Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

A complete proof of the Feigenbaum conjectures

The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map . This map acts on functions that are jointly analytic in a position variable (t) and in the parameter () that controls the period doubling phenomenon. A fixed point ^* for this map is found. The usual renormalization group doubling operatorN acts on this function ^* simply by multiplication of with the universal Feigenbaum ratio ^*= 4.669201..., i.e., (N ^*(,t)= ^*( ^* ,t). Therefore, the one-parameter family of functions, ^* , ^* (t)=( ^*(,t), is invariant underN. In particular, the function ₀ ^* is the Feigenbaum fixed point ofN, while ^* represents the unstable manifold ofN. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.     

Sato's universal Grassmannian and group extensions

An extension of the symmetry group GL of Sato's universal Grassmannian GM is constructed. The extension plays a similar role to that of the central extension in the approach of Segal and Wilson to functions and KP hierarchy. Our group G contains GL_res as a subgroup and the associated function is a deformation of the usual function, leading to a deformed KP hierarchy. A relation to current algebra of Yang-Mills theory in 3+1 dimension is discussed.     

Braided Clifford Algebras as Quantum Deformations

We present a general algebraic framework for the study of quantum/braided Clifford algebras. We allow that the quadratic form g on the base vector space takes values from a noncommutative algebra . Clifford algebra is understood as a Chevalley—Kähler deformation of the braided exterior algebra built from V, , and the initial braid operator : . The new product is canonically associated to g, , and , and it is constructed by applying Rota's and Stein Cliffordization.     

Periodic inverse problem for a new hierarchy of coupled evolution equations

We study the inverse problem with periodic boundary condition for a new class of integrable nonlinear evolution equations. The multiphase periodic solutions for the nonlinear fields (p, q, r) are expressed in terms of the Riemann theta function, which is obtained via the linearization of the flows of the set of auxiliary variables _j on a Riemann surface. An explicit case is evaluated to obtain the form of the algebraic curve on which the variables _j move.     

SU q (2) covariant\hat R-matrices for reducible representations

We consider SU _q (2) covariant -matrices for the reducible3 1 representation. There are three solutions to the Yang-Baxter equation. They coincide with the previously known -matrices for SO _q (3) and SO _q (3, 1). Also, they are the three -matrices which can be constructed by using four different SU _q (2) doublets. Only two of the three -matrices allow a differential structure on the reducible four-dimensional quantum space.     

Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even-dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fieldsF . We derive a topological bound on R⁸, , wherep1 is the first Pontryagin class of the SO(n) Yang-Mills bundle, andk is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.     

Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface

Given n2, we put r=min . Let be a compact, C ^r -smooth surface in ⁿ which contains the origin. Let further be a family of measurable subsets of such that as . We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator in L ²( ⁿ ), where is a positive constant, as . An analogous result is given also for geometrically induced bound states due to a interaction supported by an infinite planar curve.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C (M) and ( (E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C ^*-algebras. We also discuss the semi-classical geometry arising from these deformations.     

Minimal affinizations of representations of quantum groups: the irregular case

Let be a finite-dimensional complex simple Lie algebra and U_q( ) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of U_q( ), an affinization ofV is an irreducible representationVV of the quantum affine algebra U_q( ) which containsV with multiplicity one and is such that all other irreducible U_q( )-components ofV have highest weight strictly smaller than the highest weight ofV. There is a natural partial order on the set of U_q( ) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if is of typeA, B, C, F orG, the minimal affinization is unique up to U_q( )-isomorphism; (ii) if is of typeD orE and is not orthogonal to the triple node of the Dynkin diagram of , there are either one or three minimal affinizations (depending on ). In this paper, we show, in contrast to the regular case, that if U_q( ) is of typeD ₄ and is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of .As a by-product of our methods, we disprove a conjecture according to which, if is of typeA _n,every affinization is isomorphic to a tensor product of representations of U_q( ) which are irreducible under U_q( ) (in an earlier paper, we proved this conjecture whenn=1).Both authors were partially supported by the NSF, DMS-9207701.     

Ensemble and trajectory statistics in a nonlinear partial differential equation

We have examined the influence of parametric noise on the solution behavioru(t, x) of a nonlinear initial value() problem arising in cell kinetics. In terms of ensemble statistics, the eventual limiting solution mean and variance are well-characterized functions of the noise statistics, and and depend on . When noise is continuously present along the trajectory, and are independent of the noise statistics and . However, in their evolution toward and , both _u (t, x) and _u ² (t, x) depend on the noise and.     

Computer simulation of shock waves in the completely asymmetric simple exclusion process

We study the evolution of the completely asymmetric simple exclusion process in one dimension, with particles moving only to the right, for initial configurations corresponding to average density _– ( ₊) left (right) of the origin, _{– +}. The microscopic shock position is identified by introducing a second-class particle. Results indicate that the shock profile is stable, and that the distribution as seen from the shock positionN(t) tends, as time increases, to a limiting distribution, which is locally close to an equilibrium distribution far from the shock. Moreover , withV=1– _–– ₊, as predicted, and the dispersion ofN(t), ²(t), behaves linearly, for not too small values of ₊– _–, i.e., , whereS is equal, up to a scaling factor, to the valueS _WA predicted in the weakly asymmetric case. For ₊= _– we find agreement with the conjecture .Dedicated to the memory of Paola Calderoni.     

