[en] We describe a rich family of binary variables statistical mechanics models on planar graphs which are equivalent to Gaussian Grassmann Graphical models (free fermions). Calculation of partition function (weighted counting) in the models is easy (of polynomial complexity) as reduced to evaluation of determinants of matrixes linear in the number of variables. In particular, this family of models covers Holographic Algorithms of Valiant and extends on the Gauge Transformations discussed in our previous works.

[en] We construct approximate solutions of the time-dependent Schroedinger equation iℎ∂ψ/∂t=-ℎ2Δψ/2+Vψ for small values of ℎ. If V satisfies appropriate analyticity and growth hypotheses and vertical stroke t vertical stroke ≤T, these solutions agree with exact solutions up to errors whose norms are bounded by C exp(-γ/ℎ), for some C and γ>0. Under more restrictive hypotheses, we prove that for sufficiently small T', vertical stroke t vertical stroke ≤T' vertical stroke log(ℎ) vertical stroke implies the norms of the errors are bounded by C' exp(-γ'/ℎ ^simg), for some C', γ'>0, and σ>0. (orig.)

[en] There are examples of Calogero-Sutherland models associated to the Weyl groups of type A and B. When exchange terms are added to the Hamiltonians the systems have non-symmetric eigenfunctions, which can be expressed as products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differential-difference operators introduced by the author (1989). After a description of known results, particularly from the works of Baker and Forrester, and Sahi; there is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,..,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermite-type polynomials. These can be expressed by using the generalized binomial coefficients. A complete basis of eigenfunctions of Yamamoto's B_N spin Calogero model is obtained by multiplying these polynomials by the ground state. (orig.)

[en] We study Bogomolnyi equations on R² x S¹. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using the Nahm transform, we show that periodic monopoles are in one-to-one correspondence with solutions of Hitchin equations on a cylinder with Higgs field growing exponentially at infinity. The moduli spaces of periodic monopoles belong to a novel class of hyperkaehler manifolds and have applications to quantum gauge theory and string theory. For example, we show that the moduli space of k periodic monopoles provides the exact solution of N =2 super Yang-Mills theory with gauge group SU(k) compactified on a circle of arbitrary radius. (orig.)

[en] The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A⁽ⁿ⁾ of observables ''up to n loops'', where A⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. (orig.)

[en] The paper deals with the analytic theory of the quantum q-deformed Toda chains; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L. Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin-Barnes type. For the periodic chain the two dual Baxter equations are derived. (orig.)

[en] We consider the relativistic electron-positron field interacting with itself via the Coulomb potential defined with the physically motivated, positive, density-density quartic interaction. The more usual normal-ordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and, by choosing the normal ordering in a suitable, self-consistent manner, the quadratic term can be seen to be equivalent to a renormalization of the Dirac operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in reality it is non-perturbative, for it leads to an inequivalent, fine-structure dependent representation of the canonical anticommutation relations. This non-perturbative redefinition of the electron/positron states can be interpreted as a mass, wave-function and charge renormalization, among other possibilities, but the main point is that a non-perturbative definition of normal ordering might be a useful starting point for developing a consistent quantum electrodynamics. (orig.)

[en] Motivated by the recently observed relation between the physics of D-branes in the background of B-field and the noncommutative geometry we study the analogue of the Nahm transform for the instantons on the noncommutative torus. (orig.)

[en] The aim of this paper is to offer an affirmative answer to the conjectures by A. Floer (1989) in which states that there is a module structure on the Z_2N-graded symplectic Floer cohomology for monotone symplectic manifolds. By constructing a Z-graded symplectic Floer cohomology as an integral lift of the Z_2N-graded symplectic Floer cohomology, we gain control of the holomorphic bubbling spheres. This makes a module structure on the Z-graded Floer cohomology. There is a spectral sequence with E¹_*,* given by the Z-graded symplectic Floer cohomology. Such a spectral sequence converges to the Z_2N-graded symplectic Floer cohomology. Hence we induce a module structure for the Z_2N-graded symplectic Floer cohomology by the spectral sequence and algebraic topology methods. (orig.)

[en] In this paper we present a renormalizability proof for spontaneously broken SU(2) gauge theory. It is based on flow equations, i.e. on the Wilson renormalization group adapted to perturbation theory. The power counting part of the proof, which is conceptually and technically simple, follows the same lines as that for any other renormalizable theory. The main difficulty stems from the fact that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the Slavnov-Taylor identities of SU(2) Yang-Mills theory on which the gauge invariance of the renormalized theory is based. (orig.)