2002/01/28 by Sergey Alexandrov, Sergei Alexandrov
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Classical mechanics #Combinatorics #Computer science #Covariant transformation #Hilbert space #Loop (graph theory) #Loop quantum cosmology #Loop quantum gravity #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum gravity #Quantum mechanics #Space (punctuation) #Spin foam #Theoretical physics #gr-qc
paper · pdf · doi:10.1103/physrevd.66.024028
published as Phys.Rev. D66 (2002) 024028 · 20 pages, RevTEX
arxiv created 2002/01/28 · openalex publication_date 2002/07/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We investigate Hilbert space in the Lorentz covariant approach to loop quantum gravity. We restrict ourselves to the space where all area operators are simultaneously diagonalizable, assuming that it exists. In this sector, quantum states are realized by a generalization of spin network states based on Lorentz Wilson lines projected on irreducible representations of an SO(3) subgroup. The problem of infinite dimensionality of the unitary Lorentz representations is absent due to this projection. Nevertheless, the projection preserves the Lorentz covariance of the Wilson lines so that the symmetry is not broken. Under certain conditions, the states can be thought of as functions on a homogeneous space. We define the inner product as an integral over this space. With respect to this inner product, the spin networks form an orthonormal basis in the investigated sector. We argue that it is the only relevant part of a larger state space arising in the approach. The problem of the noncommutativity of the Lorentz connection is solved by restriction to the simple representations. The resulting structure shows similarities with the spin foam approach.