Structure of superspace in supersymmetry

It is known that the scalar superspace in supersymmetry theory is the direct sum of chiral, antichiral, and isoscalar (or linear) fields, provided that the mass of the system is not zero. However, we show here that the situation changes drastically for the massless case. The whole superspace becomes reducible but not fully reducible. Moreover, it is indecomposable in a sense to be specified. The same reducible but indecomposable property is also shared by any chiral, antichiral, and isoscalar spaces. However, if we accept only unitary representations on positive-metric spaces, then only irreducible components of these spaces must be physically relevant. We demonstrate that these facts are intimately connected with the structure of the commutator algebra of the little Lie superalgebra as well as with a difference of Casimir invariants of the supersymmetry between the massive and massless cases.