On the oscillator realization of conformal U(2, 2) quantum particles and their particle-hole coherent states

Abstract

We revise the unireps. of U(2, 2) describing conformal particles with continuous mass spectrum from a many-body perspective, which shows massive conformal particles as compounds of two correlated massless particles. The statistics of the compound (boson/fermion) depends on the helicity h of the massless components (integer/half-integer). Coherent states (CS) of particle-hole pairs (“excitons”) are also explicitly constructed as the exponential action of exciton (non-canonical) creation operators on the ground state of unpaired particles. These CS are labeled by points Z (2 × 2 complex matrices) on the Cartan-Bergman domain \documentclass[12pt]{minimal}\begin{document}$\mathbb {D}_4=U(2,2)/U(2)^2$\end{document}D4=U(2,2)/U(2)2, and constitute a generalized (matrix) version of Perelomov U(1, 1) coherent states labeled by points z on the unit disk \documentclass[12pt]{minimal}\begin{document}$\mathbb {D}_1=U(1,1)/U(1)^2$\end{document}D1=U(1,1)/U(1)2. First, we follow a geometric approach to the construction of CS, orthonormal basis, U(2, 2) generators and their matrix elements and symbols in the reproducing kernel Hilbert space \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_\lambda (\mathbb {D}_4)$\end{document}Hλ(D4) of analytic square-integrable holomorphic functions on \documentclass[12pt]{minimal}\begin{document}$\mathbb {D}_4$\end{document}D4, which carries a unitary irreducible representation of U(2, 2) with index \documentclass[12pt]{minimal}\begin{document}$\lambda \in \mathbb {N}$\end{document}λ∈N (the conformal or scale dimension). Then we introduce a many-body representation of the previous construction through an oscillator realization of the U(2, 2) Lie algebra generators in terms of eight boson operators with constraints. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the many-body jargon. In particular, the index λ is related to the number 2(λ − 2) of unpaired quanta and to the helicity h = (λ − 2)/2 of each massless particle forming the massive compound.