Dynamical group of microscopic collective states. II. Boson representations in d dimensions

Abstract

The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.