Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.

A Local (Perturbative) Construction of Observables in Gauge Theories: The Example of QED

Abstract: Interacting fields can be constructed as formal power series in the framework of causal perturbation theory. The local field algebra is obtained without performing the adiabatic limit; the (usually bad) infrared behavior plays no role. To construct the observables in gauge theories we use the Kugo–Ojima formalism; we define the BRST-transformation as a graded derivation on the algebra of interacting fields and use the implementation of by the Kugo–Ojima operator Q int. Since our treatment is local, the operator Q int differs from the corresponding operator Q of the free theory. We prove that the Hilbert space structure present in the free case is stable under perturbations. All assumptions are shown to be satisfied in QED.

Differential equations and intertwining operators

Abstract: We show that if every module W for a vertex operator algebra V = ∐n∈ℤV(n) satisfies the condition dim W/C1(W) 0 V(n) and w ∈ W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reductivity conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V-modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V-modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V-modules.

The mathematical structure of the bardeen-cooper-schrieffer model

Abstract: The BCS-model for an infinitely extended superconductor is analysed mathematically. The reason why the model is solvable becomes evident in the present formulation. It is shown that the ground states with unsharp particle number belong to irreducible representations, those with sharp particle number to reducible representations of the basic operator algebra. The connection between uniqueness of the ground state, irreducibility and linked cluster decomposition is reviewed.

Introducing Cadabra: a symbolic computer algebra system for field theory problems

Abstract: Cadabra is a new computer algebra system designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification taking care of Bianchi and Schouten identities, for fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many other field theory related concepts. The input format is a subset of TeX and thus easy to learn. Both a command-line and a graphical interface are available. The present paper is an introduction to the program using several concrete problems from gravity, supergravity and quantum field theory.