New Light on Infrared Problems: Sectors,Statistics, Symmetries and Spectrum

A new approach to the analysis of the physical state space of a theory is presented within the general setting of local quantum physics. It also covers theories with long range forces, such as quantum electrodynamics. Making use of the notion of charge class, an extension of the concept of superselection sector, infrared problems are avoided by restricting the states to observables localized in a light cone. The charge structure of a theory can be explored in a systematic manner. The present analysis focuses on simple charges, thus including the electric charge. It is shown that any such charge has a conjugate charge. There is a meaningful concept of statistics: the corresponding charge classes are either of Bose or of Fermi type. The family of simple charge classes is in one-to-one correspondence with the irreducible unitary representations of a compact Abelian group. Moreover, there is a meaningful definition of covariant charge classes. Any such class determines a continuous unitary representation of the Poincaré group or its covering group satisfying the relativistic spectrum condition. The resulting particle aspects are also briefly discussed.     

On the superselection rule for electric charge

It is shown that global transformations can be included into the group of local gauge transformations. In the Dirac scheme of the quantization of gauge theories, it allows us to prove that for electric charge there is a superselection rule in electrodynamics and, even more, that the total electric charge of the open Universe is zero.     

On the Structure of the Observable Algebra for QED on the Lattice

We prove that the matter field subalgebra of the observable algebra for QED on a finite lattice is isomorphic to the enveloping algebra of the Lie algebra sl(2N, C), factorized by a certain ideal. Using this result, we give a new proof of the decomposition of the physical Hilbert space into charge superselection sectors.     

Galilei invariance and superfluidity II

Continuing the (heuristic) analysis of the mathematical structure of the Landau excitations, we find that inone dimension they may be described by a vector bundle over the base space of the boosts. The total space is a direct integral of all irreducible representations (of a given class) of the Galilei group. The existence of an energy-momentum spectrum requires the action of the boosts to be non-linear. This action can also be formulated as a superselection rule.     

Framed Vertex Operator Algebras, Codes and the Moonshine Module

For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ?, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ? are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras. Received: 14 July 1997 / Accepted: 8 September 1997     

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.     

Explicit solutions to the intrinsic generalization for the wave and sine-Gordon equations

Theta-vacua superselection for Bloch electrons, charge confinement in the Schwinger model, fermionic charges, superselection structures in conformal two-dimensional models are shown to follow from general properties of a class of nonregular representations of Weyl algebras.     

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory

We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of ? form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry. Received: 7 July 1999 / Accepted: 13 January 2001     

Object-subject split and superselection partial states

In an accompanying work it was shown that any subset of quantum mechanical observables determines an (up to equivalence) unique partial statistical theory with partial states. The crucial role of a so-called basic (e.g., pointer) observable in determining an object-subject split with a well-defined subject was made clear. In this article the subject subsystem is assumed to be quantum mechanical, but such that the basic observable is a superselection one. This leads to superselection partial states and to a different approach to the split. Advantages and disadvantages of the latter approach are discussed.     

Conformal field algebras with quantum symmetry from the theory of superselection sectors

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.     

From Path Representations to Global Morphisms for a Class of Minimal Models

We construct global observable algebras and global DHR morphisms for the Virasoro minimal models with central charge c(2,q), q odd. To this end, we pass from the irreducible highest weight modules to path representations, which involve fusion graphs of the c(2,q) models. The paths have an interpretation in terms of quasi-particles which capture some structure of non-conformal perturbations of the c(2,q) models. The path algebras associated to the path spaces serve as algebras of bounded observables. Global morphisms which implement the superselection sectors are constructed using quantum symmetries: We argue that there is a canonical semi-simple quantum symmetry algebra for each quasi-rational CFT, in particular for the c(2,q) models. These symmetry algebras act naturally on the path spaces, which allows to define a global field algebra and covariant multiplets therein.     

Theory of symmetry in the quantum mechanics of infinite systems: II. Isotropic representations of the Galilei group and its central extensions

The isotropic bundle representations of the Galilei group and its central extension are classified, and the natural cross-section, action of the group on the base manifold and the canonical cocyle are determined for all cases. Projective bundle representations of the Galilei group are defined and the extension of Bargmann's superselection rule is established. Coordinate transformations on the base space are discussed in all cases, and the notion of generalized coordinate transformations is introduced. It is then shown that the bundle representations being considered do not violate the principle of Galilean relativity as it is commonly understood. The physical interpretation of the irreducible and some reducible representations is discussed. It is found that some bundle representations might correspond to objects which can act as sources or sinks of linear and/or angular momentum.     

