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

Abstract

We propose new noncommutative models of quantum phase spaces, containing a pair of $\kappa$-deformed Poincar\'e 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 $\hat{o}(1,5)/\hat {o}(2)$ 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 $[\hat{x}_\mu,\hat{q}_\nu]$, with the standard numerical $i\hbar\eta_{\mu\nu}$ term; therefore it describes the quantum-deformed Heisenberg algebra relations which cannot be equipped with the Hopf algebra.