It is shown that the state space corresponding to a spontaneously broken global (position-independent) symmetry can be described in two complementary ways. The first is a Hilbert bundle and the second a Hilbert space. The Hilbert space, which is obtained from the bundle via a direct integral, is reducible, i.e., the vacuum state is degenerate. The degeneracy is labeled by a certain integer-valued parameter. It is noted that the above structure continues to be true for a suitably restricted broken local (position-dependent) gauge symmetry in the unitary gauge. The significance of the vacuum degeneracy is discussed.

• Received 29 May 1981

DOI:https://doi.org/10.1103/PhysRevD.24.2153