supersymmetrySupersymmetryOrSupersymmetries2008-07-30 20:56:552008-10-20 21:45:15Definitionextended quantum symmetry structuresboth local and globalSupergravityQuantum GravitySuperspaceRelativistic QFT and quantum field theories% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\hr}{{\hookrightarrow}}\begin{definition}
\emph{Supersymmetry} or Poincar\'e, (extended) quantum symmetry is usually defined as an extension of ordinary spacetime symmetries obtained by adjoining $N$ spinorial generators $Q$ whose anticommutator yields a
translation generator: $\left\{Q ,Q \right\} = \left\{P\right\}$.
\end{definition}
As further explained in ref. \cite{JSG98}:
\begin{quote}
``This \emph{(super)} symmetry...(of the \emph{superspace})... can be realized on ordinary fields (that are defined as certain functions of physical spacetime(s)) by transformations that mix bosons and fermions.
\emph{Such realizations suffice to study supersymmetry (one can write invariant actions, etc.) but are as
cumbersome and inconvenient as doing vector calculus component by component. A compact alternative to this `component field' approach is given by the \emph{superspace--superfield} approach}", which is defined next.
\end{quote}
\begin{definition}
\emph{Quantum superspace, or superspacetimes}, can be defined as an extension(s) of ordinary spacetime(s) to include
additional anticommuting coordinates, for example, in the form of $N$ two-component Weyl spinors $\theta$.
\end{definition}
\begin{definition}
\emph{(Quantum) superfields} $\Psi(x , \theta)$ are \emph{functions} defined over such superspaces, or superspacetimes.
Taylor series expansions of the superfield functions can be then performed with respect to the anticommuting coordinates $\theta$; this Taylor series has only a finite number of terms and the series expansion
coefficients obtained in this manner are the ordinary `component fields' specified above.
\end{definition}
\textbf{Remarks:}
Supersymmetry is expected to be manifested, or observable, in such superspaces, that is, the \emph{supersymmetry algebras} are represented by translations and rotations involving \emph{both} the spacetime and the anticommuting coordinates. Then, the transformations of the `component fields' can be computed from the Taylor expansion of
the \emph{translated and rotated superfields}. Especially important are those transformations that mix boson
and fermion symmetries; further details are found in ref. \cite{LS2k}.
\begin{thebibliography}{9}
\bibitem{JSG98}
J.S. Gates, Jr, et al. ``Superspace''., arxiv-hep-th/0108200 preprint (1983).
\bibitem{LS2k}
``Preprint of 1,001 Lessons in Supersymmetry.'' \PMlinkexternal{on line PDF}{http://arxiv.org/abs/hep-th/0108200}.
\end{thebibliography}