Explicit form and convergence of 1-differential formal deformations of the poisson Lie algebra

Introducing the notion of an admissible graded Lie subalgebra A of the Nijenhui-Richardson algebra A(V) of the vector space V, it is shown that each cohomology class of a subcomplex C _A of the Chevalley-Eilenberg complex (C ⁰ M), extends in a cononical way as a graded cohomology class of weight — 1 of A. Applying this when V is the space N of smooth functions of a smooth manifold M, shows that the de Rham cohomology of M is induced by the graded cohomology of weight — 1 of the Schouten graded Lie algebra of M. This allows us to construct explicitly all 1-differential, nc formal deformations of the Poisson bracket of a symplectic manifold. The construction also applies for an arbitrary Poisson manifold but leads to only part of these deformations when the structure degenerates, as shown by an example.     

A Note on the Koszul Complex in Deformation Quantization

The aim of this short note is to present a proof of the existence of an A _∞-quasi-isomorphism between the A _∞-S(V ^*)-ù(V){\wedge(V)} -bimodule K, introduced in Calaque et al. (Bimodules and branes in deformation quantization, 2009), and the Koszul complex K(V) of S(V ^*), viewed as an A _∞-S(V ^*)-ù(V){\wedge(V)} -bimodule, for V a finite-dimensional (complex or real) vector space.     

Sequential Product and Jordan Product of Quantum Effects

The quantum effects for a physical system can be described by the set E(H)\mathcal{E(H)} of positive operators on a complex Hilbert space H\mathcal{H} that are bounded above by the identity operator I. We denote the set of sharp effects by P(H){\mathcal{P(H) }}. For A,B ? E(H)A,B\in\mathcal{E(H)}, the operation of sequential product A°B=A^\frac12BA^\frac12A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}} was proposed as a model for sequential quantum measurements. Denote by A*B=\fracAB+BA2A\ast B=\frac{AB+BA}{2} the Jordan product of A,B ? E(H)A,B\in\mathcal{E(H)}. The main purpose of this note is to study some of the algebraic properties of the Jordan product of effects. Many of our results show that algebraic conditions on A∗B imply that A and B commute for the usual operator product. And there are many common properties between Jordan product and sequential product of effects. For example, if A∗B satisfies certain associative laws, then AB=BA. Moreover, A*B ? P(H)A\ast B\in{\mathcal{P(H) }} if and only if A°B ? P(H)A\circ B\in{\mathcal{P(H)}}.     

Reichenbach's common cause principle and quantum field theory

Reichenbach's principles of a probabilistic common cause of probabilistic correlations is formulated in terms of relativistic quantum field theory, and the problem is raised whether correlations in relativistic quantum field theory between events represented by projections in local observable algebrasA(V₁) andA(V₂) pertaining to spacelike separated spacetime regions V₁ and V₂ can be explained by finding a probabilistic common cause of the correlation in Reichenbach's sense. While this problem remains open, it is shown that if all superluminal correlations predicted by the vacuum state between events inA(V₁) andA(V₂) have a genuinely probabilistic common cause, then the local algebrasA(V₁) andA(V₂) must be statistically independent in the sense of C^*-independence.     

A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Let λ _i (Ω,V) be the i ^th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^nLet λ _i (Ω,V) be the i ^th eigenvalue of the Schr?dinger operator with Dirichlet boundary conditions on a bounded domain and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V _*, we prove that λ₂(Ω,V) ≤ λ₂(S ₁,V _*). Here S ₁ denotes the ball, centered at the origin, that satisfies the condition λ₁(Ω,V)=λ₁(S ₁,V _*).Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ₂(B _R , V) / λ₁(B _R , V) decreases when the radius R of the ball B _R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.R.B. was supported by FONDECYT project # 102-0844.H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT.     

