Английская Википедия:Feynman slash notation
Шаблон:Short description 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),
- <math>{A\!\!\!/} \ \stackrel{\mathrm{def}}{=}\ \gamma^0 A_0 + \gamma^1 A_1 + \gamma^2 A_2 + \gamma^3 A_3 </math>
where γ are the gamma matrices. Using the Einstein summation notation, the expression is simply
- <math>{A\!\!\!/} \ \stackrel{\mathrm{def}}{=}\ \gamma^\mu A_\mu</math>.
Identities
Using the anticommutators of the gamma matrices, one can show that for any <math>a_\mu</math> and <math>b_\mu</math>,
- <math>\begin{align}
{a\!\!\!/}{a\!\!\!/} = a^\mu a_\mu \cdot I_4 = a^2 \cdot I_4 \\
{a\!\!\!/}{b\!\!\!/} + {b\!\!\!/}{a\!\!\!/} = 2 a \cdot b \cdot I_4.
\end{align}</math>
where <math>I_4</math> is the identity matrix in four dimensions.
In particular,
- <math>{\partial\!\!\!/}^2 = \partial^2 \cdot I_4.</math>
Further identities can be read off directly from the gamma matrix identities by replacing the metric tensor with inner products. For example,
- <math>\begin{align}
\gamma_\mu {a\!\!\!/} \gamma^\mu &= -2 {a\!\!\!/} \\
\gamma_\mu {a\!\!\!/} {b\!\!\!/} \gamma^\mu &= 4 a \cdot b \cdot I_4 \\
\gamma_\mu {a\!\!\!/} {b\!\!\!/} {c\!\!\!/} \gamma^\mu &= -2 {c\!\!\!/}{b\!\!\!/} {a\!\!\!/} \\
\gamma_\mu {a\!\!\!/} {b\!\!\!/} {c\!\!\!/}{d\!\!\!/} \gamma^\mu &= 2( {d\!\!\!/} {a\!\!\!/} {b\!\!\!/}{c\!\!\!/}+{c\!\!\!/} {b\!\!\!/} {a\!\!\!/}{d\!\!\!/}) \\
\operatorname{tr}({a\!\!\!/}{b\!\!\!/}) &= 4 a \cdot b \\
\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] \\
\operatorname{tr}({a\!\!\!/}{\gamma^\mu}{b\!\!\!/}{\gamma^\nu }) &= 4 \left[a^\mu b^\nu + a^\nu b^\mu - \eta^{\mu \nu}(a \cdot b) \right] \\
\operatorname{tr}(\gamma_5 {a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/}) &= 4 i \varepsilon_{\mu \nu \lambda \sigma} a^\mu b^\nu c^\lambda d^\sigma \\
\operatorname{tr}({\gamma^\mu}{a\!\!\!/}{\gamma^\nu}) &= 0 \\
\operatorname{tr}({\gamma^5}{a\!\!\!/}{b\!\!\!/}) &= 0 \\
\operatorname{tr}({\gamma^0}({a\!\!\!/}+m){\gamma^0}({b\!\!\!/}+m)) &= 8a^0b^0-4(a.b)+4m^2 \\
\operatorname{tr}(({a\!\!\!/}+m){\gamma^\mu}({b\!\!\!/}+m){\gamma^\nu}) &= 4 \left[a^\mu b^\nu+a^\nu b^\mu - \eta^{\mu \nu}((a \cdot b)-m^2) \right] \\
\operatorname{tr}({a\!\!\!/}_1...{a\!\!\!/}_{2n}) &= \operatorname{tr}({a\!\!\!/}_{2n}...{a\!\!\!/}_1) \\
\operatorname{tr}({a\!\!\!/}_1...{a\!\!\!/}_{2n+1}) &= 0
\end{align}</math>
where:
- <math>\varepsilon_{\mu \nu \lambda \sigma}</math> is the Levi-Civita symbol
- <math>\eta^{\mu \nu}</math> is the Minkowski metric
- <math>m</math> is a scalar.
With four-momentum
This section uses the Шаблон:Math 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,
- <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 contravariant four-momentum in natural units,
- <math> p^\mu = \left(E, p_x, p_y, p_z \right) \,</math>
we see explicitly that
- <math>\begin{align}
{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{align}</math>
Similar results hold in other bases, such as the Weyl basis.
See also
References
Шаблон:Reflist Шаблон:Refbegin
de:Dirac-Matrizen#Feynman-Slash-Notation Шаблон:Quantum-stub
