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,

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

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 $b\_mu$,

$a!!!/a!!!/=a^mu\; a\_mu=a^2$

$a!!!/b!!!/+b!!!/a!!!/\; =\; 2\; a\; cdot\; b\; ,$.

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,

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

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

$operatorname\{tr\}(gamma\_5\; a!!!/b!!!/c!!!/d!!!/)\; =\; 4\; i\; epsilon\_\{mu\; nu\; lambda\; sigma\}\; a^mu\; b^nu\; c^lambda\; d^sigma$

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

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

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

where

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

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 $gamma,$'s,

as well as the definition of four momentum

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

We see explicitly that

$p!!!/\; =\; gamma^mu\; p\_mu\; ,$	$=\; gamma^0\; p\_0\; +\; gamma^i\; p\_i\; ,$

See also

• Gamma matrices
• Trace theorems

References

• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.