Lattice knot theory and quantum gravity in the loop representation

Abstract

We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern--Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chern--Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including intersections. We also discuss the issue of intersections with kinks. This paper is the first step towards setting up the loop representation in a rigorous, computable setting.

23 pages, RevTeX, 14 figures included with psfig

References in corpus (4)

• Quantum Spin Dynamics (QSD)
• Variational derivation of exact skein relations from Chern--Simons theories
• Is the third coefficient of the Jones knot polynomial a quantum state of gravity?
• The general solution of the quantum Einstein equations?

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• Bibliography of publications related to Classical and Quantum Gravity in terms of Connection and Loop Variables
• Loop Quantum Gravity effects on electromagnetic properties of charged leptons
• Mimetic discretization of the Abelian Chern-Simons theory and link invariants