Representations Of SU Research Articles

The origin of multiplets of chiral fields inSU(2)k at rational level

We study solutions of the Knizhnik–Zamolodchikov equation for discrete representations ofSU(2)k at rationallevel k+2 = p/q using a regular basis in which the braid matrices are well defined for all spins. We show that at spinJ = (j+1)p−1 for there are always a subset of 2j+1 solutions closed under the action of the braid matrices. For these fields have integer conformal dimension and all the2j+1 solutions are monodromy free. The action of the braid matrices on these can beconsistently accounted for by the existence of a multiplet of chiral fields with extraSU(2) quantumnumbers (m = −j,...,j). In thequantum group SUq(2), with q = e−i π/(k+2), there is an analogous structure and the related representations aretrivial with respect to the standard generators but transform in a spinj representationof SU(2) under the extended centre.

The coherent generation of the photon pairs by a stream of excited atoms passing through the cavity

The problem of the coherent generation of the entanglement photon pairs by an ensemble of the two-level atoms passing through the cavity is studied. The dependence of the generation rate of the photon pairs as a function of the coherent states of the atomic stream is established. Considering that the flying time of the atoms passing through the cavity is less than decay time, a new method of the elimination of atomic variables from the system is proposed. The master equation for multimode cavity electromagnetic field is obtained. Using the Fokker–Planck equations, the steady state solution of this master equation is studied. The quantum properties of the multimode electromagnetic field are analyzed, using the non-diagonal P representation for su(1,1) symmetry.

A class of vector coherent states defined over matrix domains

A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable z. In the present scheme, the variable z is replaced by matrix valued functions over appropriate domains. As particular examples, we analyze the quaternionic extensions of the canonical coherent states and the Gilmore–Perelomov and Barut–Girardello coherent states arising from representations of SU(1,1). Possible physical applications are indicated.

Observation of quantum-classical correspondence from high-order transverse patterns

We experimentally observe the formation of high-order transverse patterns in a laser resonator with a high degree of frequency degeneracy. It is found that the transverse patterns are well localized on the Lissajous orbits. The connection between the wave functions and the classical periodic orbits is analytically constructed by using the representation of SU(2) coherent states. With this connection, the observed transverse patterns are reconstructed very well. The nice reconstruction suggests that the laser resonators can be deliberately designed to attain a more thorough understanding of the quantum-classical connection.

Extended chiral algebras and the emergence of SU(2) quantum numbers in the Coulomb gas

We study a set of chiral symmetries contained in degenerate operators beyond the `minimal' sector of the c(p,q) models. For the operators h_{(2j+2)q-1,1}=h_{1,(2j+2)p-1} at conformal weight [ (j+1)p-1 ][ (j+1)q -1 ], for every 2j \in N, we find 2j+1 chiral operators which have quantum numbers of a spin j representation of SU(2). We give a free-field construction of these operators which makes this structure explicit and allows their OPEs to be calculated directly without any use of screening charges. The first non-trivial chiral field in this series, at j=1/2, is a fermionic or para-fermionic doublet. The three chiral bosonic fields, at j=1, generate a closed W-algebra and we calculate the vacuum character of these triplet models.

Vortex structure of quantum eigenstates and classical periodic orbits in two-dimensional harmonic oscillators

The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU(2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.

Odd coset quantum mechanics

The standard quantum states of n complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same ‘free-fermion’ model has a non-linearly realized SU(n|1) symmetry; it can be viewed as the mechanics of a ‘particle’ on the Grassmann-odd coset space SU(n|1)/U(n). We implement a quantization of this model for which the states with non-zero norm transform as a representation of SU(n|1), the representation depending on the U(1) charge of the wave-function. For n=2 the wave-function can be interpreted as a BRST superfield.

