  The hyperbolic spin chain is used to elucidate the notion of spontaneous
symmetry breaking for a non-amenable internal symmetry group, here SO(1,2). The
noncompact symmetry is shown to be spontaneously broken -- something which
would be forbidden for a compact group by the Mermin-Wagner theorem.
Expectation functionals are defined through the L \to \infty limit of a chain
of length L; the functional measure is found to have its weight mostly on
configurations boosted by an amount increasing at least powerlike with L. This
entails that despite the non-amenability a certain subclass of noninvariant
functions is averaged to an SO(1,2) invariant result. Outside this class
symmetry breaking is generic. Performing an Osterwalder-Schrader reconstruction
based on the infinite volume averages one finds that the reconstructed quantum
theory is different from the original one. The reconstructed Hilbert space is
nonseparable and contains a separable subspace of ground states of the
reconstructed transfer operator on which SO(1,2) acts in a continuous, unitary
and irreducible way.
