Charge conjugation transforms the positive-energy spinor into the negative-energy spinor. Charge conjugation is a mapping (an involution) ψ ψ ↦ ↦ ψ ψ c having the explicit form ψ ψ c = η η C ( ψ ψ ¯ ¯ ) T where ( ⋅ ⋅ ) T denotes the transpose, C is a 4×4 matrix, and η η is an arbitrary phase factor, η η ∗ ∗ η η = 1. The article on charge conjugation derives the above form, and demonstrates why the word "charge" is the appropriate word to use: it can be interpreted as the electrical charge. In the Dirac representation for the gamma matrices, the matrix C can be written as C = i γ γ 2 γ γ 0 = ( 0 − − i σ σ 2 − − i σ σ 2 0 ) Thus, a positive-energy solution (dropping the spin superscript to avoid notational overload) ψ ψ ( + ) = u ( p → → ) e − − i p ⋅ ⋅ x = E + m 2 m [ ϕ ϕ σ σ → → ⋅ ⋅ p → → E + m ϕ ϕ ] e − − i p ⋅ ⋅ x is carried to its charge conjugate ψ ψ c ( + ) = E + m 2 m [ i σ σ 2 σ σ → → ∗ ∗ ⋅ ⋅ p → → E + m ϕ ϕ ∗ ∗ − − i σ σ 2 ϕ ϕ ∗ ∗ ] e i p ⋅ ⋅ x Note the stray complex conjugates. These can be consolidated with the identity σ σ 2 ( σ σ → → ∗ ∗ ⋅ ⋅ k → → ) σ σ 2 = − − σ σ → → ⋅ ⋅ k → → to obtain ψ ψ c ( + ) = E + m 2 m [ σ σ → → ⋅ ⋅ p → → E + m χ χ χ χ ] e i p ⋅ ⋅ x with the 2-spinor being χ χ = − − i σ σ 2 ϕ ϕ ∗ ∗ As this has precisely the form of the negative energy solution, it becomes clear that charge conjugation exchanges the particle and anti-particle solutions. Note that not only is the energy reversed, but the momentum is reversed as well. Spin-up is transmuted to spin-down. It can be shown that the parity is also flipped. Charge conjugation is very much a pairing of Dirac spinor to its "exact opposite".