Derivation of Lorentz Invariance and Three Space Dimensions in Generic   Field Theory
  A very general quantum field theory, which is not even assumed to be Lorentz invariant, is studied in the limit of very low energy excitations. Fermion and Boson field theories are considered in parallel. Remarkably, in both cases it is argued that, in the free and lowest energy approximation, a relativistic theory with just three space and one time dimension emerges for each particle type separately. In the case of Fermion fields it is in the form of the Weyl equation, while in the case of the Bosons it is essentially in the form of the Maxwell equations. 
C. D. Froggatt and H. B. Nielsen