Minkowski Dirac operator

Definition

On four-dimensional Minkowski spacetime, choose a constant complex spinor module $S$ and gamma matrices $\gamma^\mu=c(dx^\mu)$ satisfying

\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=-2\eta^{\mu\nu}I, \qquad \eta=\operatorname{diag}(-1,1,1,1).

The Minkowski Dirac operator is

D_{\mathrm M}=c\circ d=\gamma^\mu\partial_\mu

on smooth spinor fields $\psi:\mathbb R^{1,3}\to S$ .

Square and symbol

Because the matrices are constant and partial derivatives commute,

D_{\mathrm M}^{\,2} =-\eta^{\mu\nu}\partial_\mu\partial_\nu =\Box_\eta.

Its principal symbol is $c(\zeta)$ , and

c(\zeta)^2=-\eta^{-1}(\zeta,\zeta)I.

It is invertible away from the null cone but singular at nonzero null covectors. Thus this Lorentzian operator is not elliptic; its square is normally hyperbolic.

Relation to gamma-matrix conventions

With the metric convention above, the matrices $\gamma^\mu=c(dx^\mu)$ square to $+I$ in the time direction and $-I$ in spatial directions. This is the same matrix relation often written with the mostly-minus metric $-\eta$ . Switching metric or Clifford sign conventions changes the displayed anticommutator and may insert factors of $i$ .

Curved analogue

On a Lorentzian spinor bundle, ordinary differentiation is replaced by the spin connection and $D_g=c\circ\nabla^S$ . Squaring then produces curvature terms. By contrast, the Riemannian spin Dirac operator is elliptic and belongs to a different analytic theory.

References

Helga Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, Teubner, 1981. Bibliographic record. Relevant: Chapter 3.
Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995. Publisher record. Relevant: Chapters 3–4.