1992/05/31 by John C. Baez, J C Baez
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #Noncommutative and Quantum Gravity Theories #hep-th #math.QA
paper · pdf · doi:10.1088/0264-9381/10/4/006
published as Class.Quant.Grav.10:673-694,1993 · 16 pages (with major corrections)
arxiv created 1992/06/22 · openalex publication_date 1993/04/01 · arxiv updated 2010/04/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
In the loop representation of quantum gravity in 3+1 dimensions, there is a space of physical states consisting of invariants of links in S 3 . The correct inner product on this space of states is not known, or in other words, the *-algebra structure of the algebra of observables has not been determined. In order to address this issue, the author works instead with quantum gravity on D 3 , and work with a space K spanned by tangles. A certain algebra T, the 'tangle algebra', acts as operators on K. The 'empty link' psi 0 , corresponding to the class of the empty set, is shown to be a cyclic vector for T. The author constructs inner products on quotients of K from link invariants, shows that these quotients are representations of T, and calculates the *-algebra structures of T in these representations. In particular, taking the link invariant to be the Jones polynomial (or more precisely, Kauffman bracket), the author obtains the inner product for states of quantum gravity arising from SU(2) Chern-Simons theory.