Konrad Waldorf.

Classical Gauge Theory and Matter Fields II.

No elementary particle can be described in the framework of the previous lecture. Some non-elementary particles, like π^{−}-meson, can be described in these terms.

Dirac’s starting point was that Schrödinger’s equation is first order, whereas Klein-Gordon’s equation (Δ^{ω} + m^{2})ϕ = 0 is second order.

We consider Clifford algebra Cl(p, q) of R^{p,q}. This algebra has two involutive anti-automorphisms: α is induced by v ↦ −v on R^{p,q} and ·^{t}: v_{1} ⊗ ⋯ ⊗ v_{r} → v_{r} ⊗ ⋯ ⊗ v_{1}. The Clifford algebra has a real valued metric H(v, w) = (v^{t}w)_{0}.

Spin. We have SO(p, q) and Spin(p, q) = {v_{1} ⋯ v_{2r} | v_{i} ∈ R^{p,q} ∧ ∥v_{i}∥ = 1}. Λ: Spin(p, q) → SO(p, q) is given by Λ(ϕ)(v) = α(ϕ)vϕ^{−1}. We have an exact sequence 1 → Z/(2) → Spin(p, q) → SO(p, q) → 1. Cl(p, q) ⊗_{R} C decomposes into k copies of a subrepresentation Σ. If p + q is odd, then Σ is irreducible, k = 2^{(p+q−1)/2}, if p+q is even, then k = 2^{(p+q)/2} and Σ decomposes into sum of two irreducible representations Σ^{+} ⊕ Σ^{−}. For example, if p + q = 4, then dim Σ^{±} = 2.

Spin structure. Suppose M is a spacetime with signature (p, q). F M (frame bundle of M) is an SO(p, q)- bundle. A spin structure on M is a Spin(p, q)-bundle SM with λ: SM → F M such that λ(Xϕ) = λ(X)Λ(ϕ). If θ ∈ Ω^{1}(F M, so(p, q)) is the Levi-Civita connection on F M, then Θ = dΛ^{−1}(λ^{∗}θ) ∈ Ω^{1}(SM, spin(p, q)).

Dirac operator. Let ρ: Spin(p, q) → GL(V) be a representation of Spin-group on a subrepresentation V ⊂ Cl(p, q) of Cl(p, q). The spinor bundle ΣM is defined as SM ×_{ρ} V. We have Clifford multiplication T M ⊗ΣM → ΣM. Now we can define the Dirac operator D: Γ(M, ΣM) → Γ(M, ΣM): ψ ↦ ∑_{i} e_{i}·D^{Θ}ψ(e_{i}).

Spinors. A spinor is a field for SM of matter type (V, h, ρ, f) where V is the same as above, ρ is the restruction of Clifford multiplication to Spin(p, q), h(v, w) = (H(v, w) + H(w, v))/2. Fields are sections of ΣM as usual, but we have a different action functional S(ψ) = ∫_{M} Dψ ∧_{h} ∗ψ + ∗(fψ).

Example. (Weyl spinor.) f = 0. V = Σ^{±}. dim_{C} V = 2. In the standard model we have “neutrinos”. Example. (Dirac spinor.) V = Σ, f(v) = mh(v, v)/2. dim_{C} V = 4. Standard model: “electrons”.

Euler-Lagrange equations: Dψ + imψ = 0. (Dirac equation.) Particles that satisfy Dirac equation also satisfy Klein-Gordon equation.