Canonical Quantum Gravity, Constructive QFT and Renormalisation

The canonical approach to quantum gravity has been put on a firm mathematical foundation in the recent decades. Even the quantum dynamics can be rigorously defined, however, due to the tremendously non-polynomial character of the gravitational interaction, the corresponding Wheeler-DeWitt operator valued distribution suffers from quantisation ambiguities that need to be fixed. In a very recent series of works we have employed methods from constructive quantum field theory in order to address those ambiguities. Constructive QFT trades quantum fields for random variables and measures thereby phrasing the theory in the language of quantum statistical physics. The connection to the canonical formulation is made via Osterwalder-Schrader reconstruction. It is well known in quantum statistics that the corresponding ambiguities in measures can be fixed using renormalisation. The associated renormalisation flow can thus be used to define a canonical renormalisation programme. The purpose of this article is to review and further develop these ideas and to put them into context with closely related earlier and parallel programmes.