The answer is in decoherence. for classical systems, if a subsystem breaks a symmetry, the system as a whole also breaks the symmetry. not so in quantum mechanics because of entanglement. here lies the complication.
think of zurek's pointer states. there lies the clue. i may give you a many body quantum state which literally is invariant under the symmetry in question, but if it decomposes into decoherent pointer states which aren't invariant, feel free to say the symmetry is spontaneously broken? but zurek's analysis only works for open systems.
can this work for finite closed systems? unfortunately no because of poincare recurrences. we might naively think a symmetry is spontaneously broken, but wait long enough and the slight (or not so slight) energy differences between the various energy eigenvalues corresponding to different irreps will lead to a washout in phase differences in energy eigenstates carry info on symmetry breaking.
what are zurek's pointer states? those which preserve information longest in time while minimizing dynamical generation of entanglement with the environment. sometimes, a pointer state invariant under a symmetry will generate more entanglement with the environment than one not invariant.
complications abound. take a collection of helium-4 atoms at a low temperature. superfluid phase. u(1) symmetry corresponding to number of he-4 atoms. put the atoms in a very sealed box where not even a single he-4 atom can pass but info can pass. idealized, yes, but bear with me. quantum state with a fixed specific value for number of he-4 atoms. invariant under u(1)? what are the pointer states? unfortunately, not condensate states with a superposition in number of he-4 atoms? but the dynamical generation of enviroentanglement remains small in either case anyway: fixed atom num and condensate. just that over very long periods of time, fixed atom num has slightly more entanglement. because dynamical processes sensitive to total num of he-4 atoms will dominate but only because of absolute suppression of permeability. unrealistic, no?
but loosen up. make box slightly permeable. just let only one or two he-4 atoms pass after relatively long time. voila? pointer state changes favoring condensates? confused yet? the number of he-4 atoms in the environment is in a superposition entangled with the num of he-4 atoms in the box. THE ENVIRONMENT!!! the symmetry has to be broken in the environment, not the system.
but what about the universe as a whole? it has no external environment. aah, but there are no global symmetries in quantum gravity. ok, what about gauge symmetries then. oh boy, another huge can of worms. What is spontaneous symmetry breaking in QUANTUM GAUGE systems? that is worth another s.e. question.
This post imported from StackExchange Physics at 2014-04-04 16:11 (UCT), posted by SE-user ribbit ribbit kermit