2.2 Smolin's lattice model2 Gauge-theoretic discretizations of gravity2 Gauge-theoretic discretizations of gravity

2.1 Lagrangian treatment. Introduction

This area of research was inspired by the success of non-perturbative lattice methods in treating non-abelian gauge theories [173]. To apply some of their techniques, gravity has to be brought into a gauge-theoretic, first-order form, with the pure-gravity Lagrangian

  equation35

where the so (3,1)-valued spin-connection tex2html_wrap_inline2213 (with curvature R) and the vierbein tex2html_wrap_inline2217 are considered as independent variables. The important feature of (1Popup Equation) is that tex2html_wrap_inline2213 is a gauge potential, and that the action - in addition to its diffeomorphism-invariance - is invariant under local frame rotations. Variation with respect to tex2html_wrap_inline2217 leads to the metricity condition,

  equation42

which can be solved to yield the unique torsion-free spin connection A = A [e] compatible with the tex2html_wrap_inline2225 . There are some obvious differences with usual gauge theories: i) the action (1Popup Equation) is linear instead of quadratic in the curvature two-form tex2html_wrap_inline2227 of A, and ii) it contains additional fields tex2html_wrap_inline2217 . Substituting the solution to (2Popup Equation) into the action, one obtains tex2html_wrap_inline2233, where R denotes the four-dimensional curvature scalar. This expression coincides with the usual Einstein action tex2html_wrap_inline2237 only for tex2html_wrap_inline2239 .

Most of the lattice gauge formulations I will discuss below share some common features. The lattice geometry is hypercubic, defining a natural global coordinate system for labelling the lattice sites and edges. The gauge group G is SO (3,1) or its ``Euclideanized'' form SO (4), or a larger group containing it as a subgroup or via a contraction limit. Local curvature terms are represented by (the traces of) G -valued Wilson holonomies tex2html_wrap_inline2249 around lattice plaquettes. The vierbeins are either considered as additional fields or identified with part of the connection variables. The symmetry group of the lattice Lagrangian is a subgroup of the gauge group G, and does not contain any translation generators that appear when G is the Poincaré group.

When discretizing conformal gravity (where G = SO (5,1)) or higher-derivative gravity in first-order form, the metricity condition on the connection has to be imposed by hand. This leads to technical complications in the evaluation of the functional integral.

The diffeomorphism invariance of the continuum theory is broken on the lattice; only the local gauge invariances can be preserved exactly. The reparametrization invariance re-emerges only at the linearized level, i.e. when considering small perturbations about flat space.



2.2 Smolin's lattice model2 Gauge-theoretic discretizations of gravity2 Gauge-theoretic discretizations of gravity

image 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