  We present a detailed study of quantized noncompact, nonlinear SO(1,N)
sigma-models in arbitrary space-time dimensions D \geq 2, with the focus on
issues of spontaneous symmetry breaking of boost and rotation elements of the
symmetry group. The models are defined on a lattice both in terms of a transfer
matrix and by an appropriately gauge-fixed Euclidean functional integral. The
main results in all dimensions \geq 2 are: (i) On a finite lattice the systems
have infinitely many nonnormalizable ground states transforming irreducibly
under a nontrivial representation of SO(1,N); (ii) the SO(1,N) symmetry is
spontaneously broken. For D =2 this shows that the systems evade the
Mermin-Wagner theorem. In this case in addition: (iii) Ward identities for the
Noether currents are derived to verify numerically the absence of explicit
symmetry breaking; (iv) numerical results are presented for the two-point
functions of the spin field and the Noether current as well as a new order
parameter; (v) in a large N saddle-point analysis the dynamically generated
squared mass is found to be negative and of order 1/(V \ln V) in the volume,
the 0-component of the spin field diverges as \sqrt{\ln V}, while SO(1,N)
invariant quantities remain finite.
