On the problem of reality conditions for the Ashtekar formulation

In this paper we discuss the reality conditions for Lorentzian and Euclidean gravity in the Ashtekar formulation by introducing a double conformal transformation. We generalize Marugan's results and demonstrate that the values of the double conformal factor have to be either real or double complex numbers. Either Lorentzian or Euclidean gravitational theory is up to the different values of the double conformal factor. Furthermore, the reality conditions of Lorentzian and Euclidean gravitational theory can be expressed in a unified way by use of the double complex function method.