λ→0 limit of 2+1 quantum gravity for arbitrary genus

The abstract quantum algebra of observables for 2+1 gravity is analysed in the limit of small cosmological constant. The algebra splits into two sets with an explicit phase space representation; one set consists of 6g–6commuting elements which form a basis for an algebraic manifold defined by the trace and rank identities; the other set consists of 6g–6 tangent vectors to this manifold. The action of the quantum mapping class group leaves the algebra and algebraic manifold invariant. The previously presented representation forg=2 is analysed in this limit and reduced to a very simple form. The symplectic form forg=2 is computed.     

2+1 gravity for genus >1

In [1] we analysed the algebra of observables for the simple case of a genus 1 initial data surface ² for 2+1 De Sitter gravity. Here we extend the analysis to higher genus. We construct for genus 2 the group of automorphismsH of the homotopy group ₁ induced by the mapping class group. The groupH induces a groupD of canonical transformations on the algebra of observables which is related to the braid group for 6 threads.     

Conformal field algebras with quantum symmetry from the theory of superselection sectors

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.     

On Lie groups with left-invariant symplectic or Kählerian structures

Left-invariant symplectic structure on a group G; properties of the corresponding Lie algebra g. A unimodular symplectic Lie algebra has to be solvable (see [1]). Symplectic subgroups and left-invariant Poisson structures on a group. Affine Poisson structures: an affine Poisson structure associated to g and admitting g ^* as a unique leaf corresponds to a unimodular symplectic Lie algebra and the associate group is right-affine. If G is unimodular and endowed with a left-invariant metric g, harmonic theory for the left-invariant forms. Kählerian group is metabelian and Riemannianly flat. Decomposition of a simply connected Kählerian group. A symplectic group admitting a left-invariant metric with a nonnegative Ricci curvature is unimodular and admits a left-invariant flat Kählerian structure.     

The Bohm-Aharonov effect: A seven-dimensional structural group

We realize a nonfaithful representation of a seven-dimensional Lie algebra, the extension of which to its universal enveloping algebra contains most of the observables of the scattering Aharonov-Bohm effect, as essentially self-adjoint operators: the scattering Hamiltonian, the total and kinetic angular momenta, the positions and the kinetic momenta. By restriction, we obtain the model introduced in Lett. Math. Phys. 1 (1976), 155–163.     

A Generalization of Entanglement to Convex Operational Theories: Entanglement Relative to a Subspace of Observables

We define what it means for a state in a convex cone of states on a space of observables to be generalized-entangled relative to a subspace of the observables, in a general ordered linear spaces framework for operational theories. This extends the notion of ordinary entanglement in quantum information theory to a much more general framework. Some important special cases are described, in which the distinguished observables are subspaces of the observables of a quantum system, leading to results like the identification of generalized unentangled states with Lie-group-theoretic coherent states when the special observables form an irreducibly represented Lie algebra. Some open problems, including that of generalizing the semigroup of local operations with classical communication to the convex cones case, are discussed. PACS: 03.65.Ud.     

Non-trivial extension of the (1+2)-Poincaré algebra and conformal invariance on the boundary of {\mathrm{AdS}}_3

Using recent results on strings on AdS$_3\times N^d$, where N is a d dimensional compact manifold, we re-examine the derivation of the non-trivial extension of the (1+2)-dimensional-Poincaré algebra obtained by Rausch de Traubenberg and Slupinsky. We show by explicit computation that this new extension is a special kind of fractional supersymmetric algebra which may be derived from the deformation of the conformal structure living on the boundary of AdS. The two so(1,2) Lorentz modules of spin used in building of the generalization of the (1+2) Poincaré algebra are re-interpreted in our analysis as highest weight representations of the left and right Virasoro symmetries on the boundary of AdS. We also complete known results on 2d-fractional supersymmetry by using spectral flow of affine Kac–Moody and superconformal symmetries. Finally we make preliminary comments on the trick of introducing Fth roots of g-modules to generalize the so(1,2) result to higher rank Lie algebras g. Received: 20 July 2000 / Published online: 19 September 2001     

Noncommutative Differential Forms and Quantization of the Odd Symplectic Category

There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for all functions f and g. We notice that this noncommutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relations) in terms of ₂-graded chain complexes. It is a straightforward consequence of the theory of BV operator acting on semidensities, due to H. Khudaverdian.     

