On polinomial formulations of general relativity

Abstract

As basic variables in general relativity (GR) are chosen antisymmetric connection and bivectors - bilinear in tetrad area tensors subject to appropriate (bilinear) constraints. In canonical formalism we get theory with polinomial constraints some of which are II class. On partial resolving the latter we get another polinomial formulation. Separating self- and antiselfdual parts of antisymmetric tensors we come to Ashtekar constraints including those known as "reality conditions" which connect self- and antiselfdual sectors of the theory. These conditions form second class system and cannot be simply imposed on quantum states (or taken as initial conditions in classical theory). Rather these should be taken into account in operator sence by forming corresponding Dirac brackets. As a result, commutators between canonical variables are no longer polinomial, and even separate treatment of self- and antiselfdual sectors is impossible.