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Spontaneous Symmetry Breaking in Quantum Systems: Emergence or Reduction?

Beginning with Anderson (1972), spontaneous symmetry breaking (SSB) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics, since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show SSB in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called `Earman's Principle', a genuine physical effect (viz. SSB) seems theoretically recovered only in some idealization (namely the thermodynamic limit), disappearing as soon as the the idealization is removed. We review the well-known arguments that (at first sight) no finite system can exhibit SSB, using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite- and infinite-volume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which typically forbids SSB) versus classical mechanics (which allows it). This discrepancy between formalism and reality is quite similar to the measurement problem, and hence we address it in the same way, adapting an argument of the author and Reuvers (2013) that was originally intended to explain the collapse of the wave-function within conventional quantum mechanics. Namely, exponential sensitivity to (asymmetric) perturbations of the (symmetric) dynamics as the system size increases causes symmetry breaking already in finite but very large quantum systems. This provides continuity between finite- and infinite-volume descriptions of quantum systems featuring SSB and hence restores Earman's Principle (at least in this particularly threatening case).