The theory of general relativity encodes our understanding of the classical nature of space–time, which includes gravity, but does not describe the fundamental nature of space–time. For example, close to the singularity inside a black hole, the theory ceases to be valid. It is widely expected that integrating quantum effects into the theory could enable us to describe the physics in such situations. To describe the notion of space–time in the quantum regime requires a theory of quantum gravity. However, the notion of combining general relativity and quantum mechanics faces a major obstacle in the form of the different mathematical languages and conflicting conceptual bases of each theory.

The development of a quantum gravity theory is one of the most challenging problems faced by contemporary theoretical physics, and its formulation is expected to provide remarkable developments in astrophysics and cosmology. The challenge involves not only theory, but experiment. Due to the extreme nature of the physical phenomena that are central to quantum gravity, it has been thought for many years that it would impossible to measure quantum gravitational effects. As a result, the conceptual dichotomy between quantum mechanics and gravity, together with a lack of experimental evidence, have prevented a satisfactory unification of gravity and quantum theory for more than 80 years. Three concurrent developments have paved way for a new approach to this old problem.

1.Non-commutative geometry (NCG): Following the naive idea that gravity

is characterized by geometry, and that quantum mechanics is characterized by non-commutativity, a quantum gravity theory could be encoded by a non commutative geometry, that is a geometry where coordinates do not commute. Since Snyder’s first proposal in 1947 [2], this naïve idea has been made rigorous by Connes [3] and many others during the 90’s. At the same time the notion of quantum (Lie) group, ie groups which have non commutative coordinates have been extensively studied by Majid and many others [4]. Quantum groups can also be seen as symmetry groups of NCG. The non commutativity is encoded by a combination of parameters, such as the Newton constant, the Planck constant, which sets to zero allows to recover a standard geometry.

2.Loop quantum gravity (LQG): In the 90’s, under the impulsion of Ashtekar, Rovelli, Smolin, and Thiemann among others, the loop quantization program arose as a non pertubative canonical quantization of gravity [1]. It was used to make major advances in solving some of the key technical problems in the quantization of gravity. An important outcome is that the spatial geometry becomes discrete at the quantum level. This approach does not incorporate the cosmological constant. Such constant encodes a global acceleration of the Universe expansion. Its precise measurement led to the 2011 Physics Nobel Prize. It is actually difficult to introduce this constant in the LQG scheme, so for simplicity it is usually assumed to be zero.

In the 2000’s, a path integral approach called the “spinfoam approach” was developed. It can be seen as a discretized version of Misner and Hawking’s sum over histories [1,5] and is supposed to be related to the LQG approach. In a three dimensional (3d) space- time this relationship can be identified exactly [6]. Interestingly, a cosmological constant can be introduced in the spinfoam framework, using quantum groups. This follows from some seminal works by Witten and Turaev and Viro from the 90’s [7]. The cosmological constant appears in the combination of parameters encoding the group noncommutative structure. It is then very intriguing to understand why in the spinfoam formalism there is a quantum group and not in the loop quantum gravity formalism. To have a well understood quantum gravity theory, it is mandatory to understand this issue.

3.Experimental signatures of quantum gravity: Following the LQG development, one of the key questions for the field is to determine whether there exists a semi- classical limit in which the discreteness smoothes out to a classical geometry. It has been argued that the flat semi-classical limit is effectively expressed in terms of a flat non-commutative geometry. Quantum gravity phenomenology has reached an exciting point with the opening of a new experimental window: the launching in early 2009 of the FERMI satellite, which is the first opportunity ever to observe very-high-energy gamma- ray bursts [8]. There is a general expectation that the quantum gravity semi-classical regime will be explored through the study of events such as gamma-ray bursts [9]. Semi-classical quantum gravitational effects generate modified dispersion relations, which in turn generate delays in the time of arrival of the high-energy part of the gamma- ray bursts, with respect to the lower-energy part, when traveling on a sufficiently long distance. They could be used to probe the quantum nature of space–time. The phenomenology of quantum gravity could be demonstrated; if this is the case, it will be of great significance to understand the semi-classical limit in the various quantum gravity models to systematically derive experimental predictions. Comparing such predictions with experimental data is the only way that we will be able to distinguish between these models.

My current research is mainly based on these three topics and the possible interplays between them.

References:

1.T. Thiemann, Modern canonical quantum general relativity, Cambridge University Press, 2007. C. Rovelli, Quantum Gravity, Cambridge University Press, 2004.

2.H. Snyder, Phys. Rev. 71 (1947) 38.

3. A. Connes, Non-commutative geometry, Academic press, 1994.

4.S. Majid, Foundations of quantum group theory, Cambridge University Press, 1995. V.Chari , A. Pressley, A guide to quantum groups, Cambridge University Press, 1994.

5. 5.A. Perez, Class. Quant. Grav. 20, R43 (2003).
   

6. 6.6.K. Noui and A. Perez, Class. Quant. Grav. 22, 1739 (2005).
   

7. 7.7.E. Witten, Commun. Math. Phys. 121 351 (1989). V. Turaev, O. Viro, Topology 31 (1992) 865.
   

8. 8.Briggs et al., Science 27, Vol. 323 (2009) 1688.
   

9. G.Amelino-Camelia, S. Majid, Int.J.Mod.Phys.A15 (2000) 4301-4324.