Quantum deformation of Lorentz group

A one parameter quantum deformationS L(2,) ofSL(2,) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withS U(2), whereas the solvable part is identified as a Pontryagin dual ofS U(2). It shows thatS L(2,) is the result of the dual version of Drinfeld's double group construction applied toS U(2). The same construction applied to any compact quantum groupG _c is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualG _d ofG _c and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (tensor bundles) overG _c . The theory of smooth representations ofS L(2,) is the same as that ofSL(2,) (Clebsh-Gordon coefficients are however modified). The corresponding tame bicovariant bimodules onS U(2) are classified. An application to 4D ₊ differential calculus is presented. The nonsmooth case is also discussed.     

On the Poisson Limit Theorems of Sinai and Major

Let f(ϕ) be a positive continuous function on 0 ≤ϕ≤Θ, where Θ≤ 2 π, and let ξ be the number of two-dimensional lattice points in the domain Π _R (f) between the curves r=(R+c ₁/R)f(ϕ) and r=(R+c ₂/R)f(ϕ), where c ₁<c ₂ are fixed. Randomizing the function f according to a probability law P, and the parameter R according to the uniform distribution μ _L on the interval [a ₁ L,a ₂ L], Sinai showed that the distribution of ξ under P×μ _L converges to a mixture of the Poisson distributions as L→∞. Later Major showed that for P-almost all f, the distribution of ξ under μ _L converges to a Poisson distribution as L→∞. In this note, we shall give shorter and more transparent proofs to these interesting theorems, at the same time extending the class of P and strengthening the statement of Sinai. Received: 15 June 1999 / Accepted: 11 February 2000     

Deconfinement transition for nonzero baryon density in the field correlator method

Deconfinement phase transition due to the disappearance of confining colorelectric field correlators is described using a nonperturbative equation of state. The resulting transition temperature T _c (μ) at any chemical potential μ is expressed in terms of the change of the gluon condensate ΔG ₂ and absolute value of the Polyakov loop L _fund(T _c ), which is known from the lattice and analytic data, and is in good agreement with the lattice data for ΔG ₂ ≈ 0.0035 GeV⁴; e.g., T _c (0) = 0.27, 0.19, and, 0.17 GeV for n _f = 0, 2, and 3, respectively. The text was submitted by the authors in English.     

A Condensed Matter Interpretation of SM Fermions and Gauge Fields

We present the bundle (Aff(3)⊗ℂ⊗Λ)(ℝ³), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (ℂ⊗Λ)(ℝ³) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗ℂ)(ℤ³). This space allows a simple physical interpretation as a phase space of a lattice of cells. We find the SM SU(3) _c ×SU(2) _L ×U(1) _Y action on (Aff(3)⊗ℂ⊗Λ)(ℝ³) to be a maximal anomaly-free gauge action preserving E(3) symmetry and symplectic structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations. The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with ℤ₂-degenerated vacuum state. We construct anticommuting fermion operators for the resulting ℤ₂-valued (spin) field theory. A metric theory of gravity compatible with this model is presented too.     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Probing the field-induced variation of the chemical potential in Bi2Sr2CaCu2Oy via magneto-thermopower measurements

Approximating the shape of the magneto-thermoelectric power (TEP) ΔS(T,H) measured in Bi₂Sr₂CaCu₂O_y by an asymmetric linear triangle of the form ΔS(T,H)≃S _p (H)±B ^±(H)(T _c −T) with positive B ⁻(H) and B ⁺(H) defined below and above T _c , we observe that B ⁺(H) ≃2B ⁻(H). To account for this asymmetry, we explicitly introduce the field-dependent chemical potential μ(H) of holes into the Ginzburg-Landau theory and calculate both an average ΔS _av(T,H) and fluctuation contribution ΔS _fl(T,H) to the total magneto-TEP ΔS(T,H). As a result, we find a rather simple relationship between the field-induced variation of the chemical potential in this material and the above-mentioned magneto-TEP data around T _c , viz. Δ μ(H)∝S _p (H). Zh. éksp. Teor. Fiz. 116, 257–262 (July 1999) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Coincidence theorem for the direct correlation function of hard-particle fluids

The Mayerf-function for purely hard particles of arbitrary shape satisfiesf ²(1, 2)=–f(1, 2). This relation can be introduced into the graphical expansion of the direct correlation functionc(1, 2) to obtain a graphical expression for the case of exact coincidence, in position and orientation, of two identical hard cores. The resulting expression forc(1, 1)+1 contains only graphsG fromc(1), the sum of irreducible graphs with one labeled point. Relative to its coefficient inc(1),G occurs inc(1, 1) with an additional factorR _c which is 1 for the leading graph in the expansion and of the form 2–2L(G) for all other graphs. HereL(G)=0, 1, 2,..., is a nonnegative integer. Topological analysis is used to derive an expression forL(G) in terms of the connectivity properties ofG.     

