Statistical Mechanics and Quantum Fields

统计力学关注的是原子的性质如何转化为整体物质的行为。例如，如果我们把水想象成大量的小台球，被限制在一个盒子里，相互碰撞，或者在盒子的两边反弹，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的力，比如吸引力，会不会更好呢？甚至比用计算机模拟更好的是，我们可以用数学方法证明这些假想的系统有不同的相（固体、液体、气体）。这种假想的系统被称为模型。真正的水分子不是球体，但从模型中得出的逻辑结果揭示了什么在大块物质的行为中起重要作用，什么不起作用。这一建议是一个程序，以加强数学的力量来回答这类预测问题。可以用简单的模型捕捉到的大块物质的特性被称为“普适性”。例如，一个亚基相互排斥的长链分子的端到端距离很可能是链中单位数量的普遍函数。宇宙法则将取决于周围空间的维度，而不是斥力的细节。当斥力很弱，并且周围空间有四维或四维以上时，我们可以证明这类结果。

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.