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In standard quantum field theory, the one-particle states are classified byunitary representations of the Poincar\'e group, whereas the causal fields'classification employs the finite dimensional (non-unitary) representations ofthe (homogeneous) Lorentz group. A natural question arises - why the fields arenot allowed to transform nontrivially under translations? We investigate thisissue by considering the fields that transform under the full representation ofthe Poincar\'e group. It follows that such fields can be consistentlyconstructed, although the Lagrangians that describe them necessarily exhibitexplicit dependence on the space-time coordinates. The two examples of thePoincar\'e-spinor and the Poincar\'e-vector fields are considered in details.The inclusion of Yang--Mills type interactions is considered on the simplestexample of the U(1) gauge theory. The generalization to the non-abelian case isstraightforward so long as the action of the gauge group on fields isindependent of the action of the Poincar\'e group. This is the case for all theknown interactions but gravity.

Nontrivial realization of the space-time translations in the theory of quantum fields