# What is a compact symmetry transformation?

I am taking a course on particle physics, I am not familiar with a lot of mathematical terminology. So I don't properly understand what topology actually means. So when I looked up on the web for the definition of compact or non-compact symmetry transformation, I face terms related to topology.

Is it possible to explain the concepts of compact and non-compact symmetry transformations in easier terms? May be an example of a static symmetry transformation will be helpful to me.

• To be pedantic, symmetry transformations are not compact, but the set which they form, when given a proper topology, becomes a compact/non-compact topological space. The examples below by Zero the Hero are self-explanatory. Commented Oct 8, 2017 at 20:48

Loosely speaking (to avoid topological terminology), compact transformations are expressed in terms of parameters which have finite range, v.g. the rotation angle $\theta$ so that the rotation $$R(\theta)=\left(\begin{array}{cc} \cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{array}\right)\, , \qquad 0\le \theta <2\pi\, .$$ The entries are always finite and bounded. The volume of the group, as measured by the integral $\int_0^{2\pi}\, d\theta$, is finite. In this sense the rotation group $SO(2)$ is compact. Likewise the group $SO(3)$ of rotations, with elements parametrized by 3 Euler angles, is also compact. Using the standard range of the Euler angles, the volume of the group $\int \sin\beta d\beta\, d\alpha\, d\gamma$ is finite.
Non-compact transformations are expressed in terms of parameters with infinite range, v.g. the rapidity $w=\hbox{arctanh}(\beta)\$ in a Lorentz transformation $$\Lambda(w)=\left(\begin{array}{cc} \cosh(w) & \sinh(w)\\ \sinh(w) &\cosh(w)\end{array}\right)\, , \qquad -\infty \le w\le \infty\, .$$ The entries of the matrix can be arbitrary large. The volume of the group, as measured by the integral $\int_{-\infty}^\infty\, dw$, is infinite. The group of Lorentz transformation is non-compact.