Infraparticles with Superselected Direction of Motion in Two-Dimensional Conformal Field Theory

Particle aspects of two-dimensional conformal field theories are investigated, using methods from algebraic quantum field theory. The results include asymptotic completeness in terms of (counterparts of) Wigner particles in any vacuum representation and the existence of (counterparts of) infraparticles in any charged irreducible product representation of a given chiral conformal field theory. Moreover, an interesting interplay between the infraparticle’s direction of motion and the superselection structure is demonstrated in a large class of examples. This phenomenon resembles the electron’s momentum superselection expected in quantum electrodynamics.     

Asymptotic Completeness for a Renormalized Nonrelativistic Hamiltonian in Quantum Field Theory: The Nelson Model

Scattering theory for the Nelson model is studied. We show Rosen estimates and we prove the existence of a ground state for the Nelson Hamiltonian. Also we prove that it has a locally finite pure point spectrum outside its thresholds. We study the asymptotic fields and the existence of the wave operators. Finally we show asymptotic completeness for the Nelson Hamiltonian.     

Axiomatic Conformal Field Theory

A new rigourous approach to conformal field theory is presented. The basic objects are families of complex-valued amplitudes, which define a meromorphic conformal field theory (or chiral algebra) and which lead naturally to the definition of topological vector spaces, between which vertex operators act as continuous operators. In fact, in order to develop the theory, M?bius invariance rather than full conformal invariance is required but it is shown that every M?bius theory can be extended to a conformal theory by the construction of a Virasoro field. In this approach, a representation of a conformal field theory is naturally defined in terms of a family of amplitudes with appropriate analytic properties. It is shown that these amplitudes can also be derived from a suitable collection of states in the meromorphic theory. Zhu's algebra then appears naturally as the algebra of conditions which states defining highest weight representations must satisfy. The relationship of the representations of Zhu's algebra to the classification of highest weight representations is explained. Received: 22 October 1998 / Accepted: 16 July 1999     

Fusion Rules in Conformal Field Theory

Several aspects of fusion rings and fusion rule algebras, and of their manifestations in two-dimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme.     

Introduction to Conformal Field Theory

An elementary introduction to conformal field theory is given. Topics include free bosons and fermions, orbifolds, affine Lie algebras, coset conformal field theories, superconformal theories, correlation functions on the sphere, partition functions and modular invariance.     

Asymptotic Completeness for Compton Scattering

Scattering in a model of a massive quantum-mechanical particle, an electron, interacting with massless, relativistic bosons, photons, is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of photons by the electron; but electron-positron pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed electron is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one–electron states with propagation estimates for the electron and the photons.Dedicated to Freeman Dyson on the occasion of his 80th birthdayWork partially supported by U.S. National Science Foundation grant DMS 01-00160.Acknowledgement. We thank V. Bach for his hospitality at the University of Mainz, where part of this work was done, and we are indebted to Gian Michele Graf for pointing out a serious gap in an earlier version of this paper. We also thank one of the referees for pointing out many typos and some small errors.     

Asymptotic Completeness in a Class of Massless Relativistic Quantum Field Theories

This paper presents the first examples of massless relativistic quantum field theories which are interacting and asymptotically complete. These two-dimensional theories are obtained by an application of a deformation procedure, introduced recently by Grosse and Lechner, to chiral conformal quantum field theories. The resulting models may not be strictly local, but they contain observables localized in spacelike wedges. It is shown that the scattering theory for waves in two dimensions, due to Buchholz, is still valid under these weaker assumptions. The concepts of interaction and asymptotic completeness, provided by this theory, are adopted in the present investigation.     

A Monoidal Category for Perturbed Defects in Conformal Field Theory

Starting from an abelian rigid braided monoidal category C{\mathcal{C}} we define an abelian rigid monoidal category C_F{\mathcal{C}_F} which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V){\mathcal{C} = {\rm Rep}(V)} and an object in C_F{\mathcal{C}_F} corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F{\mathcal{C}_F} an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F{\mathcal{C}_F}. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.     

Asymptotic Completeness in Quantum Field Theory: Translation Invariant Nelson Type Models Restricted to the Vacuum and One-Particle Sectors

Time-dependent scattering theory for a large class of translation invariant models, including the Nelson and Polaron models, restricted to the vacuum and one-particle sectors is studied. We formulate and prove asymptotic completeness for these models. The translation invariance imply that the Hamiltonians considered are fibered with respect to the total momentum. On the way to asymptotic completeness we determine the spectral structure of the fiber Hamiltonians, establish a Mourre estimate and derive a geometric asymptotic completeness statement as an intermediate step.     

