A quantum-mechanical central limit theorem for sums of pairwise anti-commuting representations of the canonical anti-commutation relations over a finite-dimensional space is formulated and proved.
A formulation of supersymmetric quantum mechanics is given and superunitary and generalized canonical transformations are defined acting in a module. Next it is assumed that there are operators that give an irreducible representation of the canonical commutation and anticommutation relations, respectively, and it is proved that two such representations are connected by a uniquely determined superunitary transformation, under suitable domain assumptions. This extends the well-known uniqueness theorem of von Neumann to canonical (anti−) commutation relations using anticommuting parameters.