Kinematical correlations via $\hat{I}º$-Poincaré coproducts

Abstract

We study a kinematical consequence of the Hopf-algebraic momentum composition law in $\hat{I}º$-Minkowski spacetime. The same curved momentum space can be described in different coordinates. In the bicrossproduct basis the ordered-plane-wave labels are the translation-generator eigenvalues, so the relevant map is one-to-one. In the classical basis, instead, the translation eigenvalues $P_{\hat{I}}¼$ are nonlinearly related to the ordered-plane-wave labels $p_{\hat{I}}¼$. This relation can fail to be globally one-to-one in a high-momentum region. When a given classical-basis four-momentum admits more than one real auxiliary preimage, the branch-sensitive quantity $P_{+}\equiv P_0+P_4=\hat{I}ºe^{p_0/\hat{I}º}$ enters the coproduct and resolves the branches in two-particle states. Imposing the vanishing total-momentum constraint therefore gives branch-dependent $\hat{I}º$-deformed back-to-back momentum correlations. In a single-branch regime this is just a deformed correlated product, while in a multibranch regime a state specified only by $P_{\hat{I}}¼$ can be expanded into distinct auxiliary branches. If $P_{\hat{I}}¼$ are taken as the directly meaningful momenta, the physical content is the resulting deformed correlation pattern. If the auxiliary variables $p_{\hat{I}}¼$ are assigned operational meaning, the same constrained state can be interpreted as a superposition over different auxiliary branches. We also compare this structure with standard regular self-adjoint nonrelativistic minimal-length models and find no analogous smooth local two-real-branch inversion on their physical domains.

21+7 pages, 1 figure