1994/11/02 by Carlo Rovelli, Lee Smolin · 18 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Diffeomorphism #Eigenvalues and eigenvectors #Hilbert space #Immirzi parameter #Loop quantum gravity #Mathematical analysis #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Operator (biology) #Physics #Quantum #Quantum Electrodynamics and Casimir Effect #Quantum gravity #Quantum mechanics #Spin foam #Spin network #gr-qc
paper · pdf · doi:10.1016/0550-3213(95)00150-q
published as Nucl.Phys. B442 (1995) 593-622; Erratum-ibid. B456 (1995) 753 · 36 pages, latex, 13 figures uuencoded
arxiv created 1994/11/02 · openalex publication_date 1995/05/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.