renormalization made easy , baez

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enormalization

enormalization Made Easy

ohn Baez

ugust 14, 1999

want to explain to you the basic idea of renormalization in quantum field theory, and I want to say a

out how it's related to statistical mechanics. You won't get the mathematical details here - for that,

mething like Peskin and Schroeder's book, "An Introduction to Quantum Field Theory". Instead, I j

ant to give you a gut-level understanding of this stuff.

m gonna assume you vaguely know what a Lagrangian for a quantum field theory looks like, and th

ou know how different terms in the Lagrangian correspond to different sorts of particle interactions

hich can be drawn as "vertices" of Feynman diagrams. But you won't need to understand this stuff i

y detail - if you've read Feynman's cute little popular book called "QED: Strange Theory of Matterd Light", you're probably ready for what I'm about to say.

: The Game Called "Renormalization"

kay, let's see.... let's consider a quantum field theory whose Lagrangian has a few free parameters -

asses and charges and so. Just to sound cool, let's call all of these numbers "coupling constants". N

get finite answers from this theory, we need to impose a "frequency cutoff". We do this by simply

noring all waves in our fields that have a a frequency higher than some fixed value. This works bes

ter we replace "t" by "it" everywhere in our equations, so let's do that - this is called a "Wick rotatio

y the experts. Now we're working with a theory on Euclidean spacetime, and the frequency cutoff c

so be thought of as a distance cutoff. In other words, it amounts to ignoring effects that involve fiel

rying on distance scales shorter than some distance D.

what follows, you have to keep your eye on the parameters in the theory: I'm gonna keep shuffling

em around, so to check that I'm not conning you, you have to make sure there's always the same

umber of 'em around - sort of like watching a magician playing a shell game. So make sure you see

hat we're starting with! Our Lagrangian has some numbers in it called "coupling constants", but our

eory really has one more parameter: the cutoff scale D.

ow our Lagrangian has some coupling constants in it, but it's hard to measure these directly. Even

ough they have names like "mass", "charge" and so on, these parameters aren't what you *directly*

easure by colliding particles in an accelerator. In fact, if you try to measure the charge of the electr

ay) by smashing two electrons into each other in an accelerator, seeing how much they repel each

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electrodynamics and other quantum field theories appropriate to elementary particle

physics, the cutoff would have to be associated with some fundamental graininess of

spacetime, perhaps the result of quantum fluctuations in gravity. We discuss some

speculations on the nature of this cutoff in the Epilogue. But whatever this scale is, it lies

far beyond the reach of present-day experiments. Wilson's arguments show that this this

circumstance *explains* the renormalizability of quantum electrodynamics and other

quantum field theories of particle interactions. Whatever the Lagrangian of quantum

electrodynamics was at the fundamental scale, as long as its couplings are sufficientlyweak, it must be described at the energies of our experiments by a renormalizable

effective Lagrangian.

: Ultraviolet and Infrared Fixed Points

the last section I described the "renormalization group" game. Now I want to explain "ultraviolet a

frared fixed points" of the renormalization group, but first let me summarize what I already said. W

ve a quantum field theory described by a Lagrangian with a bunch of terms multipled by numbers

lled "bare" coupling constants - we call the list of all of them C. We ignore effects going on at leng

ales smaller than some distance D called the "cutoff". And now we can compute stuff....

particular, we can compute the so-called "physical" coupling constants C' as measured at any give

ngth scale D'. And we can watch how C' changes as we slowly crank D' up. This is called the

enormalization group flow".

arious things can happen. I already said a bit about this: I said that for nonrenormalizable terms in t

agrangian, the physical coupling constants shrink as we increase D'.

fact we can say more: they scale roughly like D' to some negative power. If you're smart, you can

en guess what this power is by staring at the term in question and doing some dimensional analysis

sing Planck's constant and the speed of light you can express all units in terms of length. If a particu

re coupling constant c in front of some term in the Lagrangian has dimensions of length to the pow

then the corresponding physical constant c' will scale roughly like D' to the power -d. More precise

c ~ (D'/D)^{-d}

particular, this term will be nonrenormalizable if d is greater than zero. (For a more thorough

planation of this criterion, and also some loopholes, click here.)

f course, another way to put this is that for nonrenormalizable theories, the physical coupling const

grow* as we *decrease* D'. This is another way to see why nonrenormalizable theories are "bad" -

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rite down some Lagrangian and start playing the renormalization group game to see what happens a

e zoom out.

ou may be suspicious here: how are we ever going to guess which Lagrangian corresponds to our

iginal problem involving a crystal of iron? After all, iron is complicated stuff!

uckily, it's not so bad. At short distance scales, to get a decent approximation to our original proble

e may need to start with a really complicated Lagrangian. However, suppose we do this. Then as wom out to large distance scales, the renormalization group game says that the Lagrangian will

mplify. For example, we've already seen that nonrenormalizable terms in the Lagrangian become

rrelevant" as we go to large distance scales: the physical coupling constants in front of them go to z

ore generally, we shouldn't be at all surprised if our physical coupling constants approach an infrar

xed point as we zoom out, letting the distance scale approach infinity. This is exactly what infrared

xed points are all about! Even better, all sorts of theories with different bare coupling constants can

proach the same infrared fixed point. We say two different theories, or two different physical syste

e in the same "universality class" if they approach the same infrared fixed point as we crank up thestance scale.

or example, when we're studying what happens at the Curie temperature, lots of different ferromagn

e in the same universality class. Indeed, it turns out that you can study a lot of them using slight

riations of one of the simplest quantum field theories of all: the phi^4 theory.

here is a lot more to say, and I'm too tired to say most of it, but there's one thing I *must* tell you, j

wrap up some loose ends. Wilson's real triumph was to calculate critical exponents like the numbe

the power law for the 2-point function:

(x)s(y)> ~ 1/|x-y|^d

ow did he do it? Well, Landau already had one way to do this, which gives just the results you wou

uess using dimensional analysis. But that method didn't always give the right answers. To get the rig

swers, it helps to realize that n-point functions are closely related to physical coupling constants. In

ct, while I never actually *defined* the physical coupling constants, they are really just a way of

tracting some information about n-point functions. So if we calculate the "running of coupling

nstants" using the renormalization group game, we can work out the critical exponents.

o actually do this, it helps to use something called the "Callan-Symanzik equation", but I'm not goin

explain this - for this, you should probably read a book on quantum field theory. But don't worry

out this too much; be happy if you feel you get my main point here, which is that 2ND-ORDER

HASE TRANSITIONS CORRESPOND TO INFRARED FIXED POINTS OF THE

ENORMALIZATION GROUP!



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renormalization made easy , baez

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