Large field renormalization. I. The basic step of the ℝ operation

We construct the renormalization operation of the expressions connected with the large field regions. This operation, denoted by , removes the main obstacle to prove the ultraviolet stability of four-dimensional gauge field theories. The proof will be completed in the second part of this paper.Research supported in part by the National Science Foundation under Grant DMS-86 02207     

Multiscale representation of generating and correlation functions for some models of statistical mechanics and quantum field theory

We consider models of statistical mechanics and quantum field theory (in the Euclidean formulation) which are treated using renormalization group methods and where the action is a small perturbation of a quadratic action. We obtain multiscale formulas for the generating and correlation functions aftern renormalization group transformations which bring out the relation with thenth effective action. We derive and compare the formulas for different RGs. The formulas for correlation functions involve (1) two propagators which are determined by a sequence of approximate wave function renormalization constants and renormalization group operators associated with the decomposition into scales of the quadratic form and (2) field derivatives of the nth effective action. For the case of the block field -function RG the formulas are especially simple and for asymptotic free theories only the derivatives at zero field are needed; the formulas have been previously used directly to obtain bounds on correlation functions using information obtained from the analysis of effective actions. The simplicity can be traced to an orthogonality-of-scales property which follows from an implicit wavelet structure. Other commonly used RGs do not have the orthogonality of scales property.     

Perturbative renormalization of composite operators via flow equations I

We apply the general framework of the continuous renormalization group, whose significance for perturbative quantum field theories was recognized by Polchinski, to investigate by new and mathematically simple methods the perturbative renormalization of composite operators. In this paper we demonstrate the perturbative renormalizability of the Green functions of the Euclidean massive ₄ ⁴ theory with one insertion of a (possibly oversubtracted, in the BPHZ language) composite operator. Moreover we show that our method admits an easy proof of the Zimmermann identities and of the Lowenstein rule.Supported by the Swiss National Science Foundation     

Continuum limit of the hierarchicalO(N) non-linear σ-model

We construct rigorously the continuum limit of theO(N) non-linear model in two euclidean dimensions with a hierarchical kinetic term. Asymptotic freedom and weak coupling Wilson renormalization group flow are established.     

The short distance behavior of (φ 4)3

We consider the ₃ ⁴ quantum field theory on a torus and study the short distance behavior. We reproduce the standard result that the singularities can be removed by a simple mass renormalization. For the resulting model we give anL _p bound on the short distance regularity of the correlation functions. To obtain these results we develop a systematic treatment of the generating functional for correlations using a renormalization group method incorporating background fields.Research supported by NSF Grant DMS 9102564Research supported by NSF Grant PHY9200278.Research supported by the Natural Sciences and Engineering Research Council of Canada.     

Renormalized Field Theory of Resistor Diode Percolation

We study resistor diode percolation at the transition from the non-percolating to the directed percolating phase. We derive a field theoretic Hamiltonian which describes not only geometric aspects of directed percolation clusters but also their electric transport properties. By employing renormalization group methods we determine the average two-port resistance of critical clusters, which is governed by a resistance exponent . We calculate to two-loop order.     

The Renormalization Group Treatment of the Hubbard Model

The article presents the renormalization group treatment to the Hubbard model. To begin with, the bosonization of Hubbard model Hamiltonian is performed. We have obtained the sine-Gordon Hamiltonian. We have further approximated this Hamiltonian by the Hamiltonian of ⁴-theory. Then we utilized Wilson's results of the renormalization group method and obtained the recursion formula for the Hubbard model. Having solved these formulas we have obtained the critical indices for the Hubbard model.     

Quantum field-theory models on fractal spacetime

Continuing the analysis of a previous paper, the present work applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non-Gaussian ultraviolet renormalization group fixed points in 4- dimensions.     

Low temperature properties of the hierarchical classical vector model

We obtain low temperature properties of the classical vector model in a hierarchical formulation in three or more dimensions. We consider the lattice model in a zero or non-zero magnetic field, where the single site spin variable R ^v has a density proportional to for large . Using renormalization group methods we obtain a convergent expansion for the free energy with zero magnetic field. For non-zero fields a shift formula is used to obtain the effective action generated by the renormalization group transformation (RGT). To obtain the pure state zero field free energy and spontaneous magnetization we take the thermodynamic limit together with the zero field limit at a specified rate. The spontaneous magnetization,m, is calculated, is non-zero and the pure state free energy coincides, as expected, with the zero field free energy. Also the sequence of zero field actions does not have a limit but we show that the sequence of actions generated from the original action shifted bym does; the limiting action corresponds to a non-canonical Gaussian fixed point of the RGT.     

Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State

We present a renormalization group construction of a weakly interacting Bose gas at zero temperature in the two-dimensional continuum, both in the quantum critical regime and in the presence of a condensate fraction. The construction is performed within a rigorous renormalization group scheme, borrowed from the methods of constructive field theory, which allows us to derive explicit bounds on all the orders of renormalized perturbation theory. Our scheme allows us to construct the theory of the quantum critical point completely, both in the ultraviolet and in the infrared regimes, thus extending previous heuristic approaches to this phase. For the condensate phase, we solve completely the ultraviolet problem and we investigate in detail the infrared region, up to length scales of the order \((\lambda ^3\rho _0)^{-1/2}\) (here \(\lambda \) is the interaction strength and \(\rho _0\) the condensate density), which is the largest length scale at which the problem is perturbative in nature. We exhibit violations to the formal Ward Identities, due to the momentum cutoff used to regularize the theory, which suggest that previous proposals about the existence of a non-perturbative non-trivial fixed point for the infrared flow should be reconsidered.     

Construction of the two-dimensional sine-Gordon model for β<8π

We present a rigorous renormalization group construction of the two-dimensional massless and massive quantum sine-Gordon models in finite volume for the range 0<<8. We prove analyticity in the coupling constant , which implies the convergence of perturbation theory. The field correlation functions and their generating functional are analyzed and shown to have the short distance asymptotics of the free field theory. In the massive case the bounds are uniform in volume and we also obtain uniform estimates on the long distance decay of correlations.Research supported by NSF Grant PHY-9001178Research supported by the Natural Sciences and Engineering Research Council of Canada     

A non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories

A rigorous method is developed to handle the large field problems in the Wilson-Kadanoff renormalization group approach to critical lattice systems of unbounded spins. We use this method to study in a hierarchical approximation the non-Gaussian renormalization group fixed point which governs the infrared behaviour of critical lattice field theories in three dimensions. The method is an improvement of the analyticity techniques of Gawedzki and Kupiainen: using Borel summation techniques we are able to incorporate the large field region into the perturbative region so that the theory is completely described in terms of convergent expansions.Supported in part by the National Science Foundation under Grant No. DMS-8540879Supported in part by the National Science Foundation under Grant No. DMR81-14726Part of this work has been carried out during a visit of the author at the Courant Institute of Mathematical Sciences, NYU     

Group-theoretic description of external gauge fields

For an arbitrary external gauge field we construct an infinite group which contains all the information about the given field and describes some of its properties. We construct a field representation of the group . We show that covariant derivatives become translation generators in such a representation of the group . This allows us to interpret transformations from the group as motions in an external gauge field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 31–35, December, 1988.     

Quasi-Linear Quantum Field Theories for Maps to Groups and Their Quotients

I describe a functional integral for maps from to a Lie group or its quotient which has a simple renormalization that leads to a quantum field theory for maps from into the Lie group or its quotient whose Hamiltonian is the time translation generator for a unitary action of the n+1 dimensional Poincaré group on the quantum Hilbert space. I also explain how the renormalization provides a functional integral for maps from a Riemann surface to a compact Lie group or its quotient that exhibits many conformal field theoretic properties.Support in part by a grant from the National Science Foundation     

Equation of motion and nonlinear response near the critical point

For a one-component (Ising-like) Ginzburg-Landau field we have derived an equation of motion which governs the behaviour of the order parameter nearT _c . We present this equation in a general scaling form dictated by the renormalization group, and calculate the form explicitly to first order in=4–d. There is a memory term in the result which in order contains the full nonlinear response of the order parameter. As a special case we obtain the linear response function for a nonvanishing external field.     

On a Mathematical Definition of a Renormalization Transformation for Lattice Gauge Theories

In lattice gauge theories, the renormalization transformation and its properties are formally defined and formally proved by making use of Dirac's function and its properties. In this Letter, we shall give a mathematically rigorous definition of a renormalization transformation for lattice pure gauge field theories and show the required properties, which are use to show ultraviolet stability of lattice gauge theories.     

A renormalization group analysis of lattice models of two-dimensional membranes

We study lattice models of two-dimensional membranes of interest in statistical physics. The energy functional of a membrane is expressed as a sum of terms proportional to the surface area of the membrane, an extrinsic curvature and an intrinsic curvature quantity, respectively, but we neglect excluded volume effects. We introduce a renormalization transformation for these models which preserves the form of the energy functional, up to nonlocal terms. Our renormalization group construction is used to analyze the phase diagram and the different critical regimes of our models. We find evidence for a crumpling transition, separating a regime where surfaces are crystalline from one where the surfaces collapse to branched polymers, and we find a third genuine random-surface regime.     

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice ℤ3

We consider an Euclidean supersymmetric field theory in ℤ³ given by a supersymmetric Φ⁴ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in ℤ³. The Green’s function depends on the Lévy-Khintchine parameter with 0<α<2. For the Φ⁴ interaction is marginal. We prove for sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in ℤ³ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in ℤ³.     

Convergence of Perturbation Expansions in Fermionic Models. Part 1: Nonperturbative Bounds

An estimate on the operator norm of an abstract fermionic renormalization group map is derived. This abstract estimate is applied in another paper to construct the thermodynamic Greens functions of a two dimensional, weakly coupled fermion gas with an asymmetric Fermi curve. The estimate derived here is strong enough to control everything but the sum of all quartic contributions to the Greens functions.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschungsinstitut für Mathematik, ETH Zürich.