An Example of Loop Quantization

Summary

Motivation

The background for the investigations to be discussed below is provided byrecent studies of canonical quantum gravity in 3+1 dimensions. However, related issues arise in other physical theories that can be formulated on spaces of connections and are invariant under a corresponding group of gauge transformations. We will be interested in the so-called loop approach to the quantization of gravity, and will have a closer look at the (2+1)-dimensional theory, in the hope of gaining further understanding of the general approach.

Let us summarize in a nutshell the ideas that have gone into proposals for a quantization program for general relativity.

1. * The starting point is Ashtekar's reformulation of 3+1-dimensional Hamiltonian gravity in terms of Yang-Mills variables, namely, an sl(2, (ℂ)-valued pair (A, E) of a connection one-form and its conjugate momentum [1].

2. * On the phase space spanned by these variables, define Wilson loop variables T⁰(γ) = TrP exp ∫_γA, and momentum-dependent generalizations which together form a closed Poisson bracket algebra of loop functions.

3. * Next, “quantize” this classical structure by finding representations of the Wilson loop algebra on spaces of wave functions that are themselves labelled by spatial loops γ.

4. * Rewrite the Hamiltonian in terms of loop variables, regularize it appropriately and look for solutions of the Wheeler-DeWitt equation, i.e. wave functions that are annihilated by the quantum Hamiltonian operator.