Before we move on, we comment on the issue of units. At the same time,
we choose a convention which is convenient for our discussion.

We can only measure relative sizes. However, comparing things with
other things at random will be inconvenient. Therefore, it is always
convenient to compare thing we measure with some pre-determined
standard sizes. These standards are called units.

Of course, units are arbitrary choices made by us. Therefore, our
physical theory can never depend on the units we have chosen to talk
about things. So we could choose any unit to talk about things without
have any physical effect. It is irrelevant to our discussion here on
scale transformations (which actually rescale things).

Next, let's think about what kind of scale transformation we are
interested in. In particular, we could think about the transformation
which only scale the spatial separation, or we could also scale time
at the same time.

Einstein's relativity shows us that we should not really talk about
only space alone. We should be really dealing with a 4-dimensional
space-time. Talking about space alone only makes sense in the small
speed, non-relativistic limit. However, as we shall see later, it is
quite impossible (with possible exceptions) to have scale
invariance in a non-relativistic world. Therefore, it is natural to
begin our discussion with a scale transformation which acts, equally
as dictated by Lorentz invariance, both on space and time.

To make such a transformation more explicit, lets adopt a particular
convention of unit. We will choose the unit of time (for example), so
that the speed of light is c=1. (One way to achieve this is to take a
particular time on the light-cone, call ct the new time. )

Under this convention, space and time
have the same dimension (new time is ct under old convention).


Notice also the relativistic relation:

E^2=p^2 c^2 + m^2 c^4

becomes

E^2=p^2+ m^2

which makes E, p and m having the same dimension.

Notice also that although this is motivated by relativity, we can do
this nevertheless, even without relativity. So we will use it even in
non-relativistic cases, just because we can deal with one less unit
this way.

Now, under this convention, velocity is invariant under scale
transformation x-> \lambda x where x is a four vector (t, x1,x2,x3).

These are good points.

Maybe I will write my thoughts as well.