J. H. Lowenstein

Convergence theorems for renormalized Feynman integrals with zero-mass propagators

A general momentum-space subtraction procedure is proposed for the removal of both ultraviolet and infrared divergences of Feynman integrals. Convergence theorems are proved which allow one to define time-ordered Green functions, as tempered distributions, for a wide class of theories with zero-mass propagators.

Generalization of Zimmermann's Normal-Product Identity

Abstract Zimmermanns identity relating normal products of different subtraction degrees is generalized to include (a) arbitrary masses, (b) ultraviolet regulators and (c) Green functions with arbitrary numbers of normal products. The generalized Zimmermann identity provides the basis for the derivation of normal-product field equations in models with zero-mass propagators, as well as a simple, direct proof of the equivalence of the subtractive and counterterm approaches to renormalization in the auxiliary-mass BPHZ framework.

The Power Counting Theorem for Feynman Integrals with Massless Propagators

Dysons power counting theorem is extended to the case where some of the mass parameters vanish. Weinbergs ultraviolet convergence conditions are supplemented by infrared convergence conditions which combined are sufficient for the convergence of Feynman integrals.

On the formulation of theories with zero-mass propagators

Abstract A new renormalization scheme is proposed for theories with zero-mass propagators. For each Feynman diagram the method yields an ultraviolet and infrared convergent contribution to the Green functions. The method is first developed for the massless A4 model and then applied to the Goldstone and pre-Higgs models.

Analysis of the Bethe-ansatz equations of the chiral-invariant Gross-Neveu model

Abstract The Bethe-ansatz equations of the chiral-invariant Gross-Neveu model are reduced to a simple form in which the parameters of the vacuum solution have been eliminated. The resulting system of equations involves only the rapidities of physical particles and a minimal set of complex parameters needed to distinguish the various internal symmetry states of these particles. The analysis is performed without invoking the time-honored assumption that the solutions of the Bethe-ansatz equations, in the infinite-volume limit, are comprised entirely of strings (“bound states”). Surprisingly, it is found that the correct description of the n -particle states involves no strings of length greater than two (except for special values of the momenta).

Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off

We investigate the effects of round-off errors on the orbits of a linear symplectic map of the plane, with rational rotation number nu=p/q. Uniform discretization transforms this map into a permutation of the integer lattice Z(2). We study in detail the case q=5, exploiting the correspondence between Z and a suitable domain of algebraic integers. We completely classify the orbits, proving that all of them are periodic. Using higher-dimensional embedding, we establish the quasi-periodicity of the phase portrait. We show that the model exhibits asymptotic scaling of the periodic orbits and a long-range clustering property similar to that found in repetitive tilings of the plane. (c) 1997 American Institute of Physics.

Quadratic rational rotations of the torus and dual lattice maps

We develop a general formalism for computer-assisted proofs concerning the orbit structure of certain nonergodic piecewise affine maps of the torus, whose eigenvalues are roots of unity. For a specific class of maps, we prove that if the trace is a quadratic irrational (the simplest nontrivial case, comprising eight maps), then the periodic orbits are organized into finitely many renormalizable families, with exponentially increasing period, plus a finite number of exceptional families. The proof is based on exact computations with algebraic numbers, where units play the role of scaling parameters. Exploiting a duality existing between these maps and lattice maps representing rounded-off planar rotations, we establish the global periodicity of the latter systems, for a set of orbits of full density.

Recursive tiling and geometry of piecewise rotations by π/7

We study two piecewise affine maps on convex polygons, locally conjugate to a rotation by a multiple of ?/7. We obtain a finite-order recursive tiling of the phase space by return map sub-domains of triangles and periodic heptagonal domains (cells), with scaling factors given by algebraic units. This tiling allows one to construct efficiently periodic orbits of arbitrary period, and to obtain a convergent sequence of coverings of the closure of the discontinuity set ?. For every map for which such finite-order recursive tiling exists, we derive sufficient conditions for the equality of Hausdorff and box-counting dimensions, and for the existence of a finite, non-zero Hausdorff measure of . We then verify that these conditions apply to our models; we obtain an irreducible transcendental equation for the Hausdorff dimension involving fundamental units, and establish the existence of infinitely many disjoint invariant components of the residual set . We calculate numerically the asymptotic power law growth of the number of cells as a function of maximum return time, as well as the number of cells of diameter larger than a specified . In the latter case, the exponent is shown to coincide with the Hausdorff dimension.

Anomalous transport in a model of Hamiltonian round-off

We study the propagation of round-off error near the periodic orbits of a linear area-preserving map - a planar rotation by a rational angle - which is discretized on a lattice in such a way as to retain invertibility. We consider the round-off error probability distribution as a function of time t, and we show that for each t this is an algebraic number, which can be calculated exactly. We prove that its kth moment increases asymptotically as , where is the fractional dimension of a self-similar set related to periodic orbits of long-period, while G is a bounded function, periodic in the logarithm of t. This implies the diffusion coefficient displays bounded variations, while all higher order transport coefficients diverge, resulting in anomalous transport. This result contrasts with the case of irrational rotations, where the existence of a central limit theorem has been recently established (Vladimirov I 1996 Preprint Deakin University).

Generalization of the momentum-space subtraction procedure for renormalized perturbation theory

The momentum-space subtraction procedure for defining renormalized Feynman integrals is modified to allow for subtraction operators more general than the usual Taylor operators. The added generality permits one to assign subtraction degree less than four to some terms of the unperturbed Lagrangian.