arXiv2025
We propose new noncommutative models of quantum phase spaces, containing a pair of $κ$-deformed Poincaré algebras, with two independent double ($κ,\tildeκ$)-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 $κ$-deformed $(κ,\tildeκ)$-Yang models, with the parameters $M,R$ describing inverse space-time and four-momenta curvatures and constant four-vectors $a_μ, b_μ$ determining nine types of $(κ,\tildeκ)$-deformations. The second considered model is provided by the nonlinear doubly $κ$-deformed TSR algebra spanned by 14 coset $\hat{o}(1,5)/\hat {o}(2)$ generators. The basic algebraic difference between the two models is the following: the first one, described by $\hat{o}(1,5)$ Lie algebra can be supplemented by the Hopf algebra structure, while the second model contains the quantum phase space commutators $[\hat{x}_μ,\hat{q}_ν]$, with the standard numerical $i\hbarη_{μν}$ term; therefore it describes the quantum-deformed Heisenberg algebra relations which cannot be equipped with the Hopf algebra.