Statistical Mechanics and Quantum Fields

统计力学关注的是原子的性质如何转化为物质的行为。例如，如果我们把水想象成一个盒子里的大量小台球，它们相互之间以及盒子的侧面反弹，如果我们假设每个球的平均动能是“温度”，那么计算机模拟会发现这种想象中的物质就像冰、液态水和蒸汽一样，取决于温度吗？如果我们还假设小球体之间有其他的力，比如引力，它会更好地工作吗？ 甚至比用计算机模拟更好的是，我们可以从数学上证明这些想象的系统有不同的相（固体，液体，气体）。 这种假想的系统称为模型。 真实的水分子不是球体，但是从模型中得出的逻辑结论揭示了什么在大量物质的行为中起重要作用，什么不起重要作用。这个建议是一个计划，以加强数学的力量来回答这类预测问题。 可以被简单模型捕捉到的大块物质的性质被称为“普适”。 例如，一个长链分子的末端到末端的距离，其亚基彼此排斥，很可能是链中单元数的普适函数。 普遍定律将取决于周围空间的维度，但不取决于排斥的细节。 我们可以证明这种类型的结果时，斥力是相当弱的，周围的空间有四个或更多的维度。

Statistical mechanics is concerned with how properties of atoms translate into the behaviour of bulk matter. For example, if we imagine water to be a huge number of tiny billiard balls confined to a box, bouncing off each other and the sides of the box, and if we assume that the average kinetic energy per ball is "temperature", will a computer simulation find that this imaginary material is like ice, liquid water and steam, depending on temperature? Will it work better if we also assume that the little spheres have other forces between them, such as an attraction? Even better than using a computer to simulate, could we prove, mathematically, that these imaginary systems have different phases (solid, liquid, gas). Such imaginary systems are called models. Real water molecules are not spheres, but drawing logical consequences from models reveals what plays an important role in the behaviour of bulk matter and what does not. This proposal is a program to strengthen the power of mathematics to answer this kind of predictive problem. Properties of bulk matter that can be captured by simple models are called "universal". For example the end-to-end distance of a long chain molecule whose subunits repel each other is likely to a universal function of the number of units in the chain. The universal law will depend on the dimension of the ambient space, but not on the details of the repulsion. We can prove this type of result when the repulsion is quite weak and the ambient space has dimension four or more.