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^[1]). 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 $b_\mu$ ,

{\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}\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} (\gamma _{5}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&\equiv 4i\epsilon _{\mu \nu \lambda \sigma }a^{\mu }b^{\nu }c^{\lambda }d^{\sigma }\\\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\!\!\!/}\\\end{aligned}}

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 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 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.

References

↑ Weinberg, Steven (1995), The Quantum Theory of Fields, 1, Cambridge University Press, p. 358 (380 in polish edition), ISBN 0-521-55001-7

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

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Feynman slash notation

Identities

With four-momentum

See also

References