Knowlpedia

Theorem

Cauchy problem for the Lorentzian Dirac operator

Initial spinor data on a smooth spacelike Cauchy hypersurface determine a unique Lorentzian Dirac solution with causal propagation.

Statement

Let DgD_g be the on a spinor bundle S→MS\to M over a . Let Σ⊂M\Sigma\subset M be a smooth spacelike . For every compactly supported smooth source f∈Γc∞(S)f\in\Gamma^\infty_c(S) and compactly supported smooth initial spinor ψ0∈Γc∞(S∣Σ)\psi_0\in\Gamma^\infty_c(S|_\Sigma), there is a unique smooth spinor ψ∈Γ∞(S)\psi\in\Gamma^\infty(S) satisfying

Dgψ=f,ψ∣Σ=ψ0.D_g\psi=f, \qquad \psi|_\Sigma=\psi_0.

The solution has finite propagation speed:

supp⁡ψ⊆J ⁣(supp⁡f∪supp⁡ψ0).\operatorname{supp}\psi \subseteq J\!\left(\operatorname{supp}f\cup\operatorname{supp}\psi_0\right).

Thus the Lorentzian has one freely prescribed spinor trace on Σ\Sigma, unlike the two traces required by a second-order normally hyperbolic equation.

The same conclusion holds for Dirac operators with smooth zeroth-order terms, including a constant mass term. Such operators also have unique on a globally hyperbolic spacetime.

References
  1. Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society,
  2. Publisher record. Relevant: §3.4.