In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If A is a covariant vector (i.e., a 1-form),
{A/} \stackrel{def
where γ are the gamma matrices. Using the Einstein summation notation, the expression is simply
{A/} \stackrel{def
Using the anticommutators of the gamma matrices, one can show that for any
a\mu
b\mu
\begin{align} {a/}{a/}=a\mua\mu ⋅ I4=a2 ⋅ I4\\ {a/}{b/}+{b/}{a/}=2a ⋅ b ⋅ I4. \end{align}
where
I4
In particular,
{\partial/}2=\partial2 ⋅ I4.
Further identities can be read off directly from the gamma matrix identities by replacing the metric tensor with inner products. For example,
\begin{align} \gamma\mu{a/}\gamma\mu&=-2{a/}\\ \gamma\mu{a/}{b/}\gamma\mu&=4a ⋅ b ⋅ I4\\ \gamma\mu{a/}{b/}{c/}\gamma\mu&=-2{c/}{b/}{a/}\\ \gamma\mu{a/}{b/}{c/}{d/}\gamma\mu&=2({d/}{a/}{b/}{c/}+{c/}{b/}{a/}{d/})\\ \operatorname{tr}({a/}{b/})&=4a ⋅ b\\ \operatorname{tr}({a/}{b/}{c/}{d/})&=4\left[(a ⋅ b)(c ⋅ d)-(a ⋅ c)(b ⋅ d)+(a ⋅ d)(b ⋅ c)\right]\\ \operatorname{tr}({a/}{\gamma\mu}{b/}{\gamma\nu})&=4\left[a\mub\nu+a\nub\mu-η\mu(a ⋅ b)\right]\\ \operatorname{tr}(\gamma5{a/}{b/}{c/}{d/})&=4i\varepsilon\mua\mub\nucλd\sigma\\ \operatorname{tr}({\gamma\mu}{a/}{\gamma\nu})&=0\\ \operatorname{tr}({\gamma5}{a/}{b/})&=0\\ \operatorname{tr}({\gamma0}({a/}+m){\gamma0}({b/}+m))&=8a0b0-4(a.b)+4m2\\ \operatorname{tr}(({a/}+m){\gamma\mu}({b/}+m){\gamma\nu})&= 4\left[a\mub\nu+a\nub\mu-η\mu((a ⋅ b)-m2)\right]\\ \operatorname{tr}({a/}1...{a/}2n)&=\operatorname{tr}({a/}2n...{a/}1)\\ \operatorname{tr}({a/}1...{a/}2n+1)&=0 \end{align}
where:
\varepsilon\mu
η\mu
m
This section uses the metric signature. Often, when using the Dirac equation and solving for cross sections, one finds the slash notation used on four-momentum: using the Dirac basis for the gamma matrices,
\gamma0=\begin{pmatrix}I&0\ 0&-I\end{pmatrix}, \gammai=\begin{pmatrix}0&\sigmai\ -\sigmai&0\end{pmatrix}
as well as the definition of contravariant four-momentum in natural units,
p\mu=\left(E,px,py,pz\right)
we see explicitly that
\begin{align} {p/}&=\gamma\mup\mu=\gamma0p0-\gammaipi\\ &=\begin{bmatrix}p0&0\ 0&-p0\end{bmatrix}-\begin{bmatrix}0&\sigmaipi\ -\sigmaipi&0\end{bmatrix}\\ &=\begin{bmatrix}E&-\vec{\sigma} ⋅ \vec{p}\ \vec{\sigma} ⋅ \vec{p}&-E\end{bmatrix}. \end{align}
Similar results hold in other bases, such as the Weyl basis.