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A new presentation of the Borchers-Buchholz result of the Lorentz-invarianceof the energy-momentum spectrum in theories with broken Lorentz symmetry isgiven in terms of properties of the Green's functions of microcausal Bose andFermi-fields. Strong constraints based on complex geometry phenomenons areshown to result from the interplay of the basic principles of causality andstability in Quantum Field Theory: if microcausality and energy-positivity inall Lorentz frames are satisfied, then it is unavoidable that all stableparticles of the theory be governed by Lorentz-invariant dispersion laws: inall the field sectors, discrete parts outside the continuum as well as thethresholds of the continuous parts of the energy-momentum spectrum, withpossible holes inside it, are necessarily represented by mass-shellhyperboloids (or the light-cone). No violation of this geometrical fact can beproduced by spontaneous breaking of the Lorentz symmetry.Comment: Plain Tex, 23 pages, 4 figure

Microcausality and energy positivity in all frames imply Lorentz invariance of dispersion laws