The real dynamics

2.16 The real dynamics

To avoid problems with the non-compactness of the gauge group and the formulation of suitable “quantum reality conditions”, Barbero [25 ] advocated to use a real $SU (2)$ -connection formulation for Lorentzian continuum gravity. This can be achieved, at the price of having to deal with a more complicated Hamiltonian constraint. Loll [148 , 153 ] translated the real connection formulation to the lattice and studied some of the differences that arise in comparison with the complex approach. Adding for generality a cosmological constant term, this leads to a lattice Hamiltonian

where schematically $latt − 1∕2 2 ℋ kin = (detp ) Tr (U τ)p$ , $latt 2 6 ℋ pot = ((Tr (τ Upτ U) − p) p$ $− 5∕2 (detp)$ . This regularized Hamiltonian is well-defined on states with $deˆt p ⁄= 0$ , but its functional form is not simple. The negative powers of the determinant of the metric can be defined in terms of the spectral resolution of $ˆ det p$ . The type of representation and regularization enables one to handle this non-polynomiality.