Quantum Geometry

Exploring the Quanta of Space, Differential Forms, the Tetrad formalism of GR, Canonical Relativity, Ashtekar Variables.

This book offers solutions to the defective quantum mechanical atomic model of de Broglie, Bohr, and Schrodinger. The quantum mechanical model is actually an attempt to probe an unrecognized four-dimensional quantum geometric structure, a structure which is outlined in the book.

This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.

This graduate level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantization of strings, gravity and topological field theory.

The fundamental structure of matter and spacetime at the shortest length scales remains an exciting frontier of basic research in theoretical physics. A unifying theme in this area is the quantization of geometrical objects. The majority of lectures at the Advanced Study Institute on Quantum Ge­ ometry in Akureyri was on recent advances in superstring theory, which is the leading candidate for a unified description of all known elementary par­ ticles and interactions. The geometric concept of one-dimensional extended objects, or strings, has always been at the core of superstring theory but in recent years the focus has shifted to include also higher-dimensional ob­ jects, so called D-branes, which play a key role in the non-perturbative dynamics of the theory.

The present monograph provides a systematic and basicaIly self-eontained introduetion to a mathematieal framework eapable of ineOIporating those fundamental physical premises of general relativity and quantum meehanics which are not mutually ineonsistent, and which ean be therefore retained in the unifieation of these two fundamental areas of twentieth- eentury physics.