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.     

The CPT Group of the Dirac Field

Using the standard representation of the Dirac equation, we show that, up to signs, there exist only two sets of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T), which give the transformation of fields , and , where and . These sets are given by , , and , , . Then , and two successive applications of the parity transformation to fermion fields necessarily amount to a 2 rotation. Each of these sets generates a non abelian group of 16 elements, respectively, and , which are non isomorphic subgroups of the Dirac algebra, which, being a Clifford algebra, gives a geometric nature to the generators, in particular to charge conjugation. It turns out that and , where is the dihedral group of eight elements, the group of symmetries of the square, and 16E is a non trivial extension of by , isomorphic to a semidirect product of these groups; S₆ and S₈ are the symmetric groups of six and eight elements. The matrices are also given in the Weyl representation, suitable for taking the massless limit, and in the Majorana representation, describing self-conjugate fields. Instead, the quantum operators C, P and T, acting on the Hilbert space, generate a unique group , which we call the CPT group of the Dirac field. This group, however, is compatible only with the second of the above two matrix solutions, namely with , which is then called the matrix CPT group. It turns out that , where is the dicyclic group of 8 elements and S₁₀ is the symmetric group of 10 elements. Since , the quaternion group, and , the 0-sphere, then .     

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.     

Universal R-Matrix for Esoteric Quantum Groups

The universal R-matrix for a class of esoteric (nonstandard) quantum groups q(gl(2N+1)) is constructed as a twisting of the universal R-matrix S of the Drinfeld–Nimbo quantum algebras. The main part of the twisting cocycle is chosen to be the canonical element of an appropriate pair of separated Hopf subalgebras (quantized Borel's (N) q (gl(2N+1))), providing the factorization property of . As a result, the esoteric quantum group generators can be expressed in terms of Drinfeld and Jimbo.     

Preferred Invariant Symplectic Connections on Compact Coadjoint Orbits

Let be the Haag--Kastler net generated by the (2) chiral current algebra at level 1. We classify the SL(2, )-covariant subsystems by showing that they are all fixed points nets ^H for some subgroup H of the gauge automorphisms group SO(3) of . Then, using the fact that the net ₁ generated by the (1) chiral current can be regarded as a subsystem of , we classify the subsystems of ₁. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem .     

On the Product of Real Spectral Triples

The product of two real spectral triples and , the first of which is necessarily even, was defined by A.Connes as given by and, in the even-even case, by . Generically it is assumed that the real structure obeys the relations , , , where the -sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' >-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this -sign table. In this Letter, we propose an alternative definition of the product real structure such that the -sign table is also satisfied by the product.     

Braided Clifford Algebras as Quantum Deformations

We present a general algebraic framework for the study of quantum/braided Clifford algebras. We allow that the quadratic form g on the base vector space takes values from a noncommutative algebra . Clifford algebra is understood as a Chevalley—Kähler deformation of the braided exterior algebra built from V, , and the initial braid operator : . The new product is canonically associated to g, , and , and it is constructed by applying Rota's and Stein Cliffordization.     

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).     

Symmetries and Modular Intersections of Von Neumann Algebras

Let be von Neumann algebras acting on a Hilbert space and let be a common cyclic and separating vector. We say that have the modular intersection property with respect to if(1) -half-sided modular inclusions,(2) (If (1) holds the strong limit exists.) We show that under these conditions the modular groups of and generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2, )/Z ₂ generated by modular groups.     

A Construction of Representations and Quantum Homogeneous Spaces

A simplified construction of representations is presented for the quantized enveloping algebra q ( ), with being a simple complex Lie algebra belonging to one of the four principal series A\ell, B\ell, C\ell or D\ell. The carrier representation space is the quantized algebra of polynomials in antiholomorphic coordinate functions on the big cell of a coadjoint orbit of K where K is the compact simple Lie group with the Lie algebra – the compact form of .     

