Embeddings ofU(1)-current algebras in non-commutative algebras of classical statistical mechanics

Motivated by multiplicativeK-homology, and understanding critical phenomena in some classical statistical mechanical models, we construct actions ofGL() on the operator algebras of V. Jones and Ocneanu, and analyse these in terms of embeddings ofU(1)-current algebras.Partially supported by the Science and Engineering Research Council     

Jucys-Murphy elements for Birman-Murakami-Wenzl algebras

The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.     

The Canonical Solutions of the Q-Systems¶ and the Kirillov–Reshetikhin Conjecture

We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite $Q$-systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland–Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov–Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems. Received: 2 August 2001 / Accepted: 27 December 2001     

Nonparametric identification of nonlinear dynamic systems using a synchronisation-based method

The present study proposes an identification method for highly nonlinear mechanical systems that does not require a priori knowledge of the underlying nonlinearities to reconstruct arbitrary restoring force surfaces between degrees of freedom. This approach is based on the master–slave synchronisation between a dynamic model of the system as the slave and the real system as the master using measurements of the latter. As the model synchronises to the measurements, it becomes an observer of the real system. The optimal observer algorithm in a least-squares sense is given by the Kalman filter. Using the well-known state augmentation technique, the Kalman filter can be turned into a dual state and parameter estimator to identify parameters of a priori characterised nonlinearities. The paper proposes an extension of this technique towards nonparametric identification. A general system model is introduced by describing the restoring forces as bilateral spring-dampers with time-variant coefficients, which are estimated as augmented states. The estimation procedure is followed by an a posteriori statistical analysis to reconstruct noise-free restoring force characteristics using the estimated states and their estimated variances. Observability is provided using only one measured mechanical quantity per degree of freedom, which makes this approach less demanding in the number of necessary measurement signals compared with truly nonparametric solutions, which typically require displacement, velocity and acceleration signals. Additionally, due to the statistical rigour of the procedure, it successfully addresses signals corrupted by significant measurement noise. In the present paper, the method is described in detail, which is followed by numerical examples of one degree of freedom (1DoF) and 2DoF mechanical systems with strong nonlinearities of vibro-impact type to demonstrate the effectiveness of the proposed technique.     

Operator ideals and the statistical independence in quantum field theory

We consider analytic properties of a class of dynamical systems which are defined by the action of certain homomorphism groups on von Neumann algebras if restricted to subalgebras. In particular, the analyticity of nuclear maps in the nuclear norm is shown. Furthermore, the statistical independence will be derived from nuclearity conditions. These results give new insight in the statistical independence of commuting algebras.This paper is a result of a collaboration with H. J. Borchers.     

Weyl quantization and star products

Many non-linear classical mechanical systems arise as the symplectic reductions of linear systems. The star products on the corresponding quantized algebras can be derived from the Weyl-Moyal product on the algebras of the linear systems. An algebraic approach to Berezin quantization is sketched.     

Inherent limits on optimization and discovery in physical systems

Topological mapping of a large physical system on a graph, and its decomposition using universal measures are proposed. We find inherent limits to the potential for optimization of a given system and its approximate representations by motifs, and the ability to reconstruct the full system given approximate representations. The approximate representation of the system most suited for optimization may be different from that which most accurately describes the full system.     

Dynamical entropy of quasi-local algebras in quantum statistical mechanics

We study the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of automorphisms on quasi-local algebras in quantum statistical mechanics. We extend their Kolmogorov-Sinai type theorem for AF-algebras to quasi-local algebras which are not necessarily AF-algebras.Work supported in part by Korean Science Foundation and the Basic Science Research Program, Ministry of Education, 1991     

Why Do the Quantum Observables Form a Jordan Operator Algebra?

The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.     

Noncommutative Riemannian geometry on graphs

We show that arising out of noncommutative geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a ‘finite manifold structure’ on the set of vertices. We equip any graph with a canonical ‘Euclidean metric’ and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.     

