Working Group I:

Choice of Poisson algebra

Members: C. Meusburger, T. Thiemann

Due to the presence of fermionic matter in nature it is necessary to describe the gravitational field in terms of connections and canonically conjugate frame fields, both valued in the Lie algebra of some (compact) group. The Poisson algebra of these fields evaluated at points is singular and thus one must smear these fields. The choice of smearing of these fields and more complicated objects derived from those defines a choice of Poisson subalgebra on which the whole quantisation programme is based. In Loop Quantum Gravity one chooses an algebra based on parallel transports (holonomies) of the connection along one dimensional paths and non Abelian fluxes of the frame field through 2 dimensional surfaces. Other choices are conceivable and will lead to entirely new properties of the corresponding quantum theory. For instance, a celebrated result of Loop Quantum Gravity is the discreteness of the spectrum of geometric operators describing area and volume of surfaces and three dimensional submanifolds respectively, hinting at a discrete structure of spacetime at the Planck scale. This property is directly rooted in the choice of holonomies and fluxes as generators of the corresponding abstract operator algebra. While this is a natural and background independent choice, the results might not hold if one starts from an entirely different choice. The amount of freedom in the choice of this algebra under the guidance of the principles of background independence and gauge covariance (that is, the gauge transformations preserve the algebra) and maybe additional dynamical input are therefore worthwhile studying. This analysis is directly connected with item II. below.

On the other hand, this is a situation where insights can be gained from the study of three-dimensional gravity. In three-dimensions, this problem can be formulated rigorously in terms of different parametrisations of moduli spaces of flat connections. Moduli spaces of flat connections have been investigated extensively in mathematics and physics and arise as the phase space of gravity in three-dimensions. There exist various quantisation procedures for the moduli spaces relevant to three-dimensional gravity, some of which are based on formulation in terms of holonomies similar to the ones arising in loop quantum gravity, while there are also descriptions in terms of hyperbolic and Teichmüller geometry.

Depending on the choice of the parametrisation, different quantisation formalisms appear natural: combinatorial quantisation based on the representation theory of certain quantum groups in the first case, methods from the quantisation of Teichmüller space in the second. While the correspondence between these two descriptions is well-understood on the classical level, the relation between the associated quantisation formalisms has not been explored yet and their equivalence remains to be established. It also seems reasonable that a more precise understanding of the relation between these constructions can help to overcome obstacles in quantisation that arise from the representation theory of non-compact quantum groups. It seems likely that this would yield results that are also relevant to mathematics in their own right.

Impressum