Orthocomplementation and Compound Systems

In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice _sep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms” of the lab, and before any interaction takes place. In that case, the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore _sep cannot satisfy all Piron’s axioms. In previous works, assuming that _sep is orthocomplemented, it was argued that _sep is not orthomodular and fails to have the covering property. Here we prove that _sep cannot admit an orthocomplementation. Moreover, we propose a natural model for _sep which has the covering property. PACS: 03.65.Ta, 03.65.Ca     

Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice , which is a sublattice of the lattice of all quantum propositions and is determined by a quantum state e and a preferred determinate observable R. Topos-theoretic extension is made in the functor category of which base category is determined by R. Each true atom, which determines truth values, true or false, of all propositions in , generates also a multi-valued valuation function of which domain and range are and a Heyting algebra given by the subobject classifier in , respectively. All true propositions in are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category . Here, the base category includes all ’s as subcategories. Although has a structure apparently different from , a subobject semi-classifier of gives valuations completely equivalent to those in ’s.     

Why is \mathcal{CPT} Fundamental?

Lüders and Pauli proved the theorem based on Lagrangian quantum field theory almost half a century ago. Jost gave a more general proof based on “axiomatic” field theory nearly as long ago. The axiomatic point of view has two advantages over the Lagrangian one. First, the axiomatic point of view makes clear why is fundamental—because it is intimately related to Lorentz invariance. Secondly, the axiomatic proof gives a simple way to calculate the transform of any relativistic field without calculating , and separately and then multiplying them. The purpose of this pedagogical paper is to “deaxiomatize” the theorem by explaining it in a few simple steps. We use theorems of distribution theory and of several complex variables without proof to make the exposition elementary.     

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral <Emphasis Type="Italic">su</Emphasis>(2) WZNW Model

A zero modes’ Fock space is constructed for the extended chiral WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra at an even root of unity, and of its infinite dimensional extension by the Lusztig operators We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of A central result is the characterization of the Grothendieck ring of both and in Theorem 3.1. The properties of the fusion ring in are related to the braiding properties of correlation functions of primary fields of the conformal current algebra model.      

An Infinite Genus Mapping Class Group and Stable Cohomology

We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion . L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.     

Dirac-Hestenes Lagrangian

We formulate the variational principle of theDirac equation within the noncommutative even space-timesubalgebra, the Clifford -algebra . A fundamental ingredient in ourmultivectorial algebraic formulation is a -complex geometry, . We derive the Lagrangian for theDirac-Hestenes equation and show that it must be mapped on , where denotes an -algebra of functions.     

Feynman Graphs,Rooted Trees,and Ringel-Hall Algebras

We construct symmetric monoidal categories of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of , are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.     

(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads

For a (co)monad T _l on a category , an object X in , and a functor , there is a (co)simplex in . The aim of this paper is to find criteria for para-(co)cyclicity of Z ^*. Our construction is built on a distributive law of T _l with a second (co)monad T _r on , a natural transformation , and a morphism in . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads and on the category of R-bimodules. The functor Π can be chosen such that is the cyclic R-module tensor product. A natural transformation is given by the flip map and a morphism is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called  ×  _R -Hopf algebras, is introduced. In the particular example when T is a module coring of a  ×  _R -Hopf algebra and X is a stable anti-Yetter-Drinfel’d -module, the para-cyclic object Z _* is shown to project to a cyclic structure on . For a -Galois extension , a stable anti-Yetter-Drinfel’d -module T _S is constructed, such that the cyclic objects and are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups.     

Cardy Condition for Open-Closed Field Algebras

Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that , the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation on the space of intertwining operators of V. We then derive a graphical representation of S in the modular tensor category . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy -algebra. In the end, we give a categorical construction of the Cardy -algebra in the Cardy case.     

Non-relativistic Limit of a Dirac Polaron in Relativistic Quantum Electrodynamics

A quantum system of a Dirac particle interacting with the quantum radiation field is considered in the case where no external potentials exist. Then the total momentum of the system is conserved and the total Hamiltonian is unitarily equivalent to the direct integral of a family of self-adjoint operators acting in the Hilbert space , where is the Hilbert space of the quantum radiation field. The fiber operator is called the Hamiltonian of the Dirac polaron with total momentum . The main result of this paper is concerned with the non-relativistic (scaling) limit of . It is proven that the non-relativistic limit of yields a self-adjoint extension of a Hamiltonian of a polaron with spin 1/2 in non-relativistic quantum electrodynamics.     

