Approaches to Quantum Gravity

From quantum reference frames to deformed special relativity 511
In section 26.2 I will quickly recall the construction of observables in QG, but
also ask a number of questions that should be answered to my mind, to understand
the QG physics. It is hard of course to do so in a QG theory like LQG, so I will
illustrate the possible answers using a little toy model consisting in a universe of
spin 1 2 (qubits).
In section 26.3 I want to describe what kind of flat semiclassical spacetime we
can expect to recover. For this I will start by recalling how a modification of the
measurement theory can be seen as implementing a deformation of the symmetries.
A strong analogy holds with the toy model, an analogy that can be seen as another
heuristic argument indicating that DSR is the right QG semiclassical limit. The
deformation is usually done in the momentum space, that is the cotangent space.
I will argue then that the geometry (that is, the tangent bundle picture) associated
to this flat semiclassical spacetime can be a described by a Finsler geometry [10].
Since each type of deformed reference frame will correspond to a deformation of
the symmetries, it is natural to ask if there is a global structure that allows one to
unify the different constructions. I will show that indeed these different choices of
reference frames just correspond to different choices of gaugefixings (or choice of
observers) in an extended phase space [11]. This allows us also to specify in an
unambiguous way the symplectic form and the physical spacetime coordinates. I
will conclude with some comments on the multiparticles states.
26.2 Physics of Quantum Gravity: quantum reference frame
The symmetry group of General Relativity is the diffeomorphisms group. Invariance
under this group means that the physics should not depend on the choice of
coordinates. The coordinates x μ are parameters, they should not have any physical
meaning. To understand that was an essential step in the GR construction. It also
led to a long-standing misunderstanding. Indeed when doing physics it is natural to
use coordinates systems: there exists a reference frame (clock, rulers) that allows us
to measure a spacetime position, and so provide physical coordinates. The confusion
arose since it seems that a coordinate system must be at the same time physical
and not physical. As so often, the answer to this paradox lies in its formulation: the
measured coordinates do not have the same status as the coordinates met in the GR
mathematical definition.
To define the physical coordinates, we must use some degrees of freedom [12]:
the reference frame (that often can be confused with the measurement apparatus)
is made of matter (clock and rulers) or gravitational degrees of freedom. This is
a general feature: any physical quantity that is the outcome of some measurement
quantifies the relation between two systems (the reference frame or apparatus and
the system under study).