Gamma matrices, the Clifford algebra and trace identities

Work in SI units with x^μ = (ct, x), ∂_μ = ∂/∂x^μ = ((1/c)∂/∂t, ∇) and the metric g = diag(1, −1, −1, −1), kept throughout. Topic 4.01's equation is iħ ∂ψ/∂t = (−iħc αⁱ∂ᵢ + βmc²)ψ. Multiply on the left by β/c and use β² = 1: iħβ∂₀ψ + iħβαⁱ∂ᵢψ − mcψ = 0. Naming γ⁰ = β and γⁱ = βαᵢ gives (iħγ^μ∂_μ − mc)ψ = 0, summed over μ = 0 to 3, derived. With p_μ = iħ∂_μ, so that p₀ = E/c, it reads (p̸ − mc)ψ = 0. Dimensions: the γ^μ are pure numbers, ħ∂_μ has units J s m⁻¹ = kg m s⁻¹, and mc is a momentum, 2.731 × 10⁻²² kg m s⁻¹ for the electron, so the equation compares the spinor's variation with the inverse of λ̄_C = ħ/mc = 3.862 × 10⁻¹³ m. Nothing new was assumed. This is topic 4.01's equation under new names, reached by multiplying by a matrix, not by changing frame.

Compute each product from the α, β algebra. (γ⁰)² = β² = 1. (γⁱ)² = βαᵢβαᵢ = −β²αᵢ² = −1, because βαᵢ = −αᵢβ. For i ≠ j, γⁱγʲ + γʲγⁱ = −(αᵢαⱼ + αⱼαᵢ) = 0, and γ⁰γⁱ + γⁱγ⁰ = αᵢ + βαᵢβ = αᵢ − αᵢ = 0. All ten conditions now read (γ^μ, γ^ν) = 2g^(μν), the Clifford algebra, derived, with the metric's signs in the squares. Hermiticity changes character. γ⁰ is Hermitian, but γⁱ† = αᵢβ = −βαᵢ = −γⁱ, anti-Hermitian, and no representation can avoid it: a Hermitian matrix has real eigenvalues and cannot square to −1. One rule covers both, γ^μ† = γ⁰γ^μγ⁰, checked directly: γ⁰γ⁰γ⁰ = γ⁰ and γ⁰(βαᵢ)γ⁰ = αᵢβ = γⁱ†. In topic 4.02's Dirac representation, γ⁰ = diag(1, 1, −1, −1), and γⁱ has σᵢ in its upper-right block and −σᵢ in its lower-left, so each statement can be confirmed by block multiplication.

For any four-vector a, define a̸ = γ^μa_μ = γ⁰a⁰ − γ⋅a. Because a_μa_ν is symmetric, only the symmetric part of γ^μγν survives: a̸a̸ = ½(γ^μ, γ^ν)a_μa_ν = (a⋅a) × 1, derived. The same step gives a̸b̸ + b̸a̸ = 2(a⋅b) × 1. Now multiply (p̸ − mc)ψ = 0 by (p̸ + mc): (p⋅p − m²c²)ψ = 0, which for a plane wave is E² = p²c² + m²c⁴, Dirac's requirement recovered in one line rather than a page of matrices. For a plane wave p̸ is a numerical matrix with p̸p̸ = m²c² and, as the next section shows, Tr p̸ = 0, so its eigenvalues are +mc and −mc, two of each, and the equation puts the spinor in the +mc pair. The other two classes of vector behave differently. For a photon's k, k̸k̸ = k⋅k = 0 although k̸ is not zero: it is nilpotent, with no inverse. A spacelike vector's slash squares to a negative multiple of 1, as γ³ does, so its eigenvalues are imaginary. The figure shows all three.

Three inputs suffice: the Clifford algebra, cyclicity Tr(AB) = Tr(BA), and Tr 1 = 4, the one fact taken from the representation. Then Tr(γ^μγν) = ½Tr(γ^μ, γ^ν) = 4g^(μν), derived, so Tr(γ¹γ¹) = −4 and Tr(a̸b̸) = 4a⋅b. For odd products let Γ = γ⁰γ¹γ²γ³. Each γ^μ anticommutes with three of Γ's factors and commutes with the fourth, itself, so Γγ^μ = −γ^μΓ, and Γ is invertible because every factor is. For a product X of n gammas, ΓXΓ⁻¹ = (−1)ⁿX, while cyclicity gives Tr(ΓXΓ⁻¹) = Tr X. So Tr X = (−1)ⁿTr X, and every odd trace vanishes, derived; Unit 5 calls iΓ the matrix γ⁵. A change of representation, X → UXU⁻¹, leaves every trace unchanged, so no result can depend on which matrices were written down: Tr γ⁰ = 1 + 1 − 1 − 1 = 0 in the Dirac representation, and the Weyl γ⁰, with 1 in its off-diagonal blocks, has no nonzero diagonal entry at all.

Move γ^μ to the right through the other three, each step costing a 2g term and a sign: γ^{μγνγργσ} = 2g^(μν)γ^ργσ − 2g^(μρ)γ^νγσ + 2g^(μσ)γ^νγρ − γ^{νγργσγμ}. The last term has the same trace as the left side, by cyclicity, so twice the trace is the sum of three two-γ traces: Tr(γ^{μγνγργσ}) = 4(g^(μν)g^(ρσ) − g^(μρ)g^(νσ) + g^(μσ)g^(νρ)), derived. Read it as a pairing rule, adjacent pairs with + and the crossed pair with −. A direct check: γ⁰γ¹γ⁰γ¹ = −γ⁰γ⁰γ¹γ¹ = −(1)(−1) = 1, of trace 4, and the formula gives 4(0 − (1)(−1) + 0) = 4. Contracting indices gives γ^μγ_μ = ½g_μν(γ^μ, γ^ν) = g_μνg^(μν) = 4 and γ^μa̸γ_μ = γ^μ(2a_μ − γ_μa̸) = 2a̸ − 4a̸ = −2a̸, derived, shortcuts that keep longer traces short. Six gammas give fifteen pairings by the same reduction.

The index on γ^μ invites a reading the algebra does not support. The γ^μ are four fixed numerical matrices, the same in every frame; μ labels which matrix, as i labels σᵢ. Under a boost the coordinates change by Λ, but nobody replaces γ^μ by Λ^μ_νγ^ν. Covariance therefore asks for something the notation cannot supply: a 4 × 4 matrix S acting on the spinor with S⁻¹γ^μS = Λ^μ_νγ^ν, so that the equation keeps its form. Unit 5 proves that S exists and builds it; until then covariance is assumed. The algebra also leaves the matrices free up to equivalence, as Pauli's theorem, quoted in topic 4.02, says. What it fixes is every trace, and traces are what physics needs: topic 4.08 turns sums over unobserved spins into traces of p̸ ± mc, and topic 5.09 evaluates them into Mott's factor 1 − β²sin²(θ/2), quoted here, the baseline against which high-energy electron scattering from nuclei is measured.