2.3 Numerical implementation2 Gauge-theoretic discretizations of gravity2.1 Lagrangian treatment. Introduction

2.2 Smolin's lattice model

The first gauge-theoretic model for lattice gravity is due to Smolin [186], based on the continuum formulation of MacDowell and Mansouri [157], with de Sitter gauge group O (3,2) or O (4,1), and a Lagrangian of tex2html_wrap_inline2261 -type,

  equation66

where the components of tex2html_wrap_inline2263 are related to those of the usual curvature tensor by

  equation76

Although the underlying gauge potentials tex2html_wrap_inline2265 are o (3,2)- or o (4,1)-valued, the action is only invariant under the 6-dimensional subgroup of Lorentz transformations. The theory contains a dimensionful parameter l . The gauge potentials associated with the internal 5-direction are identified with the frame fields tex2html_wrap_inline2217, and the action can be decomposed into the usual Einstein-term (1Popup Equation) plus a cosmological constant term with tex2html_wrap_inline2275 and a topological tex2html_wrap_inline2277 -term. Smolin analyzed its lattice discretization, and found both a weak- and a strong-coupling phase, with respect to the dimensionless coupling constant tex2html_wrap_inline2279 . He performed a weak-coupling expansion about flat space and rederived the usual propagator. In the strong-coupling regime he found massive excitations and a confining property for spinors.



2.3 Numerical implementation2 Gauge-theoretic discretizations of gravity2.1 Lagrangian treatment. Introduction

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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