AIP Digital Archive
We consider *-representations of the unital complex *-algebra generated by the identity and two elements, α and ν, with ν=ν* and one relation, αν−να=α, the ultra-commutation relations (ucr). In general, we do not impose any commutation relation between α and α*. This is a very general scheme, encompassing many important physical examples, inter alia: the ccr, car, q-deformed bosons and fermions. The representations of interest in physics have a diagonal number operator π(ν) whose spectrum is contained in the positive integers (together with some other technical conditions). Our principal result is that every *-representation in this class is completely determined, up to unitary equivalence, by the sequence of numbers [n+1]=|〈Ωn+1,π(α+)Ωn〉|2 for n≥0, with =0. Here Ωn is the normalized eigenvector of π(ν) corresponding to the eigenvalue n if the dimension of that eigenspace is 1. If the carrier Hilbert space is infinite dimensional, this representation is irreducible if and only if [n]〉0 for n≥1. Finally, we consider spatial representations of some of these representations by kernels and differential operators. © 1997 American Institute of Physics.
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