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Ok, so it looks like everything I said in my last post was correct. It's inevitable that I start doubting my memory after I write things, especially when it's about stuff I worked on nearly 5 years ago. I gave http://arxiv.org/abs/hep-ph/9803479 a quick skim and I think it backs up everything I said regarding the sphaleron process. I was not confusing anything with QCD. However, there is one thing I forgot which is that most of the transition probability comes from thermal processes rather than quantum tunneling, the tunneling rate at zero temperature being extremely tiny. This could use some clarification.

When I referred to transitions between different topologically inequivalent vacuum states as "untying a knot without letting go of either end of the string" this may seem like a pretty random and weird analogy. And the fact that it can also untie the knot via thermal fluxuations makes me realize that I may have exaggerated how miraculous it is. Nevertheless, it's still pretty neat. To clarify what's going on, there are several quantum fields that get twisted up in spacetime: the gauge fields and the Higgs field. The gauge fields are the fields responsible for the "fundamental forces of nature", in this case electromagnetism and the weak nuclear force. The Higgs field is the field responsible for the Higgs particle that the LHC, the giant particle accelerator near Geneva, is looking so hard for and that the media often refers to as "the God particle"... it gives the quarks and leptons of the Standard Model their masses. If you imagine spacetime having a spherical boundary, the visual picture of what happens in one of these topologically non-trivial vacuum states is that each of these fields gets "wound up" around the boundary of spacetime.

Imagine winding a rubber-band around your wrist. You can wind it once, or you can stretch it a bit and wind it twice, or really stretch it and wind it 3 times. Each of these 3 is a stable configuration once you get it wound up, it's not going to suddenly slip off or spontaneously transition into a state with a different "winding number". To do so your hand would have to disappear or something. However, if you expend energy, you can stretch it back over your fingers, and then let it snap back to a different winding number (say, going from 3 times back down to 2 times). Stretching it is getting it over an "energy barrier" that separates the different topologically inequivalent configurations. Well, the same thing happens for the gauge fields and the Higgs field, except that they are sort of winding around the whole universe. And unlike a rubber band which is a 1-dimensional object, these fields are 3-dimensional and they are essentially wrapping themselves around a hypersphere at the boundary of spacetime. If you picture a normal ball, that's a 2-sphere. Its surface is 2-dimensional, and if you try to wrap it with something else that's 2-dimensional, you'll find that it's very hard to imagine wrapping it more than once. However, these fields are more intangible than most wrapping paper, and a lot more stretchy. So they can actually wrap around a sphere more than once... every additional time they wrap it that increases the winding number by 1. Except they are actually wrapping a hypersphere that has a 3-dimensional surface rather than a 2-dimensional surface like a ball. So visualizing exactly what's happening becomes even more difficult, although hopefully you have more of an idea now than you got from my last post which was rather vague. Apparently the official name for the number that indicates winding number, in the case of the sphaleron, is the Chern-Simons class. (I mention this because it was something vaelynphi asked about, so I looked it up).

At any rate, the point here is that there are only two ways you can get the fields to unwind. They won't unwind themselves naturally because they would have to expend energy to go into a sort of "stretched" state that they don't like to be in temporarily, before they could relax into the next topological vacuum. So there are only two ways for them to unwind. One is quantum tunneling, which I mentioned. That's the more magically seeming route, where despite the fact that they don't have enough energy to do it, they just sort of do it anyway. It's almost as if your hand temporarily becomes immaterial and the rubber band just passes right through your wrist. The second way, which is a bit more mundane, is that they could get a random energy fluxuation due to the constant background of thermal fluxuations. This could cause them to temporarily become excited and get over the energy barrier and then relax. Like the rubber band suddenly stretching out randomly, slipping over your hand, and then going back onto it in an unwound state. However, this can only happen at very high temperatures. At normal room temperature, the sphaleron process just never happens. But if you go back in time to just moments after the big bang, you'll find temperatures much much hotter than room temperature. Hot enough to randomly cause the fields to unwrap momentarily and then rewrap. The awkward state that they have to pass through between their happy relaxed wrapped states is what's officially called the "sphaleron". So the sphaleron is sort of like the hardest demon they have to battle before getting past to the new world they seek. And yet, the amazing thing is, they must have done this again, and again, and again... to give eventually give us how many atoms we have in the world today. Without the primordial gauge fields doing battle with the sphaleron to cross the energy barrier we would have no protons or neutrons around, and therefore, no atoms!

Also, I mentioned that the sphaleron was a "saddle point" rather than a maximum like most instantons. A maximum is like a hill you have to get over. But for those who don't know what a saddle point is, it's similar to a maximum in one direction but a minimum in another direction. So it's like a hill in one direction that's also a valley because it's between two hills in a perpendicular direction... the whole thing looks kind of like the saddle of a horse, hence the name. So while they were on their epic quest to get through the sphaleron on state into the next topological vacuum, the gauge fields may have said to themselves something like "as I walk through the valley of the shadow of death..." since they were both in a valley and on top of a hill at the same time, depending on which direction you're talking about. And in the case of the sphaleron, there is actually only one direction that's a valley and all the other directions are maximums. (On a 2-dimensional surface like the earth, you can only have two directions so the most you could have is "one of each" at a saddle point in the landscape, but in this case we are talking about more dimensions so there can be more hilly directions). Incidentally, it took me about a month to calculate the contribution of the sphaleron to the instanton action, but I remember using Mathematica a lot to plot out different perspectives of the saddle-shaped point and rotating it around. Ironically, after I submitted the paper that was the one thing they said I should take out, because the referee said it was a well-known calculation that most cosmologists already knew how to do in a different way... so it never made it into the final published version but it is still in the version on arxiv.org.

Also, I forgot to mention that the name for this kind of anomaly in general is a "chiral anomaly"... it shows up in a lot more types of situations than the electroweak sphaleron though. Actually, if you've ever seen the show Sliders, where the four main characters jump from one quantum world to the next in the multiverse, the very first episode has the professor lecturing and the genius kid Quinn in the class listening... then after class, Quinn goes up and says something like "Hey professor Arturo... I read your paper on chiral anomalies, and it was totally brilliant!" however the Hollywood actor (Jerry O Connell) pronounces it wrong and says "Cheeral anomalies"... it's supposed to be pronounced like "kiyral". The word means "handedness" (like lefthanded versus righthanded) although it would take quite a while to explain why that is relevant here. I used to watch the show in high school and never noticed that he pronounced it wrong, and then I thought it was hilarious when I eventually watched it again later in grad school.

Well, my part 3 was supposed to be about the conformal anomaly and how protons and neutrons get mass from dimensional transmutation. But I still haven't made it there. So that will start in part 4. Then, eventually, I'll say some things about the conformal anomaly in string theory, and why that places a constraint on the number of dimensions of spacetime.


( 1 comment — Leave a comment )
Jul. 15th, 2010 02:40 pm (UTC)
::sigh:: All the smart guys always live so far away! Heh.

I'm actually working through some of this mathematically; it's quite enjoyable, though I did a three-page calculation yesterday that I botched halfway by writing down the wrong covariant derivative!

I think I'm experiencing index soup.
( 1 comment — Leave a comment )


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