Polymer Geometry at Planck Scale and Quantum Einstein Equations

TL;DR

This paper discusses a background-independent, non-perturbative approach to quantum gravity, revealing polymer-like quantum geometries with discrete spectra for geometrical observables, and advances in quantum constraints and solutions.

Contribution

It introduces a new functional calculus on gauge connections, demonstrating polymer-like quantum geometries and progress in formulating anomaly-free quantum constraints in loop quantum gravity.

Findings

Quantum geometry exhibits one-dimensional, polymer-like excitations.

Geometrical observables such as areas and volumes have discrete spectra.

Progress in solving quantum constraints, including a Wick transformation approach.

Abstract

Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge equivalent connections. This calculus does not use any background fields (such as a metric) and is thus well-suited to a fully non-perturbative treatment of quantum gravity. Using this framework, quantum geometry is examined. Fundamental excitations turn out to be one-dimensional, rather like polymers. Geometrical observables such as areas of surfaces and volumes of regions have purely discrete spectra. Continuum picture arises only upon coarse graining of suitable semi-classical states. Next, regulated quantum diffeomorphism constraints can be imposed in an anomaly-free fashion and the space of solutions can be given a natural Hilbert space structure. Progress has also been made on the quantum Hamiltonian constraint in a number of directions. In particular, there is a recent approach based on a generalized Wick transformation which maps solutions to the Euclidean quantum constraints to those of the Lorentzian theory. These developments are summarized. Emphasis is on conveying the underlying ideas and overall pictures rather than technical details.