Two ${\theta }_{\mu \nu }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids

Abstract

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known ${\theta }_{\mu \nu }$-deformed $D=4$ Poincare-Hopf algebra $\mathbb{H}$. The first algebroid describes ${\theta }_{\mu \nu }$-deformed relativistic phase space with canonical NC space-time (constant ${\theta }_{\mu \nu }$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum ${\theta }_{\mu \nu }$-deformed Poincare-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $\Lambda$-dependent ${\Theta }_{\mu \nu }$ parameters ( $\equiv {\Lambda }_{\mu \nu }$ parametrize classical Lorentz group). The canonical ${\theta }_{\mu \nu }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $\mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.

16 pages, v2 extended and improved, with title modified

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