Statistical Mechanics and Quantum Fields

统计力学关注的是原子的性质如何转化为块状物质的行为。例如，如果我们想象水是被限制在一个盒子里的大量微小的台球，它们相互反弹，从盒子的侧面反弹，如果我们假设每个球的平均动能是“温度”，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，取决于温度吗？如果我们还假设这些小球体之间有其他力，比如引力，它会工作得更好吗？甚至比使用计算机模拟更好的是，我们能从数学上证明这些想象中的系统有不同的相(固体、液体、气体)。这种想象中的系统被称为模型。真正的水分子不是球体，但从模型中得出合乎逻辑的结果，揭示了什么在块状物质的行为中起着重要作用，什么不起作用。这项建议是一个加强数学能力来回答这类预测性问题的方案。简单模型可以捕捉到的块状物质的性质被称为“万能”。例如，亚基相互排斥的长链分子的末端到末端的距离很可能是链中单元数量的普遍函数。宇宙定律将取决于周围空间的大小，而不是排斥的细节。当斥力很弱，且周围空间有四维或更多维时，我们可以证明这种类型的结果。

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.