Clifford slash notation

Definition

Let $c:\operatorname{Cl}(V,g)\to\operatorname{End}(\Delta)$ be a Clifford module. For $v\in V$ , its Clifford slash is

\slashed v=c(v).

For a covector $\xi\in V^*$ , the metric is used to raise its index:

\slashed\xi=c(\xi^\sharp).

In a basis with gamma matrices $\gamma^a$ , this is

\slashed\xi=\gamma^a\xi_a.

Under the convention $c(v)^2=-g(v,v)$ , slash notation satisfies

\slashed\xi^{\,2} =-g^{-1}(\xi,\xi)\operatorname{id}_\Delta.

This identity is the principal-symbol calculation behind the fact that the square of a Dirac operator has metric quadratic principal symbol.

The musical isomorphism in the covector formula is essential: Clifford multiplication is defined on vectors unless the Clifford algebra has instead been constructed directly from the cotangent quadratic form.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. DOI record. Relevant: Chapter I.
Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison–Wesley, 1995. Relevant: §3.1.