Quantum Geometry and Quantum Gravity
Research Networking Programme of the European Science Foundation 

Research AreaQuantum Geometry arises when quantization techniques and principles are applied in a geometric context, such as group theory, gauge theories, particle physics and quantum gravity. The most physically profound of these applications is quantum gravity, the task of unifying Einstein's geometric conception of general relativity with quantum principles. The main focus of the network is understanding quantum geometry and using the results and techniques in the study of quantum gravity. The recent history of quantum gravity is its development in two parallel paradigms, firstly as the quantization of general relativity, in which quantum geometry plays a central role, and secondly through the study of particles physics and strings, in which quantum geometry concepts are emergent. There is now some overlap in which the concepts fuse, such as the AdS/CFT correspondence in which quantum gravity in Antide Sitter space is related to certain Conformal Field Theory, or such as noncommutative geometry that naturally appears as the geometry of open strings. The quantization of general relativity is currently studied under two headings, loop quantum gravity, which empasises the role of space and the Hamiltonian formalism, and spin foam models, which empasise the role of spacetime and the path integral formalism. Although the conceptual frameworks differ, both approaches use the same mathematical techniques of spin networks and it is widely believed that these are complementary approaches to the same underlying theory. Loop quantum gravity counts among its accomplishments the construction of a kinematical Hilbert space for quantum gravity on a spatial hypersurface of fixed time. Quantum states corresponding to black holes have been defined and there is a successful calculation of the black hole entropy. Operators measuring physical quantities such as area and volume have also been constructed and their spectra is known. The theory has some applications to scenarios where dynamics is required, such as quantum cosmology. It has also led to valuable spinoff in classical relativity with the definition of an isolated horizon. A limitation of the theory is that the dynamics is not known in the general case, and the related problem of constructing a semiclassical correspondence with general relativity is not solved, although there are encouraging partial results. Spin foam models describe 2+1 dimensional quantum gravity. Euclidean versions of this theory are welldefined and the focus of current research in this area is extracting the physics from the theory and comparing it to naïve expectations. The Lorentzian versions of the theory are incomplete and the key questions in defining the theory relate to the representation theory of noncompact quantum groups. In 3+1 dimensions there are various spin foam models which have some features of quantum gravity. Their properties are being extensively investigated but at present all proposals have some issues unresolved. However the concept of spin foam model is very flexible and the exploration of all the possibilities is still at an early stage. There are closely related mathematical construtions in topology, homotopy theory, category theory and, again, quantum groups. These provide rich avenues for exploration. One area of importance for spin foam models are the techniques of what has became known as group field theory, which is a generalization of the matrix model approach to 2d gravity. Noncommutative geometry has been applied to deformations of classical geometries and groups, the study of gauge theories and particle physics, and recently as the geometry of Dbranes of open string theory in the background of a B field. Noncommutative geometry has been proposed as an ultraviolet cutoff in field theory, thus removing the divergences of perturbation theory. And recently there has been great progress in the understanding of renormalizability issues of noncommutative field theories. An example of the noncommutative manifold of particular importance and simplicity is the socalled fuzzy sphere. Field theoretical models on fuzzy manifolds are examples of matrix models and are divergencefree. It was recently realized that some of them, like the YangMills theory, are exactly solvable. The main objective of the programme is to stimulate exchange of ideas between researchers pursuing different approaches to quantum geometry. The envisaged achievements are a better understanding of quantum geometry through crossfertilization of all the approaches listed above, and the application of quantum geometry to quantum gravity. Expected benefit from European collaboration in this area: There is a variety of European research groups pursuing different approaches to quantum geometry. It is essential that the effort is combined and the level of interaction between the groups increases. It is also essential to educate a new generation of young researchers that are aware of all the approaches. The proposed networking activity will achieve these goals. 

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