PGL2

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),

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

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 $b_{\mu }$,

$a/a/=a^{\mu }{a}_{\mu }=a^2$

$a/b/+b/a/=2a\cdot b$.

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,

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

$\mathrm{tr}(a/b/c/d/)=4[(a\cdot b)(c\cdot d)-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)]$

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

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

${\gamma }_{\mu }a/b/{\gamma }^{\mu }=4a\cdot b$

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

where

${ϵ}_{\mu \nu \lambda \sigma }$ 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,

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

as well as the definition of four momentum

$p_{\mu }=(E,-p_x,-p_y,-p_z)$

We see explicitly that

$\begin{array}{rl}p/& ={\gamma }^{\mu }{p}_{\mu }={\gamma }^0{p}_0+{\gamma }^i{p}_i\\ & =\left[\begin{array}{cc}{p}_0& 0\\ 0& -p_0\end{array}\right]+\left[\begin{array}{cc}0& {\sigma }^i{p}_i\\ -{\sigma }^i{p}_i& 0\end{array}\right]\\ & =\left[\begin{array}{cc}E& -\sigma \cdot \overrightarrow{p}\\ \sigma \cdot \overrightarrow{p}& -E\end{array}\right]\end{array}$

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

See also

• Weyl basis
• Gamma matrices

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Template:Refbegin

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534