INTEGRABLE REPRESENTATIONS OF THE ULTRA-COMMUTATION RELATIONS

We consider -representations of the unital complex *-algebra generate d by the identity and two elements, alpha and nu, with nu-nu and one relation, alpha nu-nu alpha=alpha, the ultra-commutation relations (uc r). In general, we do not impose any commutation relation between alph a and alpha. This is a very general scheme, encompassing many importa nt physical examples, inter alia: the ccr, car, q-deformed bosons and fermions. The representations of interest in physics have a diagonal n umber operator pi(nu) whose spectrum is contained in the positive inte gers (together with some other technical conditions). Our principal re sult is that every -representation in this class is completely determ ined, up to unitary equivalence, by the sequence of numbers [n+1]=\[Om ega(n+1),pi(alpha(+))Omega(n)]\(2) for n greater than or equal to 0, w ith [0]=0. Here Omega(n) is the normalized eigenvector of pi(nu) corre sponding to the eigenvalue n if the dimension of that eigenspace is 1. If the carrier Hilbert space is infinite dimensional, this representa tion is irreducible if and only if [n]>0 for n greater than or equal t o 1. Finally, we consider spatial representations of some of these rep resentations by kernels and differential operators. (C) 1997 American Institute of Physics.