Classical general relativity can be formulated in phase space form as follows [9, 40]. We fix a three-dimensional manifold M (compact and without boundaries) and consider a smooth real SU (2) connection and a vector density (transforming in the vector representation of SU (2)) on M . We use for spatial indices and for internal indices. The internal indices can be viewed as labeling a basis in the Lie algebra of SU (2) or in the three axis of a local triad. We indicate coordinates on M with x . The relation between these fields and conventional metric gravitational variables is as follows: is the (densitized) inverse triad, related to the three-dimensional metric of constant-time surfaces by
where g is the determinant of ; and
is the spin connection associated to the triad, (defined by , where is the triad). is the extrinsic curvature of the constant time three surface.
In (2), is a constant, denoted the Immirzi parameter, that can be chosen arbitrarily (It will enter the hamiltonian constraint.) [114, 115, 116]. Different choices for yield different versions of the formalism, all equivalent in the classical domain. If we choose to be equal to the imaginary unit, , then A is the standard Ashtekar connection, which can be shown to be the projection of the selfdual part of the four-dimensional spin connection on the constant time surface. If we choose , we obtain the real Barbero connection. The hamiltonian constraint of Lorentzian general relativity has a particularly simple form in the formalism; while the hamiltonian constraint of Euclidean general relativity has a simple form when expressed in terms of the real connection. Other choices of are viable as well. In particular, it has been argued that the quantum theory based on different choices of are genuinely physical inequivalent, because they yield ``geometrical quanta'' of different magnitude . Apparently, there is a unique choice of yielding the correct 1/4 coefficient in the Bekenstein-Hawking formula [134, 135, 176, 11, 178, 70], but the matter is still under discussion.
The spinorial version of the Ashtekar variables is given in terms of the Pauli matrices , or the su (2) generators , by
Thus, and are anti-hermitian complex matrices.
The theory is invariant under local SU (2) gauge transformations, three-dimensional diffeomorphisms of the manifold on which the fields are defined, as well as under (coordinate) time translations generated by the hamiltonian constraint. The full dynamical content of general relativity is captured by the three constraints that generate these gauge invariances [190, 9].
As already mentioned, the Lorentzian hamiltonian constraint does not have a simple polynomial form if we use the real connection (2). For a while, this fact was considered an obstacle to defining the quantum hamiltonian constraint; therefore the complex version of the connection was mostly used. However, Thiemann has recently succeeded in constructing a Lorentzian quantum hamiltonian constraint [206, 201, 202] in spite of the non-polynomiality of the classical expression. This is the reason why the real connection is now widely used. This choice has the advantage of eliminating the old ``reality conditions'' problem, namely the problem of implementing non-trivial reality conditions in the quantum theory.
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