The projector on physical states in loop quantum gravity

The loop approach to quantum gravity [1] based on the Ashtekar variables [2] has been successful in establishing a consistent and physically reasonable framework for the mathematical description of quantum spacetime [3]. This framework has provided intriguing results on the quantum properties of space, most notably detailed quantitative results on the discrete quanta of the geometry [4]. The nonperturbative dynamics of the quantum gravitational field, however, is not yet well understood. Two major questions are open. First, several versions of the hamiltonian constraint have been proposed [5–7], but the physical correctness of these versions has been questioned [8]. Second, a general scheme for extracting physical consequences from a given hamiltonian constraint, and for computing expectations values of physical observables is not available. In this paper, we address the second of these problems – a solution of which, we think, is likely to be a prerequisite for addressing the first problem (the choice of the correct hamiltonian constraint). The problem we address is how expectation values of physical observables can be computed, given a hamiltonian constraint operator H(x). (For an earlier attempt in this direction, see [9].) We address this problem by constructing the “projector” P on the physical Hilbert space of the theory, namely on the space of the solutions of the hamiltonian constraint equation. Formally, this projector can be written as