Partially coherent states of the real symplectic group

Abstract

In the present paper, we introduce partially coherent states for the positive discrete series irreducible representations 〈λ_d+n/2,...,λ₁+n/2〉 of Sp(2d,R), encountered in physical applications. These states are characterized by both continuous and discrete labels. The latter specify the row of the irreducible representation [λ₁λ₂⋅⋅⋅λ_d] of the maximal compact subgroup U(d), while the former parametrize an element of the factor space Sp(2d,R)/H, where H is the Sp(2d,R) subgroup leaving the [λ₁λ₂⋅⋅⋅λ_d] representation space invariant. We consider three classes of partially coherent states, respectively, generalizing the Perelomov and Barut–Girardello coherent states, as well as some recently introduced intermediate coherent states. We prove that each family of partially coherent states forms an overcomplete set in the representation space of 〈λ_d+n/2,...,λ₁+n/2〉, and study its generating function properties. We show that it leads to a representation of the Sp(2d,R) generators in the form of differential operator matrices. Finally, we relate the latter to a boson representation, namely a generalized Dyson representation in the cases of Perelomov and Barut‐Girardello partially coherent states, and a generalized Holstein–Primakoff representation in that of the intermediate partially coherent states.