Superselection rules in quantum theory: Part II. Subensemble selection

A dynamical analysis of standard procedures for subensemble selection is used to show that the state restriction violation proposal in Part I of the paper cannot be realized by employing familiar correlation schemes. However, it is shown that measurement of an observable not commuting with the superselection operator is possible, a violation of the observable restrictions. This is interpreted as supporting the position that each of these restrictions is sufficient but not necessary for the superselection rule. The results do constitute a proposal for superselection rule violation in theories requiring both restrictions, e.g., the axiomatic treatment by Bogolubov, Logunov, and Todorov. It is also concluded that superselection rules place restrictions on procedures for selective state preparations using correlations. More generally, it is conjectured that a mathematically conceivable decomposition of a given density operator does not necessarily represent a possibility for partitioning of the corresponding ensemble into subensembles by any physically realizable means.     

Fields,statistics and non-Abelian gauge groups

We examine field theories with a compact groupG of exact internal gauge symmetries so that the superselection sectors are labelled by the inequivalent irreducible representations ofG. A particle in one of these sectors obeys a parastatistics of orderd if and only if the corresponding representation ofG isd-dimensional. The correspondence between representations of the observable algebra and representations ofG extends to a mapping of the intertwining operators for these representations preserving linearity, tensor products and conjugation. Although we assume no explicit commutation property between fields, the commutation relations of fields of the same irreducible tensor character underG at spacelike separations are largely determined by the statistics parameter of the corresponding sector. For fields of conjugate irreducible tensor character the observable part of the commutator (anticommutator) vanishes at spacelike separations if the corresponding sector has para-Bose (para-Fermi) statistics.     

Local aspects of superselection rules

We study a theory of short range forces in terms of local observable quantities; among the superselection structure determined by the algebra of all local observables, to each additive independent charge we associate local observables having a meaning analogous to the regularized integrals of charge density fields over a finite volume. Among other assumptions, we require that parastatistics are absent from the theories considered.     

Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model

We present sufficient conditions that imply duality for the algebras of local observables in all Abelian sectors of all locally normal, irreducible representations of a field algebra if twisted duality obtains in one of these representations. It is verified that the Yukawa₂ model satisfies these conditions, yielding the first proof of duality for the observable algebra in all coherent charge sectors in this model. This paper also constitutes the first verification of the assumptions of the axiomatic study of the structure of superselection sectors by Doplicher, Haag and Roberts in an interacting model with nontrivial sectors. The existence of normal product states for the free Fermi field algebra and, thus, the verification of the funnel property for the associated net of local algebras are demonstrated.     

The statistical quantum number and gauge groups in parafermi field theory

Properties of a system consisting of a single parafermi field of order p are studied mainly in connection with gauge groups. Following the theory of Drühl, Haag and Roberts, the algebra of observables is classified into four cases according to the types of gauge groups, i.e., SO(p), O(p), U(p), and SU(p). A detailed study is made of irreducible representations of these gauge groups that are realized in the state-vector space of the parafermi field. Superselection operators which give rise to the corresponding superselection rules related to the gauge groups are studied, and their explicit expressions given. The statistical quantum number which we introduced before is found to be nothing other than the eigenvalues of a superselection operator for the gauge group O(p).     

A class of 6D conformal field theories: relation to 4D superconformal Yang-Mills theories?

We describe a class of six-dimensional conformal field theories that possibly are related to the tensionless string theories. They have an ADE classification, but no other discrete or continuous parameters, with the A(N-1) theory arising by factoring out the collective "center of mass" degrees of freedom from N noninteracting chiral two-forms. The Hilbert space carries an irreducible representation of the same Heisenberg group that appears in the tensionless string theories, and the "Wilson surface" observables obey the same superselection rules. When compactified on a two-torus, our theories have the same behavior under S duality as N = 4 super Yang-Mills theories.     

On the spectral function of carriers in the pseudogap state

We consider the evolution of the spectral function of charge carriers for a 2D Kondo lattice depending on the parameters of the model. A self-consistent solution is obtained for the spectral function using the formalism of irreducible Green’s functions. In the low doping level regime, the behavior of the spectral function exhibits suppression of the spectral weight of carriers in the low-frequency range, which is typical of the pseudogap state.