Microcausality and Energy-Positivity in all frames imply Lorentz Invariance of dispersion laws

TL;DR

This paper demonstrates that in quantum field theories, the combination of microcausality and energy-positivity across all frames enforces Lorentz invariance on the dispersion laws of stable particles, regardless of symmetry breaking.

Contribution

It provides a new geometric proof that Lorentz-invariance of dispersion laws is guaranteed under certain fundamental principles, even with broken Lorentz symmetry.

Findings

Lorentz-invariance of energy-momentum spectrum is enforced by causality and stability.

Stable particles' dispersion laws must be Lorentz-invariant hyperboloids or light-cone.

Breaking Lorentz symmetry cannot alter the geometric form of stable particle spectra.

Abstract

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.