Translations in Quantum Field Theory and the Poincar\'e Gauge Theory of Gravity

Abstract

In standard quantum field theory, the one-particle states are classified by the unitary representations of the Poincar\'e group, whereas the causal fields' classification employs the finite-dimensional (non-unitary) representations of the (homogeneous) Lorentz group. We investigate the possibility of constructing fields that transform under the full representation of the Poincar\'e group. We show that such fields can be consistently constructed, although the Lagrangians that describe them exhibit explicit dependence on the space-time coordinates. The inclusion of gravity within the framework of the Poincar\'e gauge theory is then discussed. A new feature that occurs is that the translational gauge fields enter the covariant derivative of matter fields. The Poincar\'e-gauge approach works still well and leads to interesting consequences. The detailed discussion of the Dirac field is presented and the relation to the earlier accounts on Poincar\'e-spinors is drawn. Another example that is considered is the Poincar\'e-vector field. The presentation has a partly didactic character and is addressed to all the readers who are interested in the rudiments of quantum field theory and the gauge description of gravity.