Working Group II:

Representation Theory

Members: K.-H. Neeb, T. Thiemann

The representation theory of the algebra underlying Loop Quantum Gravity is under very good control if one asks for a cyclic representation (that is, there is a `vacuum' vector on which the algebra generates a dense subspace of the Hilbert space) that admits a unitary representation of the (spatial) diffeomorphism group. Here spatial refers to the fact that one deals with globally hyperbolic spacetimes which admit a foliation by spacelike hypersurfaces. Specifically, there is only one such representation up to unitary equivalence. However, an unpleasant feature of this representation is that one parameter subgroups of the spatial diffeomorphism group do not act strongly continuously which results in technical difficulties in the imposition of the remaining constraints.

Directly related to this is the fact that the resulting Hilbert space is not separable. On one hand, this is not unexpected in a diffeomorphism invariant theory. On the other hand, this is a situation rather unfamiliar in standard Quantum Field Theory. It may be that representations of different algebras studied under I. have better continuity and separability properties or that more such representations of the algebra underlying Loop Quantum Gravity exist which admit only cocycle representations of the spatial diffeomorphism group. In the language of measure theory, this means that the measure underlying the Hilbert space of square integrable functions is no longer invariant under spatial diffeomorphisms but only quasi-invariant.

The construction of the type of representations underlying the Loop Quantum Gravity framework uses projective limit techniques [1], [14], [15] in order to define topology and measure theory on distributional spaces of connections, its generalised cotangent bundle and Yang-Mills type gauge transformations respectively. What is of interest for physics is the moduli space of these rather singular spaces which can be obtained purely geometrically also by projective limit techniques. This is interesting from the purely mathematical point of view because it provides a rare, non trivial example of infinite dimensional constructions in a context where the relevant spaces do not even carry a canonical manifold structure any more. It is then natural to ask whether these constructions have a precise mathematical interpretation in terms of notions familiar from geometric quantisation of finite dimensional symplectic manifolds [16] such as coadjoint orbits, momentum maps and Hamiltonian actions and thus provide a suitable generalisation to the infinite dimensional context. This example might then serve as a germ for new ideas and generalised constructions in an even wider context. For instance, the generalised Yang-Mills type gauge transformations can be identified with a mapping group from a manifold into a (compact) gauge group equipped with a natural (Tychonov) topology with respect to which it is compact.

The gauge symmetry generated by this Lie algebra has to be implemented by looking for gauge invariant vectors in the Hilbert space or a suitable dual space. If the symmetry group in question is represented unitarily then a possible approach consists in rigging techniques and direct integral decompositions of the Hilbert space. This is already far from trivial for finite dimensional groups. It is therefore the more interesting that in Loop Quantum Gravity an infinite dimensional example of a rigging map has been constructed for the Yang-Mills type of gauge symmetry. There are also results for the spatial diffeomorphism gauge symmetry but the situation here is under less control due to the lack of a natural topological and measure theoretic framework for the spatial diffeomorphism group. Again it would be desirable to extract a general mathematical framework that is applicable in sufficiently general situations.

Impressum