  A new presentation of the Borchers-Buchholz result of the Lorentz-invariance
of the energy-momentum spectrum in theories with broken Lorentz symmetry is
given in terms of properties of the Green's functions of microcausal Bose and
Fermi-fields. Strong constraints based on complex geometry phenomenons are
shown to result from the interplay of the basic principles of causality and
stability in Quantum Field Theory: if microcausality and energy-positivity in
all Lorentz frames are satisfied, then it is unavoidable that all stable
particles of the theory be governed by Lorentz-invariant dispersion laws; in
all the field sectors, discrete parts outside the continuum as well as the
thresholds of the continuous parts of the energy-momentum spectrum, with
possible holes inside it, are necessarily represented by mass-shell
hyperboloids (or the light-cone). No violation of this geometrical fact can be
produced by spontaneous breaking of the Lorentz symmetry, even if the
field-theoretical framework is enlarged so as to include short-distance
singularities wilder than distributions.
