The Extended Loop Representation of Quantum Gravity

Abstract

A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum Gravity can be realized on the state space of extended loop dependent wavefunctions. The extended representation generalizes the loop representation and contains this representation as a particular case. The resulting diffeomorphism and hamiltonian constraints take a very simple form and allow to apply functional methods and simplify the loop calculus. In particular we show that the constraints are linear in the momenta. The nondegenerate solutions known in the loop representation are also solutions of the constraints in the new representation. The practical calculation advantages allows to find a new solution to the Wheeler-DeWitt equation. Moreover, the extended representation puts in a precise framework some of the regularization problems of the loop representation. We show that the solutions are generalized knot invariants, smooth in the extended variables, and any framing is unnecessary.

27 pages, report IFFC/94-13

References in corpus (11)

• Weaving a classical geometry with quantum threads
• The physical hamiltonian in nonperturbative quantum gravity
• Jones Polynomials for Intersecting Knots as Physical States of Quantum Gravity
• How the Jones Polynomial Gives Rise to Physical States of Quantum General Relativity
• A Loop Representation for the Quantum Maxwell Field
• The Extended Loop Group: An Infinite Dimensional Manifold Associated with the Loop Space
• Loop Representations for 2+1 Gravity on a Torus
• New Loop Representations for 2+1 Gravity
• Extended Loops: A New Arena for Nonperturbative Quantum Gravity
• An analytical expression for the third coefficient of the Jones Polynomial
• Classical Loop Actions of Gauge Theories

Cited by in corpus (13)

• Discreteness of area and volume in quantum gravity
• Loop Quantum Gravity
• Quantum deformation of quantum gravity
• Closed formula for Wilson loops for $SU(N)$ Quantum Yang-Mills Theory in two dimensions
• Chern--Simons states in spin-network quantum gravity
• Lattice knot theory and quantum gravity in the loop representation
• Is the third coefficient of the Jones knot polynomial a quantum state of gravity?
• The space of states of quantum gravity in terms of loops and extended loops: some remarks
• The Kauffman bracket and the Jones polynomial in quantum gravity
• The general solution of the quantum Einstein equations?
• Extended knots and the space of states of quantum gravity
• Structure of Extended Loop Wave Function in Quantum Gravity and Operator Formalism
• A Manifestly Gauge-Invariant Approach to Quantum Theories of Gauge Fields