1) What is scale invariance?

We must be clear first what we mean by scale invariance. Scale
transformation, as we will talk about more later, is about changing
the size of things. At this point, we must realize that one cannot
talk about absolute sizes of things, since there is no such
measurement one can do. We can only measure the RELATIVE size of
things. Therefore, to judge whether a theory is scale invariant is not
simply a matter of whether the theory Lagrangian depends on position
or not (such as a Coulomb potential). It must involve constructing
physical observables which could determine whether the theory depends
on scale transformation or not. For example, in the case of Coulomb
potential, we must ask how we measure it, and so on.

To give a more or less useful definition, we begin by setting up some
terminologies. A theory, (such as Maxwell E&M, Newtonian gravity,
string theory...), is characterized by a set dynamical variables (such
as position, momentum, field strength, field, etc.) and a set of
parameters (such as masses and coupling constants). A particular
choice of the KIND (not the size) of the dynamical variables and the
KIND AND VALUE of the parameters determine a theory. For example,
electron field, photon field, fine-structure constant (=1/137) and
electron mass (= 0.511 MeV) defines the Dirac theory of electrons in EM
field.

Scale transformation first act on space: x -> \lambda x. Such a
transformation will induce a transformation on the VALUE dynamical variables:
A->A'. Such transformation, by itself, does not say anything about
whether the theory is scale invariant or not, since after this
transformation, we have only change the value, not the KIND of the
dynamical variables.

A definite test of whether a theory is scale invariant comes from
whether the VALUE of the PARAMETERS of the theory, as measured from
experiment, will change under scale transformation (which only acts
on dynamical variables) or not.

A practical way of doing this is to construct a way of measuring
parameters of the theory, in terms of a function of the dynamical
variables. Then, do the scale transformation (acting on
the dynamical variables) and see whether the parameter we extract from
experiment in the scale transformed world agrees with the old one or
not.

Let's do this in a toy (not really makes a sound physics model)
Lagrangian, just to demonstrate the procedure. We will do real
theories when we talk about them later.

Say we have a Lagranian of three particles

L= (r_1-r_2)+ \alpha (r_1 - r_3)

(r_1, r_2, r_3), positions in certain units, are dynamical
variables. \alpha is a parameter. We could measure \alpha by:

a. only have 1 and 2, take the energy of the configuration.

b. only have 1 and 3, at the same distance, take the energy of the configuration.

c. take the ratio of the measurements, this gives us \alpha.

This theory is obviously scale invariant.


Before we go on, let's add a short (and probably vague) side comment
about scale transformation
vs dimensional analysis. Scale transformation only acts on dynamical
variables. Dimensional analysis only acts on (scales)
parameters. Therefore, dimensional analysis always take one theory
into another with same set of dynamical variables but different values
of parameters. In this sense, dimensional analysis is always a
symmetry in the solution space, i.e., given exact solutions of one
theory, dimensional analysis always turns it into exact solutions of
another theory with changed parameters.

期待下一篇。看来如何测量是讨论尺度不变性问题的一个前提。
我感觉“尺度不变性问题”跟“变常数(比如光速c)问题”有密切的关联。

这两天我读了几篇有关判断量子系统是否是纠缠的文章，里面也强调了测
量是讨论纠缠的前提。如果我们只问“一个系统的某个状态是不是量子纠
缠态？”，而不给出讨论纠缠的子空间是什么，这个问题就没有意义。把
量子系统Hilbert空间分成多个子空间的直积结构，从而让我们可以讨论系
统的量子纠缠的，就是由我们所选取的测量方式引起的。

期待下一篇。

标度不变性，相变，共性不变性，这些希望 sage兄都讲到

写错了，上面应该是“共形不变性”。

感觉sage兄想谈跟重整化群相关的东西（或者相变下由于关联长度无穷大而导致的标度无关性？），记得重整化群理论那里，把所有具有某一量纲（比如质量）的量，作同样的标度变换，这种变换构成一个量纲标度变换群，此时没有量纲的参数是不变的。最后得到重整化群、跑动耦合常数这些东西。

为了让我们有一个不会概念混淆的认识，sage不妨顺便给我们科普一下跟标度不变性相关的自相似形和分形现象、相变下由于关联长度无穷大而导致的标度无关性、还有共性不变性，这些东东。可能我的这些问题的提法本身，都是有些混乱，呵呵！

Quantum many-body system at criticality is describled by corresponding
conformal field theory, actually i am not certain about the relation between scaling invariance and conformal invariance. in my sense,scaling transfromation is just a kind of conformal transformation,i don't know whether scaling transformation has some special properties.
Maybe Sage will take some time talking about this.

to yinzhangqi: the pattern of entanglement is an nice tool to investigate the nature of quantum criticality. for many-boby system it has been more and more accepted that the order of matter should to be characterized by quantum entanglement other than former order parameters( for my part order parameter is ugly and uncontrollable specially when i learned statistical mechinasm, it cant be a final description)