Lorentzian spinor bundle

Definition

Let $(M,g)$ carry a Lorentzian spin structure $P_{\mathrm{Spin}}$ , and let $\Delta_{1,n-1}$ be a real or complex module for the relevant Clifford algebra, restricted to $\mathrm{Spin}^+(1,n-1)$ . The associated Lorentzian spinor bundle is

S=P_{\mathrm{Spin}}\times_{\mathrm{Spin}^+(1,n-1)}\Delta_{1,n-1}.

It is a Clifford module bundle with multiplication

c(v)c(w)+c(w)c(v)=-2g(v,w)\operatorname{id}_S.

Associated connection

The Lorentzian Levi–Civita connection lifts to $P_{\mathrm{Spin}}$ and induces a covariant derivative $\nabla^S$ on $S$ . Clifford contraction gives the Lorentzian Dirac operator. Unlike the Dirac operator on a Riemannian spinor bundle, this operator is not elliptic.

Representation choices

The rank and additional structures of $S$ depend on dimension, signature, scalar field, and the chosen spin representation. Majorana, Weyl, and Majorana–Weyl conditions exist only in appropriate signatures and dimensions. The complex spinor bundle forgets some signature-dependent real structure even though its Clifford action still remembers the Lorentzian metric.

Spinor pairing

In indefinite signature the natural invariant spinor pairing is generally not a positive-definite Hermitian metric preserved in the same way as in Riemannian geometry. Formal adjoints and conserved currents therefore require an explicitly chosen pairing and a time orientation; they should not be imported unchanged from the Riemannian $L^2$ theory.

References

Helga Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, Teubner, 1981. Bibliographic record. Relevant: Chapters 2–3.
Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §§1.3 and 3.4.