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Oct312010

Renormalization

You are about to read about one of the most profound and beautiful principles in all of physics. It is simple, yet fundamentally important. It was formulated in the 1970's by physicist Kenneth Wilson. Wilson later on mentioned that he had thought about the problem for more than 10 years - during this period, he did not publish a single paper… the subject had consumed him. And in one year, he finally released several papers announcing his discovery and was immediately awarded the Nobel prize for the work…

Imagine you are to study a physical system that is rather complex, with many many constituents. For example, a gas in a box with numerous molecules bouncing around. You also happen to have several pairs of ACME magic eyeglasses that can make your vision immensely more powerful - each one progressively more powerful than the previous - allowing you to probe the gas in more detail. You first stare at the box of this uninteresting gas with your own limited vision and perhaps describe it with a handful of measurements: the strength by which the molecules are bouncing off each other, the mass of the molecules, and maybe a few others - all usually determined indirectly by measuring things like temperature, pressure, and density. You then write some equations that describe the system and use this new theory to make predictions. This is presumably a crude description because you could not see much details about the underlying dynamics. But maybe it's good enough?

Now, imagine you put on the progressively more powerful pairs of magic eyeglasses: you now can see more of the detailed interactions of the molecules and make new measurements accordingly. Does your previous theory of the gas change? There are two possibilities: your theory is "renormalizable", or it ain't. If the theory is renormalizable, you can then use your original equations to describe the more detailed physics you are now aware of; the only change you need to do is to tweak the handful of parameters that you originally had in your theory: the strength by which the molecules are bouncing off each other, the mass of the molecules, etcetera. The equations you wrote originally are otherwise unchanged. If you think about this for a moment, that is simply amazing! That means the details of the physical system are all tucked into a handful of parameters only… 

The other possibly is that your original theory is NOT renormalizable. This means that - as you probe more details of the system - you discover you need to modify your equations, perhaps add to them new physical parameters that you didn't need to previously to make good predictions. As you use more powerful eyeglasses, you may find that you always have to add new modifications to your theory ad infinitum - a plethora of new parameters depending on how accurate you want to describe the system. Still, for a desired level of accuracy, you can still write a bunch of equations and do physics. But this theory now is said to be "effective" - good enough to a certain precision but not beyond. This implies that you really do not understand the fundamental physics underlying the physical system; you just approximate things the best that you can. In contrast, a renormalizable theory can in principle be the ultimate most fundamental description of the physics at hand - an exact theory. 

Every physics theory in the world falls in one of these two categories (to be careful, it's actually three categories - with the addition of a possibility called superrenormalizability, but this is mostly inconsequential for our discussion). Interestingly enough, of the four known forces of Nature - electromagnetism, the weak force, the nuclear (strong) force, and gravitation - three happen to be renormalizable! All except the gravitational force… Thus, in reality, the accepted description of gravity - General Relativity - cannot be fundamental… From Wilson's amazing and general ideas, we know with certainty that we really do not fundamentally understand the oldest force law around us! We also know that our description of gravity will surely fail when we come across a pair of eyeglasses that can see down to 10 to the power minus 33 centimeters… And we know that - until then - we should be ok with using General Relativity, unless some other unforeseen mechanism (like extra dimensions… see previous post) kicks in.

The accompanying graph shows how the three parameters that quantify the strengths of the three renormalizable forces - electromagnetism, the weak force, and the nuclear force - change as we probe smaller distances (with more powerful "eyeglasses"). Higher energy on the horizontal axes corresponds to higher detail. Otherwise, the force laws do not change structural form! Adding supersymmetry to the mix (see previous post), one finds that, at some small distance of about 10 to the power minus 29 centimeters- all parameters unite in strength as shown… this is the notion of grand unification (see previous post for more) - that all three forces are different manifestations of the same force law!

 

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