Lorentzian Dirac operator

Definition

Let $(M,g)$ be a Lorentzian spin manifold, $S\to M$ its Lorentzian spinor bundle, and $\nabla^S$ its spin connection. The Lorentzian Dirac operator is Clifford contraction of the covariant derivative:

D_g=c\circ\nabla^S: \Gamma^\infty(S)\longrightarrow\Gamma^\infty(S).

If $(e_a)$ is a local pseudo-orthonormal frame and $\varepsilon_a=g(e_a,e_a)\in\{-1,+1\}$ , then

D_g\psi=\sum_a\varepsilon_a\,c(e_a)\nabla^S_{e_a}\psi.

This formula is independent of the chosen pseudo-orthonormal frame.

Principal symbol

Ignoring the conventional factor of $i$ used in some symbol conventions, the principal symbol at a covector $\xi$ is

\sigma_1(D_g)(x,\xi)=c(\xi^\sharp).

With $c(v)^2=-g(v,v)$ , it satisfies

\sigma_1(D_g)(x,\xi)^2 =-g_x^{-1}(\xi,\xi)\operatorname{id}_{S_x}.

Thus $D_g$ is nonelliptic: its nonzero characteristic covectors are exactly the null covectors. Its square has the principal symbol of a normally hyperbolic operator.

Cauchy problem

On a globally hyperbolic spacetime, compactly supported initial spinor data on a smooth spacelike Cauchy hypersurface determine a unique solution with finite propagation speed. This is the Cauchy theorem for the Lorentzian Dirac operator.

Scope

The Lorentzian Dirac operator is the signature- $(1,n-1)$ counterpart of the Riemannian spin Dirac operator, but their analytic theories differ. The flat model is the Minkowski Dirac operator. Choices of invariant spinor pairing and time orientation are required before formulating formal-adjoint or Hilbert-space statements.

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,
Publisher record. Relevant: §§1.3 and 3.4.