Spontaneous Symmetry Breaking in Quantum Systems: Emergence or Reduction?∗

Dedicated to the memory of Gérard Emch (1936–2013) 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 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 (whic