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 $Z$ ($2\times 2$ 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.

arXiv comment25 pages, no figures. arXiv admin note: substantial text overlap with arXiv:1312.3148

References in corpus (10)

• Perturbative Gauge Theory As A String Theory In Twistor Space
• Exactly solvable Richardson-Gaudin models for many-body quantum systems
• Exactly-Solvable Models Derived from a Generalized Gaudin Algebra
• The Conformal Group SU(2,2) and Integrable Systems on a Lorentzian Hyperboloid
• Quantum Field Theory on quantized Bergman domain
• Coherent states on the Grassmannian $U(4)/U(2{)}^2$: Oscillator realization and bilayer fractional quantum Hall systems
• Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space
• Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity
• Generalized W(infinity) Higher-Spin Algebras and Symbolic Calculus on Flag Manifolds
• Coherent States of Accelerated Relativistic Quantum Particles, Vacuum Radiation and the Spontaneous Breakdown of the Conformal SU(2,2) Symmetry

Cited by in corpus (5)

• Husimi function and phase-space analysis of bilayer quantum Hall systems at $\nu =2/\lambda$
• Conformal Blocks in Two and Four Dimensions from Oscillator Representations
• Groups, Jacobi functions and rigged Hilbert spaces
• Some properties of Grassmannian $U(4)/U(2{)}^2$ coherent states and an entropic conjecture
• Massive conformal particles with non-Abelian charges from free U(2N,2N)-twistor dynamics: quantization and coherent states