Global renormalization group flow and thermodynamic limit for the hierarchical fermionic model

The renormalization group transformation for the hierarchical fermionic model is presented as a rational map in the plane of the coupling constants. We investigate the global dynamics of this transformation. The existence of the thermodynamic limit in the domain under investigation is proved. The automodel limit of the constructed fields is described.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 201–212, May, 1996.Translated by L. O. Chekhov.     

A geometric introduction to the two-loop renormalization group flow

The Ricci flow has been of fundamental importance in mathematics, most famously through its use as a tool for proving the Poincaré conjecture and Thurston’s geometrization conjecture. It has a parallel life in physics, arising as the first-order approximation of the renormalization group flow for the nonlinear sigma model of quantum field theory. There recently has been interest in the second-order approximation of this flow, called the RG-2 flow, which mathematically appears as a natural nonlinear deformation of the Ricci flow. A curvature flow arising from quantum field theory seems to us to capture the spirit of Yvonne Choquet-Bruhat’s extensive work in mathematical physics, and so in this commemorative article we give a geometric introduction to the RG-2 flow. A number of new results are presented as part of this narrative: short-time existence and uniqueness results in all dimensions if the sectional curvatures K _ij satisfy certain inequalities; the calculation of fixed points for n =  3 dimensions; a reformulation of constant curvature solutions in terms of the Lambert W function; a classification of the solutions that evolve only by homothety; an analogue for RG flow of the 2-dimensional Ricci flow solution known to mathematicians as the cigar soliton, and discussed in the physics literature as Witten’s black hole. We conclude with a list of open problems whose resolutions would substantially increase our understanding of the RG-2 flow both physically and mathematically.     

Properties of renormalization group flow in the neighborhood of Gaussian fixed point

We investigate the dynamics of the renormalization group transformation in the fermionic hierarchical model, given by the Lagrangian. We construct the invariant neighborhood of the Gaussian fixed point in the upper half-plane g > 0. We describe the subsets of the points of this neighborhood tending to the Gaussian fixed point under the iterations of the renormalization group transformation from the left and from the right.     

Symmetries and cycles of the renormalization group in a fermionic hierarchical model

We study singular points and symmetries of the renormalization group mapping in a fermionic hierarchical model. This mapping taken at the renormalization group singular point and with the renormalization group parameter =1 generates cycles of arbitrary lengths.     

Soliton metrics for two-loop renormalization group flow on 3D unimodular Lie groups

The two-loop renormalization group flow is studied via the induced bracket flow on 3D unimodular Lie groups. A number of steady solitons are found. Some of these steady solitons come from maximally symmetric metrics that are steady, shrinking, or expanding solitons under Ricci flow, while others are not obviously related to Ricci flow solitons.     

Dynamics of renormalization group in the lower half-plane of the coupling constants of the fermionic hierarchical model

We consider four-component fermionic (Grassmann-valued) field on the hierarchical lattice. The Gaussian part of the Hamiltonian in the model is invariant under the block-spin renormalization group transformation with given degree of normalization factor (renormalization group parameter). The non-Gaussian part of the Hamiltonian is given by the sum of the selfinteraction forms of the 2-nd and 4-th order. The action of the renormalization group transformation in this model is reduced to the rational map in the plane of coupling constants. We investigate the global dynamics of this map in the case when the coupling constant of the 4-th order form is less than zero (lower half-plane) and the renormalization group parameter belongs to the interval [1, 3/2).     

Stability of renormalization group trajectories and the fermion flavor problem

A long-standing puzzle of the current standard model for particle physics is that both leptons and quarks arise in replicated patterns. Our work suggests that the number of fermion flavors may be directly derived from the dynamics of coupling flow equations. Specifically, we argue that the number of flavors results from demanding stability of the coupling flow about its fixed-point solution.     

The Hopf graph algebra and renormalization group equations

We study the renormalization group equations implied by the Hopf graph algebra. The vertex functions are regarded as vectors in the dual space of the Hopf algebra. The renormalization group equations for these vertex functions are equivalent to those for individual Feynman integrals. The solution of the renormalization group equations can be represented in the form of an exponential of the beta function. We clearly show that the exponential of the one-loop beta function allows finding the coefficients of the leading logarithms for individual Feynman integrals. The calculation results agree with those obtained in the parquet approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 22–32, April, 2005.     

Smooth Feshbach map and operator-theoretic renormalization group methods

A new variant of the isospectral Feshbach map defined on operators in Hilbert space is presented. It is constructed with the help of a smooth partition of unity, instead of projections, and is therefore called smooth Feshbach map. It is an effective tool in spectral and singular perturbation theory. As an illustration of its power, a novel operator-theoretic renormalization group method is described and applied to analyze a general class of Hamiltonians on Fock space. The main advantage of the new renormalization group method over its predecessors is its technical simplicity, which it owes to the use of the smooth Feshbach map.     

<Emphasis Type="Italic">p</Emphasis>-adic renormalization group solutions and the euclidean renormalization group conjectures

We discuss algebraic similarity of the Wilson’s renormalization groups in the Euclidean and p-adic spaces. Automodel Hamiltonians have identical form in both cases in the framework of perturbation theory. Fermionic p-adic model has exact renormalization group solution which generates a list of non-trivial conjectures for the Euclidean case.     

Functional self-similarity and renormalization group symmetry in mathematical physics

The results from developing and applying the notions of functional self-similarity and the Bogoliubov renormalization group to boundary-value problems in mathematical physics during the last decade are reviewed. The main achievement is the regular algorithm for finding renormalization group-type symmetries using the contemporary theory of Lie groups of transformations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 1, pp. 66–88, October, 1999.     

A two-regime traffic flow model and the consistency of its parameters

An attempt is made to demonstrate the traffic behaviour and phenomena under normal morning peak period conditions, and to examine the suitability of a two-regime traffic flow model for these conditions. This paper has three main parts. First, the consistency of flow and concentration patterns of a 9-mile freeway section is examined and provides a basis for distinguishing between the free-flow and the congested-flow regimes. This distinction clearly indicates the data points obtained from traffic flow situations, which, in time-sequence, approach maximum flow conditions, congested conditions, and through a recovery process backwards to free-flow conditions.

In the second part, a car-following model for the two-regime approach is introduced. By using the analysis of driver performance as a sensitivity measurement, model parameters are defined and evaluated. An overall comparison between the proposed and known generalized car-following models emphasizes the advantages of the proposed model, particularly its simplicity and clarity at both the micro- and macroscopic levels, for the two-regime phenomenon.

In the third part, the steady-state formulation, derived from the proposed car-following model, is evaluated by using the time-sequence data points. The consistency of the two-regime model parameters is apparently well preserved regarding data sets of 3-year period (1972–1975) with respect to three independent variables: years, workdays and locations.