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But what does it mean that global gauge symmetry (which is unobservable) is broken in a superconductor?

These days I came across the interesting classic article by Rudolph Haag, "The Mathematical Structure of the Bardeen-Cooper-Schrieffer Model", Nuovo Cimento, Vol 25(2), pp.287 (1962).

There Haag finds that the states of broken symmetry, which do not conserve particle number and lead to a diagonalization of the BCS Hamiltonian via the Bogoliubov transformation, correspond to irreducible representations of the algebra of the local field operators. There are different irreducible representations which are labeled by a parameter alpha which ranges from 0 to 2 pi. The ground states of all these irreducible representations have the same energy, but each one lives in a Hilbert space on its own.

He also constructs representations where the ground state(s) have a fixed particle number, however, these states are not irreducible, but, I think physically more relevant.

What is the physical relevance of a representation being reducible or irreducible? Why do states which unsharp particle number become so important in a problem which in principle conserves particle number?