I’ve had better weeks. For now, let’s just assume that the highlight of the last week has been my discovery that our downstairs toilet leaked throughout the entire night…My week has been that awesome.
I feel like I should be talking about physics, this is a physics blog, so I’ll tell you about something I learned on Tuesday.
Spontaneous Symmetry Breaking
As is usually the case with most complex ideas in physics, the right explanation makes the problem very simple and easy to understand. So let’s start off with the simplest spontaneous symmetry breaking (SSB), the breaking of a discrete Symmetry. For example, say you are a nice fuzzy kitten who likes walking back and forth along a one dimensional line. Our kitten’s energy is going to based solely on how far the kitten is from the tick mark that reads zero along this line. Therefore it doesn’t matter if the kitten is at x=-5 or x=5…the kitten is still the same distance away from the center. This kind of function is called an “even” function which means we could replace “x” with “-x” in our equations and nothing would change. Below I’m going to plot two examples of even functions that might explain our kitty’s energy.
Both of the Potentials above have this same discrete “symmetry”. So what’s the problem? Well here comes the physics folks, Kitty doesn’t like having a high energy and tries to minimize it. So in the case of the simple blue potential kitty just sits quietly at x=0 and everybody is happy. However in the purple potential the kitty gets stuck in one of those two low wells. Now as I often like to say, physics is a science of approximations. So if I were to go up to the kitty and ask it what it sees its explanation might be very surprising. You see the kitty assumes that wherever it finally decides to sit down is the minimum and that it thinks the potential is very different than it actually is. The Kitty is only able to see a small portion of the potential…just the point around where its sitting. In terms of the math, the kitty Taylor expands (or approximates) what the potential is by looking at the potential lines around it and their slopes and how fast their slopes are changing and so on. Well in the end the kitty tries to write down an equation for what it sees using a new x-coordinate, sigma, and it gets something like this:
Uh oh. That can’t be write can it? We started this problem with a potential energy curve that obeyed the symmetry x -> -x. Kitty wrote down something with a sigma^3, or cubic, term that does not obey that symmetry at all. Low and behold kitty unknowingly broke our symmetry. bad kitty, bad.
(Edit: The astute reader will note the sigma term which I forgot to mention. The astuter reader will note that its coefficient is automatically zero considering we are expanding around a minimum.)
So what does this have to do with physics? Well this was just a simple example but this mechanism would have gotten you a nobel prize if you had thought of this a long time ago. It turns out this Spontaneous Symmetry Breaking accounts for everything from the Higgs Mechanism to the Meissner Effect in Superconductors…instead of discrete symmetries however we’d have to start talking about Local Non-Abelian Gauge Symmetries, which are a tad more complicated.
Phew, well I hope that was inteligable, if not I apologize. At the moment I’m pretty down about stuff, but I’m still hoping things will eventually turn around.
My friend showed me this AMAZING compilation cd that has practically every artist I listen to on it, from Feist and Andrew Bird to Spoon and Sufjan Stevens. It’s called Dark Was the Night. Speaking of which I still haven’t gotten a hold of the new Andrew Bird CD, Noble Beast…it took an extremely long time for his last CD to grow on me, but recently its been getting a lot of play time.
Also you all should probably watch this video, as I can only assume this girl invented lolcats.
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