Defects in higher-dimensional quantum field theory [Elektronische Ressource] : relations to AdS-CFT-Correspondence and kondo lattices / by Robert Schmidt

Robert Schmidt

Sujets

• Physik

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2007
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 7 Mo

Extrait

Defects in Higher-Dimensional
Quantum Field Theory
Relations to AdS/CFT-Correspondence
and Kondo Lattices
Robert Schmidt
Ludwig-Maximilians-Universit at MunchenDefects in Higher-Dimensional
Quantum Field Theory
Relations to AdS/CFT-Correspondence
and Kondo Lattices
A Dissertation Presented to the
Faculty of Physics of
Ludwig-Maximilians-Universit at Munc hen
in Candidacy for the Degree of doctor rerum naturalium by
Robert Schmidt
from Jena
Munich, March 2007Referee I: Johanna Erdmenger II: Dieter Lus t
Date of oral examination: July 17, 2007Zusammenfassung
DievorliegendeArbeitbefa tsichvordemHintergrundderAdS/CFT-Korres-
pondenz mit Defekten beziehungsweise R andern in der Quantenfeldtheorie.
WiruntersuchendieWechselwirkungenvonFermionenmitaufdiesenDefekten
lokalisierten Spins. Dazu wird eine Methode weiterentwickelt, die die kanonis-
che Quantisierungsvorschrift um Re exions- und Transmissionsterme erg anzt
und fur Bosonen in zwei Raum-Zeit-Dimensionen bereits Anwendung fand.
Wir er ortern die M oglichkeiten derartiger Re exions-Transmissions-Algebren
in zwei, drei und vier Dimensionen. Wir vergleichen mit Modellen aus der
Festkorp ertheorieundderBeschreibungdesKondo-E ektesmithilfekonformer
Feldtheorie.
Wir diskutieren ferner Ans atze der Erweiterung auf Gitterstrukturen.
56Abstract
The present work is addressed to defects and boundaries in quantum eld
theory considering the application to AdS/CFT correspondence.
We examine interactions of fermions with spins localised on these boundaries.
Therefore,analgebramethodisemphasisedaddingre ectionandtransmission
terms to the canonical quantisation prescription. This method has already
been applied to bosons in two space-time dimensions before. We show the
possibilities of such re ection-transmission algebras in two, three, and four
dimensions. We compare with models of solid state physics as well as with the
conformal eld theory approach to the Kondo e ect.
Furthermore, we discuss ansatzes of extensions to lattice structures.
78Dans la boucle de l’hirondelle un orage s’informe, un jardin se
construit.
(In der Schleife des Schwalben ugs fugt sich Gewitter, gestalten
sich Garten.)
Rene Char, A la sante du serpent
Contents
Zusammenfassung 5
Abstract 7
1 Introduction 15
Main Ideas of AdS/CFT Correspondence . . . . . . . . . . . . . . . . 15
Application in Solid State Physics . . . . . . . . . . . . . . . . . . . . 18
Dealing with Defects and Boundaries . . . . . . . . . . . . . . . . . . 21
Defects and Boundaries in RT Algebra Formalism . . . . . . . . . . . 22
2 Defects in QFT and RT Formalism 25
2.1 De nition of an RT Algebra . . . . . . . . . . . . . . . . . . . . 27
2.2 RT Formalism in Bosonic Theory . . . . . . . . . . . . . . . . . 30
2.2.1 Properties of the RT Algebra . . . . . . . . . . . . . . . 30
2.2.2 Properties of the Theory in Terms of Re ection and
Transmission . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Matrix Optics for Bosons . . . . . . . . . . . . . . . . . . . . . . 39
3 Fermionicδ Defects 43
910 CONTENTS
3.1 Interaction Terms and Conventions . . . . . . . . . . . . . . . . 44
3.1.1 Lagrangean . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.2 Boundary Condition . . . . . . . . . . . . . . . . . . . . 46
3.1.3 Many-Particle Statistics . . . . . . . . . . . . . . . . . . 48
3.2 δ Defects in Two-Dimensional Fermionic Theory . . . . . . . . . 49
3.2.1 Boundary Condition . . . . . . . . . . . . . . . . . . . . 49
3.2.2 Quantisation with RT Algebra . . . . . . . . . . . . . . . 50
3.2.3 RT Coe cients . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.4 Gibbs States and Expectation Values . . . . . . . . . . . 55
3.2.5 Conserved Quantities . . . . . . . . . . . . . . . . . . . . 56
3.2.6 Comparing with the One-Impurity CFT Approach . . . . 69
3.3 RT Results in Three Dimensions . . . . . . . . . . . . . . . . . . 70
3.3.1 The Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.2 Boundary Condition and RT Coe cients . . . . . . . . . 71
3.3.3 Energy Density . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.4 Noether Currents . . . . . . . . . . . . . . . . . . . . . . 75
3.3.5 Comparison with Two Dimensions . . . . . . . . . . . . 78
3.4 Fermionicδ Defects in Four Dimensions . . . . . . . . . . . . . 78
3.4.1 The RT Algebra. . . . . . . . . . . . . . . . . . . . . . . 79
3.4.2 Boundary Condition . . . . . . . . . . . . . . . . . . . . 80
3.4.3 Spin Solutions in Four Dimensions and RT Coe cients . 80
3.4.4 Gibbs States and Currents . . . . . . . . . . . . . . . . . 81
3.4.5 Comparison with Two and Three Dimensions . . . . . . 88
3.5 Matrix Optics for Fermions . . . . . . . . . . . . . . . . . . . . 91
4 Summary and Outlook 95
Acknowledgements 101