Theory of Renormalization

You might still be bothered by many things in the previous section. When we renormalize physical quantities such as charge and mass, you might be thinking that these quantities are observable and are not infinte. So how can you get away with making them divergent and then ignoring it?! The answer to that question is actually deeper than it first seems.

First of all, there is a flaw to the skeptic's argument that the electron is not infinitely massive or carries infinite charge. In fact, according to QFT, it does! The reason we don't see it is subtle but beautiful. If the electron has infinite charge, then it has an infinite amount of energy from the electromagnetic field. This energy manifests itself by the uncertainty principle which says that the field is allowed to create and destroy particles in very short times; such particles are called ``virtual particles''. With this huge amount of energy, the field is able to produce many particles with charge all around the electron. But because these virtual particles are charged, they line up with the field and dampen the strength, analogously to dielectrics in classical electrodynamics. Hense as you go further away from the electron, its effective charge becomes weaker due to this dielectric effect, thus lowering the charge of the electron to the values we measure.

Ah, but in that case, shouldn't the electron's charge get larger and larger as we get closer and closer to it, cutting through this quantum dielectric? The answer is yes, and perhaps even more amazingly, this is precisely what happens! In the everyday world, we measure $\alpha=\frac{1}{137}$ , but at high-energy accelerators such as the Tevatron at Fermilab, we measure $\alpha=\frac{1}{128}$ - this is a real effect[5].

In the past, this effect has been calculated directly by deriving the ``Uehling Potential'' which is the quantum correction to the Coulomb potential. However, in the past thirty years or so, physicists have developed a much more powerful technique for describing these results in a beautifully elegant and intuitive way. This technique was pioneered by physicists K. Wilson, M. Fisher, L. Kadanoff and others. The technique is called the Renormalization group.

The renormalization group is a very complicated object, but I will just say a few things about it. In general, the idea is to do everything that we have been doing, only now our RC will generally depend on the scale of our experiment; call it $\mu$ . This scale is referred to as a subtraction point; its value depends on the scale of the experiment as well as the subtraction scheme. Then we can ask: ``How do our physical parameters depend on our subtraction point?'' We can write down a set of differential equations, called the ``renormalization group equations'' that try to answer how our couplings and masses evolve with changing subtraction point. This gives us the results I said above.

It is this technique that has led to the discovery of asymtotic freedom, the key quality of QCD, where the forces get weaker as the subtraction point increases. This allows for a perturbative analysis of QCD at high energies, when perturbation theory fails at low energies[4]. In addition, the renormalization group has helped to solve a number of questions in statistical mechanics, such as the behavior of magnets, liquid crystals and general phase transitions, just to name a few[6].