Studies in nonperturbative quantum gravity

Abstract

This thesis investigates how quantum effects modify the evolution of cosmological space-times and what those modifications teach us about the viability of candidate quantum-gravity frameworks. Two complementary approaches are pursued.

First, we develop a semiclassical geometrodynamics in which a fully dynamical classical metric is coupled to quantized matter with self-consistent back-reaction. For isotropic Friedmann–Robertson–Walker (FRW) universes and for the anisotropic Bianchi IX (mixmaster) model we derive Hamiltonian “classical–quantum” (CQ) equations and solve them numerically. Quantum excitations of the scalar field affect the Kasner exponents, leave a late-time energy-density remnant that can mimic dark energy, and admit exact static solutions in the isotropic case. When spacetime-discreteness corrections are included, the mixmaster dynamics changes: a sequence of bounces replaces the classical singularity, anisotropies grow and decay through each bounce, and Lyapunov exponent and fractal dimension analyses show a marked suppression of chaos compared with the purely classical system.

Second, we test the functional renormalization group programme in a symmetry-reduced setting. We recast an FRW minisuperspace with a massless scalar field and dust into harmonic-oscillator variables and track its RG flow using the exact, non-perturbative Wetterich equation. In contrast to Reuter’s four-dimensional Einstein–Hilbert truncation, we find no ultraviolet fixed points – instead, we find that at low energies, the cosmological constant flows to small values, and the gravitational constant flows to zero as energies grow high.

Taken together, the two lines of inquiry demonstrate (i) how quantum matter backreaction can leave durable imprints on large-scale, late-stage cosmology, (ii) how effective discreteness mitigates classical mixmaster chaos, curbs the initial singularity, and (iii) how the absence of fixed points in symmtery-reduced models constrains the scope of asymptotic safety. The work therefore sharpens our understanding of the accuracy and validity of minisuperspace reductions, and what obstacles remain on the path to a complete quantum theory of gravity.