Loop space representation of quantum general relativity

Abstract We define a new representation for quantum general relativity, in which exact solutions of the quantum constraints may be obtained. The representation is constructed by means of a noncanonical graded Poisson algebra of classical observables, defined in terms of Ashtekar's new variables. The observables in this algebra are nonlocal and involve parallel transport around loops in a three-manifold Σ. The theory is quantized by constructing a linear representation of a deformation of this algebra. This representation is given in terms of an algebra of linear operators defined on a state space which consists of functionals of sets of loops in Σ. The construction is general and can be applied also to Yang-Mills theories. The diffeomorphism constraint is defined in terms of a natural representation of the diffeomorphism group. The hamiltonian constraint, which contains the dynamics of quantum gravity, is constructed as a limit of a sequence of observables which incorporates a regularization prescription. We give the general solution of the diffeomorphism constraint in closed form. It is spanned by a countable basis which is in one-to-one correspondence with the diffeomorphism equivalence classes of multiple loops, which are a generalization of the link classes studied in knot theory. Then we explicitly construct, in closed form, a large space of solution of the entire set of constraints, including the hamiltonian constraint. These turn out to be classified by the ordinary knot and link classes of Σ. The space of solutions that we find is a sector of the physical states space of nonperturbative quantum general relativity. The failure of perturbation theory is thus shown to be not relevant to the problem of the existence of a nontrivial physical state space in quantum gravity. The relationship between this new loop representation and the self-dual representation of Ashtekar is illuminated by means of a functional transform between states in the two representations. Questions of the completeness of the solution space, the meaning of the physical operators and the physical inner product, are discussed, but not, so far, resolved.

[1]  Stanley Deser,et al.  Canonical variables for general relativity , 1960 .

[2]  L. Smolin.  Invariants of links and critical points of the Chern-Simon path integrals , 1989 .

[3]  Knot Invariants and the Critical Statistical Systems , 1987 .

[4]  A. Kakas,et al.  A group theoretical approach to the canonical quantisation of gravity. I. Construction of the canonical group , 1984 .

[5]  R. I. Bogdanov.  Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues , 1975 .

[6]  P. Dirac.  Fixation of coordinates in the Hamiltonian theory of gravitation , 1959 .

[7]  Yang-Mills theory and general relativity in three and four dimensions , 1989 .

[8]  L. Smolin,et al.  Exactly solvable quantum cosmologies from two killing field reductions of general relativity , 1989 .

[9]  A. Ashtekar,et al.  New variables for classical and quantum gravity. , 1986, Physical review letters.

[10]  M. Virasoro.  YANG-MILLS FIELDS AND THE OLD DUAL MODEL: AN AMBIGUOUS RELATIONSHIP , 1979 .

[11]  I. Gel'fand,et al.  REPRESENTATIONS OF THE GROUP OF DIFFEOMORPHISMS , 1975 .

[12]  M. Virasoro,et al.  The interaction among dual strings as a manifestation of the gauge group , 1980 .

[13]  P. Bergmann,et al.  The Hamiltonian of the General Theory of Relativity with Electromagnetic Field , 1950 .

[14]  V. Husain.  Intersecting-loop solutions of the hamiltonian constraint of quantum general relativity , 1989 .

[15]  R. Arnowitt,et al.  Quantum Theory of Gravitation: General Formulation and Linearized Theory , 1959 .

[16]  V. Jones.  A polynomial invariant for knots via von Neumann algebras , 1985 .

[17]  Stanley Deser,et al.  Dynamical Structure and Definition of Energy in General Relativity , 1959 .

[18]  Louis H. Kauffman,et al.  State Models and the Jones Polynomial , 1987 .

[19]  W. Furmanski,et al.  Yang-Mills vacuum: An attempt at lattice loop calculus , 1987 .

[20]  Alexander M. Polyakov,et al.  Gauge Fields And Strings , 1987 .

[21]  Y. Nambu.  QCD and the string model , 1979 .

[22]  L. Smolin,et al.  Nonperturbative quantum geometries , 1988 .

[23]  C. DeWitt-Morette,et al.  Analysis, manifolds, and physics , 1977 .

[24]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[25]  A. Ashtekar,et al.  2+1 quantum gravity as a toy model for the 3+1 theory , 1989 .

[26]  M. Henneaux,et al.  Explicit Solution for the Zero Signature (Strong Coupling) Limit of the Propagation Amplitude in Quantum Gravity , 1982 .

[27]  M. Wadati,et al.  Virasoro Algebra, von Neumann Algebra and Critical Eight-Vertex SOS Models , 1986 .

[28]  B. Dewitt.  Quantum Theory of Gravity. I. The Canonical Theory , 1967 .

[29]  J. Fröhlich.  Statistics of Fields, the Yang-Baxter Equation, and the Theory of Knots and Links , 1988 .

[30]  M. Virasoro,et al.  Gauge fields as phonon excitations in a condensate of dual strings , 1979 .

[31]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[32]  A. Migdal,et al.  Exact equation for the loop average in multicolor QCD , 1979 .

[33]  Rovelli,et al.  Knot theory and quantum gravity. , 1988, Physical review letters.

[34]  A. Polyakov.  String Representations and Hidden Symmetries for Gauge Fields , 1979 .

[35]  Paul Adrien Maurice Dirac.  Generalized Hamiltonian dynamics , 1950 .

[36]  Vaughan F. R. Jones.  Index for subfactors , 1983 .