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Feynman slash notation

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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. 1-form,

<math>A\!\!\!/\ \stackrel{\mathrm{def}}{=}\ \gamma^\mu A_\mu</math>

using the Einstein summation notation where γ are the gamma matrices.

1 Identities
2 With four-momentum
3 See also
4 References

Identities

Using the anticommutators of the gamma matrices, one can show that for any <math>a_\mu</math> and <math>b_\mu</math>,

<math>a\!\!\!/b\!\!\!/+b\!\!\!/a\!\!\!/ = 2 a \cdot b \,</math>.

In particular,

<math>\partial\!\!\!/^2=\partial^2.</math>

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

<math>\operatorname{tr}(a\!\!\!/b\!\!\!/) = 4 a \cdot b</math>

<math>\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]</math>

<math>\operatorname{tr}(\gamma_5 a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/) = 4 i \epsilon_{\mu \nu \lambda \sigma} a^\mu b^\nu c^\lambda d^\sigma</math>

<math>\gamma_\mu a\!\!\!/ \gamma^\mu = -2 a\!\!\!/ </math>.

<math>\gamma_\mu a\!\!\!/ b\!\!\!/ \gamma^\mu = 4 a \cdot b \,</math>

<math>\gamma_\mu a\!\!\!/ b\!\!\!/ c\!\!\!/ \gamma^\mu = -2 c\!\!\!/ b\!\!\!/ a\!\!\!/ \,</math>

where

<math>\epsilon_{\mu \nu \lambda \sigma} \,</math> 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 <math>\gamma\,</math>'s,

<math>\gamma^0 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix} \,</math>

as well as the definition of four momentum

<math> p^{\mu} = \left(E, p^x, p^y, p^z \right) \,</math>

We see explicitly that

<math>p\!\!\!/ = \gamma^\mu p_\mu \,</math>	<math>= \gamma^0 p_0 + \gamma^i p_i \,</math>
	<math>=\begin{bmatrix} p_0 & 0 \\ 0 & -p_0 \end{bmatrix} + \begin{bmatrix} 0 & \sigma^i p_i \\ - \sigma^i \cdot p_i & 0 \end{bmatrix} \,</math>
	<math>=\begin{bmatrix} E & \mathbf{\sigma \cdot p} \\ -\mathbf{\sigma \cdot p} & -E \end{bmatrix} \,</math>