scholarly journals Scaling Limits of Lattice Quantum Fields by Wavelets

2021 ◽  
Author(s):  
Vincenzo Morinelli ◽  
Gerardo Morsella ◽  
Alexander Stottmeister ◽  
Yoh Tanimoto
Keyword(s):  
Renormalization Group ◽  
Inductive Limit ◽  
Operator Algebras ◽  
Limit State ◽  
Free Field ◽  
Group Scheme ◽  
Quantum Field Theories ◽  
Inductive System ◽  
Lattice Quantum ◽  
The Continuum

AbstractWe present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras. The renormalization group is considered as an inductive system of scaling maps between lattice field algebras. We construct scaling maps for scalar lattice fields using Daubechies’ wavelets, and show that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field, with the continuum action of spacetime translations. In particular, lattice fields are identified with the continuum field smeared with Daubechies’ scaling functions. We compare our scaling maps with other renormalization schemes and their features, such as the momentum shell method or block-spin transformations.

scholarly journals Variational Solution of the Gross–Neveu Model: Finite N and Renormalization

1997 ◽  
Vol 12 (19) ◽  
pp. 3307-3334 ◽  
Author(s):  
C. Arvanitis ◽  
F. Geniet ◽  
M. Iacomi ◽  
J.-L. Kneur ◽  
A. Neveu
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
General Framework ◽  
Free Field ◽  
Variational Calculation ◽  
Final Point ◽  
Variational Calculations ◽  
Perturbative Renormalization ◽  
Good Agreement ◽  
Gross Neveu Model

We show how to perform systematically improvable variational calculations in the O(2N) Gross–Neveu model for generic N, in such a way that all infinities usually plaguing such calculations are accounted for in a way compatible with the perturbative renormalization group. The final point is a general framework for the calculation of nonperturbative quantities like condensates, masses, etc., in an asymptotically free field theory. For the Gross–Neveu model, the numerical results obtained from a "two-loop" variational calculation are in a very good agreement with exact quantities down to low values of N.

scholarly journals Renormalization group consistency and low-energy effective theories

2019 ◽  
Vol 6 (5) ◽  
Author(s):  
Jens Braun ◽  
Marc Leonhardt ◽  
Jan M. Pawlowski
Keyword(s):  
Renormalization Group ◽  
Mean Field ◽  
Bound State ◽  
Field Approximation ◽  
Low Energy ◽  
Mean Field Approximation ◽  
Quantum Field Theories ◽  
Model Calculations ◽  
Effective Models ◽  
Effective Theories

Low-energy effective theories have been used very successfully to study the low-energy limit of QCD, providing us with results for a plethora of phenomena, ranging from bound-state formation to phase transitions in QCD. These theories are consistent quantum field theories by themselves and can be embedded in QCD, but typically have a physical ultraviolet cutoff that restricts their range of validity. Here, we provide a discussion of the concept of renormalization group consistency, aiming at an analysis of cutoff effects and regularization-scheme dependences in general studies of low-energy effective theories. For illustration, our findings are applied to low-energy effective models of QCD in different approximations including the mean-field approximation. More specifically, we consider hot and dense as well as finite systems and demonstrate that violations of renormalization group consistency affect significantly the predictive power of the corresponding model calculations.

Estimates on Renormalization Group Transformations

1998 ◽  
Vol 50 (4) ◽  
pp. 756-793 ◽  
Author(s):  
D. Brydges ◽  
J. Dimock ◽  
T. R. Hurd
Keyword(s):  
Renormalization Group ◽  
Gaussian Measure ◽  
Ultra Violet ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Quantum Fields ◽  
Scalar Quantum ◽  
Polymer Expansion ◽  
Renormalization Group Flows

AbstractWe consider a specific realization of the renormalization group (RG) transformation acting on functional measures for scalar quantum fields which are expressible as a polymer expansion times an ultra-violet cutoff Gaussian measure. The new and improved definitions and estimates we present are sufficiently general and powerful to allow iteration of the transformation, hence the analysis of complete renormalization group flows, and hence the construction of a variety of scalar quantum field theories.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

scholarly journals New numerical methods for quantum field theories on the continuum

2000 ◽  
Vol 83-84 ◽  
pp. 938-940 ◽  
Author(s):  
P. Emirdaǧ ◽  
R. Easther ◽  
G.S. Guralnik ◽  
S.C. Hahn
Keyword(s):  
Numerical Methods ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
The Continuum