Instantons on the Quantum 4-pheres S4q

We introduce noncommutative algebras A _q of quantum 4-spheres S ⁴ _q , with q∈ℝ, defined via a suspension of the quantum group SU _q (2), and a quantum instanton bundle described by a selfadjoint idempotent e∈ Mat₄(A _q ), e ²=e=e ^*. Contrary to what happens for the classical case or for the noncommutative instanton constructed in [8], the first Chern–Connes class ch ₁(e) does not vanish thus signaling a dimension drop. The second Chern–Connes class ch ₂(e) does not vanish as well and the couple (ch ₁(e), ch ₂(e) defines a cycle in the (b,B) bicomplex of cyclic homology. Received: 12 December 2000 / Accepted: 10 March 2001     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

Extensions of Lieb’s Concavity Theorem

The operator function (A,B)→ Trf(A,B)(K ^*)K, defined in pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. We obtain, as a special case, a new proof of Lieb’s concavity theorem for the function (A,B)→ TrA ^p K ^* B ^q K, where p and q are non-negative numbers with sum p+q ≤ 1. In addition, we prove concavity of the operator function

Gleason's Theorem for Rectangular JBW-Triples

A JBW^*-triple B is said to be rectangular if there exists a W^*-algebra A and a pair (p,q) of centrally equivalent elements of the complete orthomodular lattice P(A)\mathcal{P}(A) of projections in A such that B is isomorphic to the JBW^*-triple pAq. Any weak^*-closed injective operator space provides an example of a rectangular JBW^*-triple. The principal order ideal CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} of the complete ^*-lattice CP(A)\mathcal{C}\mathcal{P}(A) of centrally equivalent pairs of projections in a W^*-algebra A, generated by (p,q), forms a complete lattice that is order isomorphic to the complete latticeI(B)\mathcal{I}(B) of weak^*-closed inner ideals in B and to the complete lattice S(B)\mathcal{S}(B) of structural projections on B. Although not itself, in general, orthomodular, CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} possesses a complementation that allows for definitions of orthogonality, centre, and central orthogonality to be given. A less familiar notion in lattice theory, that is well-known in the theory of Jordan algebras and Jordan triple systems, is that of rigid collinearity of a pair (e₂,f²) and (e₂,f²) of elements of CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)}. This is defined and characterized in terms of properties of P(A)\mathcal{P}(A). A W^*-algebra A is sometimes thought of as providing a model for a statistical physical system. In this case B, or, equivalently, pAq, may be thought of as providing a model for a fixed sub-system of that represented by A. Therefore, CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} may be considered to represent the set consisting of a particular kind of sub-system of that represented by pAq. Central orthogonality and rigid collinearity of pairs of elements of CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} may be regarded as representing two different types of disjointness, the former, classical disjointness, and the latter, decoherence, of the two sub-systems. It is therefore natural to consider bounded measures m on CP(A)_(p,q)\mathcal{C}\mathcal{P}(A)_{(p,q)} that are additive on centrally orthogonal and rigidly collinear pairs of elements. Using results of J.D.M. Wright, it is shown that, provided that neither of the two hereditary sub-W^*-algebras pAp and qAq of A has a weak^*-closed ideal of Type I², such measures are precisely those that are the restrictions of bounded sesquilinear functionals {_m on pAp 2 qAq with the property that the action of the centroid Z(B) of B commutes with the adjoint operation. When B is a complex Hilbert space of dimension greater than two, this result reduces to Gleason's Theorem.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Bottomonium γ(5<Emphasis Type="Italic">S</Emphasis>) decays into <Emphasis Type="Italic">BB</Emphasis> and <Emphasis Type="Italic">BB</Emphasis>π

Two-and three-body decays of γ(5S) into BB, BB*, B*B*, B _s B _s , B _s B _s ^*, and BB*π, B*B*π are evaluated using the theory developed earlier for dipion-bottomonium transitions. The theory contains only two parameters—vertex masses M _br and M _ω—known from the dipion spectra and width. Predicted values of Γ_tot(5S) and six partial widths Γ _k (5S), k = BB, BB*, ... are in agreement with the experiment. The decay widths Γ_5S (πBB*) and Γ_5S (πB*B*) are also calculated and found to be on the order of 10 keV. The text was submitted by the authors in English.     

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL _q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c+[`(c)]=26{c+\bar{c}=26}. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL _q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.     

