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Quantum Gravity: a Progress Report

S. Carlip

TL;DR

The work surveys the major routes to a quantum theory of gravity, contrasting covariant and canonical quantization with focusing programmatic pillars like loop quantum gravity and string theory. It analyzes foundational obstacles—diffeomorphism invariance, the problem of time, and nonrenormalizability—while outlining alternative formalisms such as BF theory and spin foams. The article assesses nonperturbative constructions, holographic dualities, and the role of simplified models, highlighting both the achievements (e.g., discrete geometric spectra, quantum cosmology insights, and gravity-embedded string dynamics) and the substantial open challenges toward a complete, predictive theory. It also sketches potential experimental and observational tests, from black hole thermodynamics to Planck-scale phenomenology, that could distinguish among competing quantum gravity pictures. The overarching message is cautious optimism: while no consensus exists, the diverse approaches collectively illuminate the path toward reconciling quantum mechanics with spacetime geometry, with substantial progress anticipated in the coming decade.

Abstract

The problem of reconciling general relativity and quantum theory has fascinated and bedeviled physicists for more than 70 years. Despite recent progress in string theory and loop quantum gravity, a complete solution remains out of reach. I review the status of the continuing effort to quantize gravity, emphasizing the underlying conceptual issues and the various attempts to come to grips with them.

Quantum Gravity: a Progress Report

TL;DR

The work surveys the major routes to a quantum theory of gravity, contrasting covariant and canonical quantization with focusing programmatic pillars like loop quantum gravity and string theory. It analyzes foundational obstacles—diffeomorphism invariance, the problem of time, and nonrenormalizability—while outlining alternative formalisms such as BF theory and spin foams. The article assesses nonperturbative constructions, holographic dualities, and the role of simplified models, highlighting both the achievements (e.g., discrete geometric spectra, quantum cosmology insights, and gravity-embedded string dynamics) and the substantial open challenges toward a complete, predictive theory. It also sketches potential experimental and observational tests, from black hole thermodynamics to Planck-scale phenomenology, that could distinguish among competing quantum gravity pictures. The overarching message is cautious optimism: while no consensus exists, the diverse approaches collectively illuminate the path toward reconciling quantum mechanics with spacetime geometry, with substantial progress anticipated in the coming decade.

Abstract

The problem of reconciling general relativity and quantum theory has fascinated and bedeviled physicists for more than 70 years. Despite recent progress in string theory and loop quantum gravity, a complete solution remains out of reach. I review the status of the continuing effort to quantize gravity, emphasizing the underlying conceptual issues and the various attempts to come to grips with them.

Paper Structure

This paper contains 36 sections, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Why quantum gravity?
  3. Why is gravity not yet quantized?
  4. Classical preliminaries
  5. Covariant formalism: second order form
  6. Canonical formalism: second order form
  7. Covariant formalism: first order form
  8. Canonical formalism: first order form
  9. Gravity as a constrained $BF$ theory
  10. Covariant quantization
  11. Lorentzian perturbation theory
  12. The Euclidean path integral
  13. Consistent histories
  14. Canonical quantization
  15. Dirac quantization and the Wheeler-DeWitt equation
  16. ...and 21 more sections

Figures (4)

  • Figure 1: The ADM decomposition expresses the Lorentzian version of the Pythagoras theorem.
  • Figure 2: A manifold $M$ with a single boundary $\Sigma$ describes the birth of a universe in the Hartle-Hawking approach to quantum cosmology.
  • Figure 3: A trivalent spin network. The intertwiners at each vertex are ordinary Clebsch-Gordan coefficients.
  • Figure 5: A closed string vertex and a one-loop process.