Quantum groups and noncommutative geometry

2000

A search for the uni cation of quantum theory and gravity has forced mathematical physicists to re-evaluate the meaning of geometry itself. The surprising answer has led to an explosion of research papers, a vast collection of examples, and to revolutions in at least three branches of pure mathematics. It o¬ers insights into the origin of the universe and the nature of physical reality.

Lecture Notes in Physics, 2000

This is an introduction for nonspecialists to the noncommutative geometric approach to Planck scale physics coming out of quantum groups. The canonical role of the 'Planck scale quantum group' C[x]◮⊳C[p] and its observable-state T-dualitylike properties are explained. The general meaning of noncommutativity of position space as potentially a new force in Nature is explained as equivalent under quantum group Fourier transform to curvature in momentum space. More general quantum groups C(G ⋆)◮⊳U (g) and U q (g) are also discussed. Finally, the generalisation from quantum groups to general quantum Riemannian geometry is outlined. The semiclassical limit of the latter is a theory with generalised non-symmetric metric g µν obeying ∇ µ g νρ − ∇ ν g µρ = 0.

2005

Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a nonlinearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.

Clifford Algebras: Applications to Mathematics, Physics, and Engineering, 2004

Foundations of Physics, 2009

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.

Journal of Geometry and Physics, 1993

This is an introduction to the old and new concepts of non-commutative (N.C.) geometry. We review the ideas underlying N.C. measure and topology, N.C. differential calculus, N.C. connections on N.C. vector bundles, and N.C. Riemannian geometry by following A. Connes' point of view.

2008

This dissertation is based on research done at the Mathematical Physics sector of the International School for Advanced Studies of Trieste, during the period from October 2003 to June 2007. It is divided into two parts: the first part (Chapters 1-2) is an account of the general theory and a collection of some general notions and results; the second part (Chapters 3-5) contains the original work, carried out under the supervision of Prof. Ludwik D abrowski and Prof. Giovanni Landi. Part of the original material presented here has been published or submitted as a preprint in the following papers:

Journal of Geometry and Physics, 1989

The structure of amanifold can be encoded in the commutative algebra of functions on the manifold it sell-this is usual-. In the case of a non com.mut.ative algebra thereis no underlying manifold and the usual concepts and tools of diffe.rential geometry (differentialforms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory.. .) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The presentpaper is an introduction for the non specialist and a review oftheprincipal results on the field.

Toward a New Understanding of Space, Time and Matter, 2009

We provide a self-contained introduction to the quantum group approach to noncommutative geometry as the next-to-classical effective geometry that might be expected from any successful quantum gravity theory. We focus particularly on a thorough account of the bicrossproduct model noncommutative spacetimes of the form [t, xi] = ıλxi and the correct formulation of predictions for it including a variable speed of light. We also study global issues in the Poincaré group in the model with the 2D case as illustration. We show that any off-shell momentum can be boosted to infinite negative energy by a finite Lorentz transformaton.

2000

The quantum-event / prime ideal in a category/ noncommutative-point alternative to classical-event / commutative prime ideal/ point is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered as mathematical tools appropriate to model quantum mechanics. The space-time framework is to be reconstructed from the spectrum of the path category of a quiver. The interference experiment is considered as an example.