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

TL;DR

This paper explores the possibility of quantum fields transforming nontrivially under space-time translations, leading to explicit coordinate dependence in their Lagrangians, and examines implications for gauge theories including gravity.

Contribution

It demonstrates that fields can transform under the full Poincaré group with explicit coordinate dependence, extending standard quantum field theory frameworks.

Findings

Fields can be consistently constructed with nontrivial translation transformations.

Lagrangians for such fields explicitly depend on space-time coordinates.

Inclusion of U(1) gauge interactions is straightforward; non-abelian generalization is possible.

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.