Ashtekar's variables for arbitrary gauge group.

A generally covariant gauge theory for an arbitrary gauge group with dimension ≥ 3, that reduces to Ashtekar’s canonical formulation of gravity for SO(3,C), is presented. The canonical form of the theory is shown to contain only first class constraints. When Ashtekar [1] managed to reformulate Einstein gravity on Yang-Mills phase space, it rekindled the old dream of finding a unified theory of gravity and Yang-Mills theory. However, it soon became clear that this Ashtekar formulation relied heavily on the use of the gauge group SO(3) (or locally isomorphic ones), and the simple structure-constant identity that exists for these groups. Without this identity the constraint algebra fails to close, and the theory is not diffeomorphism invariant, and contains second class constraints. In an attempt to find an Ashtekar formulation for an arbitrary gauge group, there are no problems with the generator of gauge transformations (Gauss’ law), or the generator of spatial diffeomorphisms (the vector constraint). They form a system of first class constraints by themselves, for arbitrary gauge group. The difficult part is the generator of diffeomorphisms off the spatial hyper-surface (the Hamiltonian constraint). This constraint is constructed with the help of the structure constants, and in the Poisson bracket between two Hamiltonian constraints, the identity, mentioned above, is needed to give a weakly vanishing result. So, the strategy to construct the general theory is :Write down a Hamiltonian constraint without the use of the structure constants, such that, when choosing the gauge group SO(3), the constraint reduces to the ordinary Ashtekar constraint. The hope is then that the construction works for an arbtrary gauge group, since one does not use any particular feature of a special gauge group any more. To do this in practice, first define a scalar with the help of the four fundamental scalar densities :ǫabcfijkΠ a iΠ b jB c k, ǫabcfijkΠ a iΠ b jΠ c k, ǫabcfijkΠ a iB b jB c k, ǫabcfijkB a i B b jB c k. Then, multiply the ordinary Ashtekar Hamiltonian constraint with this scalar, and, finally, use the structureconstant identity to eliminate all structure constants. This new Hamiltonian will then in general give a closed constraint algebra for an arbitrary gauge group. In this letter, I will show how to obtain this Ashtekar theory for an arbitrary gauge group, through a Legendre transform from a pure connection Lagrangian of the form discovered by Capovilla, Jacobson and Dell[2]. The resulting canonical theory will correspond to multiplying the Hamiltonian constraint by the determinant of the ”magnetic” field, in the strategy above. In order to find the Ashtekar-theory for a general gauge group, I will start with the generally covariant and gauge invariant CDJ-action [2].