Feynman slash notation

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^[1]). If A is a covariant vector (i.e., a 1-form),

{A\!\!\!/}\ {\stackrel {\mathrm {def} }{=}}\ \gamma ^{\mu }A_{\mu }

using the Einstein summation notation where γ are the gamma matrices.

Identities

Using the anticommutators of the gamma matrices, one can show that for any $a_\mu$ and $b_\mu$ ,

{\begin{aligned}{a\!\!\!/}{a\!\!\!/}&\equiv a^{\mu }a_{\mu }\cdot I_{4}=a^{2}\cdot I_{4}\\{a\!\!\!/}{b\!\!\!/}+{b\!\!\!/}{a\!\!\!/}&\equiv 2a\cdot b\cdot I_{4}\,\end{aligned}}

.

where $I_{4}$ is the identity matrix in four dimensions.

In particular,

{\partial \!\!\!/}^{2}\equiv \partial ^{2}\cdot I_{4}.

Further identities can be read off directly from the gamma matrix identities by replacing the metric tensor with inner products. For example,

{\begin{aligned}\operatorname {tr} ({a\!\!\!/}{b\!\!\!/})&\equiv 4a\cdot b\\\operatorname {tr} ({a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&\equiv 4\left[(a\cdot b)(c\cdot d)-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)\right]\\\operatorname {tr} (\gamma _{5}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&\equiv 4i\epsilon _{\mu \nu \lambda \sigma }a^{\mu }b^{\nu }c^{\lambda }d^{\sigma }\\\gamma _{\mu }{a\!\!\!/}\gamma ^{\mu }&\equiv -2{a\!\!\!/}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}\gamma ^{\mu }&\equiv 4a\cdot b\cdot I_{4}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}\gamma ^{\mu }&\equiv -2{c\!\!\!/}{b\!\!\!/}{a\!\!\!/}\\\end{aligned}}

where

\epsilon_{\mu \nu \lambda \sigma} \,

is the Levi-Civita symbol.

With four-momentum

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,