Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2p^th root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2p^th root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .     

Sur la construction asymptotique et l'interprétation physique d'une théorie avec dégénérescence du vide

The study of the setQ ofWightman's functionals begun in ref. [5] is continued in this paper. Haag-Ruelle asymptotic construction [1, 2] is formulated for the case when all the pureWightman's functionals contained in the decomposition of the given functional generate the same set of asymptotic states. As an example we consider a theory with the degenerate vacuum and prove that it is physically equivalent to a theory with the single vacuum. For this case we show that the transition to the theory with the degenerate vacuum is equivalent to introducing of a charged spurion in the theory with the single vacuum. The mathematically correct creation and destruction operators for this spurion are given.     

A class of nonlinear relativistic partial differential equations in elementary particle theory

In mathematical approaches to elementary particle theory, the equation [² - ²/t²]=m² ;+g ³ has been of interest [1,2]; it describes a quartically self-coupled neutral scalar meson field. This paper applies the decomposition method [3-6] to obtain accurate non-perturbative timedevelopment of the field for this equation, or variations involving other nonlinear interactions, without the use of cutoff functions or truncations.     

Physical and geometrical interpretation of the Jordan-Hahn and the Lebesgue decomposition property

The Jordan-Hahn decomposition and the Lebesgue decomposition, two basic notions of classical measure theory, are generalized for measures on orthomodular posets. The Jordan-Hahn decomposition property (JHDP) and the Lebesgue decomposition property (LDP) are defined for sections of probability measures on an orthomodular poset L. If L is finite, then these properties can be characterized geometrically in terms of two parallelity relations defined on the set of faces of . A section is shown to have the JHDP if and only if every pair of f-parallel faces is p-parallel; it is shown to have the LDP if and only if every pair of disjoint faces is p-parallel. It follows from these results that the LDP is stronger than the JHDP in the setting of finite orthomodular posets. Mielnik's convex scheme of quantum theory provides the frame for a physical interpretation of these results.     

Algebraic decomposition of the electromagnetic constitutive tensor. A step towards a pre‐metric based gravitation?

The aim of this paper is to provide an algebraic decomposition of the principal part of the electromagnetic constitutive tensor χ_μνρσ (a pre‐metric structure), in a sum of terms of the form where the Z_(i) are invariant scalars and g_μν(i)unimodular conformal metrics. If the space‐time is also equipped with a volume measure, such a decomposition is equivalent to a similar decomposition of the associated area metric, where the g_μν(i) are ordinary metrics. It has been recently argued [8] that, if such a particular decomposition should be available, it would then be possible to find a compatible connection and derived tensors.     

Nonlinear relaxation at first-order phase transitions: A Ginzburg-Landau theory including fluctuations

Kinetics of phase transitions are investigated in systems with nonconserved one-component order parameter (i.e., generalized time-dependent Ginzburg-Landau models ind dimensions). The correct static critical behavior as well as fluctuation effects on the kinetics are incorporated by a suitable adaptation of the theory on spinodal decomposition by Langer, Baron and Miller. Both the case of quenches from temperaturesT above to below the critical pointT _c and the case of magnetic field changesH from positive to negative values are treated, and both time-dependent order parameter m() and structure factorS(q, ) are obtained numerically ford=2, 3. In the case of quenches atH=0, we find that m()0 andS(q, ) —S(q, ) exp(– ^1/2/7.2) , withS(q, )q ^–2. In the case of field changes we find that forH not exceeding some critical valueH ^* the system is trapped in a metastable state with infinite lifetime. In contrast to the meanfield-spinodal, the susceptibility does not seem to diverge atH ^*. These results are compared with other treatments, in particular the Monte Carlo simulations of kinetic Ising models by Binder and Müller-Krumbhaar. While our theory describes some properties of the metastable states reasonably,H ^* distinctly exceeds the observed limit of metastability. We argue that the present theory does not take into account nucleation fluctuations, and also fails to describe correctly the domain growth in the late stages of the relaxation. Contrary to Langer et al. we suggest that universality holds for nonlinear relaxation and spinodal decomposition nearT _c .Supported in part by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 130     

Colored null flags and para-Fermi fields

An inverse problem of deriving the concept of quantized fields from a certain observable conserved current is investigated. It is found that a natural framework in which to attack the problem is provided for by what we shall call Green's ansatz of null decomposition of the current. The null decomposition naturally yields a set ofcolored null flags hoisted at each space-time point, a null flag comprizing a real null vector and an associated real null six-vector, and is invariant under all permutations of colors. From the fact that to any null flag there corresponds a two-component spinor it follows that the color permutation group is extended tocolor groups O(p) orU(p), wherep is the number of null flags considered. It is shown that para-Weyl (para-Fermi) fields of orderp2 can be deduced from the (chiral) set ofp colored null flags, and that the color groupU(p) is singled out that functions as the gauge group of para-Fermi theory.     

