On the Tensor Product Construction for q-Differential Algebras

We show that for q -1 there is no natural tensor product for q-differential algebras. In particular, the q-graded tensor product of q-differentials fails to satisfy the q-graded Leibniz rule.     

Produits tensoriels continus d'espaces et d'algèbres de Banach

The notion of tensor product of a family (A _i ) _i I of Banach algebras is generalized to the case whenI is a topological space; in this case A _i is generated by some elements x _i , the family (x _i ) being subjected to certain conditions: for instance the functioni x _i must be continuous. This notion is applied to Quantum Field Theory in the following sense: certain algebras of observables can be considered as continuous tensor products of simpler ones, namely of algebras of observables with one degree of freedom.     

Highest weight representations of quantum current algebras

We study the highest weight and continuous tensor product representations ofq-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of theq-deformed algebra sl_q(2,) is calculated in detail.Alexander von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing, P.R. China.     

Sums and products of interval algebras

Aninterval algebra is an interval from zero to some positive element in a partially ordered Abelian group, which, under the restriction of the group operation to the interval, is a partial algebra. In this paper we study interval algebras from a categorical point of view, and show that Cartesian products and horizontal sums are effective as categorical products and coproducts, respectively. We show that the category of interval algebras admits a tensor product, and introduce a new class of interval algebras, which are in fact orthoalgebras, called-algebras.     

Why Does the Geometric Product Simplify the Equations of Physics?

In the last decades it was observed that Clifford algebras and geometric product provide a model for different physical phenomena. We propose an explanation of this observation based on the theory of bounded symmetric domains and the algebraic structure associated with them. The invariance of physical laws is a result of symmetry of the physical world that is often expressed by the symmetry of the state space for the system implying that this state space is a symmetric domain. For example, the ball of all possible velocities is a bounded symmetric domain. The symmetry on this ball follow from the symmetry of the space-time transformations between two inertial systems, which fixes the so-called symmetric velocity between them. The Lorenz transformations acts on the ball Sof symmetric velocities by conformal transformations. The ball Sis a spin ball (type IV in Cartan's classification). The Lie algebra of this ball is defined a triple product that is closely related to geometric product. The relativistic dynamic equations in mechanics and for the Lorenz force is described by this Lie algebra and the triple product.     

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the Formality conjecture), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.     

Yangians,integrable quantum systems and Dorey's rule

It was pointed out by P. Dorey that the three-point couplings between the quantum particles in affine Toda field theories have a remarkable Lie-theoretic interpretation. It is also well known that such theories admit quantum affine algebras as quantum symmetry groups, and widely believed that the quantum particles correspond to the so-called fundamental representations of these algebras. This led to the conjecture that Dorey's rule should describe when a fundamental representation occurs with non-zero multiplicity in a tensor product of two other fundamental representations. The purpose of this paper is to prove this conjecture, both for quantum affine algebras and for Yangians. The result reveals a hitherto unsuspected role played by Coxeter elements (and their twisted analogues) in the representation theory of these algebras.     

Pentagon Equation Arising from State Equations of a C*-Bialgebra

The direct sum \({{\mathcal O}_{*}}\) of all Cuntz algebras has a non-cocommutative comultiplication \({\Delta_{\varphi}}\) such that there exists no antipode of any dense subbialgebra of the C*-bialgebra \({({\mathcal O}_{*},\Delta_{\varphi})}\). From states equations of \({{\mathcal O}_{*}}\) with respect to the tensor product, we construct an operator W for \({({\mathcal O}_{*},\Delta_{\varphi})}\) such that W* is an isometry, \({W(x\otimes I)W^{*}=\Delta_{\varphi}(x)}\) for each \({x\in {\mathcal O}_{*}}\) and W satisfies the pentagon equation.     

Some Properties of the Bel and Bel-Robinson Tensors

The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum. We present some new properties of these tensors which might throw some light onto this relationship. First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the pure matter super-energy tensor and the matter-gravity coupling super-energy tensor. We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the square of the usual energy-momentum tensor. This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its square gives the Bel-Robinson tensor. We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented. The properties of this new tensor are examined and some interesting explicit examples are analyzed. Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves. We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution. Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types.     

