Number operators for representations of the canonical commutation relations

Anumber operator for a representation of the canonical commutation relations is defined as a self-adjoint operator satisfying an exponentiated form of the equationNa*=a*(N+I), wherea* is an arbitrary creation operator. WhenN exists it may be chosen to have spectrum {0, 1, 2, ...} (in a direct sum of Fock representations) or {0, ±1, ±2, ...} (otherwise). Examples are given of representations having number operators, and a necessary and sufficient condition is given for a direct-product representation to have a number operator.