Background independent quantizations-the scalar field: I

We are concerned with the issue of quantization of a scalar field in a diffeomorphism-invariant manner. We apply the method used in loop quantum gravity. It relies on a specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the 'quantum' polymer *-star algebra and looking for positive linear functionals, called states. It is assumed in this paper that homeomorphism invariance allows us to determine a complete class of the states. Except one, all of them are new. In this paper we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self-adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones