Field Mass-The Inspiration from Mattress

What is mass? What is the origin of mass? These questions are hard enough for particle mechanics, and even more obscure in field theory. For particles, we say that mass describes the property of inertia; for fields,  mass describes the dispersion relation, ie, the relations between wave number and frequency of the excitation of the field. Well, that is too abstract, is there any intuitive picture for the mass of field?

Let’s first consider a mattress constituted with classical oscillators. Each oscillator has mass m. The spacial division of two adjacent masses is l and the spring constant between them is k. Then we can write down the lagrangian:

After continuation, we send ,

Redifine the field function where is the mass density of the oscillators. After redefinition of the parameters, we get a massless scalar field. with speed of wave
.

But what if we want a massive scalar field? Add one term to Lagrangian:

After continuation and redefinition, this term becomes

This is just the field mass term, with mass .

The physical meaning of this term is just that besides the springs connecting between these oscillators themselves, there are some other springs connecting each oscillator to a fixed point, so that the whole mattress can not move freely. What is this solid “wall” that hold this mattress field? It can be the vacuum of another field. And yes, this is just a natural and intuitive picture of Higgs mechanism! I like this more than the celebrity picture showed in the Higgs Cartoon.

概率问题

一个运动员在药检时结果呈阳性（即怀疑服用违禁药品）。药品检验器材的厂方提供的信息是：在阳性结果里有1%的失误率，在阴性结果里有20%的失误率。在 已经受检的20000名运动员中，有300人结果呈阳性。请问这位运动员确实服用了违禁药品的几率是多大？（提示：不是99%，我也搞不懂为什么）。

Update:

这是一个条件概率问题。关键在如何理解厂方信息。所谓“阳性结果里有1%的失误率”是什么意思？是不是说一个人如果被检验成阳性，他吃药的几率就是99%？从字面理解好像是这样，但仔细想想，这个数据不是厂方可以提供的，因为要想得到这个数据，我们需要一堆事先并不知道是否吃药的样本人群，让他们去测验，拿到结果后再跟他们是否吃药的真实情况比较，但是否吃药显然跟整个人群的吃药率有关，因此这个量不是能够独立测量的。但另一个量是只跟仪器有关，跟人群吃药率无关的，那就是一个事先确定吃药或没吃药的人的检查结果，厂方正是用这个方法来定义和测量仪器灵敏度的。用数学语言说，如果用P(A|B)表示条件B成立的情况下A的概率，那么厂方提供的是P(阳性|未吃药）=1%，P(阴性|吃药)=20%，而我们想要求的是P(吃药|阳性）。最后答案是32%。

Things I learned these days

• Neother Theorem is the Lagrangian form of Ehrenfest Theorem which is in Hamiltonian form. There are two correspondences (not very exact) :

1. The Lagrangian is invariant under some symmetry corresponds to the Hamiltonian commutes with some operator (symmetry group’s generator)
2. The Neother charge is a time constant corresponds to the expectation value of the symmetry operator is a time constant, actually, the Neother charge is just the expectation value of the generator.

• From Kaplan’s Colloquium (11.29):

1. EW theory must break down because of 4W interaction, just like 4 fermion interaction must break down, which is caused by violating unitarity. And the solution is similar, 4 fermion need a gauge boson to change 4 vertex to 3 vertex, and 4W need a Higgs (or other new particle?) to change 4 vertex to 3 vertex.
2. MSSM has more than 100 parameters! ( I heard this for the first time)
3. Higgs mechnism can be traced back to Schwinger, who proved that massive gauge bosons do not necessarily violate gauge symmetry, by introducing a scalar field, but did not mention symmetry breaking. And there were other guys’ work following this, showing symmetry breaking, until Higgs pointed out the existence of a scalar particle following the referee(highly suspected to be Schwinger)’s suggestion. So came the name ‘Higgs particle’.

• From Tom’s seminar (12.3)

SUSY can be broken explicitly at UV (elementary sector) but emerges accidentally at IR (composite sector). The symmetries at lower energy are more than that at higher energy, which seems blizzard, but not. From 5D view, IR have more gauge symmetry and so more degrees of freedom just because UV and IR are two vacuums separated by the bulk (or domain wall); from 4D view, the different symmetries become global symmetry so it will not take more degrees of freedom, and the low energy global symmetry is always broken at high energy. (Because of gravity)

Diagonalize the mass matrix

When we diagonalize the mass matrix, we use unitary bilinear transformation. Why unitary? Because we want no cross terms in the kinetic part, which can only be guaranteed by unitary transformation. Why bilinear? Because the left hand particles and the right hand particles are in principle independent, and we should not expect them to follow the same transformation. We know that the usual linear diagonalization is unique (if exists) from the secular equation. Now we can prove that the bilinear unitary diagonalization is unique in the terms of molds, but each eigenvalue can take an arbitrary phase. In the case of mass matrix, we require all the eigenvalues to be positive real numbers, so the diagonalization of the mass matrix is unique.

Why off-shell?

Why are the virtual particles off-shell? One common explanation is uncertainty of energy. My opinion is just the opposite – it’s because of the certainty of energy, for the energy-momentum must be conserved. Look at the vertex in the Feynman Diagram, if we determine the four momentum of the incoming particle and the outgoing particle, the four momentum of the exchanged virtual particle is fully determined by conservation of energy-momentum. Compare with the 2-body decay, where the final momentum magnitude and energy are exclusively determined by conservation law and on-shell condition, in the virtual particle case, the final states are arbitrary, so for most of the cases, it is impossible for the virtual particle to be on-shell.

For this intuition, the virtual particles are really unreal. I would rather interpret them as transforming of quantum numbers than particles carrying these quantum numbers? What’s the difference? Maybe not too much. Virtual particle is at least a convenient concept for calculation though it seems wired.

Update:
So what happens if we set this propagating ‘particle’ to be on-shell? By using conservation of four momentum and on-shell condition, the scattering angle is entirely determined. The result comes out to be: for Coulomb scattering, the angle is 0, for which the differential cross section blows out; for s channel process like ee->mumu or Compton scattering, the on-shell condition can never be reached. Then we see the xx-shell has nothing to do with virtual ‘particles’, and the blowing out is expected because on-shell condition is a pole for the propagator.

Further more, on-shell condition is equivalent to equation of motion for free particles. When we calculate propagator, we put a delta function on the other side of the equation, so it’s no longer free. I should have noticed this long before! off-shell is equivalent to interaction. All interacting particles are off-shell. Why outer-leg particles are on-shell? Because they are far away from each other and are taken as free. This fundamental reason leads to the direct reason for off-shell which I discussed above: transformation of momentum and energy.