A quantum field theory of crystals. I

A quantum field-theoretical treatment of crystals is presented from the view-point of the spontaneous break-down of the spatial translational invariance. Our treatment starts with the assumption of the existence of the ground-state expectation value of a molecule field. It is proved that there exist at least three gapless Goldstone bosons which are identified as acoustic phonons and that up to the first order of the phonon fields a molecule field is expanded as the linear combination of the three phonon fields multiplied by the spatial derivative of the ground-state expectation value of the molecule field.