### 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 -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 , . This
regularized Hamiltonian is well-defined on states with , 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 . The type of representation and regularization enables one to handle this
non-polynomiality.