Representations of MV-Algebras by Hilbert-Space Effects

It is shown that for every Archimedean MV-effect algebra M (equivalently, every Archimedean MV-algebra) there is an injective MV-algebra morphism into the MV-algebra of all multiplication operators between the zero and identity operator on $\ell_{2}(\mathcal{S}_{0})$ , where $\mathcal{S}_{0}$ is an ordering set of extremal states (state morphisms) on M.     

Haag Duality and the Distal Split Property for Cones in the Toric Code

We prove that Haag duality holds for cones in the toric code model. That is, for a cone ??, the algebra ${\mathcal{R}_{\Lambda}}$ of observables localized in ?? and the algebra ${\mathcal{R}_{\Lambda^c}}$ of observables localized in the complement ?? ^c generate each other??s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if ${\Lambda_1 \subset \Lambda_2}$ are two cones whose boundaries are well separated, there is a Type I factor ${\mathcal{N}}$ such that ${\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}$ . We demonstrate this by explicitly constructing ${\mathcal{N}}$ .     

Courant–Dorfman Algebras and their Cohomology

We introduce a new type of algebra, the Courant–Dorfman algebra. These are to Courant algebroids what Lie–Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant–Dorfman algebra ${(\mathcal{R}, \mathcal{E})}$ we associate a differential graded algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ in a functorial way by means of explicit formulas. We describe two canonical filtrations on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ , and derive an analogue of the Cartan relations for derivations of ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; we classify central extensions of ${\mathcal{E}}$ in terms of ${H^2(\mathcal{E}, \mathcal{R})}$ and study the canonical cocycle ${\Theta \in \mathcal{C}^3(\mathcal{E}, \mathcal{R})}$ whose class ${[\Theta]}$ obstructs re-scalings of the Courant–Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; for Courant–Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.     

Smooth Crossed Products of Rieffel’s Deformations

Assume ${\mathcal{A}}$ is a Fréchet algebra equipped with a smooth isometric action of a vector group V, and consider Rieffel’s deformation ${\mathcal{A}_J}$ of ${\mathcal{A}}$ . We construct an explicit isomorphism between the smooth crossed products ${V\ltimes\mathcal{A}_J}$ and ${V\ltimes\mathcal{A}}$ . When combined with the Elliott–Natsume–Nest isomorphism, this immediately implies that the periodic cyclic cohomology is invariant under deformation. Specializing to the case of smooth subalgebras of C^*-algebras, we also get a simple proof of equivalence of Rieffel’s and Kasprzak’s approaches to deformation.     

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z, z  =  0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z  =  0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ .     

Some Properties of Generalized Quantum Operations

Let $\mathcal{B}(\mathcal{H})$ be the set of all bounded linear operators on the separable Hilbert space  $\mathcal{H}$ . A (generalized) quantum operation is a bounded linear operator defined on  $\mathcal{B}(\mathcal{H})$ , which has the form $\varPhi_{\mathcal{A}}(X)=\sum_{i=1}^{\infty}A_{i}XA_{i}^{*}$ , where $A_{i}\in\mathcal{B}(\mathcal{H})$ (i=1,2,…) satisfy $\sum_{i=1}^{\infty}A_{i}A_{i}^{*}\leq \nobreak I$ in the strong operator topology. In this paper, we establish the relationship between the (generalized) quantum operation $\varPhi_{\mathcal{A}}$ and its dual $\varPhi_{\mathcal {A}}^{\dag}$ with respect to the set of fixed points and the noiseless subspace. In particular, we also partially characterize the extreme points of the set of all (generalized) quantum operations and give some equivalent conditions for the correctable quantum channel.     

Optimal Paths for Symmetric Actions in the Unitary Group

Given a positive and unitarily invariant Lagrangian ${\mathcal{L}}$ defined in the algebra of matrices, and a fixed time interval ${[0,t_0]\subset\mathbb R}$ , we study the action defined in the Lie group of ${n\times n}$ unitary matrices ${\mathcal{U}(n)}$ by $$\mathcal{S}(\alpha)=\int_0^{t_0} \mathcal{L}(\dot\alpha(t))\,dt, $$ where ${\alpha:[0,t_0]\to\mathcal{U}(n)}$ is a rectifiable curve. We prove that the one-parameter subgroups of ${\mathcal{U}(n)}$ are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if ${\mathcal{L}}$ is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in ${\mathcal{U}(n)}$ as well as angular metrics in the Grassmann manifold.     