BMN operators and superconformal symmetry

Implications of N=4 superconformal symmetry on Berenstein–Maldacena–Nastase (BMN) operators with two charge defects are studied both at finite charge J and in the BMN limit. We find that all of these belong to a single long supermultiplet explaining a recently discovered degeneracy of anomalous dimensions on the sphere and torus. The lowest-dimensional component is an operator of naive dimension J+2 transforming in the [0, J,0] representation of SU(4). We thus find that the BMN operators are large J generalisations of the Konishi operator at J=0. We explicitly construct descendant operators by supersymmetry transformations and investigate their three-point functions using superconformal symmetry.

Affine Toda model coupled to matter and the string tension in 2D QCD

The $sl(2)$ affine Toda model coupled to matter (ATM) is shown to describe various features, such as the spectrum and string tension, of the low-energy effective Lagrangian of QCD$_{2}$ (one flavor and $N$ colors). The corresponding string tension is computed when the dynamical quarks are in the {\sl fundamental} representation of SU(N) and in the {\sl adjoint} representation of SU(2).

SU(3) classification of p-wave ηπ and π systems

An exotic meson, the $\pi_1(1400)$ with $J^{PC}=1^{-+}$, has been seen to decay into a p-wave $\eta\pi$ system. If this decay conserves flavor SU(3), then it can be shown that this exotic meson must be a four-quark state ($q\bar q+q\bar q$) belonging to a flavor ${\bf10}\oplus{\bf\bar{10}}$ representation of SU(3). In contrast, the $\pi_1(1600)$ with a substantial decay mode into $\eta'\pi$ is likely to be a member of a flavor octet.

Localization of wave patterns on classical periodic orbits in a square billiard.

The connection between wave functions and classical periodic trajectories in a square billiard is analytically constructed by using the representation of SU(2) coherent states. The analytical function form is modified to show that the wave patterns can be apparently localized on the classical periodic trajectories by superposing a few nearly degenerate eigenfunctions. Based on the analogy between the Schrödinger and Helmholtz equations, the features of wave functions are experimentally studied from the transverse pattern formation in a laterally confined microcavity laser. The experimental transverse pattern in a square-shaped microcavity agrees very well with the constructed wave pattern concentrated along classical periodic orbits.

Classical and algebraic Hamiltonian with su(3) symmetry

We use Gelfand-Zetlin patterns to obtain the coherent state for an arbitrary symmetric irreducible representation of su(3). The semiclassical evolution of a dynamical system whose Hamiltonian contains the Casimir operators of both su(2) and so(3) subalgebras is investigated, and it is concluded that the presence of a common operator in the subalgebras induces integrability despite the absence of dynamical symmetry.

Components of spaces of representations and stable triples

We consider the moduli spaces of representations of the fundamental group of a surface of genus g⩾2 in the Lie groups SU(2, 2) and Sp(4, R ) . It is well known that there is a characteristic number, d, of such a representation, satisfying the inequality | d|⩽2 g−2. This allows one to write the moduli space as a union of subspaces indexed by d, each of which is a union of connected components. The main result of this paper is that the subspaces corresponding to d=±(2 g−2) are connected in the case of representations in SU(2, 2) , while they break up into 3·2 2 g +2 g−4 connected components in the case of representations in Sp(4, R ) . We obtain our results using the interpretation of the moduli space of representations as a moduli space of Higgs bundles, and an important step is an identification of certain subspaces as moduli spaces of stable triples, as studied by Bradlow and Garcı́a-Prada.

Cycles of Bott-Samelson type for taut representations

Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut, is either variationally complete or it is one of the following orthogonal representations (n bigger than or equal to 2): the (standard) x_R (spin) representation of SO(2)xSpin(9); or the (standard) x_C (standard) representation of U(2)xSp(n); or the (standard)^3 x_ H (standard) representation of SU(2)xSp(n). In this paper we will show how to adapt the construction of the cycles of Bott and Samelson to the orbits of these three representations. As a result, they also admit explicit cycles representing a basis of their Z_2-homology and, in particular, this provides another proof of their tautness.

