Birth of SuperAlgebra

In the nineteen sixties the top topic was group theory and the classification of elementary particles. Bunji Sakita, the inventor of the SU(6) theory, visited us at the University of Tokyo, and excited our interest in this theory.

The SU(6) theory combines particles with spin 0 and 1, or spin 1/2 and 3/2, in one representation. I noticed a beautiful parallelism between Bose particles and Fermi particles and wondered if they could be combined in one representation. Ignoring statistics, this is easy to do. All existing elementary particles can be expressed by the adjoint representation of the SU(9) group. Hirotaka Sugawara and I wrote a short note entitled “SU(9) Symmetry” published in 1965 [1].

Of course, I was not satisfied with this scheme since this only works for one particle states, i.e., in the case of Boltzmann statistics. I came to the thought of a boson-fermion mixture just before the short note in 1965. I looked for a real mathematical scheme and soon found that a hamiltonian...