Joint System Quantum Descriptions Arising from Local Quantumness

We study the entropy flux in the stationary state of a finite one-dimensional sample ${\mathcal{S}}$ connected at its left and right ends to two infinitely extended reservoirs ${\mathcal{R}_{l/r}}$ at distinct (inverse) temperatures ${\beta_{l/r}}$ and chemical potentials ${\mu_{l/r}}$ . The sample is a free lattice Fermi gas confined to a box [0, L] with energy operator ${h_{\mathcal{S}, L}= - \Delta + v}$ . The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system ${\mathcal{R}_l + \mathcal{S} + \mathcal{R}_r}$ in terms of scattering data. We study the behaviour of this steady state entropy flux in the limit ${L \to \infty}$ and relate persistence of transport to norm bounds on the transfer matrices of the limiting half-line Schrödinger operator ${h_\mathcal{S}}$ .     

On Gravitational Effects in the Schrödinger Equation

The Schrödinger  equation for a particle of rest mass $m$ and electrical charge $ne$ interacting with a four-vector potential $A_i$ can be derived as the non-relativistic limit of the Klein–Gordon  equation $\left( \Box '+m^2\right) \varPsi =0$ for the wave function $\varPsi $ , where $\Box '=\eta ^{jk}\partial '_j\partial '_k$ and $\partial '_j=\partial _j -\mathrm {i}n e A_j$ , or equivalently from the one-dimensional  action $S_1=-\int m ds +\int neA_i dx^i$ for the corresponding point particle in the semi-classical approximation $\varPsi \sim \exp {(\mathrm {i}S_1)}$ , both methods yielding the equation $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi $ in Minkowski  space–time  , where $\alpha ,\beta =1,2,3$ and $\phi =-A_0$ . We show that these two methods generally yield equations  that differ in a curved background  space–time   $g_{ij}$ , although they coincide when $g_{0\alpha }=0$ if $m$ is replaced by the effective mass $\mathcal{M}\equiv \sqrt{m^2-\xi R}$ in both the Klein–Gordon  action $S$ and $S_1$ , allowing for non-minimal coupling to the gravitational  field, where $R$ is the Ricci scalar and $\xi $ is a constant. In this case $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi $ , where $\phi ^{(\mathrm g)} =\sqrt{g_{00}}$ and $\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} $ , the correctness of the gravitational  contribution to the potential having been verified to linear order $m\phi ^{(\mathrm g)} $ in the thermal-neutron beam interferometry experiment due to Colella et al. Setting $n=2$ and regarding $\varPsi $ as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time  coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div ${{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0$ , where ${{\varvec{A}}}^{\alpha }=-A^{\alpha }$ and ${{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }$ . The quantum-cosmological Schrödinger  (Wheeler–DeWitt) equation is also discussed in the $\mathcal{D}$ -dimensional  mini-superspace idealization, with particular regard to the vacuum potential $\mathcal V$ and the characteristics of the ground state, assuming a gravitational  Lagrangian   $L_\mathcal{D}$ which contains higher-derivative  terms up to order $\mathcal{R}^4$ . For the heterotic superstring theory  , $L_\mathcal{D}$ consists of an infinite series in $\alpha '\mathcal{R}$ , where $\alpha '$ is the Regge slope parameter, and in the perturbative approximation $\alpha '|\mathcal{R}| \ll 1$ , $\mathcal V$ is positive semi-definite for $\mathcal{D} \ge 4$ . The maximally symmetric ground state satisfying the field equations is Minkowski  space for $3\le {\mathcal {D}}\le 7$ and anti-de Sitter  space for $8 \le \mathcal {D} \le 10$ .     

Finite W-Superalgebras and Truncated Super Yangians

Let ${Y_{m|n}^{\ell}}$ be the super Yangian of general linear Lie superalgebra for ${\mathfrak{gl}_{m|n}}$ . Let ${e \in \mathfrak{gl}_{m\ell|n\ell}}$ be a “rectangular” nilpotent element and ${\mathcal{W}_e}$ be the finite W-superalgebra associated to e. We show that ${Y_{m|n}^{\ell}}$ is isomorphic to ${\mathcal{W}_e}$ .     

