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?

  • $\begingroup$ Your second paragraph is vague. About the third paragraph : time evolution exists in all of quantum systems, regardless of the existence of any thermal reservoirs. Another point is that the Goldstone's theorem holds for all of physical systems with a global continuous symmetry(classical or quantum systems) $\endgroup$ – Hosein Dec 4 '16 at 19:23
  • $\begingroup$ @Hosein I'm working classically here. Classically, if you have a Hamiltonian $H(q)$ that doesn't depend on the canonical momenta $p$, then the equation of motion is just $\dot{q} = 0$, and nothing happens whatsoever. $\endgroup$ – knzhou Dec 5 '16 at 0:20
  • $\begingroup$ How do you define a classical spin degree of freedom? $\endgroup$ – Hosein Dec 5 '16 at 0:30
  • $\begingroup$ @Hosein See the classical XY model I linked in the question. $\endgroup$ – knzhou Dec 5 '16 at 0:35
  • $\begingroup$ The XY model mentioned in the link is just a defined partition function. There is not such a thing as classical XY model which is really classical. The quantum XY model in the limit of large spin can be approximated by a classical XY model (yet the equations of motion for spins should be derived from quantum commutation relations) $\endgroup$ – Hosein Dec 5 '16 at 0:45

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