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Strings, loops and others: a critical survey of the present approaches to quantum gravity

Carlo Rovelli

TL;DR

This paper surveys major approaches to quantum gravity—string theory, loop quantum gravity, discrete and unorthodox methods—highlighting their distinct perspectives on spacetime quantization and background independence. It reviews concrete results such as the derivation of the Bekenstein-Hawking entropy formula $S = \frac{A}{4\hbar G}$ and the emergence of area and volume quanta in loop quantum gravity, and discusses the role of spin foam models and noncommutative geometry as new directions. It critically assesses the lack of experimental evidence and the challenges in deriving a common, predictive framework, while outlining potential routes for synthesis between approaches. The overall tone is cautious but optimistic about progress and the deep connections across frameworks.

Abstract

I review the present theoretical attempts to understand the quantum properties of spacetime. In particular, I illustrate the main achievements and the main difficulties in: string theory, loop quantum gravity, discrete quantum gravity (Regge calculus, dynamical triangulations and simplicial models), Euclidean quantum gravity, perturbative quantum gravity, quantum field theory on curved spacetime, noncommutative geometry, null surfaces, topological quantum field theories and spin foam models. I also briefly review several recent advances in understanding black hole entropy and attempt a critical discussion of our present understanding of quantum spacetime.

Strings, loops and others: a critical survey of the present approaches to quantum gravity

TL;DR

This paper surveys major approaches to quantum gravity—string theory, loop quantum gravity, discrete and unorthodox methods—highlighting their distinct perspectives on spacetime quantization and background independence. It reviews concrete results such as the derivation of the Bekenstein-Hawking entropy formula and the emergence of area and volume quanta in loop quantum gravity, and discusses the role of spin foam models and noncommutative geometry as new directions. It critically assesses the lack of experimental evidence and the challenges in deriving a common, predictive framework, while outlining potential routes for synthesis between approaches. The overall tone is cautious but optimistic about progress and the deep connections across frameworks.

Abstract

I review the present theoretical attempts to understand the quantum properties of spacetime. In particular, I illustrate the main achievements and the main difficulties in: string theory, loop quantum gravity, discrete quantum gravity (Regge calculus, dynamical triangulations and simplicial models), Euclidean quantum gravity, perturbative quantum gravity, quantum field theory on curved spacetime, noncommutative geometry, null surfaces, topological quantum field theories and spin foam models. I also briefly review several recent advances in understanding black hole entropy and attempt a critical discussion of our present understanding of quantum spacetime.

Paper Structure

This paper contains 36 sections, 13 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Directions
  3. Main directions
  4. String theory
  5. Difficulties with string theory
  6. String cosmology
  7. Loop quantum gravity
  8. Quanta of Geometry
  9. Difficulties with loop quantum gravity
  10. Traditional approaches
  11. Discrete approaches
  12. Regge calculus
  13. Dynamical triangulations
  14. Ponzano-Regge state sum models
  15. Old hopes $\rightarrow$ approximate theories
  16. ...and 21 more sections

Figures (3)

  • Figure 2: Figure 2: A node of a spin network (in bold) and its dual 3-cell (here a tetrahedron). The coloring of the node determines the quantized volume of the tetrahedron. The coloring of the links (shown in the picture) determines the quantized area of the faces via equation (\ref{['area']}). Here the vector $\vec{j}$ has a single component, because each face is crossed by one link only.
  • Figure 3: Figure 3: A branched surface with two vertices.
  • Figure 4: Figure 4: A simple vertex.