Effective Lagrangian induced by the anomalous Wess-Zumino action and the exotic resonance state withIG(JPC)=1−(1−+)in theρπ,ηπ,η′π,andK*K¯+K*Kchannels

A simple model for the exotic waves with $I^G(J^{PC})=1^-(1^{-+})$ in the reactions $VP\to VP$, $VP\to PP$, and $PP\to PP$ is constructed beyond the scope of the quark-gluon approach. The model satisfies unitarity and analyticity and uses as a "priming" the "anomalous" nondiagonal $VPPP$ interaction which couples together the four channels $\rho\pi$, $\eta\pi$, $\eta'\pi$, and $K^*\bar K+\bar K^*K$. The possibility of the resonancelike behavior of the $I^G(J^{PC})=1^-(1^{-+})$ amplitudes belonging to the $\{10\}- \{\bar{10}\}$ and $\{8\}$ representations of SU(3) as well as their mixing is demonstrated explicitly in the $1.3-1.6$ GeV mass range which, according to the current experimental evidence, is really rich in exotics.

Dynamical symmetry approach to periodic Hamiltonians

We show that dynamical symmetry methods can be applied to Hamiltonians with periodic potentials. We construct dynamical symmetry Hamiltonians for the Scarf potential and its extensions using representations of su(1,1) and so(2,2). Energy bands and gaps are readily understood in terms of representation theory. We compute the transfer matrices and dispersion relations for these systems, and find that the complementary series plays a central role as well as nonunitary representations.

Holography, Quantum Geometry, and Quantum Information Theory

We interpret the Holographic Conjecture in terms of quantum bits (qubits). N-qubit states are associated with surfaces that are punctured in N points by spin networks' edges labelled by the spin-½ representation of SU(2), which are in a superposed quantum state of spin "up" and spin "down". The formalism is applied in particular to de Sitter horizons, and leads to a picture of the early inflationary universe in terms of quantum computation. A discrete micro-causality emerges, where the time parameter is being defined by the discrete increase of entropy. Then, the model is analysed in the framework of the theory of presheaves (varying sets on a causal set) and we get a quantum history. A (bosonic) Fock space of the whole history is considered. The Fock space wavefunction, which resembles a Bose-Einstein condensate, undergoes decoherence at the end of inflation. This fact seems to be responsible for the rather low entropy of our universe.

The holomorphic effective action inN=2D=4 supergauge theories with various gauge groups

We consider the theory of hypermultiplets in arbitrary representations of arbitrary semisimple gauge groups coupled to gauge superfields. Using the N=2 harmonic superspace formulation of these models, we find the general structure of the holomorphic effective action depending on the gauge superfield with values in the Cartan subalgebra of the gauge algebra. We find explicit expressions for the effective actions in the cases where the hypermultiplets are in the fundamental and adjoint representations of SU(n), SO(n), and Sp(2n).

Universal Covering Group of U(n) and Projective Representations

Using fiber bundle theory, we construct the universal covering group of U(n),U(n), and show that U(n) is isomorphic to the semidirect product SU(n) ∝.We give a bijection between the set of projective representations of U(n) and theset of equivalence classes of certain unitary representations of SU(n) ∝.Applying Bargmann's theorem, we give explicit expressions for the liftings ofprojective representations of U(n) to unitary representations of SU(n) ∝. Forcompleteness, we discuss the topological and group theoretic relations betweenU(n), SU(n), U(t), and Zn.

Mixing Angle and Glashow Algebra

Considering transformations in the basis of fundamental fields on a principalfiber bundle, without modification in the space-time sector, we construct analgebra GA, which we call Glashow algebra. The structure constants of thisalgebra depend on a mixing angle. The Lagrangian of the gauge theory ofelectroweak interactions without masses is obtained using a representation of GAwhich is the transformed of the adjoint representation of SU(2)⊗ U(1), anddoes not coincide with the adjoint representation of GA. The mixing angle isautomatically present in the theory if GA is used.