Quantum Field Theory
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1 Quantum Field Theory Competitive Models Bearbeitet von Bertfried Fauser, Jürgen Tolksdorf, Eberhard Zeidler 1. Auflage Buch. xix, 436 S. Hardcover ISBN Format (B x L): 16,5 x 23,5 cm Gewicht: 847 g Weitere Fachgebiete > Physik, Astronomie > Physik Allgemein > Theoretische Physik, Mathematische Physik schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
2 v CONTENTS Preface... xii Bertfried Fauser, Jürgen Tolksdorf and Eberhard Zeidler Constructive Use of Holographic Projections... 1 Bert Schroer 1. Historical background and present motivations for holography 1 2. Lightfront holography, holography on null-surfaces and the origin of the area law 4 3. From holography to correspondence: the AdS/CFT correspondence and a controversy Concluding remarks 22 Acknowledgements 23 References 23 Topos Theory and Neo-Realist Quantum Theory Andreas Döring 1. Introduction What is a topos? Topos theory and physics A formal language for physics The context category V(R) and the topos of presheaves Set V(R)op Representing L(S) in the presheaf topos Set V(R)op Truth objects and truth-values Generalised elements as generalised states The construction of truth objects Truth objects and Birkhoff-von Neumann quantum logic The assignment of truth-values to propositions Conclusion and outlook 45 Acknowledgements 46 References 46 A Survey on Mathematical Feynman Path Integrals: Construction, Asymptotics, Applications Sergio Albeverio and Sonia Mazzucchi 1. Introduction The mathematical realization of Feynman path integrals Applications Quantum mechanics Quantum fields 60 Acknowledgements 62
3 vi References 62 A Comment on the Infra-Red Problem in the AdS/CFT Correspondence...67 Hanno Gottschalk and Horst Thaler 1. Introduction Functional integrals on AdS Two generating functionals The infra-red problem and triviality Conclusions and outlook 79 Acknowledgement 80 References 80 Some Steps Towards Noncommutative Mirror Symmetry on the Torus...83 Karl-Georg Schlesinger 1. Introduction Elliptic curves Noncommutative elliptic curves Exotic deformations of the Fukaya category Conclusion and outlook 91 Acknowledgements 92 References 92 Witten s Volume Formula, Cohomological Pairings of Moduli Spaces of Flat Connections and Applications of Multiple Zeta Functions Partha Guha 1. Introduction Background about moduli space Volume of the moduli space of SU(2) flat connections Asymptotic analysis and computation of volume of moduli spaces Volume of the moduli space of flat SU(3) connections Cohomological pairings of the moduli space Review of Donaldson-Thaddeus-Witten s work on SU(2) moduli space Cohomological pairings for SU(3) connections Flag manifolds and cohomology Computation of the intersection pairings Concrete examples 113 References 114 Noncommutative Field Theories from a Deformation Point of View Stefan Waldmann 1. Introduction Noncommutative space-times Matter fields and deformed vector bundles Deformed principal bundles The commutant and associated bundles 130
4 vii Acknowledgement 133 References 133 Renormalization of Gauge Fields using Hopf Algebras Walter D. van Suijlekom 1. Introduction Preliminaries on perturbative quantum field theory Gauge theories The Hopf algebra of Feynman graphs Renormalization as a Birkhoff decomposition The Hopf algebra of Green s functions 147 Appendix A. Hopf algebras 152 References 153 Not so Non-Renormalizable Gravity Dirk Kreimer 1. Introduction The structure of Dyson Schwinger Equations in QED The Green functions Gauge theoretic aspects Non-Abelian gauge theory Gravity Summary of some results obtained for quantum gravity Comments Gauss-Bonnet Two-loop counterterm Other instances of gravity powercounting 161 Acknowledgements 161 References 161 The Structure of Green Functions in Quantum Field Theory with a General State Christian Brouder 1. Introduction Expectation value of Heisenberg operators QFT with a general state Nonperturbative equations Generalized Dyson equation Quadrupling the sources An algebraic proof of Hall s equation Determination of the ground state Conclusion 172 Acknowledgement 173