Covariant (hh′)-Deformed Bosonic and Fermionic Algebras as Contraction Limits of q-Deformed Ones

GL_h(n) ×GL_h(m)-covariant (hh)-bosonic[or (hh)-fermionic] algebras are built in terms of thecorresponding R_h and -matrices by contracting theGL_q(n) × -covariant q-bosonic (or q-fermionic) algebras , = 1, 2.When using a basis of wherein theannihilation operators are contragredient to thecreation ones, this contraction procedure can be carried out for any n, m values. Whenemploying instead a basis wherein the annihilationoperators, like the creation ones, are irreducibletensor operators with respect to the dual quantumalgebra U_q(gl(n)) , a contraction limit only exists forn, m {1, 2, 4, 6, . . .}. For n = 2, m = 1, andn = m = 2, the resulting relations can be expressed interms of coupled (anti)commutators (as in the classical case), by usingU_h(sl(2)) [instead of s1(2)] Clebsch-Gordancoefficients. Some U_h(sl(2)) rank-1/2irreducible tensor operators recently constructed byAizawa are shown to provide a realization of (2, 1).     

Quantum Conjugacy Classes of Simple Matrix Groups

Let G be a simple complex classical group and its Lie algebra. Let be the Drinfeld-Jimbo quantization of the universal enveloping algebra . We construct an explicit -equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers. Dedicated to the memory of Joseph Donin This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center “Group Theoretic Methods in the study of Algebraic Varieties” of the Israel Science foundation, by the EPSRC grant C511166, and by the RFBR grant no. 06-01-00451.     

Quantization of Semi-Classical Twists and Noncommutative Geometry

A problem of defining the quantum analogues for semi-classical twists in U()[[t]] is considered. First, we study specialization at q = 1 of singular coboundary twists defined in Uq ())[[t]] for g being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when = and obtain q-deformation of the semiclassical twist introduced by Connes and Moscovici in noncommutative geometry. Mathematics Subject Classification: 16W30, 17B37, 81R50     

How to Describe the Space-Time Structure with Nets of C*-Algebras

The major subject of algebraic quantum fieldtheory is the study of nets of local C*-algebras, i.e.,maps ( ) assigning to each open,relatively compact region of space-time (M, g) aC*-algebra ( ), whose self-adjoint elements describe localobservables measurable in the region . A question discussed recently in a number ofpapers is how much information about the geometricstructure of the underlying space-time (M, g) is encoded in the algebraicstructure of the net ( ). Followingthese ideas, it is demonstrated in this paper howspace-time-related concepts like causality and observerscan be described in a purely algebraic way, i.e., using only thelocal algebras ( ).These results are then used to show how the space-time(M, g) can be reconstructed from the set _loc := { ( )| M open, compact} of local algebras.     

Time Flow in a Noncommutative Regime

We develop an approach to dynamical and probabilistic properties of the model unifying general relativity and quantum mechanics, initiated in the paper (Heller et al. (2005) International Journal Theoretical Physics 44, 671). We construct the von Neumann algebra of random operators on a groupoid, which now is not related to a finite group G, but is the pair groupoid of the Lorentzian frame bundle E over spacetime M. We consider the time flow on depending on the state . The state defining the noncommutative dynamics is assumed to be normal and faithful. Then the pair is a noncommutative probabilistic space and can also be interpreted as an equilibrium thermal state, satisfying the Kubo-Martin-Schwinger condition. We argue that both the “time flow” and thermodynamics have their common roots in the noncommutative unification of dynamics and probability.     

Finitary Spacetime Sheaves of Quantum Causal Sets: Curving Quantum Causality

A locally finite, causal, and quantal substitute for a locally Minkowskian principal fiber bundle of modules of Cartan differential forms over a bounded region X of a curved C -smooth spacetime manifold M with structure group G that of orthochronous Lorentz transformations L ⁺ := SO(1,3), is presented. is usually regarded as the kinematical structure of classical Lorentzian gravity when the latter is viewed as a Yang-Mills type of gauge theory of a sl(2, {})-valued connection 1-form . The mathematical structure employed to model this replacement of is a principal finitary spacetime sheaf of quantum causal sets with structure group G _n, which is a finitary version of the continuous group G of local symmetries of General Relativity, and a finitary Lie algebra g _n-valued connection 1-form on it, which is a section of its subsheaf . is physically interpreted as the dynamical field of a locally finite quantum causality, whereas its associated curvature as some sort of finitary and causal Lorentzian quantum gravity.     

Ion Mobility Measurements and Ion Chemical Reaction Studies at Heavy Elements in a Buffer Gas Cell

Drift time measurements of ions in a buffer gas cell filled with argon have been performed from which changes of the ion mobility and ionic radii for various heavy elements and their compounds were determined. The ionic radius of americium shrinks by (3.1 1.3)% with respect to that of plutonium, and an increase of the radius by (28 2)% of plutonium oxide with respect to plutonium was found. Ion chemical reactions of erbium ions were studied online in an argon buffer gas cell to which the reaction gases oxygen (O) and methane (CH) were added. The erbium ions were implanted into the buffer gas cell with an energy of 50 MeV. The online measured reaction constant = (3.2 0.4) 10 cm/(molecule s) for the reaction Er + O ErO + O agrees with a reference measurement = (3.6 0.4) 10 cm/(molecule s), performed with a Fourier-Transform-Mass-Spectrometer.