I wondered if someone could help me understand spontaneous symmetry breaking (SSB) in classical mechanics, quantum mechanics and quantum field theory. Consider a Higgs-like potential, with a local maximum surrounded by a degenerate ground state - a pencil balanced on its point, for example.
Classical Mechanics (CM) exhibits spontaneously symmetry breaking if and only if the system is perturbed.
Quantum mechanics (QM) exhibits no symmetry breaking, because the ground state is a superposition of the degenerate vacua.
Quantum field theory (QFT) At infinite volume, spontaneous symmetry breaking occurs. Because the degenerate vacua are orthogonal, $$ \langle \theta^\prime | \theta \rangle = \delta(\theta^\prime-\theta), $$ a ground state is chosen.
Q1 Is it true that QM never exhibits SSB? Some sources suggest otherwise. But I can't see a way round the basic argument.
Q2 In QFT, is it correct that a conceptual difference with CM is that the system needn't be perturbed? I guess this is the case, because we simply look at the expectation of the field $\langle 0 | \phi | 0 \rangle$. But how can I convince someone that the field cannot just sit at the local maxima?
Q3 I find it strange that SSB disappears from QM to CM, then reappears in QFT. Are there other phenomena that have this feature? Is there a nice way to understand this?