Nuclearity and Thermal States in Conformal Field Theory

We introduce a new type of spectral density condition, that we call L ²- nuclearity. One formulation concerns lowest weight unitary representations of and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the conformal Hamiltonian L ₀, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L ₀ is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a β-KMS state for the translation dynamics on the net of C^*-algebras for every inverse temperature β > 0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L ²-nuclearity is satisfied for the scalar, massless Klein-Gordon field. Dedicated to László Zsidó on the occasion of his sixtieth birthday Supported by MIUR, GNAMPA-INDAM and EU network “Quantum Spaces–Non Commutative Geometry” HPRN-CT-2002-00280     

Noninteraction of Waves in Two-dimensional Conformal Field Theory

In higher dimensional quantum field theory, irreducible representations of the Poincaré group are associated with particles. Their counterpart in two-dimensional massless models are ??waves?? introduced by Buchholz. In this paper we show that waves do not interact in two-dimensional M?bius covariant theories and in- and out-asymptotic fields coincide. We identify the set of the collision states of waves with the subspace generated by the chiral components of the M?bius covariant net from the vacuum. It is also shown that Bisognano-Wichmann property, dilation covariance and asymptotic completeness (with respect to waves) imply M?bius symmetry. Under natural assumptions, we observe that the maps which give asymptotic fields in Poincaré covariant theory are conditional expectations between appropriate algebras. We show that a two-dimensional massless theory is asymptotically complete and noninteracting if and only if it is a chiral M?bius covariant theory.     

The Associative Algebras of Conformal Field Theory

Modulo the ideal generated by the derivative fields, the normal ordered product of holomorphic fields in two-dimensional conformal field theory yields a commutative and associative algebra. The zero mode algebra can be regarded as a deformation of the latter. Alternatively, it can be described as an associative quotient of the algebra given by a modified normal ordered product. We clarify the relation of these structures to Zhu's product and Zhu's algebra of the mathematical literature.     

(p,q)-Analog of Two-Dimensional Conformal Field Theory. The Ward Identities and Correlation Functions

Abstract

A (p, q)-analog of two-dimensional conformally invariant field theory based on the quantum algebra U_pq (su(1, 1)) is proposed. The representation of the algebra U_pq (su(1, 1)) on the space of quasi-primary fields is given. The (p, q)-deformed Ward identities of conformal field theory are defined. The two- and three-point correlation functions of quasi-primary fields are calculated.     

Topological Sectors and a Dichotomy in Conformal Field Theory

Let be a local conformal net of factors on S¹ with the split property. We provide a topological construction of soliton representations of the n-fold tensor product that restrict to true representations of the cyclic orbifold We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then is not completely rational iff the symmetrized tensor product has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of have a conjugate sector then either is completely rational or has uncountably many different irreducible sectors. Thus is rational iff is completely rational. In particular, if the -index of is finite then turns out to be strongly additive. By [31], if is rational then the tensor category of representations of is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.Supported in part by GNAMPA-INDAM and MIURSupported in part by NSF     

Conformal Invariance and Gravitational Coupling in Quantum Field Theory

We study the quantum constraints of a conformalinvariant action for a scalar field. For this purpose webriefly present a reformulation of the duality principleadvanced earlier in the context of generally covariant quantum field theory, and apply it toexamine the finite structure of the quantum constraints.This structure is shown to admit a dimensional coupling(a coupling mediated by a dimensional coupling parameter) of states to gravity. Invariancebreaking is introduced by defining a preferredconfiguration of dynamical variables in terms of thelargescale characteristics of the universe. In thisconfiguration a close relationship between the quantumconstraints and the Einstein equations isestablished.     

A Criterion for Asymptotic Completeness in Local Relativistic QFT

We formulate a generalized concept of asymptotic completeness and show that it holds in any Haag–Kastler quantum field theory with an upper and lower mass gap. It remains valid in the presence of pairs of oppositely charged particles in the vacuum sector, which invalidate the conventional property of asymptotic completeness. Our result can be restated as a criterion characterizing a class of theories with complete particle interpretation in the conventional sense. This criterion is formulated in terms of certain asymptotic observables (Araki–Haag detectors) whose existence, as strong limits of their approximating sequences, is our main technical result. It is proven with the help of a novel propagation estimate, which is also relevant to scattering theory of quantum mechanical dispersive systems.     

Two-Term Dilogarithm Identities Related to Conformal Field Theory

We study 2 × 2 matrices A such that the corresponding thermodynamic Bethe ansatz (TBA) equations yield in the form of the effective central charge of a minimal Virasoro model. Certain properties of such matrices and the corresponding solutions of the TBA equations are established. Several continuous families and a discrete set of admissible matrices A are found. The corresponding two-term dilogarithm identities (some of which appear to be new) are obtained. Most of them are proven or shown to be equivalent to previously known identities.