[Paper Review] Quantum gravity with a positive cosmological constant

Lee Smolin|ArXiv.org|Sep 9, 2002

Noncommutative and Quantum Gravity TheoriesPhysics and Astronomy92 references128 citations

TL;DR

This paper proposes that loop quantum gravity provides a consistent quantum theory of gravity with a positive cosmological constant, using the Kodama state as an exact physical solution that semiclassically describes de Sitter spacetime. It demonstrates that long-wavelength excitations of this state reproduce gravitons and quantum field theory on de Sitter spacetime, with computable corrections to energy-momentum relations that may be experimentally testable.

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.

Motivation & Objective

• To establish loop quantum gravity as a viable quantum theory of gravity in the presence of a positive cosmological constant.
• To demonstrate that the Kodama state serves as an exact physical solution with a semiclassical limit corresponding to de Sitter spacetime.
• To show that perturbations of the Kodama state reproduce gravitons and quantum field theory on de Sitter spacetime.
• To derive computable corrections to matter field energy-momentum relations from the Wheeler-DeWitt equation.
• To provide a background-independent explanation of the Bekenstein bound and the N-bound using horizon boundary conditions.

Proposed method

• Use of the Kodama state as an exact solution to the quantum constraints of loop quantum gravity with Λ > 0.
• Application of the spin network representation, which becomes quantum deformed due to the cosmological constant.
• Construction of boundary Hilbert spaces via Chern-Simons theory on timelike or spacelike boundaries.
• Perturbative expansion of the Kodama state to study long-wavelength excitations and their correspondence to gravitons and QFT on de Sitter.
• Derivation of modified energy-momentum relations of the form E² = p² + m² + αlPlE³ + … from the Wheeler-DeWitt equation.
• Use of the loop transform to map the Kodama state into a diffeomorphism-invariant representation and analyze its excitations.

Experimental results

Research questions

• RQ1 Does loop quantum gravity admit an exact physical state that semiclassically describes de Sitter spacetime when the cosmological constant is positive?
• RQ2 Can long-wavelength excitations of the Kodama state reproduce the spectrum of gravitons on a de Sitter background?
• RQ3 Do quantum corrections to matter field energy-momentum relations emerge from the theory, and are they computable and potentially testable?
• RQ4 Can the Bekenstein bound and the N-bound be derived from a background-independent formulation of quantum gravity with Λ > 0?
• RQ5 How can quantum gravity be consistently formulated in the presence of horizons or timelike boundaries using Chern-Simons theory?

Key findings

• The Kodama state is an exact solution to all quantum constraints and has a semiclassical limit corresponding to de Sitter spacetime.
• Small, long-wavelength perturbations of the Kodama state reproduce the spectrum of gravitons on a de Sitter background.
• When coupled to matter, perturbations of the Kodama state reproduce quantum field theory on de Sitter spacetime.
• The theory yields corrections to the energy-momentum relation of matter fields, E² = p² + m² + αlPlE³ + …, with α a computable dimensionless constant.
• The Bekenstein bound is automatically satisfied due to the structure of the boundary Hilbert space constructed via Chern-Simons theory.
• The N-bound of Banks is given a background-independent explanation through the same boundary theory framework.