Thursday, May 1, 2014

Renormalization Group

Dear Diary,

Let's go back to our stick in the ground analogy for coherence lengths.  Recall that we likened a coherence length to the shadow of a stick stuck vertically in the ground.  The coherence length measures how much day or night there is and when we see a phase transition between day and night, the coherence length goes to infinity.

Now let's pretend we're blind, somehow, and all  we see is shadows.  We'd like to measure them because maybe there's a different "y" shaped shadow that looks new.  Why would we like to measure them?  We want to know something about the objects they come from or maybe because we're worried that we're losing our minds and need to make sure the shadows act regularly.  No surprises, sneaky shadows!  I won't let you get the jump on me by measuring you!

The trouble is, how do we measure it?

We need another stick, a measuring stick.  So, let's make one that is exactly one coherence length long.  But at what time of day should we choose the length of the stick?  The answer is that we can choose!  But it's best to always choose the time of day that you want to measure!

So, consider the measuring stick's length that is made at 12:01pm (just after noon) so that it's very short.  It would take a long time to measure a shadow at 5pm because the length is longer!  Say the shadow is many times longer, you'd have to use your ruler many times to see what length it is!  If you'd used your 5pm stick to measure something at 5pm, it would take exactly one measurement to get the same answer!

When measuring something in nature, the answer better not change if you measure it in two different ways.*  Quickly, pick something nearby on your desk or out the window.  If you measure it's length and someone else measures it's length, you better get the same answer...but picking how you chose to measure the object doesn't matter!  Go ahead, ask someone with a lot of time on their hands to measure it with you!

This idea of picking a different ruler for different situations is what is called renormalization group, in the condensed matter sense.  There's another permutation/consequence of renormalization group that is related, but we'll get to that later.

*-There's a notable conceptual hurdle to justify this statement in some places like relativity. Maybe we'll revisit this.

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