supersymmetry

Definition 0.1.

Supersymmetry or Poincaré, (extended) quantum symmetry is usually defined as an extension of ordinary spacetime symmetries obtained by adjoining $N$ spinorial generators $Q$ whose anticommutator yields atranslation generator: $\{Q,Q\}=\left\{P\right\}$.

As further explained in ref. [1]:

“This (super) symmetry…(of the superspace)… can be realized on ordinary fields (that are defined as certain functions of physical spacetime(s)) by transformations that mix bosons and fermions.Such realizations suffice to study supersymmetry (one can write invariant actions, etc.) but are ascumbersome and inconvenient as doing vector calculus component by component. A compact alternative to this ‘component field’ approach is given by the superspace–superfield approach”, which is defined next.

Definition 0.2.

Quantum superspace, or superspacetimes, can be defined as an extension(s) of ordinary spacetime(s) to includeadditional anticommuting coordinates, for example, in the form of $N$ two-component Weyl spinors $\theta$.

Definition 0.3.

(Quantum) superfields $\mathrm{\Psi }(x,\theta )$ are 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 expansioncoefficients obtained in this manner are the ordinary ‘component fields’ specified above.

Remarks:Supersymmetry is expected to be manifested, or 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 ofthe translated and rotated superfields. Especially important are those transformations that mix bosonand fermion symmetries; further details are found in ref. [2].