0901.4098 (Eyo Eyo Ita III)

Physical states in four dimensional gravity for arbitrary values of the
Immirzi parameter    [PDF]

Eyo Eyo Ita III

One of the virtues of the Ashtekar variables is the simplification of the initial value constraints for gravity. In the case of self-dual variables this entails a complexification of the phase space which comes at the expense of having to implement reality conditions in the Lorentzian signature case. A reformulation of the theory in terms of real variables eliminates this difficulty, albeit at the expense of having to deal with a more complicated Hamiltonian constraint. The set of available gravitational theories classically equivalent to Einstein's is parametrized by a parameter $\beta$, known as the Immirzi parameter. The construction of its physical Hilbert space of states has until now remained an unsolved problem. In this paper we show that one can have the best of both worlds, by constructing physical states of quantum gravity for arbitrary values of $\beta$. We use results from the affine group representation formalism, which casts General Relativity and its quantization into a group theoretical context. The implications of this for volume eigenstates of LQG are examined.

View original: http://arxiv.org/abs/0901.4098