Abstract :
In the supersymmetric classical or quantal mechanics of particle motion on the sphere S6, an SO(7)-invariant theory, the existence of a totally antisymmetric tensor eijk (defined by the product law of octonions) invariant under the G2 sub-group of SO(7) allows the construction of a Killing–Yano tensor and the supercharge Q′ of a hidden supersymmetry of the theory. The canonical bracket {Q′,Q′} (or quantally the corresponding anticommutator) yields not the Hamiltonian of the theory but instead an operator determined by the quadratic Casimir operator of G2. The situation is compared with the theory of a spin-12 particle moving in the field of a Dirac monopole. Use of the tensor eijk to break the SO(7)-invariance of the original theory of motion on S6 to G2 invariance is discussed.
