Charge conjugation and the positron wavefunction

Start from Unit 4's minimally coupled equation, [γ^μ(iħ∂_μ − qA_μ) − mc]ψ = 0, with A^μ = (φ/c, A) real and q = −e for the electron. The charge lives only in the relative sign of iħ∂_μ and qA_μ, both in kg m s⁻¹. Complex conjugation reverses the first and not the second: [(γ^μ)*(−iħ∂_μ − qA_μ) − mc]ψ* = 0, which is −[(γ^μ)*(iħ∂_μ + qA_μ) + mc]ψ* = 0. Inside the bracket the charge now reads −q. Two things are still wrong: the matrices are (γ^μ)* rather than γ^μ themselves, and the mass term has the wrong sign against the derivative term. A constant matrix M with M(γ^μ)*M⁻¹ = −γ^μ for every μ would repair both at once, since multiplying by M turns the bracket into −[γ^μ(iħ∂_μ + qA_μ) − mc]Mψ*. Notice what the conjugation does to a stationary state before any matrix acts: w e^(−iEt/ħ) becomes w* e^(+iEt/ħ), so the energy reverses as well. Charge and energy flip together, and that pairing is the physics of this topic.

In the Dirac representation γ⁰ = diag(1, 1, −1, −1), and γ^k has off-diagonal blocks σₖ and −σₖ. σₓ and σ_z are real and σ_y is imaginary, so γ⁰, γ¹ and γ³ are real while (γ²)* = −γ². M must therefore anticommute with the three real matrices and commute with γ². The Clifford algebra supplies one: γ² anticommutes with every other γ^μ and commutes with itself, so γ²(γ^μ)*(γ²)⁻¹ = −γ^μ for all four μ, derived. Two such matrices differ by one that commutes with every γ^μ; topic 5.02 showed that such a matrix is a multiple of the identity, so M = ηγ² with η a constant. The convention η = i makes iγ² real and symmetric with (iγ²)² = 1, and gives the component rule ψ^c = (ψ₄*, −ψ₃*, −ψ₂*, ψ₁*). Apply it twice: (ψ^c)^c = iγ²(iγ²)*ψ = ψ, derived, and true for any η of modulus 1. Dimensions check because iγ² is a pure number: ψ^c carries the m^(−3/2) of ψ, and every term of its equation is a momentum times ψ.

Many texts define ψ^c = Cψ̄^T with C = iγ²γ⁰. The two agree, derived: ψ̄^T = (ψ†γ⁰)^T = (γ⁰)^T ψ* = γ⁰ψ* because γ⁰ is real and diagonal, so Cψ̄^T = iγ²γ⁰γ⁰ψ* = iγ²ψ*. Written out, C has the entries −1, +1, −1, +1 down its anti-diagonal from the top row; it is real, C^T = −C and C² = −1, so C⁻¹ = C^T = −C. Its defining property C(γ^μ)^T C⁻¹ = −γ^μ follows from (γ^μ)^T = γ⁰(γ^μ)*γ⁰ and the γ² identity. That is the form to carry into other representations. In the Weyl representation γ² is again the only imaginary matrix and the recipe survives; in the Majorana representation all four γ's are imaginary, and there ψ^c is simply ψ*, up to a phase. The C form also shows that ψ^c is built from ψ̄, and a two-line check with Unit 5's generators, iγ²S*(iγ²)⁻¹ = S, confirms that ψ^c transforms under rotations and boosts exactly as ψ does.

Write a negative-energy electron solution as ψ = w e^(+i|E|t/ħ) e^(+ip⋅x/ħ): energy −|E|, momentum p, spin read from Σ. The star inverts both exponentials, so ψ^c = iγ²w* e^{(−i|E|t/ħ)} e^{(−ip⋅x/ħ)} has energy +|E| and momentum −p. For spin, iγ²Σ*iγ² = −Σ, from σ_yσₖ*σ_y = −σₖ, so (ψ^c)†Σψ^c = −ψ†Σψ, while helicity, a product of two reversed quantities, survives. With the reversed charge this is topic 8.04's hole: the vacancy left by an electron with −|E|, p, s and −e is a positron with +|E|, −p, −s and +e, and ψ^c is its wavefunction. In the non-relativistic limit the large components move from the lower pair to the upper, so the positron obeys Pauli's equation with charge +e and the same g = 2 (Unit 6's reduction with q → −q). For a static field the same steps give a spectral rule, derived: H(q)w = Ew implies H(−q)(iγ²w*) = −E(iγ²w*). Hydrogen's ground level at mc² − 13.6 eV becomes a negative-energy level of the +e equation at −mc² + 13.6 eV, and no level sits just below +mc²: the proton binds no positron. Measured: e⁺ and e⁻ g factors agree to (−0.5 ± 2.1) × 10⁻¹².

Unit 4 wrote v(p, s) = √((E + mc²)/c)(cσ⋅p ηₛ/(E + mc²); ηₛ) with ηₛ = −iσ_yχₛ* and called the choice a label convention. Conjugation derives it. Apply iγ², whose blocks are iσ_y above and −iσ_y below, to u(p, s)* = √((E + mc²)/c)(χₛ*; cσ*⋅p χₛ*/(E + mc²)). The lower pair becomes −iσ_yχₛ* = ηₛ, and the upper pair becomes iσ_y(σ*⋅p)χₛ* = (σ⋅p)(−iσ_yχₛ*), using σ_yσₖ*σ_y = −σₖ again, so v(p, s) = iγ²u(p, s)* exactly. Take cp = 1.000 MeV along z and s = ↑: E = 1.123 MeV and cp/(E + mc²) = 0.612, so u = 1.278(1, 0, 0.612, 0) and v = 1.278(0, −0.612, 0, 1) in √(MeV/c). Checks: v†v = 2.246 MeV/c = 2E/c, and v̄v = −1.022 MeV/c = −2mc. The free limit settles the labels Unit 4 borrowed. v(p, s) exp(ip_μx^μ/ħ) has energy −E, momentum −p and spin −s, and its conjugate is u(p, s) exp(−ip_μx^μ/ħ), a positron with energy E, momentum p and spin s.

The map is antilinear, (λψ)^c = λ*ψ^c; it also leaves the density alone: (ψ^c)†ψ^c = ψ^T(iγ²)(iγ²)ψ* = ψ†ψ. The current survives too. (ψ^c)†αₖψ^c = ψ^T iγ²αₖ iγ² ψ* is a single number, so transposing it is free for commuting components and gives ψ†(iγ²αₖ*iγ²)ψ = ψ†αₖψ: the probability current of a c-number ψ^c equals that of ψ, derived. At cp = 2.00 MeV along +z, ψ and ψ^c both move at −0.969c. The electric current qj therefore reverses only because q was switched by hand; nothing in the one-particle spinor carries the sign. Unit 9's field operators anticommute, ψₐψ_b† = −ψ_b†ψₐ + δ_ab δ³(0), so the same transposition costs a minus sign and ψ̄^c γ^μ ψ^c = −ψ̄ γ^μ ψ after normal ordering (quoted). The limit q = 0 allows a spinor equal to its own conjugate, ψ^c = ψ, which in components reads (a, b, −b*, a*): four real parameters, not eight. These Majorana spinors are only named here. Whether neutrinos are Majorana is open: searches for neutrinoless double-β decay have set only lower limits on its half-life, above 10²⁶ years.