PlanetPhysics/Supersymmetry

Supersymmetry or Poincar\'e, (extended) [[../HilbertBundle/|quantum symmetry]] is usually defined as an extension of ordinary [[../SR/|spacetime]] symmetries obtained by adjoining $N$ spinorial [[../Generator/|generators]] $Q$ whose anticommutator yields a translation generator: $\{Q,Q\}=\left\{P\right\}$.

As further explained in ref. :

"This (super) symmetry...(of the [[../MathematicalFoundationsOfQuantumTheories/|superspace]])... can be realized on ordinary [[../CosmologicalConstant/|fields]] (that are defined as certain [[../Bijective/|functions]] of physical spacetime(s)) by transformations that mix [[../QuarkAntiquarkPair/|bosons]] and [[../QuarkAntiquarkPair/|fermions]]. \emph{Such realizations suffice to study supersymmetry (one can write invariant actions, etc.) but are as cumbersome and inconvenient as doing [[../Vectors/|vector]] calculus component by component. A compact alternative to this `component field' approach is given by the superspace--superfield approach}", which is defined next.

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 [[../ECartan/|spinors]] $\theta$.

(Quantum) [[../AntiCommutationRelations/|superfields]] $\mathrm{\Psi }(x,\theta )$ are functions defined over such superspaces, or superspacetimes. [[../TaylorFormula/|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.

Remarks: Supersymmetry is expected to be manifested, or [[../QuantumSpinNetworkFunctor2/|observable]], in such superspaces, that is, the supersymmetry algebras are represented by translations and rotations involving both the spacetime and the anticommuting coordinates. Then, the transformations of the `component fields' can be computed from the Taylor expansion of the translated and rotated superfields . Especially important are those transformations that mix boson and fermion symmetries; further details are found in ref. .