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),

${A\!\!\!/}\ {\stackrel {\mathrm {def} }{=}}\ \gamma ^{1}A_{1}+\gamma ^{2}A_{2}+\gamma ^{3}A_{3}+\gamma ^{4}A_{4}$

where γ are the gamma matrices. Using the Einstein summation notation, the expression is simply

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

Identities

Using the anticommutators of the gamma matrices, one can show that for any $a_{\mu }$ and $b_{\mu }$,

${\begin{aligned}{a\!\!\!/}{a\!\!\!/}&\equiv a^{\mu }a_{\mu }\cdot I_{4}=a^{2}\cdot I_{4}\\{a\!\!\!/}{b\!\!\!/}+{b\!\!\!/}{a\!\!\!/}&\equiv 2a\cdot b\cdot I_{4}.\end{aligned}}$

where $I_{4}$ is the identity matrix in four dimensions.

In particular,

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

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

${\begin{aligned}\gamma _{\mu }{a\!\!\!/}\gamma ^{\mu }&\equiv -2{a\!\!\!/}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}\gamma ^{\mu }&\equiv 4a\cdot b\cdot I_{4}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}\gamma ^{\mu }&\equiv -2{c\!\!\!/}{b\!\!\!/}{a\!\!\!/}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/}\gamma ^{\mu }&\equiv 2({d\!\!\!/}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}+{c\!\!\!/}{b\!\!\!/}{a\!\!\!/}{d\!\!\!/})\\\operatorname {tr} ({a\!\!\!/}{b\!\!\!/})&\equiv 4a\cdot b\\\operatorname {tr} ({a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&\equiv 4\left[(a\cdot b)(c\cdot d)-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)\right]\\\operatorname {tr} ({a\!\!\!/}{\gamma ^{\mu }}{c\!\!\!/}{\gamma ^{\nu }})&\equiv 4\left[a^{\mu }c^{\nu }+a^{\nu }c^{\mu }-\eta ^{\mu \nu }(a\cdot c)\right]\\\operatorname {tr} (\gamma _{5}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&\equiv 4i\varepsilon _{\mu \nu \lambda \sigma }a^{\mu }b^{\nu }c^{\lambda }d^{\sigma }\\\operatorname {tr} ({\gamma ^{\mu }}{a\!\!\!/}{\gamma ^{\nu }})&\equiv 0\\\operatorname {tr} ({\gamma ^{5}}{a\!\!\!/}{b\!\!\!/})&\equiv 0\\\operatorname {tr} ({a\!\!\!/}_{1}...{a\!\!\!/}_{2n})&\equiv \operatorname {tr} ({a\!\!\!/}_{2n}...{a\!\!\!/}_{1})\\\operatorname {tr} ({a\!\!\!/}_{1}...{a\!\!\!/}_{2n+1})&\equiv 0\end{aligned}}$

where $\varepsilon _{\mu \nu \lambda \sigma }$ is the Levi-Civita symbol and $\eta ^{\mu \nu }$ the Minkowski metric.

With four-momentum

This section uses the (+ − − −) 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,

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

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

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

we see explicitly that

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

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.