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., 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 $b_{\mu }$,

$a\!\!\!/a\!\!\!/=a^{\mu }a_{\mu }\cdot I_{4\times 4}=a^{2}\cdot I_{4\times 4}$

$a\!\!\!/b\!\!\!/+b\!\!\!/a\!\!\!/=2a\cdot b\cdot I_{4\times 4}\,$.

where $I_{4\times 4}$ is the identity matrix in four dimensions. In particular,

$\partial \!\!\!/^{2}=\partial ^{2}\cdot I_{4\times 4}.$

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

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

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

$\operatorname {tr} (\gamma _{5}a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/)=4i\epsilon _{\mu \nu \lambda \sigma }a^{\mu }b^{\nu }c^{\lambda }d^{\sigma }$

$\gamma _{\mu }a\!\!\!/\gamma ^{\mu }=-2a\!\!\!/$.

$\gamma _{\mu }a\!\!\!/b\!\!\!/\gamma ^{\mu }=4a\cdot b\cdot I_{4\times 4}\,$

$\gamma _{\mu }a\!\!\!/b\!\!\!/c\!\!\!/\gamma ^{\mu }=-2c\!\!\!/b\!\!\!/a\!\!\!/\,$

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,

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

as well as the definition of four momentum

$p_{\mu }=\left(E,-p_{x},-p_{y},-p_{z}\right)\,$

We see explicitly that

${\begin{aligned}p\!\!/&=\gamma ^{\mu }p_{\mu }=\gamma ^{0}p_{0}+\gamma ^{i}p_{i}\\&={\begin{bmatrix}p_{0}&0\\0&-p_{0}\end{bmatrix}}+{\begin{bmatrix}0&\sigma ^{i}p_{i}\\-\sigma ^{i}p_{i}&0\end{bmatrix}}\\&={\begin{bmatrix}E&-\sigma \cdot {\vec {p}}\\\sigma \cdot {\vec {p}}&-E\end{bmatrix}}\end{aligned}}$

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

See also

• Weyl basis
• Gamma matrices

References

• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.{{cite book}}: CS1 maint: multiple names: authors list (link)