Traffic Jams in a Lattice-Gas Model

We propose a simple lattice-gas model characterized by two states of atoms, the thermalized state, which is the same as in the standard lattice-gas model, and the running state, where the atoms jump in one direction only. The model exhibits the existence of traffice jams (bunching of thermalized atoms in compact groups), the nonlinear dependence of mobility on the jump probability, and the hysteresis.     

Conformal Transformations as Observables

C denotes either the conformal group in 3+1 dimensions, PSO(4, 2), or in one chiral dimension, PSL(2, ). Let U be a unitary, strongly continuous representation of C satisfying the spectrum condition and inducing, by its adjoint action, automorphisms of a von Neumann algebra . We construct the unique inner representation of the universal covering group of C implementing these automorphisms. satisfies the spectrum condition and acts trivially on any U-invariant vector. This means in particular: Conformal transformations of a field theory having positive energy are weak limit points of local observables. Some immediate implications for chiral subnets are given. We propose the name Borchers–Sugawara construction.     

Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

We give a rigorous proof of power-law falloff in the Kosterlitz-Thouless phase of a two-dimensional Coulomb gas in the sense that there exists a critical inverse temperaturegb and a constant >0 such that for all> and all external charges R we have , whereG (x) is the two-point external charges correlation function,=dist(, Z), and for 0$$ " align="middle" border="0"> . In the case of a hard-core or standard Coulomb gas with activityz, we may choose=(z) such that(z)24 asz0.     

Yet another formulation of the Dirac equation

The 2-by-2 Pauli matrix algebra is used to write the 1-by-4 Dirac field in anequivalent 2-by-2 matrix . The current 4-vectors and *_µ are then compared and the latter is shown to not be easily interpretable as a probability density, and also tocontain .     

Ergodic states in a non commutative ergodic theory

Using the Godement mean of positive-type functions over a groupG, we study -abelian systems { , } of aC*-algebra and a homomorphic mapping of a groupG into the homomorphism group of . Consideration of the Godement mean off(g)U _g withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) ( _g (A)) withA and a covariant representation of the system { , } for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over . Finally we investigate the discrete spectrum of covariant representations of { , } (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).On leave of absence from Istituto di Fisica G. Marconi Piazzale delle Scienze 5 — Roma.     

Mass Generation by Weyl Symmetry Breaking

A massless electroweak theory for leptons is formulated in a Weyl space, W₄, yielding a Weyl invariant dynamics of a scalar field , chiral Dirac fermion fields _L and _R, and the gauge fields , A, Z, W, and W , allowing for conformal rescalings of the metric g and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields, , representing the D(1) or dilatation gauge fields. The local group structure of this Weyl electroweak (WEW) theory is given by —or its universal coverging group for the fermions—with denoting the electroweak gauge group SU(2)_W × U(1)_Y. In order to investigate the appearance of nonzero masses in the theory the Weyl symmetry is explicitly broken by a term in the Lagrangean constructed with the curvature scalar R of the W₄ and a mass term for the scalar field. Thereby also the Z and W gauge fields as well as the charged fermion field (electron) acquire a mass as in the standard electroweak theory. The symmetry breaking is governed by the relation D ² = 0, where is the modulus of the scalar field and D denotes the Weyl-covariant derivative. This true symmetry reduction, establishing a scale of length in the theory by breaking the D(1) gauge symmetry, is compared to the so-called spontaneous symmetry breaking in the standard electroweak theory, which is, actually, the choice of a particular (nonlinear ) gauge obtained by adopting an origin, , in the coset space representing , with being invariant under the electromagnetic, gauge group U(1)_e.m.. Particular attention is devoted to the appearance of Einstein's equations for the metric after the Weyl symmetry breaking, yielding a pseudo-Riemannian space, V₄, from a W₄ and a scalar field with a constant modulus . The quantity affects Einstein's gravitational constant in a manner comparable to the Brans-Dicke theory. The consequences of the broken WEW theory are worked out and the determination of the parameters of the theory is discussed.     

Asymptotically abelian systems

We study pairs { , } for which is aC*-algebra and is a homomorphism of a locally compact, non-compact groupG into the group of *-automorphisms of . We examine, especially, those systems { , } which are (weakly) asymptotically abelian with respect to their invariant states (i.e. |A _g (B) — _g (B)A 0 asg for those states such that ( _g (A)) = (A) for allg inG andA in ). For concrete systems (those with -acting on a Hilbert space andg _g implemented by a unitary representationg U _g on this space) we prove, among other results, that the operators commuting with and {U _g } form a commuting family when there is a vector cyclic under and invariant under {U _g }. We characterize the extremal invariant states, in this case, in terms of weak clustering properties and also in terms of factor and irreducibility properties of { ,U _g }. Specializing to amenable groups, we describe operator means arising from invariant group means; and we study systems which are asymptotically abelian in mean. Our interest in these structures resides in their appearance in the infinite system approach to quantum statistical mechanics.     

Free boson representation ofU_q (\widehat{sl}_3 )

A representation of the quantum affine algebra of an arbitrary levelk is constructed in the Fock module of eight boson fields. This realization reduces the Wakimoto representation in theq 1 limit. The analogues of the screening currents are also obtained. They commute with the action of modulo total differences of some fields.On leave from Department of Physics, University of Tokyo, Tokyo 113, Japan.