On the algebraic structure ofN=2 string theory

InN=2 string theory the chiral algebra expresses the generations and anti-generations of the theory and the Yukawa couplings among them and is thus crucial to the phenomenological properties of the theory. Also the connection with complex geometry is largely through the algebras. These algebras are systematically investigated in this paper. A solution for the algebras is found in the context of rational conformal field theory based on Lie algebras. A statistical mechanics interpretation for the chiral algebra is given for a large family of theories and is used to derive a rich structure of equivalences among the theories (dihedralities). The Poincaré polynomials are shown to obey a resolution series which cast these in a form which is a sum of complete intersection Poincaré polynomials. It is suggested that the resolution series is the proper tool for studying allN=2 string theories and, in particular, exposing their geometrical nature.     

On the relation between a conformal structure in spacetime and nets of local algebras of observables

In this Letter, we study the question of how to reconstruct the conformal structure (causality relation) of spacetime from a net of local algebras of observables on the underlying manifold.Partly supported by DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

On the Exact Solution of Models Based¶on Non-Standard Representations

The algebraic Bethe ansatz is a powerful method to diagonalize transfer-matrices of statistical models derived from solutions of (graded) Yang Baxter equations, connected to fundamental representations of Lie (super-)algebras and their quantum deformations respectively. It is, however, very difficult to apply it to models based on higher dimensional representations of these algebras in auxiliary space, which are not of fusion type. A systematic approach to this problem is presented here. It is illustrated by the diagonalization of a transfer-matrix of a model based on the product of two different four-dimensional representations of . Received: 24 November 1998 / Accepted: 26 March 1999     

The role of type III factors in quantum field theory

One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e. factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III, factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e. the way they are embedded into each other     

Anyons,deformed oscillator algebras and projectors

We initiate an algebraic approach to the many-anyon problem based on deformed oscillator algebras. The formalism utilizes a generalization of the deformed Heisenberg algebras underlying the operator solution of the Calogero problem. We define a many-body Hamiltonian and an angular momentum operator which are relevant for a linearized analysis in the statistical parameter ν. There exists a unique ground state and, in spite of the presence of defect lines, the anyonic weight lattices are completely connected by the application of the oscillators of the algebra. This is achieved by supplementing the oscillator algebra with a certain projector algebra.     

Radix-R > 2 Quantum Computation

A quantum mechanical system is presented forwhich a multiple-valued quantum algebra and logic arederivable. The system is distinguished from previousquantum computational proposals by the definition of higher order quantum algebras and logicsderived from multilevel quantum spin systems.     

Time-dependent mechanical symmetries and extended Hamiltonian systems

Time-dependent mechanical symmetries are discussed in the framework of an extended Hamiltonian system. The Lie-algebraic structure of the time-dependent symmetry is made clear by introducing an extended Poisson bracket. Moreover, the relationship between the symmetry algebras of the classical and the quantum system is established.     

Hartree-fock approximation for inverse many-body problems

A new method is presented to reconstruct the potential of a quantum mechanical many-body system from observational data, combining a nonparametric Bayesian approach with a Hartree-Fock approximation. A priori information is implemented as a stochastic process, defined on the space of potentials. The method is computationally feasible and provides a general framework to treat inverse problems for quantum mechanical many-body systems.     

Characters of Demazure modules and solvable lattice models

We study the path realization of Demazure crystals related to solvable lattice models in statistical mechanics. Various characters are represented in a unified way as the sums over one-dimensional configurations which we call unrestricted, classically restricted and restricted paths. As an application, characters of Demazure modules are obtained in terms of q-multinomial coefficients for several level-1 modules of classical affine algebras.     

Oscillator Algebra With Reflecting Boundary and Generalized Statistics

The oscillator algebra with reflecting boundary is constructed together with its Fock space,and is generalized to the cases with generalized statistics and multicomponent.Such oscillators depend manifestly on the reflection factor and the statistical(exchange)factor.By construction,the Fock space of such oscillator algebras can be obtained by certain projection operation.from that of the usual bosonic oscillator without reflection condition.