On vertex parts of p-adic Feynman amplitudes

We reduce the calculation of the vertex parts of p-adic Feynman amplitudes to a recursive procedure for evaluating singular parts of certain integrals. We propose an algorithm for calculating these integrals in the general form. As an example, we consider vertex parts of amplitudes in the ? ⁴ theory.     

Renormalization group in a fermionic hierarchical model in projective coordinates

We study the renormalization group action in a fermionic hierarchical model in the space of coefficients determining the Grassmann-valued density of the free measure. This space is interpreted as the two-dimensional projective space. The renormalization group map is a homogeneous quadratic map and has a special geometric property that allows describing invariant sets and the global dynamics in the whole space.     

Critical phenomena in the fermionic hierarchical model

The dynamics of the renormalization-group transformation in the coupling-constant space of the fermionic hierarchical model are discussed. The critical behavior of this model is described in terms of the complex behavior of the Grassmann-valued mean-spin distribution density with the proper normalization. Some critical indices are calculated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 471–488, December, 1998.     

Zone Structure of the Renormalization Group Flow in a Fermionic Hierarchical Model

The Gaussian part of the Hamiltonian of the four-component fermion model on a hierarchical lattice is invariant under the block-spin transformation of the renormalization group with a given degree of normalization (the renormalization group parameter). We describe the renormalization group transformation in the space of coefficients defining the Grassmann-valued density of a free measure as a homogeneous quadratic map. We interpret this space as a two-dimensional projective space and visualize it as a disk. If the renormalization group parameter is greater than the lattice dimension, then the unique attractive fixed point of the renormalization group is given by the density of the Grassmann delta function. This fixed point has two different (left and right) invariant neighborhoods. Based on this, we classify the points of the projective plane according to how they tend to the attracting point (on the left or right) under iterations of the map. We discuss the zone structure of the obtained regions and show that the global flow of the renormalization group is described simply in terms of this zone structure.     

Functional integral via functional equation

Discretization ofp-adic Grassmann-valued -model leads to a hierarchical model with the Hamtilonian given by a nontrivial functional integral over the Grassmann variables. Using renormalization group arguments, we reduce the calculation of this integral to a functional equation. The problem of the convergence of the perturbation expansion of this integral, realized as a small-divisors problem, is investigated.     

An algorithm for studying the renormalization group dynamics in the projective space

The renormalization group dynamics is studied in the four-component fermionic hierarchical model in the space of coefficients that determine the Grassmann-valued density of the free measure. This space is treated as a two-dimensional projective space. If the renormalization group parameter is greater than 1, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann δ-function. Two different invariant neighborhoods of this fixed point are described, and an algorithm is constructed that allows one to classify the points on the plane according to the way they tend to the fixed point.     

Functional equations and renormalizations inp-adic field theories

The procedure of constructing the discrete approximation of p-adic ⁴-theory on a hierarchical lattice is reduced to the solution of an integral functional equation. The renormalization procedure and -functions are constructed in terms of this solution.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No 1, pp. 3–16, October, 1996.     

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).     

Dimensional renormalization inp-adic models of field theory

We define fractional-dimensional p-adic Feynman amplitudes and construct a dimensional renormalization with minimum subtractions. In the fermionic model case, another dimensional renormalization procedure is defined as the inversion of the normalizing transformation at the trivial stable point for the hierarchical renormalization group transformation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 462–475, June, 2000.