POE problems

R: Nuclear Magnetic Resonance. Numerical estimation of the resonant frequency.
R: Ferromagnet. Why are not the spins in one magnetic domain exactly in the same direction? (Something about spin wave)
G: Possible spin J of a particle combined with s=1 and s=0 particles with angular momentum L=1?
Component of J=2, Jz=-1state.
G: Infinite potential well. The probability for the ground state of 2a-width well going to the ground and first excited state of 2b-width well.
R: Problem of moving a piston between water and ideal gas. (more details omitted)
G: frequency of an oscillator connected to the walls with two springs on both sides.

Higgs Cartoon

A very intuitive portray of Higgs by comparing with condensed matter physics. Interesting, isn’t it?
From http://www.hep.ucl.ac.uk/~djm/higgsa.html

A quasi-political Explanation of the Higgs Boson;
for Mr Waldegrave, UK Science Minister 1993.

---

1. The Higgs Mechanism

Imagine a cocktail party of political party workers who are uniformly distributed across the floor, all talking to their nearest neighbours. The ex-Prime- Minister enters and crosses the room. All of the workers in her neighbourhood are strongly attracted to her and cluster round her. As she moves she attracts the people she comes close to, while the ones she has left return to their even spacing. Because of the knot of people always clustered around her she acquires a greater mass than normal, that is, she has more momentum for the same speed of movement across the room. Once moving she is harder to stop, and once stopped she is harder to get moving again because the clustering process has to be restarted. In three dimensions, and with the complications of relativity, this is the Higgs mechanism. In order to give particles mass, a background field is invented which becomes locally distorted whenever a particle moves through it. The distortion – the clustering of the field around the particle – generates the particle’s mass. The idea comes directly from the Physics of Solids. Instead of a field spread throughout all space a solid contains a lattice of positively charged crystal atoms. When an electron moves through the lattice the atoms are attracted to it, causing the electron’s effective mass to be as much as 40 times bigger than the mass of a free electron. The postulated Higgs field in the vacuum is a sort of hypothetical lattice which fills our Universe. We need it because otherwise we cannot explain why the Z and W particles which carry the Weak Interactions are so heavy while the photon which carries Electromagnetic forces is massless.

The Higgs vacuum
(Distributed in the universe like aether, but must be covariant – scalar)

Particles coupled with Higgs gain their masses
(The stronger the coupling is, the more massive the particle is)

2. The Higgs Boson.

Now consider a rumour passing through our room full of uniformly spread political workers. Those near the door hear of it first and cluster together to get the details, then they turn and move closer to their next neighbours who want to know about it too. A wave of clustering passes through the room. It may spread out to all the corners, or it may form a compact bunch which carries the news along a line of workers from the door to some dignitary at the other side of the room. Since the information is carried by clusters of people, and since it was clustering which gave extra mass to the ex-Prime Minister, then the rumour-carrying clusters also have mass. The Higgs boson is predicted to be just such a clustering in the Higgs field. We will find it much easier to believe that the field exists, and that the mechanism for giving other particles mass is true, if we actually see the Higgs particle itself. Again, there are analogies in the Physics of Solids. A crystal lattice can carry waves of clustering without needing an electron to move and attract the atoms. These waves can behave as if they are particles. They are called phonons, and they too are bosons. There could be a Higgs mechanism, and a Higgs field throughout our Universe, without there being a Higgs boson. The next generation of colliders will sort this out.

A fluctuation in Higgs vacuum

Condensation of Higgs Boson

from David J. Miller, Physics and Astronomy, University College London.
(cartoons courtesy of CERN).

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.

LHC failure

This title is a bit exaggerating. Actually, it was the magnet that failed and was damaged. See here.

I just learned a little thing about the structure of general accelerators yesterday. Experimental physics is tough, at least for me. It is too complicated and serious due to the close relationship with reality. Physics is dealing with reality, but I would like those simple things behind reality rather than reality itself. That’s why I choose theoretical physics.

Best wishes to LHC. The whole world is waiting for you. I am not. I have learned too little to ride on the tide. I am just going with my own pace.