We draw parallels between the recently introduced ``Immirzi ambiguity'' of the Ashtekar-like formulation of canonical quantum gravity and other ambiguities that appear in Yang-Mills theories, like the $\theta$ ambiguity. We also discuss ambiguities in the Maxwell case, and implication for the loop quantization of these theories.Comment: 5 pages, revtex, no figure

We propose a model for noncommutative quantum cosmology by means of a deformation of minisuperspace. For the Kantowski-Sachs metric we are able to find the exact wave function. We construct wave packets and show that new quantum states that "compete" to be the most probable state appear, in clear contrast with the commutative case. A tunneling process could be possible among these states.PACS numbers: 04.60. Kz, 11.10.Lm, 11.25.Sq, 98.80.Hw The old proposal of noncommutativity in space-time [1] has been recently subject of renewed interest (see reviews [2,3,4]). This is a consequence of the developments in M(atrix) theory [5] and string theory, from which noncommutativity arises in the low energy effective field theory on a D-brane in a constant background B-field [6,7,8,9].One of the most exciting recent applications of the idea of a minimal size to field theory, is that concerning the description of Yang-Mills instantons in four dimensional noncommutative spacetimes. It has been shown that in these spaces instantons acquire an effective size in terms of the noncommutativity parameter θ. As a consequence, the moduli space of noncommutative instantons no longer has the singularities corresponding to small instantons [10]. This effect has a nice stringy interpretation [9].Noncommutative gravity has been considered in [11,12,13]. In particular in reference [13] a deformed Einstein gravity is constructed by using the Seiberg-Witten map [9], gauging the noncommutative ISO(3,1) group. Some aspects of noncommutative 2 + 1-dimensional Chern-Simons gravity have been also studied [14,15].One of the puzzles in quantum gravity is the measurement of length, which seems to be limited to distances greater than the Planck length L P , because to locate a particle we would need an energy greater than the Planck mass M P . The corresponding gravitational field will have an horizon given by the Schwarzschild radius R = 2GMP c 2 = 2L P and, whatever happens inside, this radius is shielded and therefore a minimal size should exist for quanta of space and time configuration.Consequently, at very early times of the universe, before the Planck time, nontrivial effects of noncommutativity can be expected.In the study of homogeneous universes, the metric depends only on the time parameter. Thus, the space dependence can be integrated out in the action and a model with a finite dimensional configuration space arises, called also minisuperspace, whose variables are the three-metric components. These theories have been considered by themselves, and their quantization is performed following the rules of quantum mechanics.The minisuperspace construction is a procedure to define quantum cosmology models in the search to describe the quantum behavior of the very early stages of the universe [16,17]. By defining these models one necessarily freezes out degrees of freedom, so that these are only simple and probably approximate models of full quantum gravity at Planckian times. Formally, a kind of Klein Gordon equation arises in these models, ...

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Using N=1 supergravity as the natural square root of gravity, the authors study the quantum cosmology of Bianchi type I cosmological models. This approach gives us a natural interpretation of the components of the state vector of the universe that was lacking in previous work on the square root of quantum cosmology.

Starting from a self-dual formulation of gravity, we obtain a noncommutative theory of pure Einstein theory in four dimensions. In order to do that, we use Seiberg-Witten map. It is shown that the noncommutative torsion constraint is solved by the vanishing of commutative torsion.Finally, the noncommutative corrections to the action are computed up to second order. * Electronic address: compean@fis.cinvestav.mx † Electronic address: octavio@ifug3.ugto.mx ‡ Permanent address: