consult RG related concepts
lbrits said:
FYI, renormalization ceased to be "mathematically dubious" a few decades ago. Unfortunately, the stigma remains.
Also, unfortunately, FunkyDwarf's reply doesn't seem to be the same renormalization that you are asking about, and is usually just called "normalizing the wavefunction".
A very good explanation of renormalization is given in Zee's quantum field theory book. Renormalization is something that happens when you take the "continuum limit", i.e., when you consider a system with many degrees of freedom and you take the limit where many becomes infinite. This is usually the case in condensed matter physics and quantum field theory.
For instance, let's say you consider the scattering of two particles with nearly zero energy, and you call the amplitude of that process g. When you calculate the scattering amplitude, you end up having to do integrals in momentum space that usually blow up (their integrands grow like some positive power of momentum, for instance). Naively you say that there is something wrong. However, taking a cue from condensed matter physics, you might reason that your theory only gives a good "long distance" or "low energy" description of nature, and that at "short distances" or "high energies" it should be replaced by something else. It turns out that some theories are insensitive to what happens at "high energies", and those theories are called renormalizable. In essence, instead of pretending that your theory is good up to arbitrary high momenta, you say that it is good up to some value \Lambda, where \Lambda is large, and (importantly), unknown. The important feature of renormalizable theories is that you can experimentally measure g at one energy scale, and the theory will tell you how it changes with energy. The actual value of \Lambda doesn't enter your final calculation.
The problem people had with renormalization had to do with getting rid of these seemingly infinite functions of \Lambda in a consistent way. If you believe that your theory is complete, then, yes, you are on shaky grounds. But if you think of it as a low energy effective theory (which was explained by Kenneth Wilson), then you're ok.
I really appreciated your nice explanation.
However, I am also reading about the renormalisation part of QFT recently. There are some problems I can't figure out in renormalisation procedure. That is, the physical prediction and the arbitrariness of the renormalisation procedure. Here are my questions to consult,
(1) The finite parts of the counter terms are actually
arbitrary, to fix these counter terms, we have to use some renormalisation prescription( or scheme). Once we chose one renormalisation prescription, we have definite renormalized parameters, however, these renormalized parameters are in general NOT the physical quantities we measured in laboratory, right? Only if we choose the "physical subtraction" prescription, we get the physical measurable quantities. And, those renormalized parameters in certain momentum-subtraction prescription are
running with an arbitrary mass scale, \mu, right? Even the measurable physical quantities which we obtained from the physical-subtraction would run with the scale.
Is my concept correct?
(2) The reason why we need those many prescriptions is that, choosing different finite parts of counter terms is equivalent to rearrange the perturbation series. So, in some cases, if we chose the renormalisation prescription appropriately, we would get a perturbation theory which has large contributions from first few terms of the perturbation series, and higher order terms are just tiny correction. Is my concept correct?
(3) Even we fixed the renormalisation prescription, we would still have the renormalized parameters depend on the arbitrary mass scale, \mu, and the differential equation which the Green function satisfies with respect to the change in \mu is called the Renormalisation Group Equation (RGE), right?
But, in the case of MS-scheme, the finite parts of the counter terms are all zero, hence the renormalized parameters don't run with scale?
Thanks so much for everyone who discuss with me or correct my concept.