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

## TL;DR

This paper explores two types of $ heta_{
u
ho }$-deformed covariant relativistic quantum phase spaces modeled as Poincare-Hopf algebroids, highlighting their algebraic structures and covariance properties under quantum Poincare transformations.

## Contribution

It introduces two Hopf algebroid frameworks for $ heta_{
u
ho }$-deformed quantum phase spaces, emphasizing their covariance and potential for describing multiparticle quantum structures.

## Key findings

- The first algebroid models $ heta_{
u
ho }$-deformed relativistic phase space with canonical noncommutative spacetime.
- The second algebroid incorporates dual quantum group algebra with $ heta_{
u
ho }$-deformed translations dependent on Lorentz parameters.
- Both frameworks are covariant under quantum Poincare transformations.

## 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 ($% \Lambda $ $\equiv \Lambda_{\mu \nu }$ 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.