Boson operator realizations of su(2) and su(1,1) are obtained. Scalar products are introduced on “Fock spaces” (Verma modules) spanned by generators of the Heisenberg algebra H and by generators of su(2). These scalar products unitarize certain of the representations of H, or of su(1,1). It is shown that the Gel’fand-Dyson realization of su(1,1) implies a scalar product that unitarizes H, while the Primakoff-Holstein realizations imply a scalar product that unitarizes su(1,1). The relationship between the Gel’fand-Dyson boson operators \(a^{†}\) and the Primakoff-Holstein boson operators \(b^{†}\) is obtained making use of the two distinct scalar products. Generalized “vacuum states” are defined that are formed by polynomials in the creation-annihilation operator pairs \(a^{†}a\). A representation \(\rho\) of H and su(1,1) on the states \(a^{† m}\) and \(a^ n\) is discussed. For this representation \(a\mathbf{1} =a| 0>\neq 0\), but rather \((a^{†}a)| 0>=0\). The states of this representation space consist of boson states and boson-hole states. All the familiar results of H and su(1,1) representation theory are preserved within the representation \(\rho\).