Eur. Phys. J. Plus (2025) 140: 409
https://doi.org/10.1140/epjp/s13360-025-06312-1

Regular Article

Towards new relativistic doubly $\kappa$-deformed D = 4 quantum phase spaces

¹ Institute of Theoretical Physics, Wrocław University, pl. Maxa Borna 9, 50-205, Wrocław, Poland
² Division of Theoretical Physics, Rudjer Bošković Institute, Bijenička c.54, 10002, Zagreb, Croatia
³ Dipartimento di Matematica, Università di Cagliari, via Ospedale 72, 09124, Cagliari, Italy
⁴ INFN, Sezione di Cagliari, 09042, Monserrato, Italy
⁵ Department of Microsystems, University of South-Eastern Norway, Campus Vestfold, Borre, Norway

^a Anna.Pachol@usn.no

Received: 11 January 2025
Accepted: 10 April 2025
Published online: 20 May 2025

Abstract

We propose new noncommutative models of quantum phase spaces, containing a pair of $\kappa$-deformed Poincaré algebras, with two independent double ($\kappa ,\tilde{\kappa }$)-deformations in space-time and four-momenta sectors. The first such quantum phase space can be obtained by contractions $M,R\to \infty$ of recently introduced doubly $\kappa$-deformed $(\kappa ,\tilde{\kappa })$-Yang models, with the parameters M, R describing inverse space-time and four-momenta curvatures and constant four-vectors $a_{\mu },b_{\mu }$ determining nine types of $(\kappa ,\tilde{\kappa })$-deformations. The second considered model is provided by the nonlinear doubly $\kappa$-deformed TSR algebra spanned by 14 coset $\widehat{o}(1,5)/\widehat{o}(2)$ generators. The basic algebraic difference between the two models is the following: the first one, described by $\widehat{o}(1,5)$ Lie algebra can be supplemented by the Hopf algebra structure, while the second model contains the quantum phase space commutators $[{\widehat{x}}_{\mu },{\widehat{q}}_{\nu }]$, with the standard numerical $iħ{\eta }_{\mu \nu }$ term; therefore, it describes the quantum-deformed Heisenberg algebra relations which cannot be equipped with the Hopf algebra.