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 Poincare invariant quantization by a singular one. This singular positive linear functional is constructed as mimicking the singular limit of the complex structure of the Poincare invariant Fock quantization. The resulting symmetry group of such polymer quantization is the subgroup SDiff(R-4) which is a subgroup of Diff(R-4) formed by spatial volume preserving diffeomorphisms. In consequence, this yields an entirely different irreducible representation of the canonical commutation relations, nonunitary equivalent to the standard Fock representation. We also compared the Poincare invariant Fock vacuum with the polymer Fourier vacuum.