Renormalisation of hierarchically interacting Cannings (original) (raw)
The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, 2010
As applied to quantum theories, the program of renormalization is successful for 'renormalizable models' but fails for 'nonrenormalizable models'. After some conceptual discussion and analysis, an enhanced program of renormalization is proposed that is designed to bring the 'nonrenormalizable models' under control as well. The new principles are developed by studying several, carefully chosen, soluble examples, and include a recognition of a 'hard-core' behavior of the interaction and, in special cases, an extremely elementary procedure to remove the source of all divergences. Our discussion provides the background for a recent proposal for a nontrivial quantization of nonrenormalizable scalar quantum field models, which is briefly summarized as well. Dedication: It is a pleasure to dedicate this article to the memory of Prof. Alladi Ramakrishnan who, besides his own important contributions to science, played a crucial role in the development of modern scientific research and education in his native India. Besides a number of recent informative discussions during his yearly visits to the University of Florida, the present author had the pleasure much earlier of hosting Prof. Alladi during his visit and lecture at Bell Telephone Laboratories.
Renormalization and the two-parameter theory
Macromolecules, 1984
A simple approximate method is given for converting two-parameter (TP) calculations in three dimensions to renormalization group (RG) expressions. These results are provided in a representation similar to the TP theory, and they describe the whole range of excluded volume interaction. We call the new representation the renormalized two-parameter (RTP) theory. Expressions in this notation are given for as2, ap-', CYH, a,,, h, (Rir), (IR,,l-'), A 2 M / [ q ] , and 9 for the linear polymer and for as2, h, and \k for rings, regular uniform combs, and regular stars based upon information derived from a general RG analysis and well-known first-order TP calculations. Since available dynamical TP calculations are performed in the preaveraging approximation and in the non-free-draining limit, our calculations likewise reflect these assumptions. The absence of dynamical TP calculations for stars and combs, even with these approximations, leads us to introduce a semiempirical approach to calculating a,,, aH, and A&/ [?] which uses available good solvent data for these branched polymers. Comparisons are presented between theory and experiment for the good solvent limit. A future paper will deal with the intermediate "crossover" regime.
Studies in History and Philosophy of Modern Physics, 2019
In this paper, I propose a general framework for understanding renormalization by drawing on the distinction between effective and continuum Quantum Field Theories (QFTs), and offer a comprehensive account of perturbative renormalization on this basis. My central claim is that the effective approach to renormalization provides a more physically perspicuous, conceptually coherent and widely applicable framework to construct perturbative QFTs than the continuum approach. I also show how a careful comparison between the two approaches: (i) helps to dispel the mystery surrounding the success of the renormalization procedure; (ii) clarifies the various notions of renormalizability; and (iii) gives reasons to temper Butterfield and Bouatta's claim that some continuum QFTs are ripe for metaphysical inquiry (Butterfield and Bouatta, 2014).
Three topics in renormalization and improvement
Modern Perspectives in Lattice QCD: Quantum Field Theory and High Performance ComputingLecture Notes of the Les Houches Summer School: Volume 93, August 2009, 2011
Introduction 2 Basics 2.1 The Wilson lattice action and its symmetries 2.2 Quark mass renormalization 2.3 Quark propagator 2.4 Quark bilinear operators (composite fields) 3 Lattice Ward identities 3.1 Lattice Vector Ward identities 3.2 Lattice Axial Ward identities 3.3 Hadronic Ward identities for Z V and Z A 3.4 Hadronic Ward identity for the ratio Z S /Z P 3.5 Singlet scalar and pseudoscalar operators 3.6 The chiral condensate 4 Momentum subtraction (MOM) schemes 4.1 The RI/MOM scheme 4.2 Goldstone pole contamination 4.3 RI/MOM scheme and Ward Identities 4.4 The RI/SMOM scheme 5 Twisted mass QCD (tmQCD), renormalization and improvement 5.1 Classical tmQCD 5.2 Lattice tmQCD 5.3 Renormalization with tmQCD 5.4 "Automatic" improvement in maximally twisted QCD 6 Conclusions 7 Appendix: Spurionic chiral symmetry 8 Appendix: Lattice discrete symmetries 9 Appendix: Regularization dependent scheme References The present set of lecture notes are based on a short course, delivered at the XCIII Les Houches Summer School (August 2009). The exciting atmosphere of the School and the students' enthusiasm have encouraged me to write up a significantly expanded version of the original course. I thank my colleagues at Les Houches, students and lecturers alike, for their stimulating and supportive attitude. I also thank Ben Svetitsky for his constant (pure malt) spiritual support during the preparation of my lectures at Les Houches. I am grateful to Giancarlo Rossi, Rainer Sommer and Chris Sachrajda for carefully reading the manuscript, for their numerous useful suggestions, and for their constructive criticism. I am especially indebted to Massimo Testa, for his patience and constant advise during several long encounters, throughout the various phases of preparation of this manuscript. If these lectures prove useful to young lattice researchers, it is largely thanks to the help provided by these colleagues. Naturally, responsibility for any remaining shortcomings rests entirely with the author. 2 Basics Although we understand that the student is familiar with the basics of quark mass and composite operator renormalization, we recapitulate them here for completeness and in order to fix our notation. We also collect several useful definitions; although this is somewhat tedious, it is important to spell out the not-so-standard notation right from the beginning. Concerning notation, we have preferred economy to mathematical rigour. Since the various bare quantities discussed below are defined in the lattice regularization, integrals like say, d 4 x 1 d 4 x 2 of eq. (2.28), are really sums (a 8 x1,x2), which run over all lattice sites, labelled by x 1 and x 2 etc. The occasional use of integrals instead of sums, partial derivatives instead of finite differences, and Dirac functions instead of Kronecker symbols, simplifies notation, hopefully avoiding any confusion. Moreover, a space-time function and its Fourier-transform will be indicated by the same symbol with a different argument (e.g. f (x) and f (p)); again mathematical rigour is being sacrificed in favour of notational economy. 2.1 The Wilson lattice action and its symmetries We opt for the lattice regularization scheme, proposed by Wilson, which consists of a gluonic action 1 (Wilson, 1974),
Phenomenological renormalization group methods
Brazilian Journal of Physics, 1999
Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of the model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more applied an extended to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also addressed.
