Renteln and Smolin [180] were the first to set up a continuous-time lattice
discretization along the lines of Hamiltonian lattice gauge
theory. Their basic configuration variables are the link
holonomies
U
(l) of the spatial Ashtekar connection along the edges. The lattice
analogues of the canonically conjugate pairs
are the link variables
, with Poisson brackets
with the
SU
(2)-generators satisfying
. Lattice links
are labelled by a vertex
n
and a lattice direction
. These relations go over to the usual continuum brackets in the
limit as the lattice spacing
a
is taken to zero. In this scheme, they wrote down discrete
analogues of the seven polynomial first-class constraints, and
also attempted to interpret the action of the discretized
diffeomorphism and Hamiltonian constraints in terms of their
geometric action on lattice Wilson loop states.
![]() |
Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |