Why only d = (3+1) carries non-zero momenta

Explain the mechanism by which fields in the presented theory acquire non-zero momenta exclusively in d = (3 + 1), while internal spaces are higher-dimensional, and provide a consistent dynamical account of this selection.

Background

Throughout the paper, fields are assumed to have non-zero momentum only in ordinary spacetime d = (3+1), with internal degrees of freedom living in higher dimensions. The authors highlight the need to account for how this selection arises.

A resolution would connect the algebraic internal-space construction to cosmological or dynamical mechanisms that privilege 3+1-dimensional kinematics.