Quantization and representation theory of finiteW algebras

In this paper we study the finitely generated algebras underlyingW algebras. These so called finiteW algebras are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings ofsl ₂ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finiteW algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finiteW symmetry. In the second part we BRST quantize the finiteW algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finiteW algebras in one stroke. Examples are given. In the last part of the paper we study the representation theory of finiteW algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finiteW algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finiteW algebras.     

Remarks on bicovariant differential calculi and exterior Hopf algebras

We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.     

Remarks on spectra of modular operators of von Neumann algebras

It is shown that if is an invariant state of an asymptotically abelianC* algebra , then the spectrum of modular operator for is contained in the spectrum of any other modular operator for the von Neumann algebra .It is also shown that a modular operator can not have an isolated spectrum with a finite multiplicity at 1 unless the associated Hilbert space is of finite dimension. It is further shown that if a modular operator has an isolated spectrum with a finite multiplicity atx 1, then the von Neumann algebra is a direct sum of ₁ and ₂ where ₁ is represented on a finite dimensional Hilbert space and the modular operator for ₂ does not have its spectrum atx.Applications to Connes invariant are given.On leave from Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan.     

Remarks on A-entropy

We study three possibilities of defining the entropy of a density matrix with respect to an observable and show that most of the properties of the usual entropy remain valid cum grano salis.     

Remarks on the exoticU-meson

In expectation of imminent result from the new hyperon beam experiment at CERN concerning the exoticU-meson at 3.1 GeV, we propose a detailed program of experimental tests to check the suggestion thatU is a \(qq\bar q\bar q\) “M-diquonium” state. Apart from some very characteristic decay modes, theU is expected to occur together with several analogous states with various quantum numbers to which it is intimately related.     

Remarks on superconformal bosonization

An alternative bosonization of the superconformal ghosts of the NSR superstring is presented. Unlike the standard approach, the new framework is shown to have BRST invariant scattering amplitudes in both the large and small algebras. The effect on picture changing is discussed. It is also shown that the new approach is required in order to obtain a manifestly supersymmetric bosonization.     

Remarks on Higgs inflation

We discuss models where the Higgs boson of the electroweak standard model plays the role of the inflaton. We focus on the question of the violation of perturbative unitarity due to the coupling of the Higgs boson either to the Ricci scalar or to the Einstein tensor and discuss the background dependence of the unitarity bounds. Our conclusion is that the simplest model which restricts itself to the standard model Higgs boson without introducing further degrees of freedom has a serious problem. However, in the asymptotically safe gravity scenario, the Higgs boson of the standard model could be the inflaton and no physics beyond the standard model is required to explain both inflation and the spontaneous breaking of the electroweak symmetry of the standard model.     

Remarks on Symplectic Connections

This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to build examples of Ricci-flat connections. For a more extensive survey, see Bieliavsky et al. [Int. J. Geom. Methods Mod. Phys. 3, 375–420 2006]. This note also includes a moment map for the action of the group of symplectomorphisms on the space of symplectic connections, an algebraic construction of a large class of Ricci-flat symmetric symplectic spaces, and an example of global reduction in a non-symmetric case.     

Remarks on nonlinear electrodynamics

Dark energy models with a slowly rolling cosmological scalar field provide a popular alternative to the standard, time-independent cosmological constant model. We study the simultaneous evolution of background expansion and growth in the scalar field model with the Ratra–Peebles self-interaction potential. We use recent measurements of the linear growth rate and the baryon acoustic oscillation peak positions to constrain the model parameter \(\alpha \) that describes the steepness of the scalar field potential.     

Remarks on quantum groups

We give a Poisson-bracket realization of SL _q (2) in the phase space ². We then discuss the physical meaning of such a realization in terms of a modified (regularized) toy model, the nonregularized version of which is due to Klauder.Some general remarks and suggestions are also presented in this Letter.     

Remarks on halo nuclei

On the ground of a relationship between the rms radius and the separation energy, we compare halo nuclei to diffuse diatomics. It underlines the essential difference between these two kinds of weakly bound systems: whereas the two-body approximation seems well justified in the case of diatomics, it becomes questionable in the nuclear case when the separation energy approaches zero. Because of this particular situation, we conjecture that the Efimov states have less chances to be observed in nuclear than in molecular cases. Discussing possibilities of measuring accurately the rms radius of halo wave function, we propose a strategy based on the parallel momentum distribution measured in dissociation experiments, together with the use of an Abel transform.     

Remarks on quantum symmetry

We discuss the concept of Quantum Symmetry in quantum field theory, and in particular the role of the gauge principle. We present a scheme how quantum symmetries can be realized in a Hilbert space, and sketch its construction from the theory of superselection sectors of the gauge invariant (observable) quantities. The approach is independent of (Drinfeld's) quantum groups.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

Remarks on conformal invariance

The existence of unitary representations for the special Conformal Group, is investigated for free fields in any dimension, and the connection between the correct transformation properties of the fields and weak conformal invariance pointed out.Partially supported by the Brazilian National Research Council, and K.F.A. Jülich, Germany.On leave of absence from University of Bielefeld, Bielefeld (Germany).     

Remarks on relativistic thermodynamics

The relativistic transformation equations for the pressure, volume, energy, and momentum of a gas in a closed container are discussed. Considerations are given in support of pressure noninvariance, volume dilatation, and the fact that the energy and momentum of a gas form a four-vector. It is shown that the relativistic transformation formula for temperature resulting from the Lorentz invariance requirement of the equation of state of an ideal gas coincides with the Ott formula.     

Remarks on nonrenormalizable interactions

It is argued that certain quantum field theories, including conventional nonrenormalizable theories, do not reduce to free theories when the interaction is turned off. The limiting, “pseudo-free” theories are formally characterized and suggested as alternative “unperturbed” theories possibly suitable for a meaningful perturbation analysis.     

Remarks on Two-Slit Probabilities

The probability pattern emerging in two-slit experiments is a typical quantum feature whose essential ingredients are examined by translating them into the spin- formalism. In view of the existence of extensions of quantum theory preserving some classical structure, we discuss how the two-slit probabilities behave under such extensions. We consider a generalization of the standard classical probability theory, to be called operational probability theory, that turns out to host the so called quantum probabilities.