Hamiltonian and Diffeomorphism Constraints Generalized for Timelike and Spacelike 3+1 Foliation

Abstract

The form of Hamiltonian and Diffeomorphism constraints in Sen-Ashtekar-Barbero-Immirzi variables is well known for the spacelike 3+1 ADM foliation. It is also known that Sen-Ashtekar-Barbero-Immirzi connection can be introduced only in 3 dimensional space and does not work for $D > 3$. The reason it works in $D = 3$ is due to existence of isomorphism between $so(3)$ algebra and $R^3$ space with vector product. It turns out that similar isomorphism exists between $so(2,1)$ algebra and $R^3_{2,1}$ space algebra with respect to its vector product. By using this isomorphism we find both analog of Sen-Ashtekar-Barbero-Immirzi connection for timelike 3+1 foliation and corresponding forms of Gauss, Diffeomorphism and Hamiltonian constraints. We then combine spacelike and timelike foliation constraints into the generalized form of the Hamiltonian and Diffeomorphism constrains using generalized Sen-Ashtekar-Barbero-Immirzi connection variables. We prove that Immirzi parameter is covariant with respect to timelike-spacelike ADM foliation change as in both cases in self-dual Ashtekar case it disappears in Hamiltionian constraint keeping it polynomial.