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At least 1,477 records · Page 82Linked to original sources
arXiv1997

Central Extensions of the Quasi-orthogonal Lie algebras

We determine the central extensions of a whole family of Lie algebras, obtained by the method of graded contractions from so(N+1), N arbitrary. All the inhomogeneous orthogonal and pseudo-orthogonal algebras are members of this family, as well as a large number of other non-semisimple algebras, all of which have at least a semidirect structure (in some cases two or more). The dimensions of their second cohomology groups H^2(G,R) and the explicit expression of their central extensions are given.

q-alg↗
arXiv1997

Representations and Clebsch-Gordan coefficients for the Jordanian quantum algebra U_h(sl(2))

Representation theory for the Jordanian quantum algebra $U=U_h(sl(2))$ is developed. Closed form expressions are given for the action of the generators of U on the basis vectors of finite dimensional irreducible representations. It is shown how representation theory of U has a close connection to combinatorial identities involving summation formulas. A general formula is obtained for the Clebsch-Gordan coefficients $C^{j_1,j_2,j}_{n_1,n_2,m}(h)$ of U. These Clebsch-Gordan coefficients are shown to coincide with those of su(2) for $n_1+n_2 \leq m$, but for $n_1+n_2 > m$ they are in general a nonzero monomial in $h^{n_1+n_2-m}$.

q-alg↗
arXiv1997

Noncommutative analogues of q-special polynomials and q-integral on a quantum sphere

The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense "q-commutative", the latter one is related to its noncommutative counterpart. We introduce also a two parameter deformation of the invariant integral on a sphere.

q-alg↗
arXiv1997

The structure of quantum Lie algebras for the classical series B_l, C_l and D_l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at $q=1$. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint ---> adjoint. We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B_l, C_l and D_l. In the quantum case also the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.

q-alg↗
arXiv1997

Hecke algebras as subalgebras of Clifford geometric algebras of multivectors

Clifford geometric algebras of multivectors are introduced which exhibit a bilinear form which is not necessarily symmetric. Looking at a subset of bi-vectors in CL(K^{2n},B), we proof that theses elements generate the Hecke algebra H_{K}(n+1,q) if the bilinear form B is chosen appropriately. This shows, that q-quantization can be generated by Clifford multivector objects which describe usually composite entities. This contrasts current approaches which give deformed versions of Clifford algebras by deforming the one-vector variables. Our example shows, that it is not evident from a mathematical point of view, that q-deformation is in any sense more elementary than the undeformed structure.

q-alg↗
arXiv1997

On the Witten-Reshetikhin-Turaev representations of mapping class groups

We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representation over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus and there are no colored points, the representations have finite image.

q-alg↗
arXiv1996

Universal Superpositions of Coherent States and Self-Similar Potentials

A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator $A=P(d/dx+x)/\sqrt2$, where $P$ is the parity operator. Such $A$ arises naturally in the $q\to -1$ limit for a symmetry operator of a specific self-similar potential obeying the $q$-Weyl algebra, $AA^\dagger-q^2A^\dagger A=1$. Coherent states for this and other reflectionless potentials whose discrete spectra consist of $N$ geometric series are analyzed. In the harmonic oscillator limit the surviving part of these states takes the form of orthonormal superpositions of $N$ canonical coherent states $|ε^kα\rangle$, $k=0, 1, \dots, N-1$, where $ε$ is a primitive $N$th root of unity, $ε^N=1$. A class of $q$-coherent states related to the bilateral $q$-hypergeometric series and Ramanujan type integrals is described. It includes a curious set of coherent states of the free nonrelativistic particle which is interpreted as a $q$-algebraic system without discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural $q$-analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.

quant-ph↗
arXiv1996

A method for solve integrable $A_2$ spin chains combining different representations

A non homogeneous spin chain in the representations $ \{3 \}$ and $ \{3^*\}$ of $A_2$ is analyzed. We find that the naive nested Bethe ansatz is not applicable to this case. A method inspired in the nested Bethe ansatz, that can be applied to more general cases, is developed for that chain. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations different from that for a homogeneous chain. A conjecture about the form of the solutions for more general chains is presented. PACS: 75.10.Jm, 05.50+q 02.20 Sv

hep-th↗
arXiv1996

Exact two-spinon dynamical correlation function of the Heisenberg model

We derive the exact contribution of two spinons to the dynamical correlation function of the spin-1/2 Heisenberg model. For this, we use the isotropic limits of the exact form factors that have been recently computed through the quantum affine symmetry of the anisotropic Heisenberg model $XXZ$

q-alg↗
arXiv1996

$SU_q(2)$ Lattice Gauge Theory

We reformulate the Hamiltonian approach to lattice gauge theories such that, at the classical level, the gauge group does not act canonically, but instead as a Poisson-Lie group. At the quantum level, it then gets promoted to a quantum group gauge symmetry. The theory depends on two parameters - the deformation parameter $λ$ and the lattice spacing $a$. We show that the system of Kogut and Susskind is recovered when $λ\rightarrow 0$, while QCD is recovered in the continuum limit (for any $λ$). We thus have the possibility of having a two parameter regularization of QCD.

