Real Ashtekar variables for Lorentzian signature space-times.

I suggest in this letter a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual SO(3) Yang-Mills phase space I show that it is possible to describe space-times with Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum theory. The key issue in the new formulation is how to deal with the more complicated Hamiltonian constraint that must be used in order to avoid the introduction of complex fields. The purpose of this letter is to suggest a new strategy to deal with the problem of the reality conditions in the Ashtekar approach to classical and quantum gravity. At the present moment there is some consensus about the reasons behind the success of the Ashtekar variables program [1]. One of them is the geometrical nature of the new variables. In particular, the fact that the configuration variable is a connection is specially interesting because this allows us to use loop variables both at the classical and quantum level [2]. Another advantage of the formalism is the simplicity of the constraints –specially the Hamiltonian constraint– that has been very helpful in finding solutions to all of them. There are, however, some difficulties in the formalism that must be solved and are not present in the traditional ADM scheme [3]. The most conspicuous one is the fact that complex variables must be used in order to describe Lorentzian signature space-times. This is often put in relation with the fact that the definition of self-duality in these space-times demands the introduction of imaginary coefficients. The now accepted way to deal with this issue is the introduction of reality conditions. They impose some consistency requirements on the scalar product in the Hilbert space of physical states. In fact, the hope is that this scalar product can be selected by the reality conditions. There are, however some difficulties with this approach too. Specifically it is very difficult to implement the reality conditions in the loop variables scheme. Only recently some positive results in this direction have been reported [6]. The main point of this letter is to consider the geometrical nature of the Ashtekar variables as the most important asset of the formalism. With this idea in mind, it is easy to see that the introduction of complex variables is necessary only if one wants to have an specially simple form for the Hamiltonian constraint. If we accept to live with a more complicated Hamiltonian constraint in the Ashtekar phase space we can use real variables. An interesting consequence of this, as emphasized by Rovelli and Smolin, is that all the results obtained within the loop variables approach (existence of volume and