Differential Geometrical Methods in Theoretical Physics

ISBN-13: 9789048184590 / Angielski / Miękka / 2010 / 471 str.

K. Bleuler; M. Werner

Opis

After almost half a century of existence the main question about quantum field theory seems still to be: what does it really describe? and not yet: does it provide a good description of nature? J. A. Swieca Ever since quantum field theory has been applied to strong int- actions, physicists have tried to obtain a nonperturbative und- standing. Dispersion theoretic sum rules, the S-matrix bootstrap, the dual models (and their reformulation in string language) and s the conformal bootstrap of the 70 are prominent cornerstones on this thorny path. Furthermore instantons and topological solitons have shed some light on the nonperturbati ve vacuum structure respectively on the existence of nonperturbative "charge" s- tors. To these attempts an additional one was recently added', which is yet not easily describable in terms of one "catch phrase". Dif- rent from previous attempts, it is almost entirely based on new noncommutative algebraic structures: "exchange algebras" whose "structure constants" are braid matrices which generate a ho- morphism of the infini te (inducti ve limi t) Artin braid group Boo into a von Neumann algebra. Mathematically there is a close 2 relation to recent work of Jones * Its physical origin is the resul t of a subtle analysis of Ei nstein causality expressed in terms of local commutati vi ty of space-li ke separated fields. It is most clearly recognizable in conformal invariant quantum field theories.

Recenzje

`Anybody ... will find in this book a lot of new original information and inspiration for further research.'
Class Quantum Grav, 7, 1990

Spis treści

The Impact of Physics on Geometry.- The Impact of Physics on Geometry.- Mathematical Contributions.- Weyl’s Program and Modern Physics.- Elliptic Genera of Level N for Complex Manifolds.- The Principle of Triality and A Distinguished Unitary Representation of SO(4,4).- On E 10.- Special Aspects of String Theory.- Operator Methods in String Theory.- On Strings and J.Douglas Variational Principle.- Groupoids and Lie Bigebras in Gauge and String Theories.- Conformal Invariance and Integrable Systems.- The Definition of Conformal Field Theory.- Statistics and Monodromy in Two- and Three-Dimensional Quantum · Field Theory.- Quantum Groups (YBZF Algebras) and Integrable Theories: An Overview.- Integrable Systems and Conformal Invariance.- Algebraic Aspects of Non-Perturbative Quantum Field Theories.- Two Classical Domains: — Symplectic Structures in Physics — General Relativity.- The pairing method and bosonic anomalies.- A Multisymplectic Approach to the KdV Equation.- Singular Points in Level Sets of the Momentum Map and Quantum Theory.- Killing Spinors and Universality of the Hijazi Inequality.- Towards a Renormalizable Theory of Quantum Gravity.- Gauge Theory for Diffeomorphism Groups.- Quantum Field Theory in Curved Spacetime.- Supersymmetric Structures.- Supertheories.- The Integrability Of N = 16 Supergravity.- Conformal Invariant Supersymmetric Theories in Four Dimensions: A Practical Application of BRS Cohomology.- Supermanifolds, Supermanifold Cohomology, and Super Vector Bundles.- Representations of Lie Superalgebras an Introduction.- Concluding Remarks.- Concluding Remarks.- Observations of Two of Our Brightest Stars.- Observations of Two of Our Brightest Stars.