Departures from Special Relativity beyond Effective Field Theories

Lorentz invariance is a fundamental ingredient of our present description of Nature, which is given in terms of relativistic quantum field theories (RQFTs). However, the idea that Lorentz invariance might be an approximate, low-energy, symmetry, has begun to emerge in the last few years essentially from quantum gravity developments1, but also from results in the fields of string theory 2, noncommutative geometry 3, or other ideas such as varying couplings4. There are other motivations to think on limitations of the RQFT framework: the strong difficulties in obtaining a RQFT containing gravitation, the large mismatch between the expected and measured value of the energy density of vacuum (cosmological constant problem), or the consideration that in a quantum theory of gravity the maximum entropy of any system should be proportional to the area and not to the volume5, contrary to what happens in RQFT6. There exist also some phenomenological results that could find a simple explanation in terms of a Lorentz invariance violation (LIV). The observation of cosmic rays with energies above 5×10 eV seems to violate the GZK cutoff7, whose existence is implied by relativistic kinematics. On the other hand, CPT symmetry might also be violated if Lorentz symmetry were not exact, which could provide the key to explain the matterantimatter asymmetry of our Universe8. The framework of effective field theories provides a conservative approach to incorporate departures from special relativity. Here Lorentz symmetry is spontaneously broken, which produces the apparition of Lorentz non-invariant terms in an effective Lagrangian. This approach has been extensively studied in the last years 9. It assumes that one can incorporate the corrections order by order in the effective theory, so that, for a certain level of precision, the effective Lagrangian has always a finite number of terms. Tests of special relativity have put strong bounds on different terms of the effective Lagrangian10. In this paper we want to point out that the effective field theory framework may be too tight to include corrections coming from a LIV, giving explicit examples of theoretical schemes beyond that framework.