is the densitized "electric" field or triad (densitized as ). What do these variables have to do with gravity? The densitized triads can be used to reconstruct the spatial metric via The densitized triads are not unique, and in fact one can perform a local in space rotation with respect to the internal indices . This is actually the origin of the gauge invariance. The connection can be used to reconstruct the extrinsic curvature. The relation is given by where is related to the spin connection, , by and . In terms of Ashtekar variables the classical expression of the constraint is given by, where field strength tensor of the gauge field . Due to the factor this is non-polynomial in the Ashtekar's variables. Since we impose the condition we could consider the densitized Hamiltonian instead, This Hamiltonian is now polynomial the Ashtekar's variables. This development raised new hopes for the canonical quantum gravity programme. Although Ashtekar variables had the virtue of simplifying the Hamiltonian, it has the problem that the variables become complex. When one quantizes the theory it is a difficult task ensure that one recovers real general relativity as opposed to complex general relativity. Also there were also serious difficulties promoting the densitized Hamiltonian to a quantum operator. A way of addressing the problem of reality conditions was noting that if we took the signature to be , that is Euclidean instead of Lorentzian, then one can retain the simple form of the Hamiltonian for but for real variables. One can then define what is called a generalized Wick rotation to recover the Lorentzian theory. Generalized as it is a Wick transformation in phase space and has nothing to do with analytical continuation of the time parameter . Expression for real formulation of Ashtekar variables Thomas Thiemann addressed both the above problems. He used the real connection In real Ashtekar variables the full Hamiltonian is where the constant is the Barbero–Immirzi parameter. The constant is -1 for Lorentzian signature and +1 for Euclidean signature. The have a complicated relationship with the densitized triads and causes serious problems upon quantization. Ashtekar variables can be seen as choosing to make the second more complicated term was made to vanish (the first term is denoted because for the Euclidean theory this term remains for the real choice of ). Also we still have the problem of the factor. Thiemann was