THOMPSON's METHOD APPLIED TO THE BROWNIAN AND NON-BROWNIAN DIFFUSION-CONTROLLED REACTIONS OF TYPE kA → lA (l < k)

2002 ◽  
Vol 16 (17) ◽  
pp. 601-613 ◽  
Author(s):  
CLÁUDIO NASSIF ◽  
P. R. SILVA
Keyword(s):  
Renormalization Group ◽  
Good Description ◽  
Wave Length ◽  
Group Scheme ◽  
Single Species ◽  
Critical Dimension ◽  
Stationary Regime ◽  
Brownian Diffusion ◽  
Diffusion Controlled ◽  
Long Time

The diffusion-controlled reactions of type kA → lA, with l < k, including the case of the annihilation reaction kA → 0, are studied by using Thompson's method, both in the case of Brownian (γ = 2) as in the case of anomalous (γ ≠ 2) diffusion conditions. These reactions are known to be strongly dependent on fluctuations below some critical dimension dc. We find dc = γ/(k-1) and the asymptotic behavior of density [Formula: see text]. At d = dc, the density goes as <∊> ~ ( ln t/t)1/k-1. For γ = 2, the scaling results obtained here are in agreement with the renormalization group calculations of Lee15 and a more recent work of Oliveira.8 We go further by also studying the case of an external homogeneous source (h) of single species particles A in the case of the stationary regime. Then we obtain the critical exponent δ in [Formula: see text], the dynamical exponent for the relaxation time Δ′ in (τh ~ hΔ′) and the exponent for the concentration decay ξ in (∊ ~ τ- ξ), with all these quantities evaluated in the limit h → 0. We get relations of scaling among critical indexes, and, in the special case of γ = 2, we recover those results previously obtained by Rácz.30 Thompson's method is a simple alternative way to the renormalization group scheme and has been shown to be a good description for the long-time (long wave-length) regime.

scholarly journals RENORMALIZATION WITHOUT INFINITIES

2005 ◽  
Vol 20 (06) ◽  
pp. 1336-1345 ◽  
Author(s):  
GERARD 'T HOOFT
Keyword(s):  
Renormalization Group ◽  
Conceptual Understanding ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Renormalization Group Equations

Most renormalizable quantum field theories can be rephrased in terms of Feynman diagrams that only contain dressed irreducible 2-, 3-, and 4-point vertices. These irreducible vertices in turn can be solved from equations that also only contain dressed irreducible vertices. The diagrams and equations that one ends up with do not contain any ultraviolet divergences. The original bare Lagrangian of the theory only enters in terms of freely adjustable integration constants. It is explained how the procedure proposed here is related to the renormalization group equations. The procedure requires the identification of unambiguous "paths" in a Feynman diagrams, and it is shown how to define such paths in most of the quantum field theories that are in use today. We do not claim to have a more convenient calculational scheme here, but rather a scheme that allows for a better conceptual understanding of ultraviolet infinities.

RENORMALIZATION OF THE SINE-GORDON MODEL AND NONCONSERVATION OF THE KINK CURRENT

1991 ◽  
Vol 06 (03) ◽  
pp. 409-429 ◽  
Author(s):  
KERSON HUANG ◽  
JANOS POLONYI
Keyword(s):  
Renormalization Group ◽  
Phase Structure ◽  
Continuum Limit ◽  
Renormalization Group Flow ◽  
Small Coupling ◽  
Vortex Density ◽  
Euclidean Lattice ◽  
Sine Gordon Model ◽  
Group Flow ◽  
The Continuum

We renormalize the (1+1)-dimensional sine-Gordon model by placing it on a Euclidean lattice, and study the renormalization group flow. We start with a compactified theory with controllable vortex activity. In the continuum limit the theory has a phase in which the kink current is anomalous, with divergence given by the vortex density. The phase structure is quite complicated. Roughly speaking, the system is normal for small coupling T. At the Kosterlitz-Thouless point T=π/2, the current can become anomalous. At the Coleman point T=8π, either the current becomes anomalous or the theory becomes trivial.

Attempered renormalization-group scheme for the SU(2)-Higgs model

1987 ◽  
Vol 35 (12) ◽  
pp. 4031-4033 ◽  
Author(s):  
David J. E. Callaway ◽  
Randall C. Furlong ◽  
Roberto Petronzio
Keyword(s):  
Renormalization Group ◽  
Group Scheme ◽  
Higgs Model
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