Negative-energy components and Zitterbewegung

Units are SI and the metric is (+, −, −, −). Take Unit 4's spin-up rest spinor ξ = (1, 0, 0, 0)ᵀ and confine it: ψ(x, 0) = G(x)ξ, with G a Gaussian of rms width d. At momentum p the eigenvectors of H(p) = cα⋅p + βmc² with energy +E have their lower pair equal to cσ⋅p/(E + mc²) times the upper, and those with −E have the upper pair −cσ⋅p/(E + mc²) times the lower, so ξ is an eigenvector only at p = 0. Projecting it gives a negative-energy amplitude cp/(E + mc²) times the positive one, derived, and a negative fraction w₋(p) = (1 − mc²/E)/2. A wide packet holds only small p, and averaging w₋ ≈ p²/4m²c² over the Gaussian gives a total negative weight (3/16)(ħ/mcd)², derived: 1.9 × 10⁻³ at d = 10ħ/mc = 3.9 pm. Summed numerically, it is 0.11 at d = ħ/mc and 0.22 at d = ħ/2mc, tending to ½ as d → 0. Below the reduced Compton wavelength, 386 fm, one sign of energy is no longer enough.

Quoted from Quantum Mechanics II: dA/dt = (i/ħ)[H, A]. For H = cα⋅p + βmc², p commutes with H, and dx/dt = (i/ħ)[H, x] = cα, Unit 4's velocity operator. Its own derivative needs [H, α]. Because (αᵢ, αⱼ) = 2δᵢⱼ and (β, αᵢ) = 0, the anticommutator is simple: (H, α) = 2cp, derived. So αH = 2cp − Hα and dα/dt = (2i/ħ)(Hα − cp). Since H² = p²c² + m²c⁴ ≥ m²c⁴, H has an inverse that commutes with p, and η = α − cpH⁻¹ absorbs the constant: dη/dt = (2i/ħ)Hη, so η(t) = e^(2iHt/ħ)η(0) and cα(t) = c²pH⁻¹ + c e^(2iHt/ħ)η(0), derived. Two facts carry the rest. η anticommutes with H, since (H, η) = 2cp − 2cp = 0, so it turns an energy-E state into an energy −E state. And c²pH⁻¹ commutes with H, has commuting components, and on a plane wave of energy ±E takes the value ±c²p/E, smaller than c.

Integrating cα(t) gives x(t) = x(0) + c²pH⁻¹t − (iħc/2)H⁻¹(e^(2iHt/ħ) − 1)η(0), derived; with ηH = −Hη the last term is Hermitian, as a position must be. The second term is uniform motion. The third oscillates at 2E/ħ, with amplitude ħc/2E times the matrix elements of η, which are at most 1. Dimensions: ħc/E is J s × m s⁻¹ ÷ J = m, and 2E/ħ is s⁻¹. Limits: at rest the amplitude is ħ/2mc = 193 fm and the angular frequency 2mc²/ħ = 1.55 × 10²¹ s⁻¹; as p grows the beat quickens and ħc/2E shrinks, so fast electrons barely tremble. The combination X(t) = x(t) + (iħc/2)H⁻¹η(t) moves exactly uniformly, X(t) = X(0) + c²pH⁻¹t, derived: remove the tremble and a drift coordinate is left. No free electron's tremble has been observed, since 193 fm at 10²¹ s⁻¹ is beyond any probe that leaves the electron single; a 2010 trapped-ion simulation of the Dirac equation recorded it, quoted.

η anticommutes with H, so its matrix elements join +E only to −E. In a packet built from positive-energy spinors alone, ⟨ψ₊|H⁻¹(e^(2iHt/ħ) − 1)η|ψ₊⟩ = 0 at every t, and ⟨x⟩ moves at ⟨c²p/E⟩ with no oscillation, derived. The tremble is interference. At one momentum and helicity the two branches form a two-level system, H = E n⋅τ with n = (cp/E, 0, mc²/E), α∥ acting as τₓ and β as τ_z: a positive-energy state is a Bloch vector along n and stands still, while any admixture precesses about n at 2E/ħ, which gives the two-branch formula. Atoms are almost pure. Projected on the free spinors, Unit 7's exact ground state has negative-energy weight 3.2 × 10⁻¹² in hydrogen and 2.9 × 10⁻³ in U⁹¹⁺, computed, mostly from momenta above 0.3mc, within about ħ/mc of the nucleus. Uranium's 13% small component belongs almost entirely to positive-energy spinors.

Unit 4 found that cα has eigenvalues ±c along every axis and components that fail to commute, while no packet moved faster than light. The solution separates the two claims. The drift part c²pH⁻¹ is what a measured displacement over a time much longer than ħ/mc² = 1.29 × 10⁻²¹ s returns, and on positive energies it is the group velocity c²p/E: 0.707c at p = mc. The eigenvalues ±c belong to the instantaneous velocity, whose eigenstates mix both branches. In the two-level picture the α∥ = +1 eigenstate has w₋ = (1 − cp/E)/2: one half at rest, 1/26 at p = 2.4mc. To catch the electron moving at c you would have to resolve the 193 fm tremble within 10⁻²¹ s, which takes probes with momentum transfer above mc and energy transfer above 2mc² = 1.02 MeV, where pairs are created and the experiment is no longer about one electron. What remains, a velocity of ±c that only a pair-making probe could catch, is real in the one-particle theory and is dissolved, not solved, by the many-particle reading Units 8 and 9 build.

δ = x − X = −(iħc/2)H⁻¹η equals ε[ε, x]/2 with ε = H/|H|: it is the odd part of x, the part joining the two branches, and X is the even part. Its mean vanishes in a one-sign state, but its square does not: δ⋅δ = (ħ²c²/4)H⁻²(3 − c²p²H⁻²), derived, so a slow electron of either sign has ⟨δ²⟩ = (3/4)(ħ/mc)² = 1.12 × 10⁻²⁵ m², an rms 334 fm, shared equally among the axes. Treat δ as a random displacement and a slowly varying potential is felt as ⟨V(X + δ)⟩ ≈ V + (1/6)⟨δ²⟩∇²V = V + (ħ²/8m²c²)∇²V, which is Unit 6's Darwin term with its coefficient. It stays a picture: δ's components do not commute with one another or with X, and only the Foldy-Wouthuysen reduction derives the term. The measurement agrees with the coefficient: in hydrogen it raises 2S₁/₂ by α⁴mc²/16 = 21.9 GHz, closing the gap to 2P₁/₂ to within the measured Lamb shift, 1.058 GHz.