The choice of test functions in gauge quantum field theories

We discuss the problem of the choice of test functions in gauge quantum field theories. Analysis of explicitly soluble models suggests that the test function spaces which are suitable for local and covariant formulation of gauge theories are the Gelfand and Shilov spaces , +>1. We also discuss a possible generalization of the spectral condition.     

The indecomposable representation of SO0(2, 2) on the one-particle space of the massless field in 1 + 1 dimension

In the continuum O(3) sigma model in two spatial dimensions, there are topological solitons whose size can be stabilized by adding Skyrme and potential terms. This Letter describes a lattice version, namely a natural way of modifying the 2D Heisenberg model to achieve topological stability on the lattice.     

Eigensystem and full character formula of theW 1+∞ algebra withc = 1

By using the free field realizations, we analyze the representation theory of theW ₁₊ algebra withc = 1. The eigenvectors for the Cartan subalgebra ofW ₁₊ are parametrized by Young diagrams, and explicitly written down byW ₁₊ generators. Moreover, their eigenvalues and full character formula are also obtained.     

Completeness of inner product spaces and quantum logic of splitting subspaces

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which EE =S holds, is a quantum logic, that is, an orthocomplemented orthomodular -orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys. 11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces.     

States on families of subspaces of pre-Hilbert spaces

We present six families of closed subspaces of a pre-Hilbert space and we show that if arbitrary, one of them possesses at least one completely additive state, then the pre-Hilbert space is complete.     

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.     

A signed measure completeness criterion

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e. of all subspaces M for which M+M =S holds, possesses at least one nonzero completely additive signed measure or, equivalently, iff S possesses at least one nonzero frame function. Moreover, we show a new and simple proof that S is complete iff E(S) contains the join of any sequence of orthogonal one-dimensional subspaces.     

On the uniqueness of the equilibrium state for an interacting fermion gas at high temperatures and low densities

Starting from bounded local perturbations, we release the spatial cutoff and prove the uniqueness of the KMS state at high temperatures and low densities for the continuous fermion system with pair-interaction proposed by Narnhofer and Thirring.     

Krichever-Novikov formulation of topological conformal field theory

The Krichever-Novikov (KN) global operator formalism is applied to construct a topological conformal field theory on a compact Riemann surface from an N=2 super-conformal field theory. The topological version of the KN algebra is derived and the BRST charge is shown to be genus-dependent in this formulation. This leads to an interesting cohomology structure for the physical subspace of the Hilbert space.     

A nuclearity condition for charged states

A nuclearity condition for charged superselection sectors is formulated. An analyticity property of nuclear maps is proved in order to show the validity of the split property under this new nuclearity condition.     

Towards a generalized distribution formalism for gauge quantum fields

We prove that the distributions defined on the Gelfand-Shilov spacesS with < 1 and, hence, more singular than hyperfunctions, retain the angular localizability property. Specifically, they have uniquely determined support cones. This result enables one to develop a distribution-theoretic technique suitable for the consistent treatment of quantum fields with arbitrarily singular ultraviolet and infrared behavior. The proof covering the most general and difficult case = 0 is based on the use of the theory of plurisubharmonic functions and Hörmander'sL ²-estimates.This work was supported in part by a Soros Humanitarian Foundation Grant awarded by the American Physical Society.     

The exponential of the two-dimensional massless scalar field as an infrared Jaffe field

The definition of the Wick exponential of the massless scalar field in two dimensions as an operator-valued distribution is discussed in the Krein space realization of the field. It is shown that the exponential Wick power series converges strongly on a suitably dense domain, provided that it is smeared with test functions which satisfy some growth conditions in configuration space. Such conditions are similar to the high energy bounds previously introduced by Jaffe for the definition of strictly localizable fields. In this case it is found that the Wick exponential is strictly localizable in momentum space.     

