In the context of nonrelativistic physics (e.g. condensed matter), Goldstone's theorem tells us that spontaneous symmetry breaking leads to gapless excitations, i.e. excitations with arbitrarily low energy above the ground state. Since $E \sim \omega$ in quantum mechanics, this tells us classically that there should exist modes with vanishing frequency.

However, Goldstone's theorem is also applied to 'purely thermodynamic' systems, such as the classical XY model, which have no dynamics of their own! That is, if you time-evolve with the XY model Hamiltonian, absolutely nothing happens, because there is no canonical momentum anywhere in sight. The system just sits there.

In practice, time evolution would happen due to coupling to a thermal reservoir, but that's not written into the Hamiltonian and certainly doesn't lead to a unique $\omega$ for a mode, or even oscillations at all. So it's hard to define any sort of "mode" for such a system.

In this case, what is the formal statement of Goldstone's theorem for such systems, and how does it relate to the usual statement of Goldstone's theorem?

This post imported from StackExchange Physics at 2016-12-09 18:26 (UTC), posted by SE-user knzhou