Polymer-Fourier quantization of the scalar field revisited

The Polymer Quantization of the Fourier modes of the real scalar field is studied within algebraic scheme. We replace the positive linear functional of the standard Poincar\'e invariant quantization by a singular one. This singular positive linear functional is constructed as mimicking the singular limit of the complex structure of the Poincar\'e invariant Fock quantization. The resulting symmetry group of such Polymer Quantization is the subgroup $\mbox{SDiff}(\mathbb{R}^4)$ which is a subgroup of $\mbox{Diff}(\mathbb{R}^4)$ formed by spatial volume preserving diffeomorphisms. In consequence, this yields an entirely different irreducible representation of the Canonical Commutation Relations, non-unitary equivalent to the standard Fock representation. We also compared the Poincar\'e invariant Fock vacuum with the Polymer Fourier vacuum.