Abstract
In this paper, I investigate the quantisation of length in euclidean quantum gravity in three dimensions. The starting point is the classical hamiltonian formalism in a cylinder of finite radius. At this finite boundary, a counter term is introduced that couples the gravitational field in the interior to a two-dimensional conformal field theory for an SU(2) boundary spinor, whose norm determines the conformal factor between the fiducial boundary metric and the physical metric in the bulk. The equations of motion for this boundary spinor are derived from the boundary action and turn out to be the two-dimensional analogue of the Witten equations appearing in Witten's proof of the positive mass theorem. The paper concludes with some comments on the resulting quantum theory. It is shown, in particular, that the length of a one-dimensional cross section of the boundary turns into a number operator on the Fock space of the theory. The spectrum of this operator is discrete and matches the results from loop quantum gravity in the spin network representation.
Highlights
One of the key open issues for loop quantum gravity is to check that the fundamental quantum discreteness of space that we see in the theory is compatible with the known physics in the continuum
The goal of this paper is to demonstrate that such a boundary field theory exists and can be constructed in terms of an SU (2) boundary spinor coupled to the gravitational field in the bulk
We studied euclidean gravity in an infinite cylinder of finite radius
Summary
One of the key open issues for loop quantum gravity is to check (or prove it impossible) that the fundamental quantum discreteness of space that we see in the theory is compatible with the known physics in the continuum. I will turn this question around, and show that in three dimensions the loop gravity quantisation of space can be understood already from the theory in the continuum without ever introducing spin networks or triangulations of space. The evaluation of the amplitudes for such boundary states defines an effective boundary action eiSeff[ξ] ∼ ZPR[Ψξ], whose critical points define a classical lattice model for the boundary spinors That such a theory should exist at finite boundaries in the continuum is motivated by another development in the field: during the last couple of years a new representation was developed for four-dimensional loop quantum gravity in terms of SL(2, C) spinors [1, 13,14,15,16]. The entire derivation happens at the level of the continuum theory, and no spin networks or triangulations of space are ever required for deriving this result
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