Curvature tensors unified field equations on SEXn

We study the curvature tensors and field equations in then-dimensional SE manifold SEX_n. We obtain several basic properties of the vectorsS andU and then of the SE curvature tensor and its contractions, such as a generalized Ricci identity, a generalized Bianchi identity, and two variations of the Bianchi identity satisfied by the SE Einstein tensor. Finally, a system of field equations is discussed in SEX_n and one of its particular solutions is constructed and displayed.     

Markov chains with exponentially small transition probabilities: First exit problem from a general domain. II. The general case

In this paper we consider aperiodic ergodic Markov chains with transition probabilities exponentially small in a large parameter . We extend to the general, not necessarily reversible case the analysis, started in part I of this work, of the first exit problem from a general domainQ containing many stable equilibria (attracting equilibrium points for the = dynamics). In particular we describe the tube of typical trajectories during the first excursion outsideQ.     

From Effect Algebras to Algebras of Effects (Sum Brouwer—Zadeh Algebras)

The algebraic structures arising in the axiomatic framework of unsharp quantummechanics based on effect operators on a Hilbert space are investigated. It isstressed that usually considered effect algebras neglect the unitary Brouwerianmap of complementation, and the main results based on this complementationare collected, showing the enrichment produced into the theory by its introduction.In particular, in these structures two notions of sharpness can be considered: K-sharpness induced by the usual complementation of effect algebrasand B-sharpness induced by this new complementation. Quantum (resp., classical) SBZalgebras are then characterized by the condition of B-coherence (resp., B-coherence plusB-compatibility), showing that in this case the poset of all B-sharp elements is orthomodular (resp., Boolean algebra). In the unsharp contextof effect operators, the finite dimensionality of the Hilbert space or the finitenessof a von Neumann algebra are both characterized by a de Morgan property ofthe Brouwer complementation. Moreover, since effect operators on a pre-Hilbertspace give rise to a standard model of effect algebras, a characterization ofcompleteness of pre-Hilbert spaces is given making use of the Brouwercomplement.     

Exchange interaction of weakly delocalized multielectron systems

The polar model of a metal developed by Bogolyubov [1] is extended to the case of a group of equivalent electrons on each of the atoms that form a crystal. A procedure is suggested for treating the second-quantized Hamiltonian that includes the interconfigurational interaction with the help of the double tensor operators W _q . The relationship between these operators and the unitary operators ^a , which act separately in the space of the spin and orbital functions, is pointed out. The conditions for the applicability of the Heisenberg model are analyzed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 80–87, May, 1973.In conclusion I wish to thank B. V. Karpenko for the statement of this problem and for numerous consultations and discussions.     

Full Field Algebras

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras V ^L and V ^R , is naturally a full field algebra and we introduce a notion of full field algebra over . We study the structure of full field algebras over using modules and intertwining operators for V ^L and V ^R . For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over and an invariant bilinear form on this algebra.     

Spinor Representations of $${U}_q (\hat {\mathfrak{o}}(N))$$ )

This Letter concerns an extension of the quantum spinor construction of . We define quantum affine Clifford algebras based on the tensor category and the solutions of q-KZ equations, and construct quantum spinor representations of .     

The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes

We consider the Klein-Gordon equation (m0) on the double Schwarzschild wedge of the Kruskal spacetime, and construct the Hartle-Hawking state _H as a thermal state relative to the Boulware quantization. We prove that, on the double wedge, _H is a pure state, and in the corresponding representation, the left- and right-wedgeC* algebras each have the Reeh-Schlieder property, while the corresponding von-Neumann algebras are typeIII ₁ factors which are dual to (i.e. commutants of) each other. We discuss the extent to which these properties may generalize to non-quasi-free field theories.Pursuing the Rindler-Fulling-Unruh analogy with the Klein-Gordon equation (m>0) in (d-dimensional) flat spacetime, we establish an explicit formula for the Minkowski vacuum on a spacelike double wedge as a thermal state relative to the Fulling quantization. We also treat the cased=2,m=0 of this formula since this is essential input for a paper with Dimock on scattering theory for the quantum Klein-Gordon equation on the Schwarzschild metric.Research supported in part by the Schweizerischer Nationalfonds     

