Anomalous moments: Pauli's term and g = 2(1 + a)

Units are SI with x^μ = (ct, x) and the metric (+, −, −, −), as in Unit 5. Gauge invariance lets the potential enter only through D_μ = ∂_μ + (iq/ħ)A_μ or through F_(μν) = ∂_μA_ν − ∂_νA_μ, and covariance asks for a Lorentz scalar. Among Unit 5's bilinears, terms linear in the field with at most one derivative of A leave two candidates that conserve parity: ψ̄γ^μD_μψ, which is minimal coupling, and the tensor bilinear contracted with the field, ψ̄σ^(μν)ψF_(μν). The Dirac-Pauli equation, (iħγ^μD_μ − mc − (μ′/2c)σ^(μν)F_(μν))ψ = 0, keeps both. Its coefficient is new: μ′/2c must carry momentum per tesla, so μ′ is a magnetic moment in J T⁻¹, and neither symmetry ties it to q or m. A third invariant, ψ̄iσ^(μν)γ⁵ψF_(μν), gives an electric dipole moment; it is odd under parity and time reversal and is named only. So covariance and gauge invariance allow any g; Dirac's g = 2 is the content of coupling through D_μ alone.

Multiply the equation by cγ⁰. With γ⁰ = β and γⁱ = βαᵢ, σ⁽⁰ⁱ⁾ = iγ⁰γⁱ = iαᵢ and σ^(ij) = εᵢⱼₖΣₖ, while F_(0i) = Eᵢ/c and F_(ij) = −εᵢⱼₖBₖ. So σ^(μν)F_(μν) = 2σ⁽⁰ⁱ⁾F_(0i) + σ^(ij)F_(ij) = −2(Σ⋅B − iα⋅E/c), and H = cα⋅π + βmc² + qφ − μ′β(Σ⋅B − iα⋅E/c), derived. The new piece is Hermitian, since (iβαᵢ)† = −iαᵢβ = iβαᵢ, and both halves are energies: μ′B is, and so is μ′E/c, because V m⁻¹ divided by m s⁻¹ is a tesla. Sort it into even and odd parts, as topic 6.06 will sort everything. −μ′βΣ⋅B commutes with β, so it is even and acts on the large component directly. iμ′βα⋅E/c anticommutes with β, so it is odd and links large to small components, as cα⋅π does. The odd half is what a magnetic moment becomes when it moves through an electric field, and covariance forces it in: a boost turns part of E into B, so a term in B alone would not take the same form in every frame.

On the upper block β = 1, so the even half adds −μ′σ⋅B to the large component's Hamiltonian. Topic 6.02's (σ⋅π)²/2m supplied −(qħ/2m)σ⋅B, so the spin energy is −(qħ/2m + μ′)σ⋅B. Writing μ′ = a(qħ/2m) makes it −(1 + a)(qħ/2m)σ⋅B, a moment (g/2)(q/m)S with g = 2(1 + a), derived; only the sum is measurable, so g − 2 measures μ′. Limits: a = 0 returns Dirac, and q = 0 removes the Dirac moment and leaves μ′ alone, the neutron's case. In a uniform B at p_z = 0, βΣ_z commutes with the whole Hamiltonian, so the even half shifts each of topic 6.03's exact levels by exactly ±aħω_c/2, with ω_c = |q|B/m and the sign set by the level's eigenvalue of βΣ_z, at every field strength: the result that topic quoted. Measured, quoted: a = 0.00115965218, so g = 2.00231930436. Schwinger's a = α/2π = 0.0011614, quoted, is 0.15% high, and the next order, −0.328(α/π)² = −1.77 × 10⁻⁶, also quoted, leaves a gap of only 1.5 × 10⁻⁸.

Now O = cα⋅π + iμ′βα⋅E/c, whose upper-right block is cσ⋅π + (iμ′/c)σ⋅E and lower-left block its adjoint. Every reduction, topic 6.01's elimination or 6.06's transformation, starts from βO²/2mc², whose upper block is (cσ⋅π + (iμ′/c)σ⋅E)(cσ⋅π − (iμ′/c)σ⋅E)/2mc². Its part linear in μ′ is (iμ′/2mc²)[(σ⋅E)(σ⋅π) − (σ⋅π)(σ⋅E)]. The Pauli identity, kept in order, turns the bracket into E⋅π − π⋅E + iσ⋅(E×π − π×E), and E⋅π − π⋅E = iħ∇⋅E. So the anomalous terms are −(aqħ/4m²c²)[σ⋅(E×π − π×E) + ħ∇⋅E], derived: 2a times the spin-orbit and Darwin terms topic 6.06 derives, so the totals carry 1 + 2a. Dimensions: aqħ²∇⋅E/m²c² is C J² s² V m⁻² ÷ kg² m² s⁻² = J. Nothing else enters at this order: the double commutator gives a-dependent terms only at relative order (v/c)², and renormalising the large component gives none linear in a. For spin-orbit, 1 + 2a = g − 1: g from the moment in the rest-frame field, −1 from Thomas precession, which is kinematics.

Minimal coupling made orbit n with spin moment along B degenerate with orbit n − 1 with spin moment against B. The even half opens every such pair by aħω_c and leaves spin-keeping transitions alone, so the trap offers two frequencies: ν_c = |q|B/2πm between pairs, and the anomaly frequency νₐ = aν_c within one. Equivalently νₛ = (g/2)ν_c flips the spin and νₐ = νₛ − ν_c. Their ratio a = νₐ/ν_c contains no B, so the electron is its own magnetometer, and measuring the difference directly buys a factor 1/a ≈ 862: a to one part in 10⁶ needs νₐ and ν_c to about 10⁻⁶ each, but νₛ/ν_c to 1.2 × 10⁻⁹. The relativistic bending of the ladder, about a part in 10⁹ of ν_c, cancels from νₐ, as topic 6.03 showed. The trap's electrostatic quadrupole, absent from this model, adds axial and magnetron motions and shifts the observed cyclotron frequency; the Brown-Gabrielse invariance theorem that undoes this, and the magnetic-bottle detection of single quantum jumps, are quoted, as is the 2023 result a = 0.00115965218059 ± 1.3 × 10⁻¹³.

Quantum Mechanics II's expectation values turn the 2a terms into energies. For s states spin-orbit vanishes and Dirac's Darwin energy α⁴mc²/2n³ becomes, times 2a, aα⁴mc²/n³: 2S_(1/2) rises by aα⁴mc²/8. For p states the Darwin term vanishes with ψ(0), and the spin-orbit energy (α⁴mc²/4n³)[j(j + 1) − l(l + 1) − ¾]/[l(l + ½)(l + 1)] is −α⁴mc²/48 for 2P_(1/2), so the anomaly lowers it by aα⁴mc²/24. The two levels Unit 7's Dirac energies keep degenerate therefore part by aα⁴mc²/6, derived. With α⁴mc²/h = 350.38 GHz and the measured a, that is 50.8 MHz up plus 16.9 MHz down, 67.7 MHz in all, or 67.8 MHz with Schwinger's a, when it reads α⁵mc²/12π. Against the measured Lamb shift of 1057.8 MHz, quoted, it is 6.4%, with the right sign: 2S_(1/2) on top. The dominant part is the self-energy Unit 13 computes, about 1040 MHz in Bethe's estimate; vacuum polarisation, about −27 MHz, and the covariant remainder are quoted there. Recoil, neglected here, lowers the 67.7 MHz by about 0.15%: the S share scales as (μ/m)³ and the P share as (μ/m)².