Primitive ideals of C q [SL(3)]

The primitive ideals of the Hopf algebraC _q [SL(3)] are classified. In particular it is shown that the orbits in PrimC _q [SL(3)] under the action of the representation groupH C ^*×C ^* are parameterized naturally byW×W, whereW is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals ofC _q [SL(3)] and symplectic leaves of the associated Poisson algebraic groupSL(3,C).Partially supported by a grant from the N.S.A.     

A characterization of dilatation analytic integral kernels

We characterize integral operators belonging to B(L ² (ℝ³))which are dilatation analytic in the Cartesian product of two sectors S _a ⊂ ℂ as analytic functions from S _a×S_a into B(L ²(Ω)), the space of bounded operators on square integrable functions on the unit sphere Ω, which satisfy certain norm estimates uniformly on every subsector.     

Group duality and the Kubo-Martin-Schwinger condition

We consider clusteringG-invariant states of aC*-algebraU endowed with an action of a locally compact abelian groupG. Denoting as usual byF _AB,G _AB, the corresponding two-point functions, we give criteria for the fulfillment of the KMS condition (w.r.t. some one-parameter subgroup ofG) based upon the existence of a closable mapT such thatTF _AB =G _AB for allA,BU. Closability is either inL (G),B(G), orC (G), according to clustering assumptions. Our criteria originate from the combination of duality results for the groupG (phrased in terms of functions systems), with density results for the two-point functions.Supported in part by the National Science Foundation     

Quantum ’t Hooft Loops of SYM $${{\mathcal N}=4}$$ as Instantons of YM2 in Dual Groups SU(N) and SU(N)/ZN

A relation between circular 1/2 BPS ’t Hooft operators in 4d N=4{{\mathcal N}=4} SYM and instantonic solutions in 2D Yang-Mills theory (YM₂) has recently been conjectured. Localization indeed predicts that those ’t Hooft operators in a theory with gauge group G are captured by instanton contributions to the partition function of YM₂, belonging to representations of the dual group ^L G. This conjecture has been tested in the case G = U(N) =  ^L G and for fundamental representations. In this paper, we examine this conjecture for the case of the groups G = SU(N) and ^L G = SU(N)/Z _N and loops in different representations. Peculiarities when groups are not self-dual and representations not “minimal” are pointed out.     

Mott-insulator-superfluid-liquid transition in a one-dimensional bosonic Hubbard model: Quantum Monte Carlo method

The values of the insulator gap Δ in one-dimensional systems of interacting bosons described by the Hubbard Hamiltonian are calculated at low temperatures by the quantum world-line Monte Carlo algorithm. The dependence of Δ on the size of the system, the temperature, and the parameters of the model is investigated. It is shown that a chain with N _a=50 sites is already sufficient to estimate the thermodynamic value of the critical quantity (t/U)_c for which a transition from the insulator into the superfluid state occurs in a commensurate system. To within the computational error, this value, (t/U)_c=0.300±0.005, agrees with the value (t/U)_c=0.304±0.002 obtained previously by the combined “exact diagonalization + renormalization-group analysis” method. The characteristic Kosterlitz-Thouless behavior of the insulator gap is demonstrated near the critical region: Δ∼exp[−b(1−t/t _c)^−1/2]. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 2, 92–96 (25 July 1996)     

The extended phase space of the BRS approach

The origin of the classical BRS symmetry is discussed for the case of a first class constrained system consisting of a 2n-dimensional phase spaceS with free action of a Lie gauge groupG of dimensionm. The extended phase spaceS _ext of the Fradkin-Vilkovisky approach is identified with a globally trivial vector bundle overS with fibreL*(G)L(G), whereL(G) is the Lie algebra ofG andL*(G) its dual. It is shown that the structure group of the frame bundle of the supermanifoldS _ext is the orthosymplectic group OSp(m,m; 2n), which is the natural invariance group of the super Poisson bracket structure on the function spaceC (S _ext). The action of the BRS operator is analyzed for the caseS=R ²ⁿ with constraints given by pure momenta. The breaking of the osp(m,m; 2n)-invariance down to sp(2n–2m) occurs via an intermediate osp(m; 2n–m). Starting from a (2n+2m)-dimensional system with orthosymplectic invariance, different choices for the BRS operator correspond to choosing different 2n-dimensional constraint supermanifolds inS _ext, which in general characterize different constrained systems. There is a whole family of physically equivalent BRS operators which can be used to describe a particular constrained system.     

