Discrete and Continuum Quantum Gravity

TL;DR

The paper compares covariant continuum and Regge-based lattice formulations of quantum gravity, highlighting how both approaches use the Feynman path integral to probe quantum gravitational dynamics. It explains the spin-2 graviton framework, perturbative renormalization issues, and the conformal instability of Euclidean gravity, while showing how higher-derivative terms or supersymmetry can alter UV behavior. The review then shifts to nonperturbative methods, notably 2+\epsilon expansions and lattice Regge calculus, which reveal a nontrivial ultraviolet fixed point and a two-phase structure in gravity, with a lattice formulation offering a controlled route to a continuum limit via critical scaling. The discussion culminates in the lattice construction of observables such as the gravitational Wilson loop and the Regge path integral, outlining the prospects for connecting discrete and continuum quantum gravity and for extracting universal, dimensionless quantities from first-principles simulations.

Abstract

I review discrete and continuum approaches to quantized gravity based on the covariant Feynman path integral approach.

Figures (42)

• Figure 1: Lowest order diagrams illustrating the gravitational analog to Compton scattering. Continuous lines indicate a matter particle, short dashed lines a graviton. Consistency of the theory requires that the two bottom diagrams be added to the two on the top.
• Figure 2: Lowest order diagrams illustrating modifications to the classical gravitational potential due to graviton exchange. Continuous lines denote a spinless heavy matter particle, short dashed lines a graviton and the long dashed line the ghost loop. The last diagram shows the scalar matter loop contribution.
• Figure 3: Quantum mechanical amplitude of transitioning from an initial three-geometry described by $g$ at time $t_{initial}$ to a final three-geometry described by $g'$ at a later time $t_{final}$. The full amplitude is a sum over all intervening metrics connecting the two bounding three-surfaces, weighted by $\exp (i I / \hbar)$ where $I$ is a suitably defined gravitational action.
• Figure 4: One-loop diagrams giving rise to coupling and field renormalizations in the non-linear $\sigma$-model. Group theory indices $a$ flow along the thick lines, dashed lines should be contracted to a point.
• Figure 5: The $\beta$-function for the non-linear $\sigma$-model in the large-$N$ limit for $d>2$.