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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.

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  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.

  2. Concretely the isometry group of Minkowski space is the Poincare group. Concerning representations of the Poincare group in relativistic field theories, see e.g. this & this Phys.SE posts and links therein.

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  • $\begingroup$ I mean rotations and translations make sense for scalar fields, as well as vector fields like Electromagnetic field. $\endgroup$ – user183683 Jan 14 at 16:05

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