NILPOTENT QUANTUM MECHANICS

Abstract

We develop a generalized quantum mechanical formalism based on the nilpotent commuting variables (η-variables). In the nonrelativistic case such formalism provides natural realization of a two-level system (qubit). Using the space of η-wavefunctions, η-Hilbert space and generalized Schrödinger equation we study properties of pure multiqubit systems and also properties of some composed, hybrid models: fermion–qubit, boson–qubit. The fermion–qubit system can be truly supersymmetric, with both SUSY partners having identical spectra. It is a novel feature that SUSY transformations relate here only nilpotent object. The η-eigenfunctions of the Hamiltonian for the qubit–qubit system give the set of Bloch vectors as a natural basis.