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 (nonperturbative) approaches to quantum gravity, and their possible implications for more phenomenological theories of physics that are directly related to observation. Here I describe some of my most recent topics of interest.
Cosmology from Fundamental Quantum Gravity
Any candidate theory of quantum gravity must ultimately be able to connect to the physics at scales that we can observe in order to provide testable predictions. The most natural place to look for such predictions is in cosmology, where we can in principle see signatures from the very earliest stages of the universe right at the Big Bang. A major first step towards making this connection is to be able to describe, in a given quantum gravity model, states that describe macroscopic, homogeneous or approximately homogeneous universes like the one around us, and to be able to do calculations within quantum gravity to derive the effective dynamics of such states. With Daniele Oriti and Lorenzo Sindoni at the Albert Einstein Institute, I have recently managed to complete these steps in the group field theory approach to quantum gravity [arXiv:1303.3576, PRL, see also the PI press release]. We show that certain states whose structure is analogous to condensate states that appear in BoseEinstein condensation describe macroscopic spatially homogeneous geometries, and that in a semiclassical regime and in the isotropic case, the dynamics they satisfy is given precisely by the Friedmann equation for pure GR. Extensions of the formalism to matter fields and inhomogeneities are currently investigated. This work builds to an extent on previous work [arXiv:1201.4151, CQG] with Gianluca Calcagni and Daniele Oriti on describing a "thirdquantized" field theory formalism for loop quantum cosmology (LQC).
Lorentz Invariance and Observers in Canonical (Quantum) Gravity
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 blackhole states seem to depend on having a compact gauge group. With Derek Wise, I have been exploring this symmetry breaking from a geometric angle using the language of Cartan geometry. In the paper [arXiv:1111.7195, PRD], we show 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. Observer space, the space of all such local choices of time direction, is the analog of the space of all choices of foliations in canonical gravity, and as such links the covariant and canonical approaches to gravity (see our essay on this). We then took this idea one step further and worked out a formulation of general relativity where the fundamental arena is no longer spacetime, but observer space [arXiv:1210.0019, JMP]. Not only does this work give a new elegant geometric formulation of general relativity in which the spontaneous symmetry breaking occuring in the AshtekarBarbero connection formulation is most naturally understood, but it also provides a general geometric framework for all kinds of extensions of general relativity, such as theories with a preferred foliation [arXiv:1301.1692] or extensions of Lorentzian geometry such as Finsler geometry (as Manuel Hohmann has shown).