5 viii References 173 The Quantum Action Principle in the Framework of Causal Perturbation Theory Ferdinand Brennecke and Michael Dütsch 1. Introduction The off-shell Master Ward Identity in classical field theory Causal perturbation theory Proper vertices The Quantum Action Principle Formulation of the Master Ward Identity in terms of proper vertices The anomalous Master Ward Identity - Quantum Action Principle Algebraic renormalization 194 Acknowledgement 195 References 195 Plane Wave Geometry and Quantum Physics Matthias Blau 1. Introduction A brief introduction to the geometry of plane wave metrics Plane waves in Rosen and Brinkmann coordinates: heuristics Curvature of plane waves Geodesics, lightcone gauge and harmonic oscillators From Rosen to Brinkmann coordinates (and back) The Heisenberg isometry algebra of a generic plane wave Geodesics, isometries, and conserved charges Synopsis The Lewis Riesenfeld theory of the time-dependent quantum oscillator Description of the problem Outline of the Lewis Riesenfeld procedure Deducing the procedure from the plane wave geometry A curious equivalence between two classes of Yang-Mills actions Description of the problem A classical mechanics toy model The explanation: from plane wave metrics to Yang-Mills actions 214 References 215 Canonical Quantum Gravity and Effective Theory Martin Bojowald 1. Loop quantum gravity Effective equations Quantum back-reaction General procedure A solvable model for cosmology Interactions 229
6 ix 4. Effective quantum gravity 231 Acknowledgement 232 References 232 From Discrete Space-Time to Minkowski Space: Basic Mechanisms, Methods, and Perspectives Felix Finster 1. Introduction Fermion systems in discrete space-time A variational principle A mechanism of spontaneous symmetry breaking Emergence of a discrete causal structure A first connection to Minkowski space A static and isotropic lattice model Analysis of regularization tails A variational principle for the masses of the Dirac seas The continuum limit Outlook and open problems 257 References 258 Towards a q -Deformed Quantum Field Theory Hartmut Wachter 1. Introduction q-regularization Basic ideas of the mathematical formalism What are quantum groups and quantum spaces? How do we multiply on quantum spaces? What are q-deformed translations? How to differentiate and integrate on quantum spaces Fourier transformations on quantum spaces Applications to physics Plane-wave solutions to the free-particle Schrödinger equation The propagator of the free q -deformed particle Scattering of q -deformed particles Conclusion 281 Acknowledgement 281 References 281 Towards a q -Deformed Supersymmetric Field Theory Alexander Schmidt 1. Introduction Fundamental Algebraic Concepts q -Deformed Superalgebras q -Deformed Superspaces and Operator Representations 293 Appendix A. q -Analogs of Pauli matrices and spin matrices 298
7 x Acknowledgement 300 References 300 L -algebra Connections and Applications to String- and Chern-Simons n-transport Hisham Sati, Urs Schreiber and Jim Stasheff 1. Introduction The setting and plan L -algebras and their string-like central extensions L -algebras L -algebras from cocycles: string-like extensions L -algebra differential forms L -algebra Cartan-Ehresmann connections g-bundle descent data Connections on n-bundles: the extension problem Higher string and Chern-Simons n-transport: the lifting problem Statement of the main results Differential graded-commutative algebra Differential forms on smooth spaces Examples Homotopies and inner derivations Examples Vertical flows and basic forms Examples L -algebras and their string-like extensions L -algebras Examples L -algebra homotopy and concordance Examples L -algebra cohomology Examples L -algebras from cocycles: string-like extensions Examples L -algebra valued forms Examples L -algebra characteristic forms Examples L -algebra Cartan-Ehresmann connections g-bundle descent data Examples Connections on g-bundles: the extension problem Examples Characteristic forms and characteristic classes Examples 379
8 xi 6.4. Universal and generalized g-connections Examples Higher string- and Chern-Simons n-bundles: the lifting problem Weak cokernels of L -morphisms Examples Lifts of g-descent objects through string-like extensions Examples Lifts of g-connections through string-like extensions Examples L -algebra parallel transport L -parallel transport Examples Transgression of L -transport Examples Configuration spaces of L -transport Examples Physical applications: string-, fivebrane- and p-brane structures 414 Appendix A. Explicit formulas for 2-morphisms of L -algebras 417 Acknowledgements 420 References 420 Index
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