Why is the structure group for lengths $\mathbb{R}^+$ and not automorphisms in $\mathbb{R}^+$?

Question

While reading what is a gauge to understand what a gauge is, I got stuck at a point where Terry Tao wrote:

This isomorphism group is called the structure group (or gauge group) of the class of geometric objects. For example, the structure group for lengths is $\mathbb{R}^+$... the structure group for lines is the affine group...

Now I can perfectly understand that the structure group for lines is the affine group since an affine group contains all invertible affine transformations to itself. And, in the same vein, the structure group for lengths should be all transformations from $\mathbb{R}^+$ to itself, right ? What am I missing ?

Answer 1

Here's my take: $\mathbb{R}^+$, if we assume it is the additive semigroup of positive real numbers, cannot be the structure group. The most compelling reason for this is that $\mathbb{R}^+$ is not a group.

Earlier in the post, he talks about rescaling. The set of all allowable rescalings is also $\mathbb{R}^+$, but it is a group under multiplication. These are clearly automorphisms, and I suspect (perhaps naively) that they are the only automorphisms. This is the only way I see that could possibly justify that notation.

Regardless, the answer is "yes." It should be the automorphism group of $\mathbb{R}^+$.