Polymer quantization of the free scalar field and its classical limit

Building on prior work, a generally covariant reformulation of free scalar field theory on the flat Lorentzian cylinder is quantized using Loop Quantum Gravity (LQG) type `polymer' representations. This quantization of the {\em continuum} classical theory yields a quantum theory which lives on a discrete spacetime lattice. We explicitly construct a state in the polymer Hilbert space which reproduces the standard Fock vacuum- two point functions for long wavelength modes of the scalar field. Our construction indicates that the continuum classical theory emerges under coarse graining. All our considerations are free of the "triangulation" ambiguities which plague attempts to define quantum dynamics in LQG. Our work constitutes the first complete LQG type quantization of a generally covariant field theory together with a semi-classical analysis of the true degrees of freedom and thus provides a perfect infinite dimensional toy model to study open issues in LQG, particularly those pertaining to the definition of quantum dynamics.