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). If A is a covariant vector, i.e. 1-form,

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

[edit] Identities

Using the anticommutators of the gamma matrices, one can show that for any $a μ$ ,

$a\!\!\!/a\!\!\!/=a^\mu a_\mu=a^2.$

In particular,

$\partial\!\!\!/^2=\partial^2.$

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

$\operatorname{tr}(a\!\!\!/b\!\!\!/) = 4 a \cdot b$

$\operatorname{tr}(a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/) = 4 \left[(a\cdot b)(c \cdot d) - (a \cdot c)(b \cdot d) + (a \cdot d)(b \cdot c) \right]$

$\operatorname{tr}(\gamma_5 a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/) = 4 i \epsilon_{\mu \nu \lambda \sigma} a^\mu b^\nu c^\lambda d^\sigma$

$\gamma_\mu a\!\!\!/ \gamma^\mu = -2 a\!\!\!/$ .

$\gamma_\mu a\!\!\!/ b\!\!\!/ \gamma^\mu = 4 a \cdot b \,$

$\gamma_\mu a\!\!\!/ b\!\!\!/ c\!\!\!/ \gamma^\mu = -2 c\!\!\!/ b\!\!\!/ a\!\!\!/ \,$

where

$\epsilon_{\mu \nu \lambda \sigma} \,$ is the Levi-Civita symbol.

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 $\gamma\,$ 's,

$\gamma^0 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix} \,$

as well as the definition of four momentum

$p^{\mu} = \left(E, p^x, p^y, p^z \right) \,$

We see explicitly that

$p\!\!\!/ = \gamma^\mu p_\mu \,$	$= \gamma^0 p_0 - \gamma^i p_i \,$
	$=\begin{bmatrix} p_0 & 0 \\ 0 & -p_0 \end{bmatrix} - \begin{bmatrix} 0 & \sigma^i p_i \\ - \sigma^i \cdot p_i & 0 \end{bmatrix} \,$
	$=\begin{bmatrix} E & -\mathbf{\sigma \cdot p} \\ \mathbf{\sigma \cdot p} & -E \end{bmatrix} \,$