nLab Feynman slash notation

Beware that the notation discussed on this page – a symbol like “$D$” or “$\partial$” with a slash “/” drawn across it – is not properly rendered by some browsers, notably Chrome currently fails to draw the slash at all, making some formulas below appear meaningless.

Idea

In the context of spin geometry, the Feynman slash notation is a notation popular in physics texts, specifically in quantum field theory, where it was introduced by Richard Feynman, for the Clifford algebra-element associated with a given vector.

Abstractly, given an inner product space $(V, \langle -\rangle)$ with associated Clifford algebra $Cl_{(V,\langle -,-\rangle)}$, then there is a canonical linear map

$$f\;\colon\;V \longrightarrow Cl_{(V,\langle -,-\rangle)}$$

such that $f(v)\cdot f(v) = \langle v,v\rangle \cdot 1 \in Cl_{(V,\langle -,-\rangle)}$.

Now let $\left( x^\mu\right)_{\mu = 1}^{dim(V)}$ be a linear basis for $V$, so that every vector $D \in V$ may be expanded, using the Einstein summation convention, as

$$D \,=\, D_\mu x^\mu \,,$$

and let $\left( \gamma^\mu \coloneqq f(x^\mu)\right)$ be the corresponding generators of the Clifford algebra. Then this map is given by

$$D \,\mapsto\, D_\mu \gamma^\mu \,.$$

This particular class of instances of the Einstein summation convention on the right is further abbreviated with a slash as

$$\slash{D} \,\coloneqq\, D_\mu \gamma^\mu \,.$$

This is the Feynman slash notation.

Concretely, in quantum field theory, the Dirac operator on Minkowski spacetime may be expanded as $\mathrm{i} \gamma^\mu \frac{\partial}{\partial x^\mu}$ and this is then similarly abbreviated as

$$\mathrm{i} \slash{\partial} \;\coloneqq\; \mathrm{i} \gamma^\mu \frac{\partial}{\partial x^\mu} \,.$$

Related concepts

• Einstein summation convention

References

• Wikipedia, Feynman slash notation