Editing Feynman slash notation (section)

==Identities==
Using the [[anticommutator]]s of the gamma matrices, one can show that for any <math>a_\mu</math> and <math>b_\mu</math>,
:<math>\begin{align}
                         {a\!\!\!/}{a\!\!\!/} =   a^\mu a_\mu \cdot I_4 = a^2 \cdot I_4 \\
  {a\!\!\!/}{b\!\!\!/} + {b\!\!\!/}{a\!\!\!/} = 2 a \cdot b \cdot I_4.
\end{align}</math>

where <math>I_4</math> is the identity matrix in four dimensions.

In particular,
:<math>{\partial\!\!\!/}^2 = \partial^2 \cdot I_4.</math>

Further identities can be read off directly from the [[Gamma matrices#Identities|gamma matrix identities]] by replacing the [[metric tensor]] with [[inner product]]s. For example,
:<math>\begin{align}


\gamma_\mu {a\!\!\!/} \gamma^\mu &= -2 {a\!\!\!/} \\

\gamma_\mu {a\!\!\!/} {b\!\!\!/} \gamma^\mu &= 4 a \cdot b \cdot I_4 \\

\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\!\!\!/}) &= 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}({a\!\!\!/}{\gamma^\mu}{b\!\!\!/}{\gamma^\nu }) &=  4 \left[a^\mu b^\nu + a^\nu b^\mu - \eta^{\mu \nu}(a \cdot b) \right] \\

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

\operatorname{tr}({\gamma^\mu}{a\!\!\!/}{\gamma^\nu}) &= 0 \\

\operatorname{tr}({\gamma^5}{a\!\!\!/}{b\!\!\!/}) &=  0 \\

\operatorname{tr}({\gamma^0}({a\!\!\!/}+m){\gamma^0}({b\!\!\!/}+m)) &= 8a^0b^0-4(a \cdot b)+4m^2 \\

\operatorname{tr}(({a\!\!\!/}+m){\gamma^\mu}({b\!\!\!/}+m){\gamma^\nu}) &=
4 \left[a^\mu b^\nu+a^\nu b^\mu - \eta^{\mu \nu}((a \cdot b)-m^2) \right] \\

\operatorname{tr}({a\!\!\!/}_1...{a\!\!\!/}_{2n}) &=  \operatorname{tr}({a\!\!\!/}_{2n}...{a\!\!\!/}_1) \\

\operatorname{tr}({a\!\!\!/}_1...{a\!\!\!/}_{2n+1}) &= 0

\end{align}</math>

where:
*<math>\varepsilon_{\mu \nu \lambda \sigma}</math> is the [[Levi-Civita symbol]]
*<math>\eta^{\mu \nu}</math> is the [[Minkowski metric]]
*<math>m</math> is a scalar.