Symmetry properties of fermionic bilinears in lattice QCD

The 256 fermionic bilinears $\text{ψ}\text{γTψ}$ appearing in the Kogut-Susskind/Kähler-Dirac theory are classified into representations of the hypercubic rotation reflection group; charge conjugation and U(1)_A are also considered. Applications to the study of the normalization of lattice currents are suggested.     

On the defining relations of quantum superalgebras

In defining quantum superalgebras, extra relations need to be added to the Serre-like relations. They are obtained for sl _q (m, n) and osp _q (m, 2n) usingq-oscillator representations.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

B2羣和基本粒子强相互作用

计算了B₂羣不可约表示的直接乘积4×4,4×5,4×10和5×5分解中的约化系数。给出了它在基本粒子强相互作用对称性方面的应用的若干例子:推导了散射截面关系和低维数表示的质量关系,讨论了共振态的填充和衰变情况以及R反演不变性等。     

Studies on the cell structure of quasicrystals

The geometric and crystallographic ideas for cell models and tilings in non-periodic ordered structures are outlined. The basic concepts, among them the cell geometry and duality in a lattice are explained for the example of a one dimensional section through the root lattice A ₂. The corresponding constructions for 3D icosahedral sections through the 6D face-centered hypercubic lattice, equivalent to the root lattice D ₆, are described. Two different tilings, one with equivalent, one with three inequivalent vertex positions, are derived and discussed.     

Migdal-Kadanoff renormalization group for theO(n) model

We perform a Migdal-Kadanoff renormalization group calculation on anO(n) symmetric model on ad-dimensional hypercubic lattice,d=2, 3. We find that in two dimensions the critical fixed point disappears asn=n _KT1.96, which is in good agreement with the exact valuen _KT=2. In three dimensions the fixed point persists much longer, albeit not all the way up to infinity. Surface critical phenomena in a semiinfiniteO(n) model are also considered.     

Lorentz covariance and kinetic charge

There is a one-to-one correspondence between inequivalent covariant displaced Fock representations of the free relativistic field and the 1-cohomology of the Poincaré group with coefficients in the 1-particle space.Representations with positive energy are obtained from cocycles with finite energy which have particle-like properties and are interpreted as condensed states of matter without a sharply defined mass.The 1-cohomology groups ofP ₊ are calculated. These are trivial in 3- or 4-dimensional space-time, or if the mass is non-zero. Non-trivial cocycles for subgroups lead to representations in whichP-invariance is spontaneously broken. We recoverP-invariance in a direct integral representation possessing a gauge group, and a superselection structure labelled by the velocities of the condensed states of matter which are the cocycles determining each irreducible component of the representation. A model in 4-dimensional space-time is constructed.     

Density of states and dispersion of quasiparticles in the Emery model: Nonlocal path-integral Monte Carlo algorithm

The spectral density of states for a 108-site Cu₃₆O₇₂ cluster in the two-dimensional three-band Emery model is reconstructed with the aid of a path integral Monte Carlo algorithm. Dispersion relations are obtained for quasiparticles in the upper Hubbard band and in the correlated-states band; this corresponds to electron and hole doping of high-T _c superconductors. The form of the isoenergy surfaces is close to the experimentally observed form and confirms the existence of singularities in the density of states near the Fermi level. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 11, 860–865 (10 June 1996)     

Application of symmetry groups of four-dimensional space in spectroscopy of crystals

A method of classification of many-electron states that involves symmetry groups of four-dimensional space is developed. Taking into account the spatial and temporal symmetry, the selection rules for electric and magnetic dipole transitions are investigated. The reduction relations for the irreducible representations of the orthogonal group O ₄ and the group R ₄ of pure rotations of the four-dimensional space on the groups and O _h ⁴ and O ⁴ of the four-dimensional cube and of the octahedron on the three-dimensional groups O _h and D _6h of the cubic and hexagonal systems, respectively, are obtained. The four-dimensional classification of the levels of the spin-orbit interaction of rare earth ions for the intermediate reduction on the group of the four-dimensional cube is performed. With the help of the irreducible representations of the O ₄ group, the selection rules of magnetic and forbidden electric dipole transitions, as well as of intercombination transitions, are investigated.     

Investigation of the substituent specific cross‐interaction effects on 13C NMR of the CN bridging group in substituted benzylidene anilines

The substituent effect on ¹³C NMR of the C?N in benzylidene anilines XPhCH?NPhY was investigated, in which the substituents X and Y are in p‐position or in m‐position of the two aromatic rings. The substituent effects including the inductive effects of X and Y, the conjugative effects of X and Y, and the substituent specific cross‐interaction effect were put into one model to quantify the ¹³C NMR chemical shift δ_C(C?N) of the C?N in XPhCH?NPhY. A penta‐parameter correlation equation with correlation coefficient 0.9975 and standard error 0.17 ppm was obtained for 80 samples of compounds. The result shows that the substituents X and Y have an opposite effect on the δ_C(C?N). The electron‐withdrawing effects of X decrease the δ_C(C?N); while the electron‐donating effects of X increase the δ_C(C?N). In contrast, the electron‐withdrawing effects of Y increase the δ_C(C?N); while the electron‐donating effects of Y decrease the δ_C(C?N). A new substituent specific cross‐interaction effect parameter Δσ² was proposed, which indicates that the most substituent specific cross‐interaction effect exists in the pair of max electron‐withdrawing group (EWG) and max electron‐donating group (EDG) or the pair of max EDG and max EWG. Further to verify the obtained correlation equation, 15 samples of model compounds were prepared and their δ_C(C?N) was measured in this work. The predicted δ_C(C?N) values with the obtained equation are in good agreement with the measured ones for these prepared compounds, which confirmed the reliability of the obtained equation. Copyright © 2010 John Wiley & Sons, Ltd.     

