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Rodolfo Gambini Universidad de la Republica Email verificata su fisica. Stephen Fairhurst Cardiff University Email verificata su astro. Professor of Physics, University of Warsaw. It is usually believed that gravitational effects black hole formation implies that the Planck length l P is a minimal distance and that the geometry is quantized in quanta of size l P. It can not be a maximal mass since macroscopic systems have a rest energy much larger than E P. Obviously, it can not be a minimal mass either.
An easy way to characterize it is as the maximal energy or mass content of a cell of size l P. More precisely, let us consider a system of characteristic size L. Then black holes in general relativity imply a maximal bound on the mass of the system:. On the other hand, quantum effect implies mass fluctuations of the system of order:. Then the mass bound is m P.
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Furthermore, there is another quantity relevant to quantum gravity. It is the cosmological constant L , who provides in more local terms an energy density or a local mean curvature. However generically one can have more subtle mathematical limits having some Planck units running to 0 while other ones remain finite. I define an effective theory in a generic regime when the considered scales are much larger compared to the quantum gravity scales.
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This still allows quantum effects and general relativistic effects. More operationally, I consider an observer, provided with a resolution d l obs. Gravity imposes a minimal bound on the observer's resolution, d l l P , but the observer doesn't necessarily saturate it.
Then there is the scale L of the observed phenomena. As explained above, the two relevant mass or energy scales relevant here are a lower bound imposed by quantum effects and related to the resolution d l and a maximal bound imposed by general relativity and related to the system scale L :. Then, using the equivalence principle of general relativity, I translate the fact that all manifolds are locally flat in an second assumption that the observer will describe the spacetime as being flat.
The goal of the effective theory is to describe quantum gravity effects as on that flat spacetime arena. From this point of view, working out an effective theory doesn't need to work in a semi-classical limit, but that limit will still need to be considered in the derived effective framework.
I would like to show that deformed special relativity[1,2] provides an arena in which to build an effective theory for quantum gravity. These results are explained in details in . I consider the cosmological constant L as coming from quantum gravity corrections to the flat spacetime. Then we are trying to describe a curved spacetime, with constant curvature L , seen by an observer will finite resolution d l as a flat spacetime.
Now, assuming the observer is working in a flat spacetime, small movements around him can not be described infinitesimally but are necessarily larger than d l.
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This finite displacements are embedded in the original curved spacetime: the "tangent space" to our effective flat spacetime is curved. Thus we get a curved momentum space on a flat spacetime. The curvature of the momentum space is related to a maximal moment norm, which is given by the minimal length scale d l on the original manifold.
On the other hand, the maximal length scale l C on the original spacetime defines a minimal moment scale. Then we are the mathematical structure of a smooth curved "tangent space". The coordinates on the flat spacetime need to be reconstructed as the translation generators on the moment space. Now the moment space is curved, so the coordinate becomes non-commutative:. And we recover the mathematical set-up of deformed special relativity DSR. Diagonalizing the coordinate operators, we get a discrete space structure and a continuous time.
Another choice would be to consider an AntiDeSitter space, which would lead to the reverse structure of a continuous space and a discrete time. However, one can always question the experimental relevance of such spectra. Indeed, a quantized time would necessary implied phenomenologically a discrete space structure since we traditionally use time-of-flight experiments to measure spacelike distances. Moreover, any clock used to measure time intervals will have discrete ticks and time will always be operationally quantized.
The resulting uncertainty relations read as:. It is possible to generalize this construction to an arbitrary initial metric, which gets translated into an arbitrary metric on the moment space, and thus take into account some dynamical effects of quantum gravity . Then one needs to take into account the finite resolution of the momentum space which gets translated into a commutation relation:. This leads to a doubly deformed special relativity.
Now that we have shown how the non-commutative spacetime of deformed special relativity provides a motivated arena to study effective quantum gravity, the issue is to make sense of the DSR physics and check whether we can formulate a consistent kinematics. Intuitively, it is the same thing as when we deformed the flat velocity space into the mass-shell hyperboloid in special relativity in order to introduce a maximal speed c : deforming the moment space into a 4-hyperboloid introduces the maximal rest mass k.
However the existence of a maximal rest mass is not consistent with the existence of macroscopic system. The solution is to consider that one copy of the algebra describes a "fundamental" system or one-particle system for the observer. To describe a two-particle states, we would need two copies of the DSR algebras. Then the algebra describing the kinematics of the center of mass would be the diagonal part of the direct sum of these two algebras.
It turns out that if the two original algebras have k as deformation parameter, then the center of the mass algebra has 2 k as deformation parameter . Thus, let us start at the Planck length with the Planck mass bound: considering systems larger than the Planck length, i.
Following this logic,  proposes a law of addition of momenta using a five-component energy-momentum and a generalized relativity principle which states that this new extended 5-momentum is conserved during scattering processes in any reference frames. The deformation parameter k then becomes an intrinsic property of the considered system, in some sense the maximal mass for the system could have. As we have seen in the previous section, non-commutative geometries are a natural set-up for effective theories of quantum gravity.
In fact, already special relativity can be considered as a non-commutative geometry.
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Indeed the curved space of velocities induces the non-commutativity of the space coordinates if we look at special relativity from a purely 3d space point of view. More precisely, for a given massive system,  derives the following commutation relation:. More generally, as argued in , requiring Lorentz invariance at the same time as some discrete spacetime structure leads naturally to geometry length and time been defined as quantum operators.
In such a framework, a necessary step towards talking about a semi-classical limit with a smooth spacetime manifold is to introduce a notion of localized spacetime points: it is necessary to build coherent states of geometry describing a notion of "semi-classical points". Let us look at the simplest case: we consider 3d space coordinates forming a algebra.
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This can actually be seen as 3d Loop Quantum Gravity. The length spectrum is obviously derived:. These are the usual SU 2 coherent states, as defined by Peremolov, and provide the notion of semi-classical points with minimal spread. Let us conclude this example by pointing out that we derive here the uncertainty relation:. A field which becomes more and more relevant to the study of quantum gravity is quantum information.
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