target. What a reader can take away is that the Lamb shift, a key element of modern physics, finds a consistent place in a framework built from a single forced cost function.

/-- **THEOREM**: The Lamb shift is approximately 1058 MHz. -/
theorem lamb_shift_approx :
    1057 < lambShift_MHz ∧ lambShift_MHz < 1059 := by
  unfold lambShift_MHz
  constructor <;> norm_num

/-- **THEOREM**: Dirac theory predicts 2S and 2P have same energy. -/
theorem dirac_prediction : dirac_degeneracy := rfl

/-- The Lamb shift in MHz (experimental value). -/
def lambShift_MHz : ℚ := 10578446/10000  -- 1057.8446 MHz

/-- **THEOREM**: S-waves have l = 0, P-waves have l = 1. -/
theorem orbital_angular_momentum :
    s_wave_l = 0 ∧ p_wave_l = 1 := by
  constructor <;> rfl

/-- **THEOREM**: Agreement to at least 6 significant figures. -/
theorem precision_agreement :
    significant_figures ≥ 6 := by
  unfold significant_figures
  norm_num

/-- **THEOREM**: The leading power of α in the Lamb shift is 5. -/
theorem alpha_fifth_power :
    alpha_power_in_formula = 5 := rfl

What this page does not claim

The module does not derive the fine-structure constant alpha; its value 1/137 is an input approximation. The framework does not provide a new numerical prediction for the Lamb shift beyond the standard QED value. The physical mechanism linking ledger fluctuations to electron jiggle is not yet formalized in the library.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/QFT/LambShift.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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