Approaches to Quantum Gravity

From quantum reference frames to deformed special relativity 525
the inspiration following the non-linear realization: we transform the P i back to the
linear momenta P i , add them and then transform them back, taking into account
the change of representation or de Sitter radius:
P 1 ⊕ P 2 = U κ
(
U
−1
MP (P 1) + U −1
MP (P 2) ) ,
where, to emphasize that P lives on the de Sitter space of radius ρ, I used the
notation P = U −1
ρ (P). 26.4 Conclusion
Since, in a few years, we expect some data on possible QG effects, it is urgent to
understand the semiclassical limit of QG. In particular, one needs to understand
the QG physics that is supposed to mix both quantum mechanical and gravitational
effects. One key feature for understanding physics in this context is the notion
of reference frame. As I recalled using a toy model, the notion of a Quantum
Reference Frame leads to interesting physics: the notion of quantum coordinates,
possibly a non-linear realization of the symmetries and a modification of the multiparticles
states. These features are expected to appear also in the QG semiclassical
limit. DSR naturally incorporates these features as a modified measurement procedure
and can be seen as the effective description of a flat semiclassical spacetime.
From the geometric point of view, DSR could be seen as a generalization of the
Riemannian geometry, where the metric is not given in terms of a scalar product
anymore. It involves therefore in a non-trivial way the full tangent bundle structure:
the notion of symmetry, curvature and so on have to be understood once again.
There is contact now with a large mathematical theory that is left to explore from
the physics perspective, promising new exciting developments.
As I argued as well, there are many different types of DSR due mainly to the
freedom in reconstructing spacetime. Most of them can be unified under a common
framework. With this respect, DSR could be compared to Maxwell’s electromagnetism
theory when Lorentz introduced his symmetries at the end of the nineteenth
century. All the theoretical ingredients were there but it was not until Einstein came
up with some new physical principles (axiomatic) and some operational guidance
that the theory was fully understood. At this time DSR still lacks these fundamental
principles to be definitely understood. This is clear, for example, when we see that
we have no clue to decide which momentum is physical and how it should add.
At this stage, according to me, an axiomatic derivation of DSR is necessary before
going to any quantum field theory: the modified notion of reference frame should
definitely matter and provide guidance to these new physical principles.
To conclude, the quest to understand the notion of semiclassical spacetime
allows us to relate to deep mathematical theories like Finsler geometry or quantum