hep-th↗
arXiv1995

A rigorous solution of the quantum Einstein equations

We show that the second coefficient of the Conway knot polynomial is annihilated by the Hamiltonian constraint of canonically quantized general relativity in the loop representation. The calculations are carried out in a fully regularized lattice framework. Crucial to the calculation is the explicit form of the skein relations of the second coefficient, which relate it to the Gauss linking number. Contrary to the lengthy formal continuum calculation, the rigorous lattice version can be summarized in a few pictures.

gr-qc↗
arXiv1996

Minimal Length Uncertainty Relation and Ultraviolet Regularisation

Studies in string theory and quantum gravity suggest the existence of a finite lower limit $Δx_0$ to the possible resolution of distances, at the latest on the scale of the Planck length of $10^{-35}m$. Within the framework of the euclidean path integral we explicitly show ultraviolet regularisation in field theory through this short distance structure. Both rotation and translation invariance can be preserved. An example geometry is studied in detail.

hep-th↗
arXiv1996

Lattice knot theory and quantum gravity in the loop representation

We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern--Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chern--Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including intersections. We also discuss the issue of intersections with kinks. This paper is the first step towards setting up the loop representation in a rigorous, computable setting.

gr-qc↗
arXiv1997

``Sum over Surfaces'' form of Loop Quantum Gravity

We derive a spacetime formulation of quantum general relativity from (hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the 3-geometry in proper time. We show that the perturbation expansion of this operator is finite and computable order by order. By giving a graphical representation a' la Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) 2d surfaces in 4d. The contribution of one surface to the sum is given by the product of one factor per branching point of the surface. Therefore branching points play the role of elementary vertices of the theory. Their value is determined by the matrix elements of the hamiltonian constraint, which are known. The formulation we obtain can be viewed as a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Ooguri-Crane-Yetter 4d topological field theory, with a few key differences that illuminate the relation between quantum gravity and TQFT. Finally, we suggests that certain new terms should be added to the hamiltonian constraint in order to implement a ``crossing'' symmetry related to 4d diffeomorphism invariance.

gr-qc↗
arXiv1995

Exact perturbative solution of the Kondo problem

We explicitly evaluate the infinite series of integrals that appears in the "Anderson-Yuval" reformulation of the anisotropic Kondo problem in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo problem of arbitrary spin with the boundary sine-Gordon model, which describes impurity tunneling in a Luttinger liquid and in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.

cond-mat↗
arXiv1995

Real Space Renormalization Group Methods and Quantum Groups

We apply real-space RG methods to study two quantum group invariant Hamiltonians, that of the XXZ model and the Ising model in a transverse field defined in an open chain with appropiate boundary terms. The quantum group symmetry is preserved under the RG transformation except for the appearence of a quantum group anomalous term which vanishes in the classical case. We obtain correctly the line of critical XXZ models. In the ITF model the RG-flow coincides with the tensor product decomposition of cyclic irreps. of $SU_q(2)$ with $q^4=1$.

cond-mat↗
arXiv1996

Nonperturbative Relations in N=2 SUSY Yang-Mills and WDVV equation

We find the nonperturbative relation between $\langle {\rm tr} ϕ^2 \rangle$, $\langle {\rm tr} ϕ^3\rangle$ the prepotential ${\cal F}$ and the vevs $\langle ϕ_i\rangle$ in $N=2$ supersymmetric Yang-Mills theories with gauge group $SU(3)$. Nonlinear differential equations for ${\cal F}$ including the Witten -- Dijkgraaf -- Verlinde -- Verlinde equation are obtained. This indicates that $N=2$ SYM theories are essentially topological field theories and that should be seen as low-energy limit of some topological string theory. Furthermore, we construct relevant modular invariant quantities, derive canonical relations between the periods and investigate the structure of the beta function by giving its explicit form in the moduli coordinates. In doing this we discuss the uniformization problem for the quantum moduli space. The method we propose can be generalized to $N=2$ supersymmetric Yang-Mills theories with higher rank gauge groups.

hep-th↗