ABSTRACT
The isotropic Heisenberg ferromagnet is the prototype of a system whose "symmetry is broken". The connection between a symmetry of the Hamiltonian which is broken by the physical state, and the existence of a collective mode of low frequency for large wavelength, is of sufficient importance that the theme can stand an additional variation. The chapter establishes the connection between broken symmetry, long-ranged correlations and long-lived modes, which were first enunciated by J. Goldstone who showed that in relativistic quantum field theory, there must exist a Bose particle with vanishing rest mass whenever a continuous symmetry of the Lagrangian is spontaneously broken. Most of the attention which the Goldstone theorem has received in the literature has been in connection with elementary excitations. The chapter analyzes, somewhat tentatively, just how convincingly the existence of hydrodynamic Goldstone modes can be established.