Categorified Symplectic Geometry and the Classical String

A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.     

Constrained current algebras and g/u(1) conformal field theories

Operator quantization of the WZNW theory invariant with respect to an affine Lie algebra with a constrained subalgebra is performed using Dirac's procedure. Upon quantization the initial energy-momentum tensor is replaced by the g/u(1)^d coset construction. The WZNW theory with a constrained current is equivalent to the su(2)/u(1) conformal field theory.     

Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields

The superselection sectors of two classes of scalar bilocal quantum fields in D ≥ 4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D−2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory.     

Kochen–Specker Theorem for von Neumann Algebras

The Kochen–Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type I_n factor as algebra of observables, including I_∞. Afterwards, we give a proof of the Kochen–Specker theorem for an arbitrary von Neumann algebra without summands of types I₁ and I₂, using a known result on two-valued measures on the projection lattice . Some connections with presheaf formulations as proposed by Isham and Butterfield are made.     

Grafting and Poisson Structure in (2+1)-Gravity with Vanishing Cosmological Constant

We relate the geometrical construction of (2+1)-spacetimes via grafting to phase space and Poisson structure in the Chern-Simons formulation of (2+1)-dimensional gravity with vanishing cosmological constant on manifolds of topology , where S _g is an orientable two-surface of genus g>1. We show how grafting along simple closed geodesics λ is implemented in the Chern-Simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on S _g .We prove that this action is generated via the Poisson bracket by a gauge invariant observable associated to the holonomy of λ. We deduce a symmetry relation between the Poisson brackets of observables associated to the Lorentz and translational components of the holonomies of general closed curves on S _g and discuss its physical interpretation. Finally, we relate the action of grafting on the phase space to the action of Dehn twists and show that grafting can be viewed as a Dehn twist with a formal parameter θ satisfying θ² = 0.     

The Nambu-Goto theory of closed bosonic strings moving in 1 + 3–dimensional Minkowski space: the construction of the quantum algebra of observables up to degree five

The quantum algebra of observables of closed bosonic strings moving in 1 + 3–dimensional Minkowski space is constructed up to degree five. All independent relations of degree four are computed; they involve three as yet undetermined parameters. Definitions and symbols are used as introduced in the above-mentioned article.     

Remarks on Effect Algebras

Erik M. Alfsen and Frederic W. Shultz had recently developed the characterisation of state spaces of operator algebras. It established full equivalence (in the mathematical sense) between the Heisenberg and the Schr?dinger picture, i.e. given a physical system we are able to construct its state space out of its observables as well as to construct algebra of observables from its state space. As an underlying mathematical structure they used the theory of duality of ordered linear spaces and obtained results are valid for various types of operator algebras (namely C ^*, von Neumann, JB and JBW algebras). Here, we show that the language they developed also admits a representation of an effect algebra.     

Hilbert space representation of an algebra of observables forq-deformed relativistic quantum mechanics

Using a representation of theq-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the massp ² is diagonal.     

Distortion of the Poisson Bracket by the Noncommutative Planck Constants

In this paper we introduce a kind of “noncommutative neighbourhood” of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a “distortion” of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow.     

On G-Trace and G-Index in Deformation Quantization

For a compact group G of symplectomorphisms we define a G-trace on the algebra of quantum observables by postulating its properties. We give an explicit construction of such a trace and prove a G-index theorem similar to the Atiyah–Segal–Singer equivariant index theorem for elliptic operators.     

Prospects for discovering supersymmetry at the LHC

Supersymmetry is one of the best-motivated candidates for physics beyond the standard model that might be discovered at the LHC. There are many reasons to expect that it may appear at the TeV scale, in particular because it provides a natural cold dark-matter candidate. The apparent discrepancy between the experimental measurement of g _μ −2 and the standard-model value calculated using low-energy e ⁺ e ⁻ data favours relatively light sparticles, accessible to the LHC. A global likelihood analysis including this, other electroweak precision observables and B decay observables suggests that the LHC might be able to discover supersymmetry with 1/fb or less of integrated luminosity. The LHC should be able to discover supersymmetry via the classic missing-energy signature, or in alternative phenomenological scenarios. The prospects for discovering supersymmetry at the LHC look very good. CERN-PH-TH/2008-208.