Hadronic weak decays of heavy mesons and nonfactorization

The parametersχ _1,2, which measure nonfactorizable soft gluon contributions to hadronic weak decays of mesons, are updated by extracting them from the data ofD, B→PP, VP decays (P: pseudoscalar meson,V: vector meson). It is found thatχ ₂ ranges from −0.36 to −0.60 in the decays fromD→Kπ toD ⁺→φπ ⁺,D→K*π, while it is of order 10% with a positive sign inB→ψK, Dπ, D*π, Dρ decays. Therefore, the effective parametera ₂ is process dependent in charm decay, whereas it stays fairly stable inB decay. This is in accordance with the picture that nonfactorizable soft gluon effects become stronger when the relative momentum of the decay particles becomes smaller. As forD, B→VV decays, the presence of nonfactorizable terms in general prevents a possible definition of effectivea ₁ anda ₂. This is reinforced by the observation of a large longitudinal polarization fraction inB→ψK* decay, implying nonfactorizable effects contributing differently toS-, P- andD-wave amplitudes. We found thatA ₁ ^nf /A ₁>0>A ₂ ^nf /A ₂,V ^nf/V (nf standing for nonfactorization) forB→ψK* decay and 0>A ₁ ^nf /A ₁>A ₂ ^nf /A ₂,V ^nf/V forD→K*ρ decay. A measurement of longitudinally and transversely polarized decay rates Γ _L and Γ _T in color-suppressed decay modesB ⁰→D*⁰ ρ ⁰,D*⁰ ω andD ⁺→φρ ⁺ is urged.     

Asymptotic Properties of Solutions to 3-Particle Schrödinger Equations

We construct a generalized Fourier transformation ℱ(λ) associated with the 3-body Schr?dinger operator H=−Δ+Σ ^a V ^a (x ^a ) and characterize all solutions of (H−λ)u= 0 in the Agmon–H?rmander space ℬ^* as the image of ℱ(λ)^*. These stationary solutions admit asymptotic expansions in ℬ^* in terms of spherical waves associated with scattering channels. Received: 20 September 2000 / Accepted: 20 May 2001     

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ?^{( n )} of observables “up to n loops”, where ?⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. Received: 9 February 2000 / Accepted: 21 March 2000     

Vortex pinning in YBa2Cu3O7?x films

Evidence that pinning on linear or planar defects dominates the vortex dynamics in YBa₂Cu₃O_7−x (YBCO) films is provided by complex impedance measurements at temperature 8 K<T<T _c and magnetic field 0<B<6 T. Below the vortex lattice melting transition B_g(T) but above a threshold field B_p≈8(1-T/T _c ) T, the inductance of vortices increases as B², much less rapidly than predicted for collective pinning of vortices by point defects. Above the vortex melting line, critical scaling persists over the region B_g(T<B<B^*(T) where the vortex correlation length ξ exceeds a characteristic length scale ξ^*≡ξ(B=B^*)≈450?. The value of ξ^* is not sensitive to Al-doping in the Cu sites in the lattice and is close to the size of twin domains in the film. The nature of the observed crossovers is discussed in terms of available theoretical models for a glass-liquid transition at B_g.     

Associativity of Quantum Multiplication

We proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold V for which c ₁(A)>1 for any effective class . The same proof also works for any positive compact symplectic manifold with c ₁(A)>1. Received: 17 November 1994 / Accepted: 8 May 1997     

Quantum Invariants

In earlier work, we derived an expression for a partition function ?^(λ), and gave a set of analytic hypotheses under which ?^(λ) does not depend on a parameter λ. The proof that ?^(λ) is invariant involved entire cyclic cohomology and K-theory. Here we give a direct proof that . The considerations apply to non-commutative geometry, to super-symmetric quantum theory, to string theory, and to generalizations of these theories to underlying quantum spaces. Received: 12 January 1998 / Accepted: 1 May 1999     

Noncommutative lattices as finite approximations

Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper we discuss an approximation scheme due to Sorkin (1991) which correctly reproduces important topological aspects of continuum physics. The approximating topological spaces are partially ordered sets (posets), the partial order encoding the topology. Now, the topology of a manifold M can be reconstructed from the commutativè C^*algebra C(M) of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordinary quantum physics on M. The latter also serves to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C^*-algebra A. This fact makes any poset a genuine ‘noncommutative’ (‘quantum’) space, in the sense that the algebra of its ‘continuous functions’ is a noncommutative C^*-algebra. We therefore also have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing quantum physics using A.