**OBSERVERS IN QUANTUM GRAVITY**

**Angel Ballesteros**

**Burgos University**

**Title**

From quantum groups to noncommutative spacetimes and worldlines

**Abstract**

The construction of noncommutative Lorentzian spacetimes and their associated noncommutative spaces of worldlines is reviewed. In the commutative case both spaces are constructed as homogeneous spaces of the (A)dS and PoincarĂ© groups, and their noncommutative analogues arise as quantizations of the corresponding Poisson homogeneous spaces that are invariant under the appropriate (co)action from (A)dS or PoincarĂ© quantum groups. In this construction, a new duality between reductive homogeneous spaces and coisotropic Lie bialgebras can be envisaged. In particular, the noncommutative spaces arising from the well-known kappa-(A)dS and PoincarĂ© quantum groups are explicitly given, and the role of the cosmological constant is outlined. Moreover, a proposal for the description of quantum observers as coordinates on the non-commutative space of time-like worldlines is sketched.