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Quantum gravity with a positive cosmological constant

Lee Smolin

TL;DR

Smolin argues that loop quantum gravity with a positive cosmological constant Λ>0 yields a consistent quantum description of spacetime, anchored by the Kodama state which is exact and semiclassically deSitter. Long-wavelength perturbations around this state reproduce gravitons and quantum field theory on deSitter, while Planck-scale corrections to dispersion relations offer potential experimental tests. The formalism naturally incorporates a boundary Chern-Simons theory, the Bekenstein bound, and the N-bound via a quantum-deformed SU_q(2) framework at level κ, linking horizon thermodynamics to background-independent quantum gravity. Together, these results position LQG as a viable quantum gravity framework with horizons and suggest concrete avenues for phenomenology and quantum cosmology.

Abstract

A quantum theory of gravity is described in the case of a positive cosmological constant in 3+1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the Wheeler-deWitt equation, Planck scale, computable corrections to the energy-momentum relations for matter fields. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.

Quantum gravity with a positive cosmological constant

TL;DR

Smolin argues that loop quantum gravity with a positive cosmological constant Λ>0 yields a consistent quantum description of spacetime, anchored by the Kodama state which is exact and semiclassically deSitter. Long-wavelength perturbations around this state reproduce gravitons and quantum field theory on deSitter, while Planck-scale corrections to dispersion relations offer potential experimental tests. The formalism naturally incorporates a boundary Chern-Simons theory, the Bekenstein bound, and the N-bound via a quantum-deformed SU_q(2) framework at level κ, linking horizon thermodynamics to background-independent quantum gravity. Together, these results position LQG as a viable quantum gravity framework with horizons and suggest concrete avenues for phenomenology and quantum cosmology.

Abstract

A quantum theory of gravity is described in the case of a positive cosmological constant in 3+1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the Wheeler-deWitt equation, Planck scale, computable corrections to the energy-momentum relations for matter fields. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.

Paper Structure

This paper contains 25 sections, 151 equations.

Table of Contents

  1. Introduction
  2. Classical gravity and its relation to topological field theory
  3. Gravity as a gauge theory
  4. The deSitter solution as a gauge field
  5. Hamilton-Jacobi theory, deSitter spacetime and Chern-Simons theory
  6. General Relativity as a constrained topological field theory
  7. Boundaries with $\Lambda >0$ and Chern-Simons Theory
  8. The quantum theory
  9. The Kodama State
  10. A brief review of quantization
  11. The Kodama state
  12. The recovery of QFT on deSitter spacetime.
  13. Gravitons from perturbations around the Kodama state
  14. Corrections to energy momentum relations
  15. The thermal nature of quantum gravity with $\Lambda > 0$.
  16. ...and 10 more sections