No, he doesn't reckon they are charged. He's talking about what happens if they were charged. This basically means that you are working with a gauge group whose Lie algebra has \(f^{a}_{bc} \not=0\) so that the boson interaction term in my previous post is non-zero. This is why gluons self-interact. This is also why photons don't.
Whoops, yes a closer read reveals that he's talking about how the other Yang-Mills fields (which can be paired), describe photons with charge. Not sure if that's a reference to the W bosons, but anyway. If photons were charged, or had positive and negative charges, they could form atoms of light, which doesn't happen. t'Hooft comments about where the term "gauge" came from - one Hermann Weyl, who in the 1920s was constructing a theory that attempted to unite EM and GR, and had to specify local contractions and dilatations of spacetime at every point. He compared the selection of local values with the gage blocks used by engineers as measurement standards. t'Hooft says the term should be "phase angles", not "length scales", in Weyl's paper. "Eich Invarianz" was translated as "calibration invariance", but later re-translated as "gauge invariance", and we're stuck with it.
Good, this looks more appropriate to my level of understanding I am struggling with this slightly, but I shan't trouble you with questions for now. Yes, it's a big ask, I know. You guys spend months of full-time study to get comfortable with this stuff. and yet I expect to understand it via a couple of paragraphs a day; how arrogant am I! This is a good idea. Tell you what, do that, and when I run into difficulty, as inevitably I shall, I will post questions on this site, but framed in such a way that those who have not read the book may get some benefit from our exchanges. How does that sound (by the way, I did read the remainder of your post, several times. I thank you for your trouble!)
Well, another way to see this is in Maxwell's equations. You should rembmer that \(\vec{B} \equiv \nabla \times \vec{A}\) You should also remember from your vector calculus that \(\nabla \times \nabla \Lambda(x) = 0\). Then clearly, the magnetic field (the PHYSICAL thing, that you actually MEASURE) is invariant under \(\vec{A}\rightarrow \vec{A}+\nabla \Lambda(x)\) A is called the ``vector potential'' and can (except for some important exceptions !) not be measured. Another way to say it (attempting to use the vernacular) is that the space of A's and the space of B's is not 1 to 1.
Ben, I can see you're used to typing in Latex, with the ``quotations" rather than "quotations" Please Register or Log in to view the hidden image!
heh...yeah. I always felt that the ` got a shit deal, so consider this equal opportunity. Plus, my laptop keyboard is so small in relation to my hands that my smallest finger is practically at the top corner of the board anyway.
Just to keep you guys honest, IMO this person wrote 12 pages of stuff that explains the state of QM back in 1980 and how it got there, without using a single equation (there are a few diagrams and tables, though). So, after reading it again (it's been a while), I now believe that he is referring to the W bosons after all: the consequences of "imposing" an electric charge on a photon are that it gains mass, and couples to the EM field differently than an "uncharged" massless photon. He explains how the Higgs "mechanism" (the field coupling) rescued the "pure" Yang-Mills theory, well, sort of.
Another way to say it is the Higgs mechanism breaks the massless symmetry (purity) of the Yang-Mills fields. Or mass is an asymmetry.
This is just not right. If the gauge bosons get mass, then the symmetry is spontaneously broken. This is a very important point, and is not a matter of symantics!
OK, maybe it's "safer" to say the mass of the IV bosons is because of the symmetry-breaking. The asymmetry explains how mass exists (because of an interaction with a field), but not how different particles end up with different masses, i.e. why massive bosons don't all have the same mass, or "coupling energy".
Just a quick question. I am accustomed to thinking of \(\Lambda\) as an element of the Lorentz group. (EDIT: Or rather, I should say, that this symbol is usually(?) reserved for that purpose). Is that what you mean here?
Oh shit, wrong again! OK, so I have a lot more work to do here. Talk among yourselves while I read a bit. I may be gone for some time........
Prefix: a casual reader of this thread might get the idea that "gauge", associated as it is with the words "choice, freedom, invariance, field, group," and so on, means some kind of measurement, and that's what it is, really, a reference point, or a standard "width". But gauge theories are about symmetry, specifically global and local symmetry. So, for another pot at this, and bearing in mind that a certain physicist believes the term should be "phase angle", not "length scale", my pick is that a gauge theory is a theory of global/local symmetry of phase, or phase invariance. A gauge choice is a choice of (phase) symmetry as a local symmetry, and a gauge group is a symmetry group, of rotations. And now, back to normal transmission... Now that asymmetric coupling to the Higgs has been more or less established, can we get back to the symmetries: C, P and T? Or is that CPT? Haven't seen parallel transport mentioned yet (or Berry phase), which must have T in it, at least. Can someone say a few pithy words about fibre bundles, and path lifting, for the benefit of the ignerrant masses? P.S. My high-school version of C, P, and T is analogistic: If two teams are playing a game of rugby, and they switch to opposite ends (like in the second half), that's parity (P), or a "left-handed v. right-handed" mirror-symmetry; charge is analogous to the teams swapping jerseys (team colours), which preserves the "game" - there are still two opposing teams; and T is playing the game forwards or backwards in time, a game played "backwards" is still a game. The "gameness" of rugby is preserved by any transform {C,P,T}.
Who said that a gauge is a length scale? I don't think that Ben or I have said that. Infact, it's clear from things like creating a U(1) local gauge invariant theory shows that it's a phase, not a scale. A theory which is scale invariant is a 'conformal theory', because it's couplings do not run when you change the energy (and energy and lengths are related in relativistic models). Theories with \(\beta=0\) are conformal since \(\beta\) defines the coefficents in the renormalisation group flow equations. C, P and T are different from CPT. CPT is the application of all 3 discrete symmetries. But then pretty much every classical system satisfies the discrete symmetry invariance. It's the naive concept that they should be satisfied on a quantum scale which lead to such suprises about violations found in the weak force etc which explain baryogenesis. But how useful would it be? Without a point of reference for discussion, me churning out how to go from a representation of a vector bundle to it's equivalent principle bundle or how to use a connection to seperate the tangent space of a bundle into it's fibre's tangent space and the uplifted path section from the base manifold would be pointless. Are you familiar with Berry phases? You mention high school. Are you in high school or university? There's no universal 'reservedness' for symbols, it depends entirely on the context. A mathematician will use 'c' as the additive constant in integration but a physicist doing relativity wouldn't do that in an integral involving quantities relating to the speed of light, due to confusion. All symbols are free unless traditionally used to mean something specific in the area in question. Relativists don't use c and \(\Lambda\) as general variables. Quantum field theoriests don't use \(\gamma\) as a free variable.
