Steven Weinberg’s massive work (“Quantum Theory of Fields”, V. 1 to 3) is, of course, the ultimate. Zee’s book helps one to fall back on simpler tracks provided one has already spent some effort to understand the foundations in some depth.

if you browse around in this site, you will find more information on particle physics from an experimentalist point of view. The post above is some kind of exception, in the sense that it is devoted to a purely theoretical concept. It takes me much more effort to write about theory, but I do it time and again.

I am afraid I have no time to discuss the ins and outs of the Higgs mechanism (if that is what you were asking for) in the short period. I will, but it may take a while. My best advice is to hang around – there is a lot of Higgs physics in my blog.

In any case, thank you for the visit…

Cheers,
T.

Cheers,
T.

Cheers,
T.

Later.

Cheers,
T.

1) Add the pure electromagnetic Lagrangian to the Goldstone Lagrangian.

2) Couple the two, i.e. gauge the Goldstone Lagrangian, by changing the plain derivatives in it to covariant derivatives: derivative + gauge field term. Physically, this means that the complex scalar now carries electric charge and interacts with the electromagnetic field. (Without the Mexican hat potential, this model is just scalar electrodynamics, i.e. the complex scalars are like electrons and positrons without spin).

3) Do the same shift of scalar field coordinates as before, to some arbirary point in the potential valley, and notice what this does to the covariant derivative terms: terms containing the electromagnetic field squared times the constant scalar part squared pop out. Those are your new gauge field mass terms. (You also get other, messy interaction terms, but let’s not worry about those now.)

So you see, there is nothing arbitrary about how the complex scalar combines with the electromagnetic field. It does so because it carries electric charge. More generally, a scalar field carrying some gauge charge can give mass to the associated gauge field, if you give it a potential with multiple ground states.

And that’s the Higgs mechanism. Which probably means that I just ruined TD’s next post in this series…

Qubit, I am confused by your analogies no less than you have been by my post. How about declaring a draw and leaving it at that ?

Hi Coin, your questions are meaningful, and I thank you for submitting them. I am not the best person to answer them, but what I can say is that if a lagrangian density describes some physical process, it does not matter what transformations you do on it: the physics will stay the same. We know that a quadratic term in a field corresponds to a mass term. We also know how to recognize terms that describe the coupling strength of three or four particles joining at a vertex. If we change coordinates unwittingly, we may hide the physics from plain view, but we won’t change it….

Cheers all,
T.

This actually clarifies a lot, thanks so much.

I guess the one thing that’s still confusing me is that we seem to be behaving really nonchalantly with regards to taking these fields apart and putting them back together. I’ve been kind of assuming that I’m supposed to be keeping two different notions of “field” separate, one the abstract, mathematical idea of “field” as a just a number associated with every point in space, and the other the particle idea of “field” which I’d just assumed described something more concrete, something that “exists”. I know that you can take a generic mathematical field and analyze it such that it behaves like a particle field, like these “phonon” things, but I assumed that this was just a formalism and that the phonon has no physical reality.

But, here you’re taking something that really is just a “mathematical” field, in the sense that it’s just a value which exists at every point in space– that is, the U(1) value taken by that degree of freedom at the bottom of the (I hate this term) “mexican hat”– and somehow we can act as if that value-at-every-point is an actual particle field. Even weirder, then we suddenly decide not to consider this U(1) value by itself, but consider it together with the vector values of a boson field, and say “hey, that’s the same set of degrees of freedom as a massive boson, let’s act like this is a massive boson”. Unless I’m totally missing something, it’s really surprising and interesting that quantum field theory is flexible enough to let you do that. If you can paste together degrees of freedom originating from different fields this way and say the result is a proper particle, then that seems to say phonons have as much physical reality as electrons…

Am I really understanding this right? Or is there some mathematics going on where I can’t see it that decides “why” the U(1) combines with the gauge boson? Thinking about it it seems like there must be, since it seems like there must be SOME reason why the U(1) symmetry decided to become the extra degree of freedom of the WZ gauge boson and not some other gauge boson. What determines which gauge symmetry “eats” the goldstone boson once the goldstone boson is hanging around?

Thanks again…

I meant to say; “when the ballpoint is inside the pen!”

The other picture is this, http://bp1.blogger.com/_-mE1mHUrF8A/RzoWFwDC0FI/AAAAAAAAAC8/AYSYKwdlLRQ/s1600-h/pen+on+side.jpg
This is representation of the left hand picture of your two pictures that are together, the other being the right.

Qubit

There is no difference to the symmetries of each picture, even though one is laid on the table and the other is on its end, because the potential is the same for both states of the pen, unless on you are viewing it from the outside. Then one has a greater potential to be stood on its end, while the other has the potential to do the same, but the odds are the almost impossible.

So you could say that; there was not a real big bang, but rather one that was done to make you believe there was a big bang, when really it was done with birotechnics J