  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.
