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 [148153] 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
∑ √ -- ∘ -------- Hlatt = N latt(n) [ℋlatt + ℋlatt+ λG det p(n)], (11 ) n kin pot
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.
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