Dirac operator - Knowlpedia

Let $M$ be an oriented Riemannian manifold without boundary equipped with a spin structure, $S\to M$ its spinor bundle, and $\nabla^S$ its spin connection. The spin Dirac operator is the Clifford contraction of the covariant derivative:

\not D=c\circ\nabla^S:\Gamma_c^\infty(S)\longrightarrow\Gamma_c^\infty(S),\qquad \not D\psi=\sum_{j=1}^{\dim M}c(e^j)\nabla^S_{e_j}\psi,

where $(e_j)$ is any local orthonormal frame. The expression is independent of that frame. Its principal symbol is Clifford multiplication, so $\not D$ is a formally self-adjoint Dirac-type operator.

Analytic realization

With the Riemannian measure and Hermitian spinor metric, $\not D$ is an unbounded symmetric operator on $L^2(M,S)$ . If $M$ is complete, $\not D$ is essentially self-adjoint, so its closure is self-adjoint. If $M$ is closed, this realization has compact resolvent and discrete real spectrum of finite multiplicity.

Geometric identities and variants

The Lichnerowicz formula

\not D^{\,2}=(\nabla^S)^*\nabla^S+\frac{1}{4}\operatorname{Scal}

relates the operator to scalar curvature. Twisting $S$ by a Hermitian bundle with connection yields a twisted Dirac operator with an additional curvature term in its square. In even dimensions the spinor bundle is $\mathbb Z/2$ -graded and $\not D$ interchanges the two graded pieces.

Conventions and scope

This page is specifically Riemannian: its principal symbol is invertible away from the zero section, so $\not D$ is elliptic, and its analytic realization uses a positive-definite $L^2$ inner product. On a Lorentzian spinor bundle, Clifford contraction instead gives a nonelliptic operator whose characteristic covectors are null. Its flat model is the Minkowski Dirac operator, and its evolution equation is the Dirac equation. These operators share a Clifford-geometric construction but belong to different analytic categories.

References

H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: chapter II, §5, spinor bundles and the Dirac operator.
J. Roe, Elliptic Operators, Topology and Asymptotic Methods, 2nd ed., Chapman & Hall/CRC, 1998. Publisher record. Relevant: chapters 3 and 5, Clifford operators and analytic properties.