Simplified components of the spin-other-orbit interaction near the middle of the atomic f shell

To honour the memory of Brian Garner Wybourne, an analysis is presented of three components of the spin-other-orbit interaction for f electrons using the kind of Lie groups he would have been familiar with. The components have been named z ₆, z ₈ and z ₁₀. They all belong to the irreducible representation (IR) (30) of Racah’s group G₂. Near the middle of the f shell it is often found that fewer independent blocks of numbers are needed to express their matrix elements than the Wigner–Eckart theorem, generalized to the IRs U of G₂, would indicate. Each block corresponds to a given U and U?′, and possesses rows and columns labelled by the angular momenta L and L′. The number of independent blocks would be expected to be given by Racah’s multiplicity function c(UU?′ (30)); but near the middle of the shell the number c(UU?′ (20)) (or less) often suffices. For this to occur, z ₈ and z ₁₀ have to be replaced by linear combinations corresponding to IRs of the types (20)×(10) and (21)×(10) of the direct product group G_2A×G_2B, where A and B refer to electrons with their spins up (A) and spins down (B). A detailed example is provided by the IR (31) of G₂, which occurs in the configurations f ⁵ through f ⁹. In addition, two antiHermitian operators (z _a6 and z _a7) that also belong to the IR (30) of G₂ are discussed.     

Muons in type-II superconductors: μ+ diffusion in ultra-pure niobiumdiffusion in ultra-pure niobium

The diffusivityD _μ of positive muons (μ⁺) in the mixed state of superconducting high-purity, high-perfection niobium single crystals is investigated by measurements of the relaxation of the transverse muon spin polarization (μ⁺SR). The method makes use of the strong magnetic field gradients existing in the mixed state of Type-II superconductors and monitorsD _μ through the variation of the magnetic field felt by the μ⁺ during their diffusion through the crystals. For μ⁺ near the centres of the flux lines inNb it givesD _μ(4.6 K)=(8±2)·10⁻¹¹m²S⁻¹. The positive temperature coefficient ofD _μ indicates that at liquid-helium temperatures the diffusivity of μ⁺ inNb is mainly due to phonon-assisted tunnelling processes.     

Quantum coadjoint orbits in gl(n)*

We present a doubleU _h(gl(n, ℂ))-equivariant quantization on semisimple coadjoint orbits of the group GL(n, ℂ) as a quotient of the extended reflection equation algebra by relations which are given explicitly. Such a quantization is a two-parameter family including an explicit GL(n)-equivariant quantization of the Kirillov-Kostant-Souriau Poisson bracket. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Levi-Civita Connections on the Quantum Groups SLq(N), Oq(N) and Spq(N)

For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on SL _q (N), O _q (N) and Sp _q (N) from the classification in [18] there exist unique Levi-Civita connections. Received: Received: 28 February 1996 / Accepted: 1 October 1996     

Bicovariant differential calculus on quantum groupsSU q (N) andSO q (N)

Following Woronowicz's proposal the bicovariant differential calculus on the quantum groupsSU _q (N) andSO _q (N) is constructed. A systematic construction of bicovariant bimodules by using the matrix is presented. The relation between the Hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules, and theq-deformed universal enveloping algebras is given. Imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined. As an application the Maurer-Cartan equations and theq-analogue of the structure constants are formulated.Address after 1 Dec. 1990, Institute of Theoretical Physics, University of München.     

On the irreducible representations of groups that contain a subgroup of index 2

The objects under consideration are a groupG containing a subgroupN of index 2 and an irreducible multiplier representationU ofG by semiunitary (=unitary or antiunitary) operators on a complex Hilbert space of arbitrary dimension. It is assumed thatU(g) is unitary for allg belonging toN. Then the following assertion is proved. The representation ofN that is obtained by restrictingU toN is either irreducible or an orthogonal sum of two irreducible representations.