Chains of twists and induced deformations

Chains of extended twists are composed of factors . The set of Jordanian twists { } can be applied to the initial Hopf algebra . In this case the remaining (transformed) factors of the chain can serve as extensions for such a multijordanian twist. We study the properties of these generalized extensions and the spectra of deformations of the corresponding Heisenberg-like algebras. The results are explicitly demonstrated for the case when .     

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If , and is a finite (nonabelian) group, then is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of . We characterize when MCA are group endomorphisms of , and show that MCA on inherit a natural structure theory from the structure of . We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.     

Energy Levels of Interacting Fields in a Box

We study the influence of boundary conditions on energy levels of interacting fields in a box and discuss some consequences when we hange the size of the box. In order to do this we calculate the energy levels of bound states of a scalar massive field nteracting with another scalar field through the Lagrangian = > in a one-dimensional box on which we impose Dirichlet boundary conditions. We find that the gap between the bound states changes with the size of the box in a nontrivial way. For the case where the masses of the two fields are equal and for large box the energy levels of Dashen-Hasslacher-Neveu (DHN model) are recovered and we have a kind of boson condensate for the ground state. Below a critical box size the ground-state level splits, which we interpret as particle-antiparticle production under small perturbations of box size. Below other critical sizes, and , of the box, the ground state and firstexcited state merge in the continuum part of the spectrum.     

Gauge Theories: Geometry and Cohomological Invariants

We develop a geometrical structure of themanifolds and associated, respectively, withgauge symmetry and BRST symmetry. Then, we show that , where is the group of BRST transformations, is endowed with the structureof a principal fiber bundle over the base manifold. Furthermore, in this geometricalsetup, due to the nilpotency of the BRST operator, weprove that the effective action of a gauge theory is aBRST-exact term up to the classical action. Then, weconclude that the effective action where only the gaugesymmetry is fixed is cohomologically equivalent to the action where the gauge and the BRSTsymmetries are fixed.     

Hopf Algebras in Analysis

We present a construction of the Hopf algebra °, dual of the polynomials in one variable, that uses rational functions. This construction illustrates how basic concepts of the theory of bialgebras can be used in analysis. We describe several spaces of interest in analysis that are isomorphic to °. Some of the results presented here were motivated by problems posed by Rota in 1994.     

Vertex Operators for Boundary Algebras

We construct embeddings of boundary algebras into ZF algebras . Since it is known that these algebras are the relevant ones for the study of quantum integrable systems (with boundaries for and without for ), this connection allows to make the link between different approaches of the systems with boundaries. The construction uses the well-bred vertex operators built recently, and is classified by reflection matrices. It relies only on the existence of an R-matrix obeying a unitarity condition, and as such can be applied to any infinite dimensional quantum group.     

Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

Fractional Spin Through Quantum Affine Algebras with Vanishing Central Charge

We study the fractional decomposition of the quantum enveloping affine algebras and with vanishing central charge in the limit . This decomposition is based on the bosonic representation and can be related to fractional supersymmetry and k-fermionic spin. The quantum affine algebras and the classical ones are equivalent in the fermionic realization.     

On the Algebra of Arrow Diagrams

The purpose of this Letter is to develop further the Gauss diagram approach to finite-type link invariants. In this context we introduce Gauss diagrams counterparts to chord diagrams, 4T relation, weight systems arising from Lie algebras, and an algebra of unitrivalent graphs modulo the STU relation. The counterparts, respectively, are arrow diagrams, 6T relation, weights arising from the solutions of the classical Yang–Baxter equation, and an algebra of acyclic arrow graphs (modulo an oriented version of the STU relation). The algebra encodes, in a graphical form, the main properties of Lie bialgebras, similarly to the well-known relation of the algebra of unitrivalent graphs to Lie algebras. We show that the oriented and relations hold, and that is isomorphic to the algebra of arrow diagrams. As an application, we consider an appropriate link-homotopy version of the algebra . Using this algebra, we construct a Gauss diagram invariants of string links up to link-homotopy, with values both in the algebra and in . We observe that this construction gives the universal Milnor's link-homotopy -invariants.