Non-perturbative renormalization

303 Pages·2008·4.037 MB·English

by  Vieri Mastropietro

NON-PERTURBATIVE
RENORMALIZATION
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
NON-PERTURBATIVE
RENORMALIZATION
Vieri Mastropietro
Università di Roma “Tor Vergata”, Italy
World Scientific
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NON-PERTURBATIVE RENORMALIZATION
Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-279-239-6
ISBN-10 981-279-239-2
Printed in Singapore.
Benjamin - Non-perturbative.pmd 1 12/27/2007, 12:05 PM
December24,2007 10:57 WorldScienti(cid:12)cBook-9inx6in libro
Preface
The notion of renormalizationis at the coreof severalspectacular achieve-
mentsofcontemporaryphysics;originatedinthecontextofQuantumField
Theory,whereit appearedin orderto solvethe problem of the \ultraviolet
divergences",it becomeslateron(partlyinaformknownas\renormaliza-
tion group") centralin manyother areas,likein the analysisof the critical
properties close to phase transitions in classical statistical mechanics or in
the theory of quantum liquids in condensed matter.
Renormalizationisgenerallypresentedinapurelyperturbativecontext
(with no control of convergence of the series expansions), but in the last
years, new mathematical techniques have been developed, allowing to put
it on a (cid:12)rm mathematical basis. The aim of this book is to provide an in-
troduction to rigorousnon-perturbativerenormalizationin QuantumField
Theory, Statistical Physics and Condensed Matter. With respect to pre-
vious books on renormalization, the focus is mainly on fermionic (rather
than bosonic) functional integrals, whose theory has been developed more
recently and for which the structure of renormalization is not obscured by
too many technicalities. Another important novelty is the implementation
of Ward Identities based on local symmetries in the context of multiscale
analysis,whichallowstherigorousanalysisofmodelswith nontrivial(cid:12)xed
pointsandanomalousbehaviour. Thebookisdevotedeithertomathemati-
cians and physicists aiming to enter into contact with the modern theory
of renormalization; prerequisites are then limited to a minimum.
We start with an introduction to renormalizationin physics and to the
mathematicaltechniquesfortreatingfermionicfunctionalintegrals,includ-
ing multiscale decomposition techniques, tree expansions and determinant
v
December24,2007 10:57 WorldScienti(cid:12)cBook-9inx6in libro
vi NON-PERTURBATIVERENORMALIZATION
bounds. Such methods allow a uni(cid:12)ed treatment of models coming from
QuantumField Theory,Statistical Physicsand Condensedmatter. In par-
ticular,the(cid:12)rstpartofthisbookisdevotedtoconstructiveQuantumField
Theory,providingamathematicalconstructionofmodelsatlowdimensions
and discussing the removal of the ultraviolet and infrared cut-o(cid:11), the veri-
(cid:12)cation of the axioms and the validity of Ward Identities with the relative
anomalies. The second part is devoted to lattice 2d Statistical Physics,
analyzing in particular the theory of universality in perturbed Ising mod-
els and the computation of the non-universal critical indices in Vertex or
Ashkin-Teller models. Finally in the third part, the theory of Quantum
liquids like Luttinger or Fermi liquids is developed, considering models of
interest in Condensed Matter like the Hubbard model in 1d or 2d or the
Heisenberg spin chain.
Mostofthematerialpresentedinthisbookgrewoutfromcommonwork
with G. Benfatto and G. Gallavotti, and with the researchers composing
theRomanschoolofrigorousrenormalization,namelyF.Bonetto,P.Falco,
G.Gentile,G.Giuliani,A.ProcacciandB.Scoppola. Ihavealsobene(cid:12)tted
from important discussions, which stronglyin(cid:13)uenced my point of view on
renormalization,withJ.MagnenandV.RivasseauinParis,withT.Spencer
in Princeton and with K. Gawedzki in Lion.
Vieri Mastropietro
December24,2007 10:57 WorldScienti(cid:12)cBook-9inx6in libro
Contents
Preface v
Introduction to Renormalization 1
1. Basic Notions 3
1.1 Relativistic quantum (cid:12)eld theory . . . . . . . . . . . . . . 3
1.1.1 Quantum (cid:12)elds . . . . . . . . . . . . . . . . . . . 3
1.1.2 Functional integrals . . . . . . . . . . . . . . . . . 5
1.1.3 Perturbative renormalization . . . . . . . . . . . . 8
1.2 Classical statistical mechanics . . . . . . . . . . . . . . . . 12
1.2.1 Phase transitions . . . . . . . . . . . . . . . . . . 12
1.2.2 Universality and non-universality . . . . . . . . . 14
1.3 Condensed Matter . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Electrons in a crystal . . . . . . . . . . . . . . . . 16
1.3.2 The free Fermi gas. . . . . . . . . . . . . . . . . . 19
1.3.3 Fermi liquids . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Luttinger liquids and BCS superconductors . . . . 23
2. Fermionic Functional Integrals 27
