Stability, causality, and Lorentz andCPTviolation

Stability and causality are investigated for quantum field theories incorporating Lorentz and $\mathrm{CPT}$ violation. Explicit calculations in the quadratic sector of a general renormalizable Lagrangian for a massive fermion reveal that no difficulty arises for low energies if the parameters controlling the breaking are small, but for high energies either energy positivity or microcausality is violated in some observer frame. However, this can be avoided if the Lagrangian is the sub-Planck limit of a nonlocal theory with spontaneous Lorentz and $\mathrm{CPT}$ violation. Our analysis supports the stability and causality of the Lorentz- and $\mathrm{CPT}$-violating standard-model extension that would emerge at low energies from spontaneous breaking in a realistic string theory.