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

TL;DR

This paper revisits the unitary irreducible representations of the conformal group U(2,2), constructing coherent states of particle-hole pairs using geometric and oscillator methods, revealing massive particles as bound states of massless ones.

Contribution

It introduces a new oscillator realization of U(2,2) generators and constructs particle-hole coherent states labeled by complex matrices, linking mathematical structures to physical interpretations.

Findings

Constructed coherent states labeled by points in a complex domain.

Linked the index λ to helicity and particle composition.

Provided a geometric and oscillator framework for conformal particles.

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 $Z$ ($2\times 2$ complex matrices) on the Cartan-Bergman domain $\mathbb D_4=U(2,2)/U(2)^2$, and constitute a generalized (matrix) version of Perelomov $U(1,1)$ coherent states labeled by points $z$ on the unit disk $\mathbb D_1=U(1,1)/U(1)^2$. Firstly 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 $\mathcal H_\lambda(\mathbb D_4)$ of analytic square-integrable holomorphic functions on $\mathbb D_4$, which carries a unitary irreducible representation of $U(2,2)$ with index $\lambda\in\mathbb N$ (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 $\lambda$ is related to the number $2(\lambda-2)$ of unpaired quanta and to the helicity $h=(\lambda-2)/2$ of each massless particle forming the massive compound.