TITLE: What is a compact symmetry transformation?
QUESTION [3 upvotes]: 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.

REPLY [4 votes]: 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.
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