Feynman slash notation

Feynman slash notation

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),

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

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

Contents

Identities

Using the anticommutators of the gamma matrices, one can show that for any aμ and bμ,

a\!\!\!/a\!\!\!/=a^\mu a_\mu=a^2
a\!\!\!/b\!\!\!/+b\!\!\!/a\!\!\!/ = 2 a \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,

\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}(\gamma_5 a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/) = 4 i \epsilon_{\mu \nu \lambda \sigma} a^\mu b^\nu c^\lambda d^\sigma
\gamma_\mu a\!\!\!/ \gamma^\mu = -2 a\!\!\!/ .
\gamma_\mu a\!\!\!/ b\!\!\!/ \gamma^\mu = 4 a \cdot b \,
\gamma_\mu a\!\!\!/ b\!\!\!/ c\!\!\!/ \gamma^\mu = -2 c\!\!\!/ b\!\!\!/ a\!\!\!/ \,
where
\epsilon_{\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 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix} \,

as well as the definition of four momentum

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

We see explicitly that

\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}

See also

References

  • Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2. 

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Feynman-Slash-Notation — Während seiner Studien zu Dirac Feldern in der Quantenfeldtheorie erfand Richard Feynman die bequeme Feynman Slash Notation (weniger bekannt als Dirac Slash Notation). Sei A ein kovarianter Vektor, dann gilt: unter Verwendung der Einsteinschen… …   Deutsch Wikipedia

  • Slash (punctuation) — / Slash Punctuation apostrophe ( ’ …   Wikipedia

  • Slash — may refer to:Music* Slash (musician) (born Saul Hudson in 1965), a Velvet Revolver guitarist and former Guns N Roses guitarist ** Slash (autobiography), a book written by Slash with Anthony Bozza * Nash the Slash, a Canadian progressive rock… …   Wikipedia

  • Feynman — may refer to: * Richard Feynman ** Feynman diagram ** Feynman graph ** Feynman Kac formula ** The Feynman Lectures on Physics ** Feynman( s) (path) integral, see Path integral formulation ** Feynman parametrization ** Feynman checkerboard **… …   Wikipedia

  • Richard Feynman — Feynman redirects here. For other uses, see Feynman (disambiguation). Richard P. Feynman Richard Feynman at Fermilab Bor …   Wikipedia

  • Gamma matrices — In mathematical physics, the gamma matrices, {γ0,γ1,γ2,γ3}, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra… …   Wikipedia

  • Chord chart — For Feynman s slash notation in Quantum Field Theory, see Feynman slash notation A chord chart.  Play …   Wikipedia

  • Dirac-Matrizen — Die Dirac Matrizen (auch Gamma Matrizen genannt) sind vier Matrizen, die der Dirac Algebra genügen. Sie treten in der Dirac Gleichung auf. Inhaltsverzeichnis 1 Definition 2 Eigenschaften 3 Zusammenhang zu Lorentztransformationen …   Deutsch Wikipedia

  • Dirac-Gleichung — Die Dirac Gleichung beschreibt in der Quantenmechanik die Eigenschaften und das Verhalten von Fermionen (Spin 1/2 Teilchen, zum Beispiel Elektronen) und berücksichtigt dabei die spezielle Relativitätstheorie. Sie wurde 1928 von Paul Dirac… …   Deutsch Wikipedia

  • Propagator — This article is about Quantum field theory. For plant propagation, see Plant propagation. Quantum field theory …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”