The six-point families of exceptional representations of the conformal group

Six-point families of topological reducible (indecomposable) representations of the conformal group of space-time M ₄ are studies. The structure of invariant subspaces is analysed and the full set of equivalence relations among the suprepresentations is derived.     

Vertex operators in the conformal field theory on P 1 and monodromy representations of the braid group

Vertex operators (primary fields) are constructed for the conformal field theory on P ¹ by means of A ₁ ⁽¹⁾ modules. The commutation relations of vertex operators induce monodromy representations of the braid group on the spaces of vacuum expectations of compositions of vertex operators.     

Coherent states for quantum compact groups

Coherent states are introduced and their properties are discussed for simple quantum compact groupsA _l, B_l, C_l andD _l. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit. The coherent state is interpreted as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compactR-matrix formulation (generalizing this way theq-deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel-Weil construction) is described using the concept of coherent state. The relation between representation theory and non-commutative differential geometry is suggested. Dedicated to Professor L.D. Faddeev on his 60th birthday     

Schloegl’s Second Model for Autocatalysis on a Cubic Lattice: Mean-Field-Type Discrete Reaction-Diffusion Equation Analysis

Schloegl’s second model for autocatalysis on a hypercubic lattice of dimension d≥2 involves: (i) spontaneous annihilation of particles at lattice sites with rate p; and (ii) autocatalytic creation of particles at vacant sites at a rate proportional to the number of diagonal pairs of particles on neighboring sites. Kinetic Monte Carlo simulations for a d=3 cubic lattice reveal a discontinuous transition from a populated state to a vacuum state as p increases above p=p _e . However, stationary points, p=p _eq (≤p _e ), for planar interfaces separating these states depend on interface orientation. Our focus is on analysis of interface dynamics via discrete reaction-diffusion equations (dRDE’s) obtained from mean-field type approximations to the exact master equations for spatially inhomogeneous states. These dRDE can display propagation failure absent due to fluctuations in the stochastic model. However, accounting for this anomaly, dRDE analysis elucidates exact behavior with quantitative accuracy for higher-level approximations.     

Ferromagnetism in Hubbard model in many-electron operator representation

The Hubbard model for a metal with strong correlations is considered in the representation of many-electron X-operators. General self-consistent expressions are obtained for the one-particle Green function taking into account fluctuation corrections. The stability regions of the saturated and unsaturated ferromagnetism in the ground state on the n-U plane (n is the electron concentration and U is the Coulomb interaction parameter) are determined for various bare densities of states (semi-elliptic band and the square, cubic, and hypercubic lattices).     

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.     

Generalized Verlinde Formulae for Some Riemann Surface Automorphisms

The generalized Verlinde formulae expressing traces of mapping classes corresponding to automorphisms of certain Riemann surfaces, and the congruence relations on allowed modular representations following from them are presented. The surfaces considered are families of algebraic curves given by suitably chosen equations, the modular curve X(11), and a factor curve of X(8). The examples of modular curves illustrate how the study of arithmetic properties of suitable modular representations can be used to gain information on automorphic properties of Riemann surfaces.     

A unified treatment of the groups SO(4) and SO(3,1)

The irreducible representations of the group SO(4) in which the SO(3) subgroup is reduced are studied by an explicit construction of the operators and the basis in the spinor representation. The basis function which is formally identical with that for the coupling of two angular momentaj ₁ andj ₂ is expressible in terms of a hypergeometric function and strongly resembles the one for the irreducible representations of the groups SO(3,1). For the Lorentz group, the bases for the unitary representations which require unphysical values ofj ₁ andj ₂ are found to be analytic continuation of those for SO(4). The realization of the unitary irreducible representations of the group SO(4) in the Hilbert space of these functions leads, for appropriate unphysical values ofj ₁,j ₂, to the Gelfand-Naimark formula for the principal and complementary series of the representations of SO(3;1). The matrix elements for finite transformations of SO(4) and SO(3,1) can be evaluated, in this approach, in a unified manner by using standard properties of the hypergeometric function. These turn out to be a finite sum of₃ F ₂-functions which, as expected, are polynomials for SO(4) and infinite series for SO(3,1). A number of special matrix elements are calculated from the general formula and these agree with the results obtained previously.The authors are deeply indebted to Professor S.Dutta Majumdar fo many important suggestions and clarifications.     

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Explicit expressions for the generators of the quantum superalgebraU _q [gl(n/m)] acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq-number identities.