Feynman slash notation

Using the anticommutators of the gamma matrices, one can show that for any $a\_\backslash mu$ and $b\_\backslash mu$,

:$\backslash begin\{align\}$

{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}

where $I\_4$ is the identity matrix in four dimensions.

In particular,

:$\{\backslash partial\backslash !\backslash !\backslash !/\}^2\; =\; \backslash partial^2\; \backslash cdot\; I\_4.$

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

:$\backslash begin\{align\}$

\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}

where:

• $\backslash varepsilon\_\{\backslash mu\; \backslash nu\; \backslash lambda\; \backslash sigma\}$ is the Levi-Civita symbol
• $\backslash eta^\{\backslash mu\; \backslash nu\}$ is the Minkowski metric
• $m$ is a scalar.

This section uses the {{math|(+ − − −)}} 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,

:

as well as the definition of contravariant four-momentum in natural units,

:$p^\backslash mu\; =\; \backslash left(E,\; p\_x,\; p\_y,\; p\_z\; \backslash right)\; \backslash ,$

we see explicitly that

:$\backslash begin\{align\}$

{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 p^i & 0 \end{bmatrix} \\

&= \begin{bmatrix} E & -\vec{\sigma} \cdot \vec{p} \\ \vec{\sigma} \cdot \vec{p} & -E \end{bmatrix}.

\end{align}

Similar results hold in other bases, such as the Weyl basis.

• Weyl basis
• Gamma matrices
• Four-vector
• S-matrix

{{reflist}}

{{refbegin}}

• {{cite book |author1=Halzen, Francis |authorlink1 = Francis Halzen |author2=Martin, Alan | authorlink2 = Alan Martin (physicist)| title=Quarks & Leptons: An Introductory Course in Modern Particle Physics |url=https://archive.org/details/quarksleptonsint0000halz |url-access=registration | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}

{{refend}}

{{Richard Feynman}}

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