Feynman slash notation

In the study of Dirac fields in quantum field theory, Richard Feynman introduced the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If A is a covariant vector (i.e., a 1-form),

$A/\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\gamma }^0{A}_0+{\gamma }^1{A}_1+{\gamma }^2{A}_2+{\gamma }^3{A}_3$

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

$A/\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\gamma }^{\mu }{A}_{\mu }$.

Identities

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

$\begin{array}{c}a/a/=a^{\mu }{a}_{\mu }\cdot I_4=a^2\cdot I_4\\ a/b/+b/a/=2a\cdot b\cdot I_4.\end{array}$

where $I_4$ is the identity matrix in four dimensions.

In particular,

${\partial /}^2={\partial }^2\cdot I_4.$

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

$\begin{array}{cc}{\gamma }_{\mu }a/{\gamma }^{\mu }& =-2a/\\ {\gamma }_{\mu }a/b/{\gamma }^{\mu }& =4a\cdot b\cdot I_4\\ {\gamma }_{\mu }a/b/c/{\gamma }^{\mu }& =-2c/b/a/\\ {\gamma }_{\mu }a/b/c/d/{\gamma }^{\mu }& =2(d/a/b/c/+c/b/a/d/)\\ \mathrm{tr}(a/b/)& =4a\cdot b\\ \mathrm{tr}(a/b/c/d/)& =4[(a\cdot b)(c\cdot d)-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)]\\ \mathrm{tr}(a/{\gamma }^{\mu }b/{\gamma }^{\nu })& =4[a^{\mu }{b}^{\nu }+a^{\nu }{b}^{\mu }-{\eta }^{\mu \nu }(a\cdot b)]\\ \mathrm{tr}({\gamma }_{5}a/b/c/d/)& =4i{\epsilon }_{\mu \nu \lambda \sigma }{a}^{\mu }{b}^{\nu }{c}^{\lambda }{d}^{\sigma }\\ \mathrm{tr}({\gamma }^{\mu }a/{\gamma }^{\nu })& =0\\ \mathrm{tr}({\gamma }^{5}a/b/)& =0\\ \mathrm{tr}({\gamma }^0(a/+m){\gamma }^0(b/+m))& =8a^0{b}^0-4(a\cdot b)+4m^2\\ \mathrm{tr}((a/+m){\gamma }^{\mu }(b/+m){\gamma }^{\nu })& =4[a^{\mu }{b}^{\nu }+a^{\nu }{b}^{\mu }-{\eta }^{\mu \nu }((a\cdot b)-m^2)]\\ \mathrm{tr}({a/}_1\mathrm{...}{a/}_{2n})& =\mathrm{tr}({a/}_{2n}\mathrm{...}{a/}_1)\\ \mathrm{tr}({a/}_1\mathrm{...}{a/}_{2n+1})& =0\end{array}$

where:

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

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,

${\gamma }^0=\left(\begin{array}{cc}I& 0\\ 0& -I\end{array}\right),{\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$

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

$p^{\mu }=(E,p_x,p_y,p_z)$

we see explicitly that

$\begin{array}{cc}p/& ={\gamma }^{\mu }{p}_{\mu }={\gamma }^0{p}^0-{\gamma }^i{p}^i\\ & =\left[\begin{array}{cc}{p}^0& 0\\ 0& -p^0\end{array}\right]-\left[\begin{array}{cc}0& {\sigma }^i{p}^i\\ -{\sigma }^i{p}^i& 0\end{array}\right]\\ & =\left[\begin{array}{cc}E& -\overrightarrow{\sigma }\cdot \overrightarrow{p}\\ \overrightarrow{\sigma }\cdot \overrightarrow{p}& -E\end{array}\right].\end{array}$

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

See also

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