Multi-graph Ensembles as Toy Models for Causal Quantum Gravity

空间、时间和物质的基本组成部分是什么？解决这个问题是寻找量子引力理论的核心。这项研究不仅仅是解决一个理论家的难题。如果我们目前的经典引力理论，广义相对论，被用来解释宇宙微波背景的观测，我们得出的结论是，宇宙始于大爆炸。在紧接着的那个时代，宇宙小得令人难以置信--实际上比原子小得多。这就引出了另一个问题;如果不使用量子力学，原子大小的东西就不能被正确理解，所以我们也期望使用广义相对论来推断大爆炸在某个时候会失败。换句话说，为了充分理解大爆炸，必须有一个量子力学的引力理论。人们已经提出了许多不同的方法来表述量子引力，这个建议涉及其中之一--因果动态三角模型，简称CDT。这个模型通过将三角形（二维）、四面体（三维）或五面体（四维）按照一定的规则组合在一起来构建一个宇宙。有许多不同的包装，每一个对应于一个不同的宇宙，具有不同的物理结果。所有包装的集合称为“ensemble”。为了测量一个物理量，我们随机选择系综中的一个成员并对该成员进行测量;因此，物理量不取确定的值，而是带有概率分布-这就是使模型“量子”的原因。不幸的是，目前最先进的是，我们所拥有的关于潜在现实的四维CDT模型的所有信息都来自非常强大的计算机上的数值模拟;三维模型也是如此。只有在二维情况下，我们才知道如何进行纸上计算。最近人们认识到，二维CDT系综本质上是一组特殊的平面三角剖分，它与一组具有增长规则的更简单的图（从技术上讲，是多重图）有关，这就是高尔顿-沃森过程，自19世纪以来，数学家就知道它，最初它是作为英国贵族世代传播的模型引入的!这简化了我们对这些模型的理解，并使某些计算更加简单。这是非常令人兴奋的是，同样的数学概念可以相关的不同的系统作为一个模型的重力和命运的family.Numerical模拟的三维和四维CDTs已经产生了一些有趣的结果，在这项研究中，我们将集中在三维的情况下，显示出高度非平凡的结构。首先，从动力学中出现的大尺度结构是三维德西特宇宙，这是广义相对论最简单的解决方案之一。更有趣的是，模拟似乎表明，就宇宙中运动的粒子而言，CDTs中的时空维度从微观距离尺度的2维到大尺度的3维不等。这种奇怪的现象被称为降维，最近引起了很多关注。我们希望将多图等价给我们的二维CDT的纸上理解扩展到更高维的版本。特别是，我们将调查是否可以通过多图图片理解的降维现象。

What are the elementary building blocks of space, time and matter? Answering this question lies at the heart of the search for a theory of quantum gravity. This search is more than solving a theorist's conundrum. If our current classical theory of gravity, General Relativity, is used to interpret observations of the cosmic microwave background we reach the conclusion that the universe started with a Big Bang. In the era immediately afterwards the universe was incredibly small -- in fact much smaller than an atom. This leads to another sort of problem; things the size of atoms cannot be understood properly without using quantum mechanics so we also expect that the extrapolation back to the Big Bang using General Relativity will fail at some point. In other words, in order to understand fully the Big Bang it is necessary to have a theory of gravitation which is quantum mechanical. Many different ways of formulating quantum gravity have been suggested and this proposal concerns one of them -- the Causal Dynamical Triangulation model, CDT for short. This model builds up a universe by packing together triangles (in 2 dimensions), tetrahedra (in 3 dimensions), or pentahedra (in 4 dimensions) according to certain rules. There are many distinct packings and each corresponds to a distinct universe with distinct physical outcomes. The set of all packings is called an 'ensemble'. To make a measurement of a physical quantity we choose one member of the ensemble at random and measure on that member; thus physical quantities don't take definite values but come with a probability distribution -- this is what makes the model 'quantum'. Unfortunately the present state-of-the-art is that all the information we have about the potentially realistic four-dimensional CDT models comes from numerical simulations on very powerful computers; this is also the case for the three-dimensional models. Only in the two-dimensional case do we know how to do pencil-and-paper calculations. Recently it has been realized that the two-dimensional CDT ensemble, which is essentially a particular set of triangulations of the plane, is related to a simpler set of graphs (technically, multi-graphs) with a growth rule, the Galton-Watson process, which has been known to mathematicians since the nineteenth century and was originally introduced as a model for the propagation of generations of the British aristocracy! This has simplified our understanding of these models and made certain calculations much more straightforward. It is very exciting that the same mathematical concepts can be relevant to such disparate systems as a model of gravity and the fate of families.Numerical simulations of three- and four-dimensional CDTs have produced several interesting results and in this research we will concentrate on the three dimensional case which shows highly non-trivial structure. Firstly the large scale structure which emerges from the dynamics is that of a three-dimensional de Sitter universe which is one of the simplest solutions of General Relativity. Even more interesting is that the simulations appear to show that, as far as a particle moving in the universe is concerned, the dimension of space-time in the CDTs varies from two at microscopic distance scales to three at large scales. This strange phenomenon is called dimensional reduction and has attracted a lot of attention recently. We want to extend the pencil-and-paper understanding that the multi-graph equivalence gives us of the two-dimensional CDT through to the higher dimensional versions. In particular we will investigate whether the dimensional reduction phenomenon can be understood through the multi-graph picture.