Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action.

Barbero recently suggested a modification of Ashtekar’s choice of canonical variables for general relativity. Although leading to a more complicated Hamiltonian constraint this modified version, in which the configuration variable still is a connection, has the advantage of being real. In this article we derive Barbero’s Hamiltonian formulation from an action, which can be considered as a generalization of the ordinary Hilbert-Palatini action. In 1986 Ashtekar presented a new pair of canonical variables for the phase spase of general relativity [1]. These variables led to a much simpler Hamiltonian constraint than that in the ADM formulation [2], but had the drawback of introducing complex variables in the phase space action—something that leads to difficulties with reality conditions which then must be imposed. A couple of years later the Lagrangian density corresponding to Ashtekar’s Hamiltonian was given independently by Samuel, and by Jacobson and Smolin [3]. That was seen simply to be the Hilbert-Palatini (HP) Lagrangian with the curvature tensor replaced by its self-dual part only. Recently Barbero pointed out that it is possible to choose a pair of canonical variables that is closely related to Ashtekars but this time real [4]. The price paid is that the simplicity of Ashtekars Hamiltonian constraint is destroyed. However, some advantages are still present with Barbero’s choice of variables. For example, they provide a real theory of gravity with a connection as configuration variable, and with the usual Gauss and vector constraint, thus fitting into the class of diffeomorphism invariant theories considered in [5] in the context of quantization. In this paper we derive Barbero’s result from an action, and since his formulation includes also that of ADM and Ashtekar via a parameter, the Lagrangian density used as starting point in this paper, also includes these cases. Hence we have found, in a sense, a generalized HP action. We now write down this action, and thereafter we will motivate that it is a good canditate for an action leading to Barbero’s formulation, which then will be explicitly derived from it: S = 1 2 ∫ eeIe β J(F IJ αβ − α ∗ F IJ αβ ) ≡ 1 2 ∫ eeIe β J(F IJ αβ − α 2 ǫKLF KL αβ ) (1) Here eαI is the tetrad, e its determinant, F IJ αβ the curvature considered as a function of the connection AαIJ , and α a (complex) parameter which will allow us to account for all the cases mentioned above. The star (∗) denotes, as is seen, the usual duality operator. Department of Physics, Stockholm University, Box 6730, S-113 85 Stockholm, Sweden. e-mail: [email protected]