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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: In this paper, it was shown that there are infinitely many DSR constructions of the energy momentum sector, each of which can be promoted to the Hopf algebra, and the co-product of this algebra and the Heisenberg double construction of κ-deformed phase space in order to derive the non-commutative space-time structure and the description of the whole of DSR phase space.
Abstract: Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there are infinitely many DSR constructions of the energy–momentum sector, each of whose can be promoted to the κ-Poincare quantum (Hopf) algebra. Then we use the co-product of this algebra and the Heisenberg double construction of κ-deformed phase space in order to derive the non-commutative space–time structure and the description of the whole of DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space–time of the DSR theory is unique and related to the theory with non-commutative space–time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space–time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space–time, its intrinsic length par...
240 citations
TL;DR: In this article, an extension of Wick's theorem to matrix elements of an arbitrary operator between two different quasi-particle vacuums is derived, which is useful for calculations which go beyond the variational Hartree-Fock-Bogoliubov methods.
Abstract: Linear transformations are considered, which preserve the (anti-) commutation rules, but not the Hermiticity relation, for (fermion) boson creation and annihilation operators; these transformations lead to Fock space representations on biorthogonal bases of the operator algebra. As an application, an extension of Wick’s theorem to matrix elements of an arbitrary operator between two different quasi-particle vacuums is derived. This theorem is useful for calculations which go beyond the variational Hartree-Fock-Bogoliubov methods (H.F.B. with projection, generator co-ordinate method, etc.). A canonical decomposition for Bogoliubov transformations is established, which proves useful, for instance in the calculation of the overlap of two different quasi-particle vacuums.
235 citations
TL;DR: In this article, the properties of analytic fields on a Riemann surface represented by a branch covering of ℂℙ1 are investigated in detail, and determinants of ∆ ∆ partial ∆ operators for Zn-symmetric surfaces and various representations for the two-loop measure in the bosonic string theory are obtained.
Abstract: The properties of analytic fields on a Riemann surface represented by a branch covering of ℂℙ1 are investigated in detail. Branch points are shown to correspond to the vertex operators with simple conformal properties. As applications we compute determinants of\(\bar \partial _j\) operators forZn-symmetric surfaces and obtain various representations for the two-loop measure in the bosonic string theory together with various identities for theta-functions of hyperelliptic surfaces. We also present an integral representation for the quantum part of the twist field correlation functions, which describe propagation of the string on the orbifold background. We also calculate the quantum part of the structure constants of the twist-field operator algebra, generalizing the results of Dixon, Friedan, Martinec, and Shenker.
234 citations
TL;DR: In this article, the authors provide a mathematically rigorous foundation for rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms.
Abstract: The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representations of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic cohomology.
232 citations