Dirac solutions at rest: positive and negative energy

Units are SI with ħ and c kept, and the metric is diag(+1, −1, −1, −1) with x^μ = (ct, x), as throughout this course. Put the particle in a box of volume V with periodic walls, so that the p = 0 mode is a normalisable, spatially uniform spinor ψ = χ(t)/√V. The momentum operator −iħ∇ annihilates it, the term cα⋅p drops out of H = cα⋅p + βmc², and Dirac's equation becomes iħ dχ/dt = βmc²χ: four first-order equations with constant coefficients and no spatial derivative left. Because β² = 1, the series for exp(−iβθ) splits into even and odd powers, cos θ − iβ sin θ, so χ(t) = [cos(mc²t/ħ) − iβ sin(mc²t/ħ)] χ(0) for every initial χ(0), derived in any representation. Dimensions check: mc²t/ħ is J × s ÷ J s, a pure number, and ψ†ψ = χ†χ/V is a density in m⁻³, as topic 4.04's ρ must be.

Stationary states are eigenvectors of β, and the algebra alone fixes the spectrum. β² = 1 forces every eigenvalue λ to obey λ² = 1, so λ = ±1, and topic 4.02 showed Tr β = 0, so in four dimensions each sign occurs twice. The rest energies are +mc² twice and −mc² twice, derived without writing a matrix element. In the Dirac representation β = diag(1, 1, −1, −1), its eigenvectors are the unit columns e⁽ʳ⁾, and the solutions are ψ⁽ʳ⁾ = e⁽ʳ⁾ e^{(−imc²t/ħ)}/√V for r = 1, 2 and e⁽ʳ⁾ e^(+imc²t/ħ)/√V for r = 3, 4: one rate, mc²/ħ = 7.76 × 10²⁰ rad s⁻¹ for the electron, in opposite senses. With the measured mc² = 0.51099895 MeV = 8.187 × 10⁻¹⁴ J the two pairs sit 2mc² = 1.022 MeV apart with nothing between them; for the proton the gap is 1.877 GeV. Unit 8 ties this gap to the measured 1.022 MeV threshold for a photon to make an electron-positron pair near a nucleus; here that is quoted only as the number to watch.

A doubly degenerate energy does not fix a state; a second operator that commutes with H must. Σ_z = diag(σ_z, σ_z), σ_z copied into both diagonal blocks, commutes with β because block-diagonal matrices multiply block by block and each block of β is ±1 times the identity. Its eigenvalue is +1 on ψ⁽¹⁾ and ψ⁽³⁾ and −1 on ψ⁽²⁾ and ψ⁽⁴⁾, and since S = (ħ/2)Σ obeys [Sₓ, S_y] = iħS_z blockwise, these are spin up and spin down along z with S_z = ±ħ/2, derived. At rest Σₓ and Σ_y commute with βmc² too, so any axis n̂ serves: put eigenspinors of σ⋅n̂ in both blocks. That freedom is special to p = 0; topic 4.07 shows Σ is no longer conserved once p ≠ 0. The non-relativistic check is exact: strip the rest-energy phase e^{(−imc²t/ħ)} from ψ⁽¹⁾ and ψ⁽²⁾ and their upper halves are the Pauli spinors (1, 0) and (0, 1) of a spin-half particle at rest. The label does not yet say which physical spin the negative pair carries; Unit 8 settles that.

In the Dirac representation positive energy means the upper two components, but only because β is diagonal there. The Weyl representation of topic 4.02, in the convention β = [[0, 1], [1, 0]] and αᵢ = diag(−σᵢ, σᵢ) in 2 × 2 blocks, is reached by ψ_W = Uψ_D with U = (1/√2)[[1, −1], [1, 1]]. Applying U to the rest solutions gives (χ, χ)/√2 for E = +mc² and (−χ, χ)/√2 for E = −mc², with χ = (1, 0) or (0, 1); check the first directly, since β swaps the halves and leaves (χ, χ) as it was. Both halves are now nonzero, so a vanishing lower half no longer marks positive energy, yet the energies, Tr Λ₊ = 2 and the spin labels are unchanged, the last because U has blocks proportional to the identity and so commutes with Σ_z. The representation-free statement of positive energy at rest is Λ₊ψ = ψ with Λ₊ = (1 + β)/2, never that the lower half vanishes. Unit 5 reads the two Weyl halves as the two chiralities.

Λ₊ = (1 + β)/2 and Λ₋ = (1 − β)/2 are projectors with Λ₊ + Λ₋ = 1 and Tr Λ₊ = (4 + Tr β)/2 = 2: the positive-energy pair fills a two-dimensional subspace of a four-dimensional space. A first-order equation accepts any column in C⁴ as its initial value, and P₊ = ψ†Λ₊ψ/ψ†ψ equals 1 only if Λ₋ψ = 0; in the Dirac representation (1, 0, 1, 0)ᵀ/√2, one of the two eigenvectors of α_z with eigenvalue +1, has P₊ = P₋ = 1/2. Sort matrices by how they treat β to see that this is generic. Even ones such as 1, β and Σ commute with it and keep the two energy signs apart; the odd αᵢ anticommute with it and swap them, so α_z e⁽¹⁾ = e⁽³⁾ and αₓ e⁽¹⁾ = e⁽⁴⁾. The αᵢ build topic 4.04's current and the coupling −qcα⋅A of topic 4.09, so a perturbation W acting on ψ⁽¹⁾ mixes in ψ⁽³⁾ with Quantum Mechanics II's first-order amplitude ⟨ψ⁽³⁾|W|ψ⁽¹⁾⟩/(2mc²). A 1.0 keV matrix element gives 9.8 × 10⁻⁴ in amplitude and 9.6 × 10⁻⁷ in probability: small because 2mc² is large, but never zero.

Take ψ = aψ⁽¹⁾ + b e^(iδ)ψ⁽³⁾ with a and b real and a² + b² = 1. The density ρ = 1/V never changes, and neither do the populations a² and b², because the free H = βmc² drives no transitions. But j_z = cψ†α_zψ = 2c Re(ψ₁*ψ₃ − ψ₂*ψ₄) multiplies two phases running in opposite senses, and j_z/ρ = 2abc cos(2mc²t/ħ + δ). It oscillates at the difference of the two rest frequencies, 2mc²/ħ = 1.55 × 10²¹ rad s⁻¹, a period πħ/mc² = 4.05 × 10⁻²¹ s for the electron. Three checks hold. Dimensions: c/V is m s⁻¹ × m⁻³ = m⁻² s⁻¹, a current density. The single-sign limit b = 0 gives j_z = 0, the group velocity pc²/E = 0 of a particle at rest. And |j_z/ρ| never exceeds c, reaching it at a = b, where ψ(0) is the α_z eigenvector of the last section. A state at rest whose density stays uniform while its current swings back and forth at 1.55 × 10²¹ rad s⁻¹ is the seed of the Zitterbewegung that Unit 8 interprets; here it is only derived.