Quantum Field Theory of Crystals and Dislocations

We present a quantum field theoretical treatment of crystals from the view point of the spontaneous breakdown of the spatial translation invariance. It is proved that at least three massless Goldstone particles exist and they are identified as the acoustic phonons. We carry out the boson transformation to show that the classical phenomenological theory of crystals can be derived. The classical momentum and stress tensor are expressed in terms of the classical displacement field. The method is extended to a microscopic derivation of the theory of crystal dislocations. It is shown that the Burgers vector is quantized and is a special case of the so‐called topological quantum number. An explicit expression of a general solution of the dislocation is obtained.