2.1 Grassmann variables . . . . . . . . . . . . . . . . . . . . . 27
2.2 Grassmann measures . . . . . . . . . . . . . . . . . . . . . 29
2.3 Truncated expectations . . . . . . . . . . . . . . . . . . . 31
2.4 Properties of Grassmann integrals . . . . . . . . . . . . . 32
2.5 Gallavotti-Nicolo(cid:19)tree expansion . . . . . . . . . . . . . . 33
2.6 Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . 39
2.7 Determinant bounds for simple expectations . . . . . . . . 42
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viii NON-PERTURBATIVERENORMALIZATION
2.8 The Brydges-Battle-Federbushrepresentation . . . . . . . 45
2.9 The Gawedzki-Kupiainen-Lesniewskiformula . . . . . . . 51
Quantum Field Theory 57
3. The Ultraviolet Problem in Massive QED2 59
3.1 Regularization and cut-o(cid:11)s . . . . . . . . . . . . . . . . . 59
3.2 Integration of the bosons . . . . . . . . . . . . . . . . . . 61
3.3 Propagatordecomposition . . . . . . . . . . . . . . . . . . 63
3.4 Renormalized expansion . . . . . . . . . . . . . . . . . . . 66
3.5 Feynman graph expansion . . . . . . . . . . . . . . . . . . 68
3.6 Convergenceof the renormalized expansion . . . . . . . . 69
3.7 Determinant bounds . . . . . . . . . . . . . . . . . . . . . 72
3.8 The short memory property . . . . . . . . . . . . . . . . . 75
3.9 Extraction of loop lines . . . . . . . . . . . . . . . . . . . 75
3.10 The 2-point Schwinger function . . . . . . . . . . . . . . . 79
3.11 The Yukawa model . . . . . . . . . . . . . . . . . . . . . . 79
4. Infrared Problem and Anomalous Behavior 81
4.1 Anomalous dimension . . . . . . . . . . . . . . . . . . . . 81
4.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Modi(cid:12)cation of the fermionic interaction . . . . . . . . . . 83
4.4 Bounds for the renormalized expansion. . . . . . . . . . . 89
4.5 The beta function at lowest orders . . . . . . . . . . . . . 96
4.6 Boundedness of the (cid:13)ow . . . . . . . . . . . . . . . . . . . 99
4.7 The 2-point Schwinger function . . . . . . . . . . . . . . . 101
5. Ward Identities and Vanishing of the Beta Function 105
5.1 Schwinger functions and running couplings . . . . . . . . 105
5.2 Ward identities in presence of cut-o(cid:11)s . . . . . . . . . . . 107
5.3 The correction identity . . . . . . . . . . . . . . . . . . . . 109
5.4 The Schwinger-Dysonequation . . . . . . . . . . . . . . . 115
5.5 Analysis of the cut-o(cid:11) corrections . . . . . . . . . . . . . . 118
5.6 Vanishing of Beta function. . . . . . . . . . . . . . . . . . 120
5.7 Non-perturbative Adler-Bardeen theorem . . . . . . . . . 122
5.8 Further remarks . . . . . . . . . . . . . . . . . . . . . . . 123
6. Thirring and Gross-NeveuModels 125
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Contents ix
6.1 The Thirring model . . . . . . . . . . . . . . . . . . . . . 125
6.2 Removing the fermionic ultraviolet cut-o(cid:11) before the
bosonic one . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Removing the bosonic ultraviolet cut-o(cid:11) before the
fermionic one . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 The Gross-Neveumodel . . . . . . . . . . . . . . . . . . . 132
7. Axioms Veri(cid:12)cation and Wilson Fermions 133
7.1 Osterwalder-Schraderaxioms . . . . . . . . . . . . . . . . 133
7.2 Lattice regularization and fermion doubling . . . . . . . . 135
7.3 Integration of the doubled fermions . . . . . . . . . . . . . 137
7.4 Lattice fermions . . . . . . . . . . . . . . . . . . . . . . . 138
8. Infraed QED4 with Large Photon Mass 143
8.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 Tree expansion . . . . . . . . . . . . . . . . . . . . . . . . 145
Lattice Statistical Mechanics 147
9. Universality in Generalized Ising Models 149
9.1 The nearest neighbor Ising model . . . . . . . . . . . . . . 149
9.2 Heavy and light Majorana fermions. . . . . . . . . . . . . 153
9.3 Generalized Ising models. . . . . . . . . . . . . . . . . . . 156
9.4 Fermionic representation of the generalized Ising model . 157
9.5 Integration of the (cid:31)-variables . . . . . . . . . . . . . . . . 159
9.6 Integration of the light fermions. . . . . . . . . . . . . . . 160
9.7 Correlation functions and the speci(cid:12)c heat . . . . . . . . . 164
10. Nonuniversality in Vertex or Isotropic Ashkin-Teller Models 165
10.1 Ashkin-Teller or Vertex models . . . . . . . . . . . . . . . 165
10.2 Fermionic representation . . . . . . . . . . . . . . . . . . . 167
10.3 Anomalous behaviour . . . . . . . . . . . . . . . . . . . . 170
10.4 Simmetry properties . . . . . . . . . . . . . . . . . . . . . 171
10.5 Integration of the light fermions. . . . . . . . . . . . . . . 175
10.6 The speci(cid:12)c heat . . . . . . . . . . . . . . . . . . . . . . . 177
11. Universality-Nonuniversality Crossover in the Ashkin-
Teller Model 181