Markov and stability properties of equilibrium states for nearest-neighbor interactions

Consider models on the lattice ^d with finite spin space per lattice point and nearest-neighbor interaction. Under the condition that the transfer matrix is invertible we use a transfer-matrix formalism to show that each Gibbs state is determined by its restriction to any pair of adjacent (hyper)planes. Thus we prove that (also in multiphase regions) translationally invariant states have a global Markov property. The transfer-matrix formalism permits us to view the correlation functions of a classicald-dimensional system as obtained by a linear functional on a noncommutative (quantum) system in (d – 1)-dimensions. More precisely, for reflection positive classical states and an invertible transfer matrix the linear functional is a state. For such states there is a decomposition theory available implying statements on the ergodic decompositions of the classical state ind dimensions. In this way we show stability properties of _ev ^d -ergodic states and the absence of certain types of breaking of translational invariance.     

Abelian decomposition method for G<Subscript>2</Subscript> gauge theory

The method of Abelian decomposition proposed by Faddeev and Niemi is used to derive the low-energy effective lagrangian of G₂ gauge theory. The G₂ algebra is studied. The commutation relations among the generators of the G₂ algebra are established, based on the framework of its regular maximal subalgebra, an SU(3) algebra. Received: 19 December 2002 / Revised version: 10 January 2003 / Published online: 14 March 2003 RID="a" ID="a" e-mail: suw@phy.ccu.edu.tw     

Three applications of the Clebsch-Gordan reduction in phenomenological crystal physics

The use of a method based on the Clebsch-Gordan reduction in terms of typical variables is described for the solution of three important problems of crystal physics and of the structural phase transition theory: 1. decomposition of any tensor into bases of irreducible representations of symmetry group, 2. construction of thermodynamic potential of given symmetry in various variables, 3. construction of gradient invariants which are used in the theory of incommensurate structures. The method is elucidated on an example of crystallographic point groupD ₄ and on the family of its isomorphs. On the basis of a table of Clebsch-Gordan products for the groupD ₄ there are given: 1. tables of tensorial bases up to fourth rank tensors for the groupsD ₄,C _4v, andD _2d, 2. the so-called extended integrity basis for the groupD ₄ in components of polarization and of strain tensor, 3. the so-called typical extended integrity basis for the isomorphic familyD ₄ On the basis of which any extended integrity basis for groups of the familyD ₄ and in any set of variables can be deduced, 4. the so-called Lifshits-type invariants and quadratic gradient invariants for all magnetic groups of the Laue classD ₄.     

Gravitation and electromagnetism

We obtain a general relativistic unification of gravitation and electromagnetism by simply(1) restricting the metric so that it admits an orthonormal tetrad representation in which the spacelike vectors are curl-free, and(2) identifying the timelike vector as the potential for an electromagnetic field whose only sources are singularities. It follows that: (A) The energy density is everywhere nonnegative, (B) the space is flat if and only if the electromagnetic field vanishes, (C) the vector potential (through which all curvature enters) admits no invariant algebraic decomposition, and satisfies the covariant Lorentz condition identically, (D) the theory is free of prior geometry, (E) the electromagnetic self-energy of a spherically symmetric point charge equalsMC ² , (F) particles deviate from geodesic motion according to the Lorentz force law with radiative reaction, and (G) particles with all electromagnetic multipole structures are included.     

