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Operator algebra
About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.
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TL;DR: Within this modeling framework one can express data clustering models, logic programs, ordinary and stochastic differential equations, branching processes, graph grammars, and stochy chemical reaction kinetics, which makes the framework particularly suitable for applications in machine learning and multiscale scientific modeling.
Abstract: We define a class of probabilistic models in terms of an operator algebra of stochastic processes, and a representation for this class in terms of stochastic parameterized grammars. A syntactic specification of a grammar is formally mapped to semantics given in terms of a ring of operators, so that composition of grammars corresponds to operator addition or multiplication. The operators are generators for the time-evolution of stochastic processes. The dynamical evolution occurs in continuous time but is related to a corresponding discrete-time dynamics. An expansion of the exponential of such time-evolution operators can be used to derive a variety of simulation algorithms. Within this modeling framework one can express data clustering models, logic programs, ordinary and stochastic differential equations, branching processes, graph grammars, and stochastic chemical reaction kinetics. The mathematical formulation connects these apparently distant fields to one another and to mathematical methods from quantum field theory and operator algebra. Such broad expressiveness makes the framework particularly suitable for applications in machine learning and multiscale scientific modeling.
39 citations
TL;DR: In this article, the twisted vertex operator equations were applied to obtain a set of twisted KZ equations for the (outer-automorphic) charge conjugation orbifolds on.
Abstract: Recently the operator algebra and twisted vertex operator equations were given for each sector of all WZW orbifolds, and a set of twisted KZ equations for the WZW permutation orbifolds were worked out as a large example. In this companion paper we report two further large examples of this development. In the first example we solve the twisted vertex operator equations in an Abelian limit to obtain the twisted vertex operators and correlators of a large class of Abelian orbifolds. In the second example, the twisted vertex operator equations are applied to obtain a set of twisted KZ equations for the (outer-automorphic) charge conjugation orbifold on .
39 citations
01 Jan 1985
TL;DR: The problem of quantum field theory has attracted the attention of both mathematicians and physicists over a period of several decades as discussed by the authors, and the most striking achievements were the calculation in the late 1940's and early 1950's of the Lamb shift and the anomalous magnetic moment of the electron together with the development of the renormalization method on which these calculations were based.
Abstract: Quantum fields, from a mathematical point of view, are highly singular. These fields are believed to describe the interactions of elementary particles. For the interaction of electrons with light (photons), the quantum field description is exact within the limits of experimental accuracy (5 significant figures). For these reasons, i.e. the mathematical difficulties and the importance to physics, the problem of formulating the mathematical foundations of quantum field theory has attracted the attention of both mathematicians and physicists over a period of several decades. On the side of the physicists, the most striking achievements were the calculation in the late 1940’s and early 1950’s of the Lamb shift and the anomalous magnetic moment of the electron together with the development of the renormalization method on which these calculations were based. Of the mathematicians, J. von Neumann was the first to realize that new mathematical theories would be required to formulate quantum field theory correctly and this realization was one of the motives for developing the theory of operator algebras.
39 citations
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TL;DR: A survey of Hopf cyclic cohomology and its application to transverse index theory on foliations is given in this article, with a detailed description of the Hopf algebra that controls the ''affine' transverse geometry of codimension $n$ foliations.
Abstract: This is primarily a survey of the way in which Hopf cyclic cohomology has emerged and evolved, in close relationship with the application of the noncommutative local index formula to transverse index theory on foliations. Being Diff-invariant, the geometric framework that allowed us to treat the `space of leaves' of a general foliation provides a `background independent' set-up for geometry that could be of relevance to the handling of the the background independence problem in quantum gravity. With this potential association in mind, we have added some new material, which complements the original paper and is also meant to facilitate its understanding. Section 2 gives a detailed description of the Hopf algebra that controls the `affine' transverse geometry of codimension $n$ foliations, and Section 5 treats the relative version of Hopf cyclic cohomology in full generality, including the case of Hopf pairs with noncompact isotropy.
39 citations
TL;DR: In this paper, it was shown that every norm-continuous generalized derivable mapping at zero point on finite CSL algebras is a generalized derivation, where the unit operator is a unit operator.
38 citations