Konrad Waldorf. Charged spinors. Definition. A spinor is a field for a spin structure SM on M of matter type (V, ρ, h, f), where (V, ρ)

is a representation of Cl(p, q) on Cl(p, q), h: V × V → R is a bilinear form, f: V → R is ρ-invariant. A field is a section ψ: M → SM ×_{ρ} V = ΣM of the spinor bundle. We have an action functional S(ψ) = ∫_{M} Dψ∧_{h}∗ψ+∗(f ∘ψ). Equation of motion is Dψ+imψ = 0 (the Dirac equation). (Here f(v) = −mh(v, v).)

We want to couple this stuff to electromagnetic field.

Bundle splicing: If G_{k} is a Lie group and P_{k} is a principal G_{k}-bundle over Π, then P_{1} ∘ P_{2} = P_{1} ×_{M} P_{2} is a principal G_{1} × G_{2}-bundle over M. If we have connections ω_{k} on P_{k}, then ω_{1} ∘ ω_{2} = p_{1}^{∗}ω_{1} ⊕ p_{2}^{∗}ω_{2} ∈ Ω^{1}(P_{1} ∘ P_{2}, g_{1} ⊕ g_{2}). If ρ_{k} is a representation of G_{k} such that ρ_{1}(g_{1}) ∘ ρ_{2}(g_{2}) = ρ_{2}(g_{2}) ∘ ρ_{1}(g_{1}), then ρ_{1} × ρ_{2} is a representation of G_{1} × G_{2}.

Setup for charged spinors. We have a spin structure SM on M with Levi-Civita connection, Yang-Mills theory (G, P) and connection ω on P. We have a representation ρ_{SM}: Spin(p, q) → GL(V) and ρ_{P}: G → GL(V), a bundle ΣP = (SM∘P)×_{ρ_{SM} ×ρ_{P}} V and a Dirac operator on this bundle D^{ω}: Γ(M, ΣP) → Γ(M, ΣP).

Definition: A charged spinor is a field for SM ∘ P of matter type (V, ρ_{SM} × ρ_{P} , h, f). Action functional is S(ψ, ω) = ∫_{M} D^{ω}(ψ) ∧ ∗ψ + ∗(f ∘ ψ) + S_{YM}(ω). Dirav equation D^{ω}ψ + imψ = 0 and D^{ω} ∗ F_{ω} = J(ψ).

Electrons: V = Σ = Σ^{+} ⊕ Σ^{−} and ρ_{P} (z)(v) = z^{n} · v, where n is an integer number (charge). Dirac equation is D^{ω}ψ + imψ = 0.