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.

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. In the recent paper [arXiv:1111.7195], Derek Wise and I argue that the use of the canonical formalism which relies on a local choice of observer or time direction directly leads to a local breaking of Lorentz invariance down to the subgroup of rotations, in a form similar to spontaneous symmetry breaking in condensed matter and particle physics. Our work provides a clear geometric understanding of a classical formulation where the gauge corresponding to the local stabiliser subgroup can be freely changed. One can view our formulation as 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. While the question of possible Lorentz violation can only be properly answered at the quantum level, our work shows that in the classical canonical setting there is no conflict between fundamental Lorentz invariance and a local breaking down to SU(2).

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.