Next, we start our analysis on a variety of physics models and access
whether they are scale invariant. We will start with classical physics
models. Although quantum theory has a far richer structure in scale
dependence (in fact one can argue this is the real meat of quantum
field theory), the study of classical theory will prepare us with the
basic language and initial intuitions.

Let me restate here that we classify the theory to be scale invariant
if all of its physical parameters (not dynamical variables) are scale
invariant. This means the same physical parameter can be used to
describe physics on different scales.

A easy way to test the scale invariance is to write physical
parameters extracted from measurements as a function of dynamical
variables with unambiguous transformation properties under scale
transformation. We shall use this method in the following analysis.

3) Classical Models

a) Classical Mechanics

Classical mechanics by itself is not a complete theory. The dynamical
equation

F=ma

leaves an external force F unspecified. Therefore, it is not clear how
to ask the question of scale invariance which is a question about a
complete Lagrangian.

We could, however, ask the scale invariance question in a particular
setup where the full Lagrangian is available (force, or more
accurately, potential, is specified). We will do that now in several
examples.

b) Classical mechanics: two charged particles

The system is defined by four parameters: q_1, q_2, m_1 and m_2.

Let's try to extract the mass m_1. Without including gravity, we will
use F=ma as a definition of mass. We also pretend that we have known q_1
and q_2. We have

m_1= [(q_1 q_2)/r_12^2]/a_1.

With our unit convention, a_1 scales like 1/distance. Therefore, the
determination of parameter m_1 is not scale invariant. Therefore, this
theory is not scale invariant.

That is: if we scale up distance (time) with this system, we will find
that it must be described by some other mass, i.e., a different
physical system. Therefore, scaling is not a symmetry for the same
dynamical system.

c) Particles in Classical gravity (Newtonian or Einstein).

Let's consider the classical gravitating system made of two point
particles. Both in Newtonian and Einstein gravity, this system is
characterized by three parameters, Newton's constant G, masses of the
two particles.

For simplicity, let's stay in Newtonian Gravity: F=G m_1 m_2/r_12^2.
A similar argument (but somewhat more involved) applies to general
relativity. GR reduce to Newtonian in small field limit anyway. We
will return to GR later on.

There are at least two independent measurement of the parameters we
can perform, G*m_1 and G*m_2, based on measurements of dynamical
variables distance and acceleration.

We have, for example

G m_1 = a_2 * r_12^2.

Following our knowledge of scaling of distance and acceleration, we
again see this extraction of the parameter is not scale invariant.

d) Charge particle couples to classical Electric field.

For simplicity, let's return to our example of two charged
particles. Only at this time, let's ignore the effect of particle's
inertia (i.e., ignore acceleration term). This is not a physical
choice. However, let do that in any case to ask a toy question: is the
remainder part of the Lagrangian scale invariant or not.

This system is characterized by two charges, q_1 and q_2. In fact, we
could set q_1 to be unit charge. The independent parameter is just
q_2.

For simplicity again, let's treat charge 2 as the source and not worry
about its motion. This part of the Lagrangian consists of two parts:

the potential energy q_1 q_2/r_12, and the self-energy of source q_2
(also q_1 of course, but let's focus on q_2 as an example).

What is the self-energy? We know that it is (electric field)^2
integrated over volume. This is divergent since we have to take into
quantum fluctuations near the origin somehow. But, let's deal with
that just with a hard cut on the smallest distance one can go, say
r_0. This will make classical physics healthy at this level.

The the answer for self-energy is proportional to

\int_{r_0} q_2^2/r^2 dr = q_2^2/r_0

Now, we can form an observable which is the ratio of those two terms
in the Lagrangian which will give (setting q_1=1)

self-energy / potential energy = q_2 r_12/r_0

If we measure those energies in a given configuration, say from
gravitational effect, we could extract q_2.

Notice that obviously this extraction is scale invariant (r_12/r_0
does not scale). We have our
first example of scale invariance. That is, I repeat, even we scale up
the size, this part of the Lagrangian can still be described using
the same coupling constant q_2.

e) Maxwell

What we have done in example f) is already essentially a proof of
scale invariance in classical Maxwell with source. A full proof is
rather boring which also involves the scaling of E and B field. I
would not repeat it here.

We have not done it for Maxwell theory in vacuum. However, that's
trivial. It is a wave equation without a coupling constant, i.e., no
parameter at all.

应该是Ⅲ了吧……下载后慢慢研究

等sage兄整个写完了，我再系统地看一下