Uses of non-commutative geometry in quantum gravity

广义相对论编码了我们对时空经典本质的理解。这包括引力，但它并没有描述时空的基本量子性质。例如，在接近大爆炸和黑洞奇点的时候，这个理论就不再有效了，因为量子效应出现了。量子引力理论的发展是当代理论物理学面临的最具挑战性的问题之一。非交换几何和环量子引力是解决这个问题的两种方法。他们利用各种数学技术为量子时空的概念提出了有希望的候选者。量子引力的挑战不仅涉及理论，现在还涉及实验。由于量子引力的核心物理现象的极端性质，多年来人们一直认为测量量子引力效应是不可能的。然而，随着一个新的实验窗口的开启，这个领域已经到了一个激动人心的时刻：2009年初，FERMI卫星的发射。它目前正在观测高能伽马射线爆发，这些伽马射线爆发能够携带量子引力机制的一些特征。这些特征可以让我们第一次间接观察到量子引力效应，并有望让我们测试和推进不同的量子引力模型。我的研究首先集中在理解非交换几何和环量子引力方法背后的物理。为此，我建议在我最近的工作中发现的这两种方法之间探索新的数学桥梁。其次，它的目标是识别和预测量子引力现象的一些具体特征，这些特征可以在一些即将到来的实验中观察到，比如费米。

The theory of general relativity encodes our understanding of the classical nature of space-time. This includes gravity, but it does not describe the fundamental quantum nature of space-time. For example, close to the Big Bang, and close to the singularity in a black hole, the theory ceases to be valid because quantum effects set in. The development of a quantum gravity theory is one of the most challenging problems faced by contemporary theoretical physics. Non-commutative geometry and loop quantum gravity are two approaches to tackle this issue. They propose promising candidates for the notion of quantum spacetime using a variety of mathematical techniques. The quantum gravity challenge involves not only theory, but also now experiments. 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. The field has reached an exciting point, however, with the opening of a new experimental window: the launching in early 2009 of the FERMI satellite. It is currently observing very-high-energy gamma-ray bursts which are able to carry some signatures of the quantum gravity regime. These signatures could give us the ability to indirectly observe for the first time quantum gravitational effects, with the prospect of allowing us to test and push forward different quantum gravity models.My research focuses firstly on understanding the physics behind the non-commutative geometry and loop quantum gravity approaches. To do so, I am proposing to explore new mathematical bridges between these two approaches that I have identified in my recent work. It aims secondly at identifying and predicting some concrete signatures of quantum gravitational phenomena that could be observed by some upcoming experiments such as FERMI.