Editing Feynman slash notation

{{Short description|Notation for contractions with gamma matrices}} 
In the study of [[Fermionic field#Dirac fields|Dirac field]]s in [[quantum field theory]], [[Richard Feynman]] introduced the convenient '''Feynman slash notation''' (less commonly known as the '''[[Paul Dirac|Dirac]]''' '''slash notation'''<ref>{{citation
 |last=Weinberg
 |first=Steven
 |authorlink=Steven Weinberg
 |year=1995
 |title=The Quantum Theory of Fields
 |volume=1
 |publisher=Cambridge University Press
 |isbn=0-521-55001-7
 |url=https://books.google.com/books?id=3ws6RJzqisQC&q=%22Dirac%20Slash%22&pg=PA358
 |page=358 (380 in polish edition)
}}</ref>). 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 ''&gamma;'' 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 [[anticommutator]]s 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 matrices#Identities|gamma matrix identities]] by replacing the [[metric tensor]] with [[inner product]]s. 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 \cdot 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 {{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==
*[[Weyl basis]]
*[[Gamma matrices]]
*[[Four-vector]]
*[[S-matrix]]

==References==
{{reflist}}
{{refbegin}}
* {{cite book |author1=Halzen, Francis |authorlink1 = Francis Halzen |author2=Martin, Alan | authorlink2 = Alan Martin (physicist)| title=Quarks & Leptons: An Introductory Course in Modern Particle Physics |url=https://archive.org/details/quarksleptonsint0000halz |url-access=registration | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}
{{refend}}

{{Richard Feynman}}


[[Category:Quantum field theory]]
[[Category:Spinors]]
[[Category:Richard Feynman]]

[[de:Dirac-Matrizen#Feynman-Slash-Notation]]
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