Non-perturbative Quantum Gravity in Fock representations

TL;DR

This paper demonstrates that background dependence and perturbation theory in quantum gravity are independent, and constructs Fock representations for the full non-perturbative metric without relying on perturbative expansions, advancing non-perturbative quantum gravity approaches.

Contribution

It introduces background-dependent Fock representations for the full non-perturbative metric, bypassing the need for perturbative expansion and challenging the assumed link between background dependence and perturbation theory.

Findings

Fock representations can be constructed for the full metric non-perturbatively.

The Hamiltonian constraint can be defined as a quadratic form in these representations.

Background dependence does not necessarily imply perturbation theory in quantum gravity.

Abstract

Perturbative quantum gravity starts from prescribing a background metric. That background metric is then used in order to carry out two separate steps: 1. One splits the non-perturbative metric into background and deviation from it (graviton) and expands the action in terms of the graviton which results in an ifinite series of unknown radius of convergence. 2. One constructs a Fock representation for the graviton and performs perturbative graviton quantum field theory on the fixed background as dictated by the perturbative action. The result is a non-renormalisable theory without predictive power. It is therefore widely believed that a non-perturbative approach is mandatory in order to construct a fundamental, not only effective, predictive quantum field theory of the gravitational interaction. Since perturbation theory is by definition background dependent, the notions of background dependence (BD) and perturbation theory (PT) are often considered as symbiotic, as if they imply each other. In the present work we point out that there is no such symbiosis, these two notions are in fact logically independent. In particular, one can use BD structures while while not using PT at all. Specifically, we construct BD Fock representations (step 2 above) for the full, non-perturbative metric rather than the graviton (not step 1 above) and therefore never perform a perturbative expansion. Despite the fact that the gravitational Lagrangean is a non-polynomial, not even analytic, function of the metric we show that e.g. the Hamiltonian constraint with any density weight can be defined as a quadratic form with dense form domain in such a representation.