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

الأصل · EN

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 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 D₁=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 Hλ(D₄) of analytic square-integrable holomorphic functions on D₄, which carries a unitary irreducible representation of U(2,2) with index λ (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.