en>Magioladitis: clean up using AWB (9746)

2013-12-01T09:35:47Z

clean up using AWB (9746)

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In the study of [[Fermionic field#Dirac fields|Dirac field]]s in [[quantum field theory]], [[Richard Feynman]] invented 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=http://books.google.com/books?id=3ws6RJzqisQC&lpg=PA358&dq=%22Dirac%20Slash%22&pg=PA358#v=onepage&q&f=false
|page=358 (380 in polish edition)
}}</ref>). If ''A'' is a [[covariant vector]] (i.e., a [[1-form]]),

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

using the [[Einstein summation notation]] where ''γ'' are the [[gamma matrices]].

==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>a\!\!\!/a\!\!\!/=a^\mu a_\mu=a^2</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 matrices#Identities|gamma matrix identities]] by replacing the [[metric tensor]] with [[inner product]]s. 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>\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 & - \sigma \cdot \vec p \\ \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]]

==References==
{{reflist}}
{{refbegin}}
* {{cite book | author=Halzen, Francis; Martin, Alan | title=Quarks & Leptons: An Introductory Course in Modern Particle Physics | publisher=John Wiley & Sons | year=1984 | isbn=0-471-88741-2}}
{{refend}}

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

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

PGL2 - Revision history

en>Magioladitis: clean up using AWB (9746)