Microlocal analysis and phase-space decompositions

A general class of phase-space decompositions f(x)=a(x,p) dp of functions f defined in ⁿ is presented. In the latter, a(x,p) depends on values of the Fourier transform F of f in a region around p whose width tends to zero as |x| increases and it decays exponentially, for each p, in all directions in x-space outside the microsupport at p of F, with a rate of exponential fall-off linked to analyticity properties of F in local tubes (in complex space) around p. A possible application in quantum-field theory is mentioned.     

Spin chains in magnetic field,non-skew-symmetric classical r-matrices and BCS-type integrable systems

We construct generalized Gaudin systems in an external magnetic field corresponding to arbitrary so(3)$\mathit{so}(3)$-valued non-skew-symmetric r-matrices with spectral parameters and non-homogeneous external magnetic fields. In the case of r  -matrices diagonal in the sl(2)$\mathit{sl}(2)$ basis we calculate the spectrum and the eigen-values of the corresponding generalized Gaudin hamiltonians using the algebraic Bethe ansatz. We explicitly consider several one-parametric families of non-skew-symmetric classical r-matrices and the corresponding generalized Gaudin systems in a magnetic field. We apply these results to fermionic systems and obtain a wide class of new integrable fermionic BCS-type hamiltonians.     

Resummation of Mass Terms in Perturbative Massless Quantum Field Theory

The neutral massless scalar quantum field Φ in four-dimensional space-time is considered, which is subject to a simple bilinear self-interaction. Is is well-known from renormalization theory that adding a term of the form to the Lagrangean has the formal effect of shifting the particle mass from the original zero value to m after resummation of all two-leg insertions in the Feynman graphs appearing in the perturbative expansion of the S-matrix. However, this resummation is accompanied by some subtleties if done in a proper mathematical manner. Although the model seems to be almost trivial, is shows many interesting features which are useful for the understanding of the convergence behavior of perturbation theory in general. Some important facts in connection with the basic principles of quantum field theory and distribution theory are highlighted, and a remark is made on possible generalizations of the distribution spaces used in local quantum field theory. A short discussion how one can view the spontaneous breakdown of gauge symmetry in massive gauge theories within a massless framework is presented.      

On the schwarz problem for the [^(\textsu)]\text2 \widehat{{\text{su}}}_{\text{2}} Knizhnik-Zamolodchikov equation

We study the monodromy representations ^k,Iof the mapping class group ₄ acting on 4-point blocks satisfying the Knizhnik-Zamolodchikov equation for the levelk su₂ current algebra. We classify all irreducible ^k,Iwhich are realized by finite groups; we also display finite irreducible components for the reducible representations corresponding tok = 10.Supported by the Federal Ministry of Science and Research, Austria.     

The Schur function realization of vertex operators

We present a realization of untwisted vertex operators in terms of operations on Schur functions. Calculations of matrix elements and traces of products of vertex operators are performed using results from the classical theory of symmetric functions. The concepts of compound, composite and supersymmetric Schur functions naturally appear in this context. Furthermore, a trace formula for a product of vertex operators turns out to be a generalization of a MacDonald identity reformulated in terms of Schur functions.     

Extended non-abelian gauge symmetries in the classical WZNW model

Higher spin extensions of the non-Abelian gauge symmetries for the classical WZNW model are considered. Both linear and nonlinear realizations of the extended affine Kac-Moody algebra are obtained. It is a characteristic property of the WZNW model that it admits a higher spin linear realization of the extended affine Kac-Moody algebra which is equivalent to the corresponding higher spin nonlinear realization of the same algebra. However, in both cases, the higher spin Noether currents do not span an invariant space with respect to their generating transformations. Here, the current space is extended to an invariant space which allows us to gauge the symmetry.Supported by Bulgarian Foundation on Fundamental Research under contract Ph-318/93-95.     

The Kazhdan-Lusztig conjecture for finiteW-algebras

We study the representation theory of finiteW-algebras. After introducing parabolic subalgebras to describe the structure ofW-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finiteW-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors and find that, forW-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a dual analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infiniteW-algebras.