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 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.

Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory

Efficiently bounding large determinants is an essential step in non-relativistic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength [Formula: see text] of the interparticle interaction. We provide, for large determinants of fermionic covariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly [Formula: see text]. In particular, the convergence of perturbation series at [Formula: see text] of any fermionic quantum field theory is ensured if the matrix entries (with respect to some fixed orthonormal basis) of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use Hölder inequalities for general non-commutative [Formula: see text]-spaces derived by Araki and Masuda [Positive cones and [Formula: see text]-spaces for von Neumann algebras, Publ. RIMS[Formula: see text] Kyoto Univ. 18 (1982) 339–411].

LUTTINGER MODEL AND LUTTINGER LIQUIDS

The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems. Nevertheless according to the Luttinger liquid conjecture a number of relations between exponents and other physical quantities, which are valid in the Luttinger model, are believed to be true in a wide class of systems, including tight binding or jellium one-dimensional fermionic systems. Recently a rigorous proof of several Luttinger liquid relations in nonsolvable models has been achieved; it is based on exact Renormalization Group methods coming from Constructive Quantum Field Theory and its main steps will be reviewed below.