Statistical Mechanics and Quantum Fields

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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.