Propagators in the continuum limit: from molecules to scalar fields

The propagator of linear molecules whose constituents interact through oscillator potentials can be obtained in a closed form for $N$ atoms as long as $N \leq 4$. We compute the propagator for arbitrary $N$ in the approximation $N \gg 1$. Taking advantage of this result it is possible to analyze the limit in which the molecule has an infinite number of constituents with infinitesimal length of sepparation, corresponding to the quantization of a string, elastic rod or the second quantization of a Klein Gordon particle. The evolution of some specific initial conditions is also studied, namely the time development of states with minimal dispersion and the effect of sudden perturbations on the vacuum of the scalar field theory.