Chemical Bond in Metal Azides and Their Reactivity

Results of experimental investigations into the processes of slow and explosive decomposition of metal azides and the corresponding phenomenological models are analyzed. The energy-band structure and the valence electron density for actual crystallographic planes as well as the hypothetical electron density of the first unoccupied band of KN₃ and AgN₃ are calculated by the density functional method with the use of pseudopotentials. It is demonstrated that the anion-cation chemical bond in KN₃ is predominantly ionic in character, whereas a significant covalent component in AgN₃ is caused by Ag d-states. From the character of distribution of the valence and hypothetical electron densities it follows that the exciton and vibron decomposition mechanisms are improbable. The chain process is also improbable for alkali metal azides, because the gap energy for them is E _g 8 eV, which is close to the energy of azide ion decomposition Q 9–12 eV; the chain process is probable for AgN₃, because E _g(AgN₃) 3 eV. The chain process in AgN₃ is also favored by intense delocalization of the valence electron density through the hybridization of the Ag d-state and the N p-state, which causes a relatively high hole mobility.     

Ground state of a spin-phonon system. III. Small-B limit

The ground state of the spin-one-half acoustic phonon system is studied in the limitB1. The technique is to combine the Brillouin-Wigner variational perturbation theory with a source canonical transformation. With theB=0 source transform the energy is calculated as a function of coupling constant through terms of orderB ². To this order there is no phase transition. The theory gives the crossover from perturbation theory to an improved classical theory with quantum fluctuations. With a source transform with a nonzero inverse length the energy estimate is further improved to next order in the coupling constant and for larger values ofB. The soft, infinite-order transition of the modified source theory is removed in the limitB1.     

Perturbation theory for kinds

In this paper we prove the validity of formal asymptotic results on perturbation theory for kind solutions of the sine-Gordon equation, originally obtained by McLaughlin and Scott. We prove that for appropriate perturbations, of size in an appropriate norm, slowly varying in time in the rest frame of the kink, the shape of the kink is unaltered in theL norm toO() for a time ofO(1/). The kink parameters, which represent its velocity and centre, evolve slowly in time in the way predicted by the asymptotics. The method of proof uses an orthogonal decomposition of the solution into an oscillatory part and a one-dimensional zero-mode term. The slow evolution of the kink parameters is chosen so as to suppress secular evolution of the zero-mode.Partially supported as a graduate student at Princeton University of NSF grant 215 6211     

Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

Chiral Charged Fermions,One-Dimensional Quantum Field Theory and Vertex Algebras

We give an explicit L ²-representation of chiral charged fermions using the Hardy–Lebesgue octant decomposition. In the pure case such a representation has already been used by M. Sato in holonomic field theory. We study both pure and mixed cases. In the compact case, we rigorously define unsmeared chiral charged fermion operators inside the unit circle. Using chiral fermions, we orient our findings towards a functional analytic study of vertex algebras as one-dimensional quantum field theory.     

States of physical systems

States of physical systems may be represented by states onB*-algebras, satisfying certain requirements of physical origin. We discuss such requirements as are associated with the presence of unbounded observables or invariance under a group. It is possible in certain cases to obtain a unique decomposition of states invariant under a group into extremal invariant states. Our main results is such a decomposition theorem when the group is the translation group in dimensions and theB*-algebra satisfies a certain locality condition. An application of this theorem is made to representations of the canonical anticommutation relations.     

The Newtonian limit in a family of metric affine theories of gravitation

A brief review of a first order theory with a quadratic LagrangianL=R+₀R² is presented. It is shown that a test particle follows a geodesic of the metric connection. The theory behaves in the Newtonian limit as the Newtonian theory with a correction which is proportional to the matter density at the field point. This behavior can be produced by a Yukawa potential with an atomic scale characteristic range and a coupling constant proportional to 1/ ². This type of potential is not excluded by the present experimental data.     

Deposition kinetics of the γ′ phase in N36T2YU2 steel

Electron microscopy is reported for the deposition in this steel; quantitative results are given on the particle size of the phase deposited during a continuous and uninterrupted decomposition. The deposition of the phase occurs in one stage, while the two stages in the variation in the mechanical properties arise from features of the interaction of dislocations with the phase particles. There appears to be only a small energy barrier to the generation of phase particles. Spherulites are formed in regions of interrupted decomposition. The effects of quenching temperature on the deposition mechanism are discussed.Translated from Izvestiya VUZ, Fizika, No. 3, pp. 41–45, March, 1973.     

Quantized affine Lie algebras and diagonalization of braid generators

Let U _q be a quantized affine Lie algebra. It is proven that the universal R-matrix R of U _q satisfies the celebrated conjugation relationR ⁺ =TR withT the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight U _q -module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin and Gould forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.