A rational framework for Group Theory in Physics | Classical to Quantum | Wild Egg Maths

Group theory is a central topic in mathematical physics, with applications in many directions. While finite groups do play an important role, particularly the symmetric group, and Weyl groups associated to root systems and simple Lie algebras, it is the theory of Lie groups and their representations that forms the main challenge for physicists to learn.

In this series, we will adopt a novel point of view towards both finite groups (mostly the symmetric groups) and Lie groups: they will be primarily groups of matrices or projective variants. This is a powerful, explicit algebraic approach that circumvents much of the usual differential geometric underpinnings of the subject, and avoids irrationalities and "real number arithmetic". It also brings into play the connections between the groups and algebras in which they are naturally embedded. In particular we are going to have to reconsider carefully the relation between a (Lie) matrix group and its Lie algebra: the Cayley transform will be the guiding light here.

Representation theory becomes more natural in this context also, and harmonic analysis can be directly connected to polynomial spaces on the associated algebras.

To begin the discussion, we will quickly survey some important applications of groups to physics. Then we will outline the case for a purely matrix oriented view to group theory. The first steps will be to understanding the single most fundamental group both in mathematics and physics: SL(2).

For us this is not an abstract group: it naturally comes with its canonical imbedding inside the Dihedron algebra. This will give us a powerful position to see this group and its representations in a new light, and will position us well to understand further more complicated groups as we proceed further into mathematical physics.