Showing papers on "Operator algebra published in 1994"

p-adic analysis and mathematical physics

Abstract: P-adic numbers p-adic analysis non-Archimedean geometry distribution theory pseudo differential operators and spectral theory p-adic quantum mechanics and representation theory quantum field theory p-adic strings.

Quantum groups, gravity, and the generalized uncertainty principle.

Abstract: We investigate the relationship between the generalized uncertainty principle in quantum gravity and the quantum deformation of the Poincar\'e algebra. We find that a deformed Newton-Wigner position operator and the generators of spatial translations and rotations of the deformed Poincar\'e algebra obey a deformed Heisenberg algebra from which the generalized uncertainty principle follows. The result indicates that in the $\ensuremath{\kappa}$-deformed Poincar\'e algebra a minimal observable length emerges naturally.

Operads, homotopy algebra and iterated integrals for double loop spaces

Abstract: This paper provides some background to the theory of operads, used in the first author's papers on 2d topological field theory (hep-th/921204, CMP 159 (1994), 265-285; hep-th/9305013). It is intended for specialists.

Batalin-Vilkovisky algebras and two-dimensional topological field theories

Abstract: By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator Δ:A⊙→A⊙+1 such that Δ2 = 0, and [Δ,a]−Δa is a graded derivation ofA for alla∈A. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads.

Local Systems of Vertex Operators, Vertex Superalgebras and Modules

Abstract: We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of ``local system of vertex operators'' for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space $M$ has a natural vertex (super)algebra structure with $M$ as a module. Then we prove that for a vertex (operator) superalgebra $V$, giving a $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of vertex operators on $M$. As applications, we prove that certain lowest weight modules for some well-known infinite-dimensional Lie algebras or Lie superalgebras have natural vertex operator superalgebra structures. We prove the rationality of vertex operator superalgebras associated to standard modules for an affine algebra. We also give an analogue of the notion of the space of linear homomorphisms from one module to another for a Lie algebra by introducing a notion of ``generalized intertwining operators.'' We prove that $G(M^{1},M^{2})$, the space of generalized intertwining operators from one module $M^{1}$ to another module $M^{2}$ for a vertex operator superalgebra $V$, is a generalized $V$-module. Furthermore, we prove that for a fixed vertex operator superalgebra $V$ and

Hilbert Space Operators in Quantum Physics

Abstract: Contents: 1. Some Notions from Functional Analysis. 2. Hilbert Spaces. 3. Bounded Operators. 4. Unbounded Operators. 5. Spectral Theory. 6. Operators Sets. 7. Operator Algebras. 8. States and Observables. 9. Position and Momentum. 10. Time Evolution. 11. Symmetries of Quantum Systems. 12. Composite Systems. 13. Second Quantization. 14. Axiomatization of Quantum Physics. 15. Schroedinger Operators. 16. Scattering Theory. Appendix A: Measure and Integration. Appendix B: Some algebraic notions.

Geometric aspects of quantum spin states

Abstract: A number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S+1)-invariant quantum spin-S chains with the interaction −P(o), whereP(o) is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of Neel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the −P(o) spin-S systems with the Potts and the Fortuin-Kasteleyn random-cluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such structure the dichotomy is implied by a topological argument, and the alternatives correspond to whether, or not, the clusters are of finite mean length.

Noncommutative Differential Calculus on the Kappa-Minkowski Space

Abstract: Following the construction of the $\kappa$-Minkowski space from the bicrossproduct structure of the $\kappa$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $\kappa$-Minkowski space and and we briefly discuss the main properties of its differential calculi.

On Schrödinger superalgebras

Abstract: Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrodinger algebra (itself a conformal extension of the Galilei algebra) were constructed. An ‘I‐type’ extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N=N++N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin‐1/2 particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, ‘exotic’ or ‘IJ‐type’ extensions arise for each pair of integers ν+ and ν−, yielding an N=2(ν++ν−) superalgebra of the type discovered recently by Leblanc et al. in nonrelativistic Chern–Simons theory. For the magnetic monopole the symmetry reduces to o(3)×osp(1/1), and for the magnetic vortex it reduces to o(2)×osp(1/2).

The Berezin transform and invariant differential operators

Abstract: The Berezin calculus is important to quantum mechanics (creation-annihilation operators) and operator theory (Toeplitz operators). We study the basic Berezin transform (linking the contravariant and covariant symbol) for all bounded symmetric domains, and express it in terms of invariant differential operators.

A powerful approach to generate new integrable systems

Abstract: In this paper, an effective algorithm to generate integrable systems is given. As a result, many new integrable equations are derived in a systematic way.

