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%%% Primary Title: supersymmetry
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%%% Filename: Supersymmetry.tex
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\begin{document}
\begin{definition}
\emph{Supersymmetry} or Poincar\'e, (extended) \htmladdnormallink{quantum symmetry}{http://planetphysics.us/encyclopedia/HilbertBundle.html} is usually defined as an extension of ordinary \htmladdnormallink{spacetime}{http://planetphysics.us/encyclopedia/SR.html} symmetries obtained by adjoining $N$ spinorial \htmladdnormallink{generators}{http://planetphysics.us/encyclopedia/Generator.html} $Q$ 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{\htmladdnormallink{superspace}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}})... can be realized on ordinary \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} (that are defined as certain \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} of physical spacetime(s)) by transformations that mix \htmladdnormallink{bosons}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} and \htmladdnormallink{fermions}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html}.
\emph{Such realizations suffice to study supersymmetry (one can write invariant actions, etc.) but are as
cumbersome and inconvenient as doing \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} 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 \htmladdnormallink{spinors}{http://planetphysics.us/encyclopedia/ECartan.html} $\theta$.
\end{definition}
\begin{definition}
\emph{(Quantum) \htmladdnormallink{superfields}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}} $\Psi(x , \theta)$ are \emph{functions} defined over such superspaces, or superspacetimes.
\htmladdnormallink{Taylor series}{http://planetphysics.us/encyclopedia/TaylorFormula.html} 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 \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html}, 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.'' \htmladdnormallink{on line PDF}{http://arxiv.org/abs/hep-th/0108200}.
\end{thebibliography}
\end{document}
