Antiparticles, charge conjugation and CPT

Write the free Dirac Hamiltonian H_D = cα⋅p + βmc² in the Dirac representation, β = diag(1, 1, −1, −1) and αᵢ carrying σᵢ off the diagonal. For every momentum p it is a 4 × 4 Hermitian matrix with eigenvalues ±Eₚ, Eₚ = √(p²c² + m²c⁴), each twice over: two positive-energy spinors u(p, s) evolving as e^{(−iEₚₜ/ħ)} and two negative-energy spinors v(p, s) evolving as e^{(+iEₚₜ/ħ)}. The four eigenvectors are complete, so any wavepacket narrower than the Compton length ħ/mc = 386 fm must contain the lower branch, and an electron coupled to the radiation field could cascade into it without limit. Dirac's answer in 1930 was to fill every negative-energy level and let Pauli exclusion block the cascade; a vacancy in the sea then behaves as a positive-energy particle of charge +e, which Anderson photographed in a cloud chamber in 1932. The sea costs an infinite negative charge density that has to be subtracted by hand, and it means nothing for bosons, whose Klein–Gordon equation has the same two branches and no exclusion principle.

The lower branch carries the phase e^{(+iEₚₜ/ħ)}, and the identity e^{(−i(−E)t/ħ)} = e^{(−iE(−t)/ħ)} says a negative-energy wave advancing in time is indistinguishable from a positive-energy wave running backward. Stückelberg (1941) and Feynman (1949) took it literally: a solution of energy −E and momentum −p propagating backward in time is an antiparticle of energy +E and momentum +p propagating forward, every additive charge reversed. On the (pc, E) plane the rule is a point reflection through the origin, drawn in the figure below. Nothing is filled and nothing subtracted, so the same reading works for the Klein–Gordon field, and in quantum field theory it becomes the statement that one field operator carries a particle annihilator and an antiparticle creator, ψ = Σ (a u e^(−ip⋅x) + b† v e^(+ip⋅x)). The price of an antiparticle is the gap. A photon in a nuclear Coulomb field needs E_γ ≥ 2m_ec² = 1.022 MeV; a proton beam on a fixed proton target needs √s ≥ 4m_pc², a kinetic energy of 6m_pc² = 5.63 GeV. The Bevatron's 6.2 GeV was chosen to clear it, and the antiproton appeared in 1955.

With ħ = c = 1, an electron of charge q in a potential A_μ obeys [iγ^μ(∂_μ + iqA_μ) − m]ψ = 0. Take the complex conjugate: [−i(γ^μ)*(∂_μ − iqA_μ) − m]ψ* = 0. In the Dirac representation γ⁰, γ¹ and γ³ are real and γ² is imaginary, and γ² anticommutes with the other three, so iγ²(γ^μ)*(iγ²)⁻¹ = −γ^μ for all four μ. Multiplying from the left by iγ² therefore gives [iγ^μ(∂_μ − iqA_μ) − m](iγ²ψ*) = 0: the same equation with q replaced by −q. The map ψ → ψᶜ = iγ²ψ* is charge conjugation, or covariantly ψᶜ = Cψ̄ᵀ with C = iγ²γ⁰, a real antisymmetric matrix with C² = −1 and C⁻¹γ^μC = −(γ^μ)ᵀ. Applied to the spin-up electron at rest, (1, 0, 0, 0)ᵀe^(−imt), it returns (0, 0, 0, 1)ᵀe^(+imt): a negative-energy, spin-down solution, which Feynman–Stückelberg reads as a spin-up positron at rest. C flips the spinor's spin and the reinterpretation flips it back, so the physical antiparticle carries the particle's spin — v(p, s) is iγ²u*(p, s) up to a phase.

C acts on every state, but it has an eigenvalue only where it returns the state itself, which needs every additive charge to vanish. The photon is the cleanest case: A_μ couples to the current j_μ, C reverses j_μ, so C|γ⟩ = −|γ⟩ and n photons carry (−1)ⁿ. A fermion–antifermion pair with orbital L and total spin S has C = (−1)^(L+S), from exchanging the two, which is what makes C a good label for flavour-neutral mesons and for positronium. The consequences are counting rules with hard numbers. π⁰ (L = S = 0, C = +1) goes to two photons 98.8% of the time, while π⁰ → 3γ would need C = −1 and is not seen, BR < 3.1 × 10⁻⁸. Para-positronium (S = 0, C = +1) decays to 2γ in 125 ps; ortho-positronium (S = 1, C = −1) must emit three, each extra vertex costing a factor of order α, and lives 142 ns — a ratio of 1140, or 8.3/α. ρ⁰ (C = −1) → π⁰γ is allowed and seen at 4.7 × 10⁻⁴, but ρ⁰ → π⁰π⁰ would put C = +1 in the final state and is absent (Bose symmetry forbids it too). None of this applies to the K⁰: C|K⁰⟩ = |K̄⁰⟩ is a different state, so the kaon carries no C-parity at all.

The weak interaction breaks C as hard as a symmetry can be broken. Only left-chiral neutrinos, projected out by P_L = (1 − γ⁵)/2, take part; C alone would turn a left-handed ν into a left-handed ν̄, a state that does not exist, and P alone into a right-handed ν, also absent. The product CP maps ν_L onto ν̄_R, which does exist, and for a while CP looked exact: π⁺ → μ⁺ν_μ emits a left-handed muon and its CP image π⁻ → μ⁻ν̄_μ emits a right-handed one at the same rate. Then in 1964 Cronin and Fitch saw the CP-odd K_L decay to the CP-even π⁺π⁻ state with branching ratio 2.0 × 10⁻³, a mixing parameter |ε| = 2.23 × 10⁻³; and the semileptonic asymmetry δ_L = 3.3 × 10⁻³, by which K_L → π⁻e⁺ν beats K_L → π⁺e⁻ν̄, defines matter against antimatter without reference to any convention. B mesons show direct CP violation an order of magnitude larger: Γ(B⁰ → K⁺π⁻) exceeds Γ(B̄⁰ → K⁻π⁺) by about 18%, an asymmetry (difference over sum) of 8%. So neither C nor CP can be the reason a particle and its antiparticle share a mass. Something else is.

Lüders and Pauli (1954–55) proved that any local, Lorentz-invariant field theory with a Hermitian Hamiltonian is invariant under the antiunitary product Θ = CPT. Because Θ commutes with H and sends a particle at rest, spin up, to its antiparticle at rest, spin down, the two states share one energy eigenvalue: m̄ = m. Unitarity carries the argument to the imaginary part of the energy, so total widths and lifetimes match, τ̄ = τ, though the sum may be shared out differently among channels, which is what direct CP violation does. Charge and magnetic moment reverse: q̄ = −q, μ̄ = −μ. Each is a tested prediction. The K⁰–K̄⁰ mass difference is below 6 × 10⁻¹⁹ of m_K, about 3 × 10⁻¹⁰ eV; the antiproton's charge-to-mass ratio matches the proton's to 1.6 × 10⁻¹¹; free-muon lifetimes give (τ_μ⁺ − τ_μ⁻)/τ = (2 ± 8) × 10⁻⁵. The muon test carries a trap: stopped μ⁻ are also captured by nuclei, at 3.8 × 10⁴ s⁻¹ in carbon, so their apparent lifetime in matter is 2.03 μs against 2.20 μs for μ⁺. That 8% is capture, not CPT violation, and must be removed before the comparison means anything.