W-Algebra W(2, 2) and the Vertex Operator Algebra $${L(\frac{1}{2},\,0)\,\otimes\, L(\frac{1}{2},\,0)}$$

In this paper the W-algebra W(2, 2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to an irreducible highest weight W(2, 2)- module or a tensor product of two simple Virasoro vertex operator algebras. Furthermore, we show that any rational, C ₂-cofinite and simple vertex operator algebra whose weight 1 subspace is zero, weight 2 subspace is 2-dimensional and with central charge c = 1 is isomorphic to . Supported by NSF grants and a research grant from the Committee on Research, UC Santa Cruz.     

Gravitational field strength and generalized Komar integral

We define a gravitational field strength in theories of the Einstein-Cartan type admitting a Killing vector. This field strength is a second rank, antisymmetric, divergence-free tensor, whose (Komar) integral over a closed 2-surface gives a physically meaningful quantity. We find conditions on the Lagrange density of the theory which ensure the existence of such a tensor, and show that they are satisfied forN=2-supergravity and for a special case of the bosonic sector ofN=4-supergravity. We discuss a possible application of the generalized Komar integral in the theory of stationary black holes. We also consider the field strength problem in Kaluza-Klein theory, where the application to black holes is particularly interesting.     

The nonsymmetric Kaluza-Klein (Jordan-Thiry) theory in the electromagnetic case

We present the nonsymmetric Kaluza-Klein and Jordan-Thiry theories as interesting propositions of physics in higher dimensions. We consider the five-dimensional (electromagnetic) case. The work is devoted to a five-dimensional unification of the NGT (nonsymmetric theory of gravitation), electromagnetism, and scalar forces in a Jordan-Thiry manner. We find interference effects between gravitational and electromagnetic fields which appear to be due to the skew-symmetric part of the metric. Our unification, called the nonsymmetric Jordan-Thiry theory, becomes the classical Jordan-Thiry theory if the skew-symmetric part of the metric is zero. It becomes the classical Kaluza-Klein theory if the scalar field=1 (Kaluza's Ansatz). We also deal with material sources in the nonsymmetric Kaluza-Klein theory for the electromagnetic case. We consider phenomenological sources with a nonzero fermion current, a nonzero electric current, and a nonzero spin density tensor. From the Palatini variational principle we find equations for the gravitational and electromagnetic fields. We also consider the geodetic equations in the theory and the equation of motion for charged test particles. We consider some numerical predictions of the nonsymmetric Kaluza-Klein theory with nonzero (and with zero) material sources. We prove that they do not contradict any experimental data for the solar system and on the surface of a neutron star. We deal also with spin sources in the nonsymmetric Kaluza-Klein theory. We find an exact, static, spherically symmetric solution in the nonsymmetric Kaluza-Klein theory in the electromagnetic case. This solution has the remarkable property of describing mass without mass and charge without charge. We examine its properties and a physical interpretation. We consider a linear version of the theory, finding the electromagnetic Lagrangian up to the second order of approximation with respect toh _v =g _v –n _v . We prove that in the zeroth and first orders of approximation there is no skewonoton interaction. We deal also with the Lagrangian for the scalar field (connected to the gravitational constant). We prove that in the zeroth and first orders of approximation the Lagrangian vanishes.     

Renormalizability and infrared finiteness of non-linear σ-models: A regularization-independent analysis for compact coset spaces

The non-linear models in two space-time dimensions corresponding to compact homogeneous coset spacesG/H are studied with particular attention to three problems: first, independence of coordinate choice and regularization, second, the physical content of the theory, and finally the regularity of the physics in the infrared limit. Concerning in particular the physical content of the theory, we construct a set of local observables whose correlation functions depend on a finite number of parameters identified among those defining the metric tensor of the coset space. For these models, we give a general proof of renormalizability based on the introduction of a nilpotent BRS operator which describes the non-linear isometries and a classical action which contains a mass term for all quantized fields. The mass term belongs to a finite dimensional representation of the groupG, which allows us to prove the conjecture that the correlation functions of local observables, i.e., the local operators invariant underG, are regular in the infrared limit.