[Paper Review] Discrete and Continuum Quantum Gravity

Herbert W. Hamber|ArXiv.org|Apr 22, 2007

Noncommutative and Quantum Gravity Theories25 references18 citations

TL;DR

This paper presents a lattice-regularized formulation of quantum gravity using Regge calculus, combining discrete and continuum approaches via the covariant Feynman path integral. It identifies a non-trivial ultraviolet fixed point in 2+ε dimensions and provides numerical evidence in four dimensions for scale-dependent gravitational couplings, suggesting a non-local effective theory and challenging the assumption of constant Newton's constant in quantum gravity.

ABSTRACT

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

Motivation & Objective

• To develop a consistent, non-perturbative lattice formulation of quantum gravity based on Regge calculus and the Feynman path integral.
• To explore the phase structure of quantum gravity in four dimensions using numerical simulations of the lattice path integral.
• To investigate whether the gravitational coupling runs with scale, motivated by results in 2+ε dimensions.
• To determine whether the lattice theory can reproduce universal scaling behavior and critical exponents consistent with analytical predictions.
• To assess the physical viability of the strong coupling phase and its implications for cosmological constant and curvature sign

Proposed method

• Formulates quantum gravity using Regge calculus, expressing gravity in terms of edge lengths, deficit angles, and simplicial geometry.
• Constructs a lattice action invariant under discrete diffeomorphisms and implements lattice Bianchi identities to preserve local curvature constraints.
• Applies the Feynman path integral with a gravitational functional measure, including higher derivative terms and matter couplings (scalar and fermionic fields).
• Performs analytical expansions such as weak field and strong coupling expansions to probe the phase structure and scaling behavior.
• Uses numerical simulations to compute invariant observables, correlation functions at fixed geodesic distance, and Wilson lines.
• Applies renormalization group techniques to extract critical exponents and determine the scaling limit in the continuum

Experimental results

Research questions

• RQ1 Does a non-trivial ultraviolet fixed point exist in four-dimensional quantum gravity, as suggested by the 2+ε expansion?
• RQ2 Can the lattice formulation reproduce universal scaling behavior and critical exponents predicted by analytical methods in 2+ε dimensions?
• RQ3 What is the nature of the phase structure in four-dimensional Euclidean lattice quantum gravity, particularly regarding curvature and correlation length?
• RQ4 Is the gravitational coupling scale-dependent, and can this explain cosmic acceleration without a cosmological constant?
• RQ5 Can the strong coupling phase of the lattice theory be physically viable, or does it imply a pathological short-distance behavior?

Key findings

• Numerical simulations in four dimensions show evidence of a non-trivial fixed point and universal scaling behavior consistent with analytical predictions from the 2+ε expansion.
• The lattice theory exhibits a renormalization group invariant gravitational correlation length, suggesting a non-local effective theory at large scales.
• In the strong coupling phase, the average curvature for small loops is negative, indicating an effective anti-de Sitter geometry at short distances.
• For large loops, the average curvature appears positive, suggesting a possible transition between phases, though the sign of curvature is not yet universally determined.
• The gravitational coupling increases at large distances, leading to gravitational anti-screening and cosmic acceleration, which may mimic a positive cosmological constant.
• The weak coupling phase, while pathological in the Euclidean formulation, may still be physical via analytic continuation, with an infinite correlation length and zero cosmological constant