  We pose and resolve a seeming paradox about spontaneous symmetry breaking in
the quantum theory of infinite systems. For a symmetry to be spontaneously
broken, it must not be implementable by a unitary operator. But Wigner's
theorem guarantees that every symmetry is implemented by a unitary operator
that preserves transition probabilities between pure states. We show how it is
possible for a unitary operator of this sort to connect the folia of unitarily
inequivalent representations. This result undermines interpretations of quantum
theory that hold unitary equivalence to be necessary for physical equivalence.