Deformations, stable theories and fundamental constants

Abstract: Models or theories that are stable, in the sense that they do not change in a qualitative manner under a small change of parameters, have a higher probability of having a wider range of validity. This also seems to be true for the fundamental theories of nature. Using the deformation theory of algebras, we review the stabilizing deformations leading from non-relativistic to relativistic and from classical to quantum mechanics. Unlike previous treatments, both deformations are carried out on a finite-dimensional algebra setting. One then finds that the resulting relativistic quantum algebra is itself unstable and admits a two-parameter stabilizing deformation. Taking into account reasonable physical constraints to identify the deformed variables, a new algebra is then proposed as the stable algebra of relativistic quantum mechanics in tangent space. This is isomorphic to the algebra of the pseudo-Euclidian group in five dimensions.

MacDonald's polynomials and representations of quantum groups

Abstract: In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining operators between certain modules over quantum sl(n). We also describe the commutative system of Macdonald's difference operators using the generators of the center of the quantum universal enveloping algebra, and use this description to prove a trace formula for generic eigenfunctions of these operators. These functions are generalized q-hypergeometric functions which are related to solutions of the quantum Knizhnik-Zamolodchikov equations.

A von Neumann Algebra Framework for the Duality of the Quantum Groups

Abstract: Classical Lie groups are important examples in the category of locally compact groups. The general theory of unitary representations is developed for these objects as harmonic analysis, which provides us a good theoretical framework for the detailed study of the unitary representations of the classical Lie groups. This is regarded as an extension of the Fourier analysis to a general context. For a locally compact group, its dual i.e. the set of all the equivalence classes of irreducible unitary representations plays an important role, and the duality established by Pontrjagin for Abelian groups, Tannaka and Krein for compact groups, Steinspring for unimodular groups, Eymard and Tatsuuma for locally compact groups is an important theoretical basis for the harmonic analysis. On the other hand, at the formal level in the framework of pure algebras, we use the notion of Hopf algebras to deal with the algebraic groups, discrete groups, or the dual of those objects at the same time. Then functional analysis is necessarily combined with the algebraic framework of Hopf algebras to have a good control with the infinite dimensional unitary representations. This theory, especially the argument utilized by Steinspring, suggests us to introduce the notion of Kac algebras in the language of von Neumann algebras. The first take off from the group or the group algebra to the Kac algebra was considered by Kac [7] and performed by Takesaki [23] by introducing the, so-called, Kac-Takesaki operator or the fundamental operator for the semifinite i.e. the unimodular case, and then completed by Enock and Schwartz [4, 20, 5] for the general case, in which the above mentioned duality was established by Takesaki, Enock and Schwartz, and others [24, 21].

Recursion operators and non-local symmetries

Abstract: A new formulation of recursion operators is presented which eliminates diffi­culties associated with integro-differential operators. This interpretation treats recursion operators and their inverses on an equal footing. Efficient techniques for constructing non-local symmetries of differential equations result.

The basis of nonlocal curvature invariants in quantum gravity theory. Third order.

Abstract: A complete basis of nonlocal invariants in quantum gravity theory is built to third order in space–time curvature and matter‐field strengths. The nonlocal identities are obtained which reduce this basis for manifolds with dimensionality 2ω<6. The present results are used in heat‐kernel theory, theory of gauge fields and serve as a basis for the model‐independent approach to quantum gravity and, in particular, for the study of nonlocal vacuum effects in the gravitational collapse problem.

A survey of certain trace inequalities

Abstract: This paper concerns inequalities like TrA ≤ TrB, where A and B are certain Hermitian complex matrices and Tr stands for the trace. In most cases A and B will be exponential or logarithmic expressions of some other matrices. Due to the interest of the author in quantum statistical mechanics, the possible applications of the trace inequalities will be commented from time to time. Several inequalities treated below have been established in the context of Hilbert space operators or operator algebras. Notwithstanding these extensions our discussion will be limited to matrices.

Representations of a q-deformed Heisenberg algebra

Abstract: We extend the symmetric operators of theq-deformed Heisenberg algebra to essentially self-adjoint operators. On the extended domains the product of the operators is not defined. To represent the algebra we had to enlarge the representation and we find a Hilbert space representation of the deformed Heisenberg algebra in terms of essentially self-adjoint operators. The respective diagonalization can be achieved by aq-deformed Fourier transformation.

Lipshitz continuity of gap boundaries for Hofstadter-like spectra

Abstract: We consider an effective HamiltonianH representing the motion of a single-band-two-dimensional electron in a uniform magnetic field. ThenH belongs to the rotation algebra, namely the algebra of continuous functions over a non-commutative 2-torus. We define a non-commutative analog of smooth functions by mean of elements of classC l,n , wherel andn characterize respectively the degree of differentiability with respect to the magnetic field and the torus variables. We show that ifH is of classC 1,3+e , the gap boundaries of the spectrum ofH are Lipshitz continuous functions of the magnetic field at each point for which the gap is open.