Formal developments in curved momentum space: the quantum field theory roadmap

TL;DR

The paper argues that curved momentum space provides a covariant framework to address foundational issues from spacetime noncommutativity and basis dependence. It introduces a generalized momentum $f^ u(p)$ and a deformed Casimir $C_D(p)$ to formulate covariant Klein–Gordon and Dirac equations in momentum space, and it develops a momentum-space quantum-field-theory structure with free-field actions $S_{ m KG}$ and $S_{ m Dirac}$. By exploring momentum-space symmetries and deformed translations, it shows how a deformed composition law $p\oplus q$ can coexist with a metric, yielding models such as Snyder and $9$-Poincaré within maximally symmetric spaces. The work lays groundwork for a full Lagrangian QFT in curved momentum space and hints at deeper connections to relative locality and potential spacetime-curvature interplay.

Abstract

We advocate that the dual picture of spacetime noncommutativity , i.e. the existence of a curved momentum space, could be a way out to solve some of the open conceptual problems in the field, such as the basis dependence of observables. In this framework, we show how to build deformed Klein--Gordon and Dirac equations. In addition, we give an outlook of how one could define quantum field theories, both free and interacting ones.