Inclusion relations for the spaces L(ℝ), \mathcal{H}\mathcal{K} (ℝ) ∩ \mathcal{B}\mathcal{V} (ℝ), and L 2(ℝ)

The inclusion relations for the spaces $ \mathcal{H}\mathcal{K} $ (I), L(I), $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I), and L ²(I) are found. On unbounded intervals, functions in $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I) need not be Lebesgue integrable.     

How to Add a Boundary Condition

Given a conformal QFT local net of von Neumann algebras ${\mathcal {B}_2}$ on the two-dimensional Minkowski spacetime with irreducible subnet ${\mathcal {A} \otimes \mathcal {A}}$ , where ${\mathcal {A}}$ is a completely rational net on the left/right light-ray, we show how to consistently add a boundary to ${\mathcal {B}_2}$ : we provide a procedure to construct a Boundary CFT net ${\mathcal {B}}$ of von Neumann algebras on the half-plane x >  0, associated with ${\mathcal {A}}$ , and locally isomorphic to ${\mathcal {B}_2}$ . All such locally isomorphic Boundary CFT nets arise in this way. There are only finitely many locally isomorphic Boundary CFT nets and we get them all together. In essence, we show how to directly redefine the C* representation of the restriction of ${\mathcal {B}_2}$ to the half-plane by means of subfactors and local conformal nets of von Neumann algebras on S ¹.     

Complex Vector Formalism of Harmonic Oscillator in Geometric Algebra: Particle Mass,Spin and Dynamics in Complex Vector Space

Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown that the particle spin plays an important role in the kinematical intrinsic or local motion of the particle. From the complex vector formalism of harmonic oscillator, for the first time, a relation between mass $m$ and bivector spin $S$ has been derived in the form $\varvec{\sigma }_3 mc^2{\mathcal {J}}_{\pm } =\lambda \Omega _{\mathbf{s}} \cdot \mathrm{{S}} {\mathcal {J}}_{\pm }$ . Where, $\Omega _{s}$ is the angular velocity bivector of complex rotations, $c$ is the velocity of light. The unit vector $\varvec{\sigma }_3$ acts as an operator on the idempotents ${\mathcal {J}}_{+}$ and ${\mathcal {J}}_{-}$ to give the eigen values $\lambda =\pm 1.$ The constant $\lambda $ represents two fold nature of the equation corresponding to particle and antiparticle states. Further the above relation shows that the mass of the particle may be interpreted as a local spatial complex rotation in the rest frame. This gives an insight into the nature of fundamental particles. When a particle is observed from an arbitrary frame of reference, it has been shown that the spatial complex rotation dictates the relativistic particle motion. The mathematical structure of complex vectors in space and spacetime is developed.     

Analysis of the {3\over 2}^{+} heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the ${3\over 2}^{+}$ heavy and doubly heavy baryon states $\varXi^{*}_{cc}$ , $\varOmega^{*}_{cc}$ , $\varXi^{*}_{bb}$ , $\varOmega^{*}_{bb}$ , $\varSigma_{c}^{*}$ , $\varXi_{c}^{*}$ , $\varOmega_{c}^{*}$ , $\varSigma_{b}^{*}$ , $\varXi_{b}^{*}$ and $\varOmega_{b}^{*}$ by subtracting the contributions from the corresponding ${3\over 2}^{-}$ heavy and doubly heavy baryon states with the QCD sum rules, and we make reasonable predictions for their masses.     

Large N approach to Kaon decays and mixing 28 years later: \Delta I=1/2 rule, \hat{B}_K,and \Delta M_K

