As mentioned above, because Ashtekar's variables are complex the resulting general relativity is complex. To recover the real theory, one has to impose what are known as the "reality conditions." These require that the densitized triad be real and that the real part of the Ashtekar connection equals the compatible spin connection (the compatibility condition being formula_210) determined by the densitized triad. The expression for compatible connection formula_47 is rather complicated and as such non-polynomial formula enters through the back door.