Renormalization group improving the effective action
Arxiv preprint hep-th/ …, 1998
The existence of fluctuations together with interactions leads to scale-dependence in the couplings of quantum field theories for the case of quantum fluctuations, and in the couplings of stochastic systems when the fluctuations are of thermal or statistical nature. In both cases the effects of these fluctuations can be accounted for by solutions of the corresponding renormalization group equations. We show how the renormalization group equations are intimately connected with the effective action: given the effective action we can extract the renormalization group equations; given the renormalization group equations the effects of these fluctuations can be included in the classical action by using what is known as improved perturbation theory (wherein the bare parameters appearing in tree-level expressions are replaced by their scale-dependent running forms). The improved action can then be used to reconstruct the effective action, up to finite renormalizations, and gradient terms.
Position-Space Renormalization Group for Directed Branched Polymers
All previous attempts to obtain accurate quantitative estimates of critical properties for directed percolation or directed animals with position-space renormalisation group (PSRG) have failed. We analyse the problems that appear in the renormalisation of directed models, and then present a PSRG that avoids these problems and gives reliable predictions. @ 1983 The Institute of Physics L375 Hornreich R M, Luban M and Shtrikman S 1975 Phys. Rev. Lett. 35 1678 Kinzel W and Yeomans J 1981 1. Phys. A: Math. Gen. 14 L163 Martin J L 1974 Phase transitions and crirical phenomena ed C Domb and M S Green vol3, p 97 Nadal J P, Derrida B and Vannimenus J 1982 J. Physique 43 1561 Oliveira P M 1983 Preprint Phani M K and Dhar D 1982 J. Phys. C: Solid Stare Phys. 15 1391 Redner S 1981 J. Phys. A: Math. Gen. 14 L349 -1982a Phys. Rev. B 25 3242 -1982b Phys. Rev. B 25 5646 Redner S and Brown A C 1981 J. Phys. A: Marh. Gen. 14 L285 Redner S and Yang Z R 1982 J. Phys. A: Math. Gen. 15 L177 Stanley H E, Reynolds P J, Redner S and Family F 1982 in Real-space renormalization ed T Burkhardt and J M J van Leeuwen (New York: Springer) p 169
Renormalization Group Analysis of the Random First-Order Transition
Physical Review Letters, 2011
We consider the approach describing glass formation in liquids as a progressive trapping in an exponentially large number of metastable states. To go beyond the mean-field setting, we provide a real-space renormalization group (RG) analysis of the associated replica free-energy functional. The present approximation yields in finite dimensions an ideal glass transition similar to that found in mean field. However, we find that along the RG flow the properties associated with metastable glassy states, such as the configurational entropy, are only defined up to a characteristic length scale that diverges as one approaches the ideal glass transition. The critical exponents characterizing the vicinity of the transition are the usual ones associated with a first-order discontinuity fixed point.
Using the renormalization group
Universality and Renormalization, 2007
In computing quantum effects, it is necessary to perform a sum over all intermediate states consistent with prescribed initial and final states. Divergences arising in the course of evaluating this sum forces one to "renormalize" parameters characterizing the system. An ambiguity inherent in this rescaling is parameterized by a dimensionful parameter µ 2 which serves to set a scale for the process. Requiring that the explicit and implicit dependence of a physical quantity on µ 2 conspire to cancel leads to the so-called "renormalization group" equation [1-10]. It has proved possible to extract a lot of useful information from this equation; we will enumerate a number of these in this report.