This may be a very blunt question but I wonder why we always use Poincare invariant Lagrangians in field theory. After all, the entire world around us is by no means homogeneous, isotropic and so on. Are we writing the Lagrangian for an ideal universe which has Poincare invariance? But we are trying to explain the real world where I do not see translational or rotational invariance. Right? Even the interacting field theories are Poincare invariant whereas in classical mechanics presence of a potential term $V(x)$ in the Lagrangian breaks translational invariance.
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The state of the universe is not homogeneous and isotropic, but the laws of physics are. For example, the speed of light propagation is the same in all directions, and the mass of the electron is not a function of position. Writing down a Lagrangian requires an assumption about the laws of physics (or more precisely, an assumption about the dynamics). There is no assumption required about the state.
In classical mechanics, it is true that an external potential breaks translation invariance. However, an interaction potential between two particles at $x_1$ and $x_2$ does not break translation invariance if it is of the form $V(x_1 - x_2)$, as it often is (e.g. Newtonian gravity). Interactions between particles in quantum field theory Lagrangians are also of this form.
answered May 12, 2015 at 16:44