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

Abstract

In standard quantum field theory, the one-particle states are classified by unitary representations of the Poincar\'e group, whereas the causal fields' classification employs the finite dimensional (non-unitary) representations of the (homogeneous) Lorentz group. A natural question arises - why the fields are not allowed to transform nontrivially under translations? We investigate this issue by considering the fields that transform under the full representation of the Poincar\'e group. It follows that such fields can be consistently constructed, although the Lagrangians that describe them necessarily exhibit explicit dependence on the space-time coordinates. The two examples of the Poincar\'e-spinor and the Poincar\'e-vector fields are considered in details. The inclusion of Yang--Mills type interactions is considered on the simplest example of the U(1) gauge theory. The generalization to the non-abelian case is straightforward so long as the action of the gauge group on fields is independent of the action of the Poincar\'e group. This is the case for all the known interactions but gravity.