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 (b
oson/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 $Z$ ($2times 2$ 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 $lambdainmathbb N$ (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.
We study coherence and entanglement properties of the state space of a composite bi-fermion (two electrons pierced by $lambda$ magnetic flux lines) at one Landau site of a bilayer quantum Hall system. In particular, interlayer imbalance and entanglem
ent (and its fluctuations) are analyzed for a set of $U(4)$ coherent (emph{quasiclassical}) states generalizing the standard pseudospin $U(2)$ coherent states for the spin-frozen case. The interplay between spin and pseudospin degrees of freedom opens new possibilities with regard to the spin-frozen case. Actually, spin degrees of freedom make interlayer entanglement more effective and robust under perturbations than in the spin-frozen situation, mainly for a large number of flux quanta $lambda$. Interlayer entanglement of an equilibrium thermal state and its dependence with temperature and bias voltage is also studied for a pseudo-Zeeman interaction.
Bilayer quantum Hall (BLQH) systems, which underlie a $U(4)$ symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian $mathbb G_2^4=U(4)/U(2)^2$, orthonormal basis, $U(4)$ generators and their matr
ix elements in the reproducing kernel Hilbert space $mathcal H_lambda(mathbb G_2^4)$ of analytic square-integrable holomorphic functions on $mathbb G_2^4$, which carries a unitary irreducible representation of $U(4)$ with index $lambdainmathbb N$. A many-body representation of the previous construction is introduced through an oscillator realization of the $U(4)$ Lie algebra generators in terms of eight boson operators. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index $lambda$ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor $ u=2$. The simpler, and better known, case of spin-$s$ CS on the Riemann-Bloch sphere $mathbb{S}^2=U(2)/U(1)^2$ is also treated in parallel, of which Grassmannian $mathbb G_2^4$-CS can be regarded as a generalized (matrix) version.
We give a quantum mechanical description of accelerated relativistic particles in the framework of Coherent States (CS) of the (3+1)-dimensional conformal group SU(2,2), with the role of accelerations played by special conformal transformations and w
ith the role of (proper) time translations played by dilations. The accelerated ground state $tildephi_0$ of first quantization is a CS of the conformal group. We compute the distribution function giving the occupation number of each energy level in $tildephi_0$ and, with it, the partition function Z, mean energy E and entropy S, which resemble that of an Einstein Solid. An effective temperature T can be assigned to this accelerated ensemble through the thermodynamic expression dE/dS, which leads to a (non linear) relation between acceleration and temperature different from Unruhs (linear) formula. Then we construct the corresponding conformal-SU(2,2)-invariant second quantized theory and its spontaneous breakdown when selecting Poincare-invariant degenerated theta-vacua (namely, coherent states of conformal zero modes). Special conformal transformations (accelerations) destabilize the Poincare vacuum and make it to radiate.
We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlin
ear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwingers Master Theorem is commented in connection with closure relations. An adaptation of the Borns Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the 8-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.
We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one on
to the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,d u_lambda) and L^2_h(C^4_+,dtilde u_lambda) of square integrable holomorphic functions with scale dimension lambda and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a lambda-extension of Schwingers Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L^2_h(D_4,d u_lambda). SMT is related to MacMahons Master Theorem (MMT) and an extension of both in terms of Loucks SU(N) solid harmonics is also provided for completeness. Convergence conditions are also studied.
