Dirac operator

Summary

A Dirac operator is a first-order differential operator $D$ whose square $D^2$ is a Laplace-type operator. It is a formal square root of the Laplacian, made possible by coefficients in a Clifford algebra. Dirac introduced such an operator to write a first-order relativistic equation for the electron, the Dirac equation. In geometry, Dirac operators on spinor bundles are elliptic, and their indices are computed by the Atiyah–Singer index theorem.