Parity for Dirac spinors: γ⁰, scalars and pseudoscalars

Units throughout: SI with x⁰ = ct, metric diag(1, −1, −1, −1), the Dirac representation, γ⁵ = iγ⁰γ¹γ²γ³ and Unit 4's u†u = 2E/c. Spatial inversion is Λ_P = diag(1, −1, −1, −1), and det Λ_P = −1. Every Λ reached in topics 5.01 to 5.04 is a product of exponentials of the generators J and K, and det e^G = e^{(tr G)} = 1 because those 4 × 4 generators are traceless. No continuous path from the identity reaches a determinant of −1, so no exponential of σ^μν can serve as S_P either: it has to be solved for. Topic 5.02's condition S⁻¹γ^μS = Λ^μ_νγ^ν never mentioned generators, so put Λ_P into it. The time row gives S_P⁻¹γ⁰S_P = γ⁰ and the space rows give S_P⁻¹γ^iS_P = −γⁱ. S_P must commute with γ⁰ and anticommute with each of γ¹, γ² and γ³, a question about 4 × 4 matrices that the Clifford algebra answers on its own.

Try γ⁰. It commutes with itself, and (γ⁰, γⁱ) = 2g⁽⁰ⁱ⁾ = 0 makes it anticommute with each γ^i, so the condition holds, derived. γ⁵ fails, because it anticommutes with γ⁰ as well. If S′ also works, S′S_P⁻¹ commutes with all four γ^μ and so with all their products; topic 5.05's sixteen products span every 4 × 4 matrix, so S′S_P⁻¹ is a multiple of 1. Hence S_P = η_Pγ⁰, and keeping ∫ψ†ψ d³x fixed forces |η_P| = 1, which is topic 5.03's S⁻¹ = γ⁰S†γ⁰ once more. The Hamiltonian form is the quickest check: x → −x turns H = cα⋅p + βmc² + V(r) into −cα⋅p + βmc² + V(r), and β = γ⁰, which anticommutes with α, turns it back, so η_Pβ with x → −x commutes with H whenever V is even. No density fixes the phase, since bilinears carry |η_P|² = 1. Asking two inversions to return the field, up to the sign a 2π rotation already allows, leaves η_P = ±1 or ±i, and the electron is given +1 by convention.

With γ⁰ = diag(1, 1, −1, −1) and ψ = (φ, χ), parity sends φ(t, x) to η_Pφ(t, −x) and χ(t, x) to −η_Pχ(t, −x): the two blocks carry opposite intrinsic parity. Unit 4's u(p, s) has lower block cσ⋅p χₛ/(E + mc²), odd in p, so γ⁰u(p, s) = u(−p, s), derived, while x → −x turns e^(ip⋅x/ħ) into e^{(−ip⋅x/ħ)} to match. The mirrored electron moves the other way with its spin unchanged, so its helicity reverses, as an axial spin and a polar momentum require. In v(p, s) the large block is the lower one, and γ⁰v(p, s) = −v(−p, s): an extra minus sign, which Unit 8 turns into the opposite intrinsic parity of electron and positron, seen in the perpendicular photon polarisations of para-positronium decay that Wu and Shaknov measured in 1950 (quoted). Limit: at 1 keV the lower block of u is 3.1% of the upper and γ⁰ barely acts, which is why Schrödinger theory never needed it.

In a central potential P commutes with H and with J, so stationary states can carry a parity as well as j and m. Give the upper block orbital l and the lower block l′, in spinor harmonics with Ω(−r̂) = (−1)^l Ω(r̂). The upper block contributes η_P(−1)^l and the lower −η_P(−1)^(l′), and a single eigenvalue needs (−1)^(l′) = −(−1)^l: l′ = l ± 1, derived. Both couple with spin ½ to the same j, so l and l′ are j − ½ and j + ½, and l′ = 2j − l. Thus 1s₁/₂ has a p-wave lower block, 2p₁/₂ an s-wave, 2p₃/₂ a d-wave and 3d₃/₂ a p-wave: the spectroscopic letter names the upper block, and η_P(−1)^l is Schrödinger's parity, the non-relativistic limit. 2s₁/₂ and 2p₁/₂ share n, j and Dirac's energy but not parity, so no P-even term mixes them; the Lamb shift moves them 1057.8 MHz apart (measured) without mixing them. Unit 7's exact 1s state (quoted) sets the size: lower ÷ upper = (1 − ε)/(Zα) with ε = √(1 − Z²α²), a probability share (1 − ε)/2 of 1.3 × 10⁻⁵ in hydrogen and 12.9% in U⁹¹⁺.

From ψ′ = η_Pγ⁰ψ(t, −x), ψ′† = η_P*ψ†(t, −x)γ⁰, so ψ̄′ = η_P*ψ̄(t, −x)γ⁰ and every bilinear obeys ψ̄′Γψ′(t, x) = ψ̄γ⁰Γγ⁰ψ(t, −x), derived, with the phase gone. Γ = 1 gives +1: ψ̄ψ is a true scalar. Γ = γ⁵ gives −1, because γ⁵ anticommutes with γ⁰: a pseudoscalar. Since (ψ̄γ⁵ψ)* = −ψ̄γ⁵ψ, the real density is iψ̄γ⁵ψ. Blocks show the rule: ψ̄ψ = φ†φ − χ†χ pairs each block with itself, so the γ⁰ signs square away, while iψ̄γ⁵ψ = −2 Im(φ†χ) pairs upper with lower and keeps one minus. The vector ψ̄γ^μψ keeps its time part and reverses its space part, a polar current. The axial vector ψ̄γ^μγ⁵ψ does the opposite: its space part survives, as a spin density must, and its time part ψ†γ⁵ψ flips. The tensor splits like the field strength, σ⁽⁰ⁱ⁾ reversing like E and σ^ij not, like B. Topic 5.05 could not tell S from P or V from A under proper transformations; parity can, and these are Unit 1's classes carried to spinors.

For a slow electron the lower row of the Dirac equation gives χ ≈ −(iħ/2mc)σ⋅∇φ. Then φ†χ − χ†φ = −(iħ/2mc)∇⋅(φ†σφ), so iψ̄γ⁵ψ ≈ (ħ/2mc)∇⋅(φ†σφ), derived: the divergence of the spin density, scaled by ħ/2mc = 1.93 × 10⁻¹³ m. Dimensions: a length times m⁻¹ times m⁻³ is m⁻³, a density like ψ̄ψ. A mirror keeps every spin and reverses ∇, carrying a hedgehog of outward spins into an inward one, so the density flips. For a packet of size L it is about ħ/(2mcL) of ψ†ψ: relativistic, and odd in any parity eigenstate, so it integrates to zero. Couplings follow the classes. gψ̄ψΦ conserves parity with a scalar field Φ and gψ̄iγ⁵ψπ with a pseudoscalar π, and the pion's parity of −1 was measured in 1951 from π⁻d → nn at rest, whose final state can only be ³P₁ (quoted). The model stops where the Hamiltonian is not P-even. The weak V − A current broke parity in ⁶⁰Co decay in 1957, and in caesium a contact term in ψ†γ⁵ψ joins 6s₁/₂ to the s-wave lower blocks of the p₁/₂ states, which gives the 6s → 7s transition a parity-violating electric-dipole amplitude, measured to 0.35% in 1997.