2.17 Summary2 Gauge-theoretic discretizations of gravity2.15 The volume operator

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

  equation251

where schematically tex2html_wrap_inline2415, tex2html_wrap_inline2417 tex2html_wrap_inline2419 . This regularized Hamiltonian is well-defined on states with tex2html_wrap_inline2421, 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 tex2html_wrap_inline2423 . The type of representation and regularization enables one to handle this non-polynomiality.



2.17 Summary2 Gauge-theoretic discretizations of gravity2.15 The volume operator

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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