Other gauge formulations

2.4 Other gauge formulations

Das et al. [84 ] lattice-discretized an $Sp (4)$ -invariant Lagrangian due to West [194 ], leading to a functional form $∘ -------------------- S [U ] = ∑ Tr ((U − 1Tr U )2) n n 4 n$ , where $U n$ denotes a sum of double-plaquette holonomies based at the vertex $n$ . Details of the functional integration were not spelled out. The square-root form of the Lagrangian does not make it very amenable to numerical investigations (see also the comments in [139

]).

Mannion and Taylor [159 ] suggested a discretization of the $(∫ e ∧ e ∧ R )$ -Lagrangian without cosmological term, with the vierbeins $e$ treated as extra variables, and the compactified Lorentz group $SO (4 )$ .

Kaku [138 ] has proposed a lattice version of conformal gravity (see also [137 , 139 ]), based on the group $O (4,2)$ , the main hope being that unitarity could be demonstrated non-perturbatively. As usual, a metricity constraint on the connection has to be imposed by hand.

A lattice formulation of higher-derivative gravity, containing fourth-order $2 R$ -terms was given by Tomboulis [189 ]. The continuum theory is renormalizable and asymptotically free, but has problems with unitarity. The motivation for this work was again the hope of realizing unitarity in a lattice setting. The square-root form of the Lagrangian is similar to that of Das et al. It is $O (4 )$ -invariant and supposedly satisfies reflection positivity. Again, the form of the Lagrangian and the measure (containing a $δ$ -function of the no-torsion constraint) is rather complicated and has not been used for a further non-perturbative analysis.

Kondo [142 ] employed the same framework as Mannion and Taylor, but introduced an explicit symmetrization of the Lagrangian. He claimed that the cluster expansion goes through just as in lattice Yang–Mills theory, leading to a positive mass gap.