Abstract
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 the 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 was to review and further develop these ideas and to put them into context with closely related earlier and parallel programmes.
Highlights
The canonical approach to quantum gravity has been initialised long time ago [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
We summarise, spell out implications of the renormalisation programme for the anomaly-free implementation of the hypersurface algebroid, and outline the steps when trying to apply the framework to interacting QFT and canonical quantum gravity such as loop quantum gravity (LQG)
The extension consisted in i. an improved derivation of the renormalisation scheme (5.29) and (5.32) from OS reconstruction using an extended minimal set of OS axioms that includes the uniqueness of the vacuum and ii. a much more systematic approach to the choice of coarse graining maps for a general QFT which are motivated by structures naturally provided already by the classical theory
Summary
The canonical approach to quantum gravity has been initialised long time ago [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. This section contains new material as compared to [54,55,56,57] in the sense that we 1) develop some systematics in the choice of coarse graining maps that are motivated by naturally available structures in the classical theory, 2) clarify the importance of the choice of random variable or stochastic process when performing OS reconstruction, and 3) improve the derivation of the Hamiltonian renormalisation flow by adding the uniqueness of the vacuum as an additional assumption ( made in the OS framework of Euclidian QFT [98,99,100]) as well as some machinery concerning degenerate contraction semi-groups and associated Kato–Trotter formulae. This article is the journal version of Ref. 196 which is organised slightly differently in the sense that Appendices F, G of this article are part of the main text of Ref. 196
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