# Poincare group and free theories

How exactly is the Poincare group related to the free relativistic theories in quantum field theory? I know Poincare group is the Lorentz group along with translations but don't see any connected why the field will influence the Poincare group.

1. This is a general phenomenon. Whenever a theory/system $$S$$ has a symmetry, it in particular means that some group $$G$$ acts $$G\times S \to S$$ on the system $$S$$. The system $$S$$ then constitutes a (possibly non-linearly realized) representation of the group $$G$$. See also my Phys.SE answer here.