Lie (Continuous) Groups

Just as there is a distinction between the (countable) set of integers and the (uncountable) set of real numbers, there is a distinction between discrete groups (where an identification can be made between group elements and the integers) and continuous groups (with an uncountably infinite number of group elements). $\mathbb{R}^*$ is a continous group, and is the basis of calculus, because it supports the idea of differentiation using a suitable limiting process such as

$$\lim_{\Delta x \to 0} \frac{\Delta\ }{\Delta x} \to \frac{d\ }{dx}$$

A Lie Group is a continuous group, which is also formally a differentiable manifold. We could easily get swept down the rabbit hole to ``real math'' at this point and explain that a differentiable manifold is any space that is locally isomorphic to a Euclidean (flat) space like $\mathbb{R}^3$ (a real space in three orthogonal dimensions) wherein differentiation is well defined. This means that a Lie group is generated by composing a large number of local ``infinitesimal transformations'' into a finite transformation. Continuous coordinate transformations in physics often form Lie groups, in particular the set of all continous rotations of a coordinate frame, SO(3).

All of this section so far, in fact, leads to this one conclusion. Coordinate transformations of interest to us in physics in general, and electrodynamics in particular, almost always end up being Lie groups (with an associated Lie algebra for the abstract group operations) generated from infinitesimal local transformations. The continous groups are often extended by a (usually small/finite) set of discrete transformations, such as inversion. Let's discuss this further.