Hints from non-commutative geometry
By Marco de Cesare, Mairi Sakellariadou, and Patrizia Vitale
It is often argued that modifications of general relativity can potentially explain the properties of the gravitational field on large scales without the need to postulate a (so far unobserved) dark sector. However, the theory space seems to be virtually unconstrained. One may then legitimately ask whether there is any guiding principle —such as symmetry— that can be invoked to build such a modified gravity theory and ground it in fundamental physics. We also know that the classical picture of spacetime as a Riemannian manifold must be abandoned at the Planck scale. The question then arises as to what kind of geometric structures may replace it, and if there are any novel gravitational degrees of freedom that they bring along. Importantly, one may ask whether there are any potentially observable effects away from the experimentally inaccessible Planck regime. These questions are crucial both from the point of view of quantum gravity and for model building in cosmology; trying to answer them will help us in the attempt to bridge the gap between the two fields, and could have far-reaching implications for our understanding of the quantum structure of spacetime.
Marco de Cesare
By Rodolfo Gambini and Jorge Pullin
Rodolfo Gambini and Jorge Pullin have been collaborating for 27 years
In the 1960’s Stanley Mandelstam set out to reformulate gravity and gauge theories in terms of observable quantities. The quantities he chose are curves, but specified intrinsically. The simplest way of understanding what does “specified intrinsically” means is to think how the trajectory of a car is specified by a GPS unit. The unit will give commands “turn right”, “advance a certain amount”, “turn left”. In this context “right” and “left” are not with respect to an external coordinate system, but with respect to your car. The list of commands would remain the same whatever external coordinate system one chooses (in the case of a car it could be a road marked in kilometres or miles, for instance). The resulting theories are therefore automatically invariant under coordinate transformations (invariant under diffeomorphisms). They can therefore constitute a point of departure for the quantization of gravity radically different from other ones. For instance, they would share in common with loop quantum gravity that both are loop-based approaches. However, in loop quantum gravity one has to implement the symmetry of the theory under diffeomorphisms. Intrinsically defined loops, on the other hand, are space-time diffeomorphism invariant, therefore such a symmetry is already implemented. It is well known that in loop quantum gravity diffeomorphism invariance is key in selecting in almost unique way the inner product of the theory and therefore on determining the theory’s Hilbert space. Intrinsically defined loops are likely to be endowed with a very different inner product and Hilbert space structure. In fact, since the loops in the Mandelstam approach are space-time ones it lends itself naturally to an algebraic space-time covariant form of quantization. Continue reading
by Kirill Krasnov and Roberto Percacci
The geometric unification of gravity with the other interactions is not currently a popular subject. It is generally believed that a unified theory can only be constructed once a quantum theory of gravity is available. The purpose of this CQG+ contribution is to advocate that it may be fruitful and even necessary to reverse the logic: instead of “quantising in order to unify” it may be necessary to “unify in order to quantise”. If the latter perspective is correct, our current approaches to quantum gravity would be similar to trying to understand the quantum theory of electricity and magnetism separately before they were unified in Maxwell’s theory.
There are several arguments for such a change of priorities.
Kirill Krasnov, Professor of Mathematical Physics, University of Nottingham
Roberto Percacci, Associate Professor of Theoretical Particle Physics at SISSA, Italy