Smolin’s lattice model

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 $2 R$ -type,

where the components of $&tidle; R μν$ are related to those of the usual curvature tensor by

Although the underlying gauge potentials $Aμ$ 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 $I eμ$ , and the action can be decomposed into the usual Einstein-term (1

) plus a cosmological constant term with $λ ∼ 1l2$ and a topological $R ∧ R$ -term. Smolin analyzed its lattice discretization, and found both a weak- and a strong-coupling phase, with respect to the dimensionless coupling constant $√G- g ∼ l$ . 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.