Ashtekar constraint surface as projection of the Hilbert-Palatini one

The Hilbert-Palatini (HP) Lagrangian of general relativity being written in terms of self-dual and anti-self-dual variables contains Ashtekar Lagrangian (which governs the dynamics of the self-dual sector of the theory on condition that the dynamics of anti-self-dual sector is not fixed). We show that nonequivalence of the Ashtekar and HP quantum theories is due to the fact that constraints which define anti-self-dual variables in terms of self-dual ones (and vice versa) do this only up to some degrees of freedom, therefore the procedure of (canonical) quantisation of such the theory turns out to be noncommutative with the procedure of excluding anti-self-dual variables.