1995/05/31 by Lee Smolin · 237 citations
Mathematics · Physics and Astronomy · #Asymptotic safety in quantum gravity #Black Holes and Theoretical Physics #Boundary (topology) #Cosmological constant #Cosmological constant problem #Cosmology and Gravitation Theories #Coupling constant #Dark energy #Mathematical analysis #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum field theory #Quantum gravity #Quantum mechanics #Theoretical physics #gr-qc #hep-th
paper · pdf · doi:10.1063/1.531251
published in Journal of Mathematical Physics 36(11), 6417-6455 (American Institute of Physics) · TEX File, Minor Changes Made, 59 pages
openalex publication_date 1995/11/01 · arxiv created 1996/01/30 · arxiv updated 2010/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Quantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory in which the pullback of the curvature to the boundary is self-dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all SU(2) Chern–Simon theories defined by all choices of punctures and representations on the spatial boundary 𝒮. The integer level k of Chern–Simons theory is found to be given by k=6π/G2Λ+α, where Λ is the cosmological constant and α is a CP breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. Given these results, it is natural to make the conjecture that the quantum states of the system are completely determined by measurements made on the boundary. One consequence of this is the Bekenstein bound, which says that once the two metric of the boundary has been measured, the subspace of the physical state space that describes the further information that may be obtained about the interior has finite dimension equal to the exponent of the area of the boundary, in Planck units, times a fixed constant. Finally, these results confirm both the categorical-theoretic ‘‘ladder of dimensions’’ picture of Crane and the holographic hypothesis of Susskind and ’t Hooft.