Quantum gravity is possibly the greatest challenge in fundamental physics. Many directions which are currently explored are showing promising partial success, but we still seem to be far from something that could be called a complete theory. My principal aim is to understand both the relations between different (non-perturbative) approaches to quantum gravity, and their possible implications for low-energy physics.

Quantum Cosmology
Dimensionally-reduced cosmological models offer the possibility to explore a candidate theory in a simplified setting, while also addressing fundamental physical questions about the origin of our Universe. In a recent paper [arXiv:1011.4290, CQG], Gianluca Calcagni, Daniele Oriti and I investigated the physical inner product in quantum cosmological models, in particular in loop quantum cosmology where the 3-dimensional volume has discrete spectrum. The physical inner product, a two-point function for the Hamiltonian constraint, is central to the physical interpretation of states in quantum gravity. We gave an overview over the possible choices, focussing on the composition laws satisfied by these. We showed that the situation is analogous to the relativistic particle: The existence of a relational time and corresponding notion of positive- and negative-frequency solutions is essential to define a physical inner product which is positive definite and satisfies a "nice" composition law. We are currently investigating a field theory model where the quantum-cosmological wavefunction is "third quantised" (for a recent review of the idea of third quantisation see arXiv:1102.2226) and the two-point function becomes a correlation function of the quantum field.

(Semi-)Classical Formulations of Gravity
Quantising gravity means quantising a certain action in certain variables, and it is not at all clear whether classically equivalent formulations will lead to equivalent quantum theories. Conversely, if one has a quantum theory, it should be describable at least in some (semiclassical) regime through an (effective) action. It is therefore worthwhile to investigate different classical actions for gravity. In spin foam models one usually formulates GR as a constrained topological ("BF") theory, where in the traditional formulation due to Plebanski the added constraints are quadratic in the "B field". In the current spin foam models, however, one usually uses linear constraints. In the paper [arXiv:1004.5371, CQG], Daniele Oriti and I gave a formulation in terms of linear constraints already at the classical level, requiring the introduction of additional dynamical variables playing the role of normals to 3-dimensional hypersurfaces. This formulation still has to be understood better at the classical level, and might also have implications for spin foam models for quantum gravity.

Lorentz Invariance in Canonical (Quantum) Gravity
The issue of possible Lorentz violation induced by quantum gravity has often been discussed as a possible experimental test of quantum gravity. There seem to be very strong bounds from observation which would rule out most proposed effects. In loop quantum gravity Lorentz invariance is an issue since the theory is based on the gauge group SU(2) instead of the full Lorentz group SL(2,C), due to partial gauge fixing ("time gauge") at the classical level. Important features such as discrete spectra of geometric observables and counting of black-hole states seem to depend on having a compact gauge group. From my point of view, one step towards Lorentz covariance would be a geometric understanding a classical formulation where the gauge can be freely changed. Such a formulation should be describable by a Cartan connection, where one has an "internal" homogeneous space G/H parametrising possible gauge choices, and a connection that can be viewed both as a G- and a H-connection. Derek Wise has written a nice review of the significance of Cartan geometry to the MacDowell-Mansouri action for GR [gr-qc/0611154, CQG], and Gary Gibbons and I have applied the same methods to find a generalisation of doubly special relativity (DSR) to arbitrary spacetimes [arXiv:0902.2001, CQG]. An intriguing possibility is if Lorentz covariance is broken to the maximal subgroup SIM(2), which would evade most experimental constraints.