Role of Dilations in Diffeomorphism-Covariant Algebraic Quantum Field Theory

A generalization of algebraic quantum field theory on differentiable manifoldsis given in terms of nets of *-algebras over open sets of the manifold. The presentinvestigations are motivated by diffeomorphism invariance and finite localizationas they appear, e.g., in quantum gravity. A possible generalization of Haag-Kastleraxioms for differentiable manifolds is discussed and a minimal framework basedon isotony, covariance, and a state-dependent GNS construction is presented.Possible adaptions of Haag's commutant duality are discussed in a specific settingof one-parameter families of finite and nondegenerate nested localization domainsof the net, with universal minimal and maximal algebras for the small and largelimits of the net, respectively. For von Neumann algebras the modular group isdiscussed. The geometric interpretation of a one-parameter subgroup of outerisomorphisms is related to dilations of the open sets of the net.     

Superselection Theory for Subsystems

An inclusion of observable nets satisfying duality induces an inclusion of canonical field nets. Any Bose net intermediate between the observable net and the field net and satisfying duality is the fixed-point net of the field net under a compact group. This compact group is its canonical gauge group if the occurrence of sectors with infinite statistics can be ruled out for the observable net and its vacuum Hilbert space is separable. Received: 23 December 1999 / Accepted: 25 November 2000     

Comparison Theory and Smooth Minimal C*-Dynamics

We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar’s Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff’s uncountable family of II₁ factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras. This research was supported in part by an NSERC Discovery Grant.     

Effect Algebras with the Riesz Decomposition Property and AF C*-Algebras

Relations between effect algebras with Riesz decomposition properties and AF C^*-algebras are studied. The well-known one-one correspondence between countable MV-algebras and unital AF C^*-algebras whose Murray-von Neumann order is a lattice is extended to any unital AF C^* algebras and some more general effect algebras having the Riesz decomposition property. One-one correspondence between tracial states on AF C^*-algebras and states on the corresponding effect algebras is proved. In particular, pure (faithful) tracial states correspond to extremal (faithful) states on corresponding effect algebras.     

The Picard Groupoid in Deformation Quantization

In this Letter we give an overview on recent developments in representation theory of star product algebras. In particular, we relate the *-representation theory of *-algebras over rings C = R(i) with an ordered ring R and i²=?1 to the *-representation theory of *-algebras over and point out some properties of the Picard groupoid corresponding to the notion of strong Morita equivalence. Some Morita invariants are interpreted as arising from actions of this groupoid     

The Picard Groupoid in Deformation Quantization

In this Letter we give an overview on recent developments in representation theory of star product algebras. In particular, we relate the *-representation theory of *-algebras over rings C = R(i) with an ordered ring R and i²=–1 to the *-representation theory of *-algebras over and point out some properties of the Picard groupoid corresponding to the notion of strong Morita equivalence. Some Morita invariants are interpreted as arising from actions of this groupoid     

Representations of Nets of C*-Algebras over S 1

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C^*-algebras over S ¹ admits faithful representations, and when the net is covariant under Diff(S ¹), it admits representations covariant under any amenable subgroup of Diff(S ¹).     

Noncommutative Finite Dimensional Manifolds II: Moduli Space and Structure of Noncommutative 3-Spheres

This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebra to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show that the above morphism corresponds to a natural ramified covering by a noncommutative 3-dimensional nilmanifold. We then compute the Jacobian of the ramified covering and obtain the answer as the product of a period (of an elliptic integral) by a rational function. We describe the real and complex moduli spaces of noncommutative 3-spheres, relate the real one to root systems and the complex one to the orbits of a birational cubic automorphism of three dimensional projective space. We classify the algebras and establish duality relations between them.     

