标题: Scale Invariance (1-3)
作者: sage
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.
II. Units
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).
III: Classical Models
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.
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.