Ashtekar formulation of general relativity and loop space nonperturbative quantum gravity: A Report

Apr 4, 1991
86 pages
Published in:
  • Class.Quant.Grav. 8 (1991) 1613-1676
Report number:
  • FPRINT-91-05

Citations per year

199119992007201520230510152025
Abstract: (IOP)
The formulation of general relativity discovered by Ashtekar (1986, 1987) and the recent results obtained in non-perturbative quantum gravity using loop-space techniques are reviewed. The new formulation is based on the choice of a set of Lagrangian (and Hamiltonian) variables, instead of the spacetime metric. In terms of these new variables, the dynamical equations are remarkably simplified and a structural identity between general relativity and the Yang-Mills theories is revealed. The formalism has proven to be useful in numerous problems in gravitational physics. In quantum gravity, the new formalism has overcome long-standing difficulties and led to unexpected results. A nonperturbative approach to quantum theory has been constructed in terms of the Wilson loops of the Ashtekar connection. This approach, denoted as loop-space representation, has led to the complete solution of the quantum diffeomorphism constraint in terms of knot states, to the discovery of an infinite-dimensional class of solutions to the quantum gravitational dynamics, and to certain surprising indications on the existence of a discrete structure of spacetime around the Planck length. These results are presented in a compact self-contained form. The basic Ashtekar formalism is presented and its applications are outlined. The loop-space representation and the non-perturbative knot states of quantum gravity are described in detail, with particular regard to their physical interpretation and to the information they may provide on the microstructure of spacetime.
  • inspirereview:I-d-6
  • review
  • general relativity
  • quantum gravity
  • Hamiltonian formalism
  • Wilson loop
  • loop space
  • nonperturbative
  • knot theory
  • transformation: diffeomorphism