States on Partial Rings

Recently, the structure theory of JB*-tripleshas received considerable attention. The reason is thatJB*-triples and those JB*-triples which are dual spaces,the JBW*-triples, not only form natural generalizations of Jordan C*-algebras and C*-algebras, andJordan W*-algebras and W*-algebras, but also provide acontext for the study of infinite-dimensional holomorphyand infinite-dimensional Lie algebras. In a JBW*-triple the tripotents play the role of the projectionsin a W*-algebra. In analogy to the projection lattice ofa W*-algebra, we investigate the partial ring oftripotents of JBW*-triple. Unlike on W*-algebras, states, i.e., positive normalized homomorphismsfrom the partial ring of tripotents of a JBW*-tripleinto the partial ring of real numbers, have not yet beendiscussed in the literature. We show that the partial ring of tripotents of a JBW*-tripleadmits a unital set of Jauch-Piron states.     

A Family of Examples with Quantum Constraints

In Rep. Math. Phys. 35 (1995), 101, the authors describe a method for constructing directly (i.e. without using explicitly any field operator nor any concrete representation of the C*-algebra) nets of local C*-algebras associated to massless models with arbitrary helicity and that satisfy Haag–Kastler's axioms. In order to specify the sesquilinear and the symplectic form of the CAR- and CCR-algebras, respectively, a certain operator-valued function is introduced. This function is shown to be very useful in proving the covariance and causality of the net and it also codes the degenerate character of massless models with respect to massive models.It is the intention of this Letter to point out that the massless bosonic examples with helicity bigger than 0 fit completely into the general theory that Grundling and Hurst used to describe systems with gauge degeneracy.     

On Definition of Skew Frames

Skew frames represent a common generalization of frames and orthomodular lattices. They could serve as Lindenbaum algebras of quantum intuitionistic logic as well as invariants of noncommutative C*-algebras. It is shown that lattices of open projections with skew (partial) operations are complete invariants of C*-algebras and that these operations are preserved by morphims of C*-algebras.     

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.     

Self-adjoint algebras of unbounded operators

Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.Work supported in part by U.S. Atomic Energy Commission under Contract AT(30-1)-2171 and by the National Science Foundation.Alfred P. Sloan Foundation Fellow.     

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.     

Yangians and finite W-algebras

Using an algebra morphism between Yangians and some finite W-algebras, we determine the (product of) finite dimensional representations of the latters. This demonstrates a correspondence between evaluation representations of Yangians and Miura representations of W-algebras.     

States of physical systems

States of physical systems may be represented by states onB*-algebras, satisfying certain requirements of physical origin. We discuss such requirements as are associated with the presence of unbounded observables or invariance under a group. It is possible in certain cases to obtain a unique decomposition of states invariant under a group into extremal invariant states. Our main results is such a decomposition theorem when the group is the translation group in dimensions and theB*-algebra satisfies a certain locality condition. An application of this theorem is made to representations of the canonical anticommutation relations.     

The Formulation of Quantum Mechanics in Terms of Nuclear Algebras

In this work we analyze the convenience ofnuclear barreled b*-algebras as a better mathematicalframework for the formulation of quantum principles thanthe usual algebraic formalism in terms of C*-algebras. Unbounded operators on Hilbert spaces have anabstract counterpart in our approach. The main resultsof the C*-algebra theory remain valid. We demonstrate anextremal decomposition theorem, an adequate functional representation theorem, and anextension of the classical GNS theorem.     

Approximations of Strongly Continuous Families of Unbounded Self-Adjoint Operators

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor \({{\mathfrak{A}}}\) : SLoc\({\to}\)S*Alg to the category of super-*-algebras, which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors \({{\mathfrak{e}\mathfrak{A}}}\) : eSLoc\({\to}\)eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess–Zumino model in 3|2-dimensions.     

Quantum Teardrops

Algebras of functions on quantum weighted projective spaces are introduced, and the structure of quantum weighted projective lines or quantum teardrops is described in detail. In particular the presentation of the coordinate algebra of the quantum teardrop in terms of generators and relations and classification of irreducible *-representations are derived. The algebras are then analysed from the point of view of Hopf-Galois theory or the theory of quantum principal bundles. Fredholm modules and associated traces are constructed. C*-algebras of continuous functions on quantum weighted projective lines are described and their K-groups computed.