We review and update our results for $K\rightarrow \pi \pi $ decays and $K^0$ – $\bar{K}^0$ mixing obtained by us in the 1980s within an analytic approximate approach based on the dual representation of QCD as a theory of weakly interacting mesons for large $N$ , where $N$ is the number of colors. In our analytic approach the Standard Model dynamics behind the enhancement of $\hbox {Re}A_0$ and suppression of $\hbox {Re}A_2$ , the so-called $\Delta I=1/2$ rule for $K\rightarrow \pi \pi $ decays, has a simple structure: the usual octet enhancement through the long but slow quark–gluon renormalization group evolution down to the scales $\mathcal{O}(1\, {\hbox { GeV}})$ is continued as a short but fast meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark–gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark–gluon evolution. While this physical picture did not yet emerge from lattice simulations, the recent results on $\hbox {Re}A_2$ and $\hbox {Re}A_0$ from the RBC-UKQCD collaboration give support for its correctness. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. Though the current–current operators dominate the $\Delta I=1/2$ rule, working with matching scales $\mathcal{O}(1 \, {\hbox { GeV}})$ we find that the presence of QCD-penguin operator $Q_6$ is required to obtain satisfactory result for $\hbox {Re}A_0$ . At NLO in $1/N$ we obtain $R=\hbox {Re}A_0/\hbox {Re}A_2= 16.0\pm 1.5$ which amounts to an order of magnitude enhancement over the strict large $N$ limit value $\sqrt{2}$ . We also update our results for the parameter $\hat{B}_K$ , finding $\hat{B}_K=0.73\pm 0.02$ . The smallness of $1/N$ corrections to the large $N$ value $\hat{B}_K=3/4$ results within our approach from an approximate cancelation between pseudoscalar and vector meson one-loop contributions. We also summarize the status of $\Delta M_K$ in this approach.     

Coset Constructions of Logarithmic (1, p) Models

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p =  2, 3, . . .) of central charge c _{1, p} =1 ? 6(p ? 1)²/p. This family includes the theories corresponding to the singlet algebras ${\mathcal{M}(p)}$ and the triplet algebras ${\mathcal{W}(p)}$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The ${W^{(2)}_n}$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of ${\widehat{\mathfrak{sl}}(n)_k}$ , generalising the Bershadsky–Polyakov algebra ${W^{(2)}_3}$ . Inspired by work of Adamovi? for p = 3, vertex algebras ${\mathcal{B}_p}$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra ${\mathcal{B}_p}$ is a quotient of ${W^{(2)}_{p-1}}$ at level ?(p ? 1)²/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra ${\mathcal{W}(p)}$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra ${\mathcal{M}(p)}$ is similarly realised inside ${\mathcal{B}_p}$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p =  2 and 3.     

Annihilation-type charmless radiative decays of B meson in non-universal Z′ model

We study charmless pure annihilation type radiative B decays within the QCD factorization approach. After adding the vertex corrections to the naive factorization approach, we find that the branching ratios of $\overline{B}^{0}_{d}\to\phi\gamma$ , $\overline{B}^{0}_{s}\to\rho^{0}\gamma$ and $\overline{B}^{0}_{s}\to\omega\gamma$ within the standard model are at the order of $\mathcal{O}(10^{-12})$ , $\mathcal{O}(10^{-10})$ and $\mathcal{O}(10^{-11})$ , respectively. The smallness of these decays in the standard model makes them sensitive probes of flavor physics beyond the standard model. To explore their physics potential, we have estimated the contribution of Z′ boson in the decays. Within the allowed parameter space, the branching ratios of these decay modes can be enhanced remarkably in the non-universal Z′ model: The branching ratios can reach to $\mathcal{O}(10^{-8})$ for $\overline{B}_{s}^{0}\to \rho^{0}(\omega)\gamma$ and $\mathcal{O}(10^{-10})$ for the $\overline{B}_{d}^{0}\to \phi \gamma$ , which are large enough for LHC-b and/or Super B-factories to detect those channels in near future. Moreover, we also predict large CP asymmetries in suitable parameter space. The observation of these modes could in turn help us to constrain the Z′ mass within the model.     

Equivariant GW Theory of Stacky Curves

We extend Okounkov and Pandharipande’s work on the equivariant Gromov–Witten theory of ${\mathbb{P}^1}$ to a class of stacky curves ${\mathcal{X}}$ . Our main result uses virtual localization and the orbifold ELSV formula to express the tau function ${\tau_\mathcal{X}}$ as a vacuum expectation on a Fock space. As corollaries, we prove the decomposition conjecture for these ${\mathcal{X}}$ , and prove that ${\tau_\mathcal{X}}$ satisfies a version of the 2-Toda hierarchy. Coupled with degeneration techniques, the result should lead to treatment of general orbifold curves.