Dimensional Regularizaiton (dimensions d)

[ChrisVer]

In general someone works with d dimensions, and at some point makes an expansion around d=4, by writing d \rightarrow 4,
This d \rightarrow 4 (or \epsilon = \frac{4-d}{2} \rightarrow 0 in most textbooks) is really confusing me in the dimensional regularization... I don't understand how you can "approach" the dimensions to 4, since they can either be 4 or 5 or 3 or some other natural number... eg 4.1 or 4.001 dimensions doesn't make sense, and so do powers of \epsilon. \epsilon can be either 0 or at next best jump, 1/2. However I haven't found any textbook that goes through that in a comprehensive way (if there is any).

Is it just a mathematical abuse?

 

[Orodruin]

I would not say that it is really writing the integral in a non-integer number of dimensions as much as finding a function which is formally equal to the desired integral when the parameter $d$ takes the value 4 or more generally the desired integral in n dimensions when d=n. If the integral is formally divergent, you can study the Laurent series around d=4. Writing it as $d^dp$ is definitely an abuse of notation.

Is it just a mathematical abuse?

Isn't everything we do mathematical abuse?

 

[ChrisVer]

So you mean that you are not dealing with the integral itself, generalize the d dimensions to d some real number, and you look at how things work when that number approaches 4, so that you can say that the integrals of the two different things should be "equal" (the one approaches the other). In that case indeed d^d p would look a weird notation, but I'll go with your last rhetorical question.
Sounds legit, thanks!

 

[George Jones]

I like the way Folland explains this, so I have attached a scanned page from "Quantum Field Theory: A Tourist Guide for Mathematicians" by Gerald Folland,

https://www.amazon.com/dp/0821847058/?tag=pfamazon01-20

Folland goes on to make some general comments on dimensional regularization, and then he does an explicit calculation for one-loop $\phi^4$ theory.

Although Folland doesn't cover as much as physics texts such as Schwartz or Peskin and Schroeder, Folland does cover a lot more physics than most rigourous math books on quantum field theory. Folland uses mathematical rigour where possible, and where physicists' quantum field theory calculations have yet to be made mathematically rigourous, Folland states the mathematical difficulties, and then works through the physicists' calculations.

I've been waiting many years for someone to write this book, but now, unfortunately, I don't have time to read it.

 

[vanhees71]

Dimensional regularization is not very intuitive from a physics point of view, but it is a regularization scheme, which for almost all practically relevant theories keeps the fundamental symmetries as Poincare invariance and (most importantly) gauge invariance valid at any step of the calculation. This is so important and makes the calculations so much more transparent that it became the key technique for modern renormalization theory, leading to the proof of the renormalizability of non-Abelian gauge theories by 't Hooft (in his PhD work!) and Veltman.

The idea is that integrals at space-time dimensions <4 (>4) are less UV (IR) divergent than in 4 space-time dimensions. The idea thus is to evaluate some standard integrals like the ones given in Appendix C of my notes on QFT as a function of the space-time dimension $d$:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

Then, using the analytical properties of the $\Gamma$ function, one analytically continuates these functions to complex values of $d$ and does a Laurent expansion around $d=4$, which enables one to separate off the divergences occurring in the limit $d \rightarrow 4$, if the original integral is divergent for $d=4$. These divergent pieces can be subtracted and the limit $d \rightarrow 4$ taken for the finite remainder. This defines a (quite unphysical) renormalization scheme, the socalled minimal-subtraction scheme (MS). It has the additional advantage of being a mass-independent renormalization scheme, which is quite important to deal also with IR divergences which occur as soon as massless particles are involved, and this is the case for (un-Higgsed) gauge theories, because the gauge bosons are massless.

A bit obscure is the way, how the most fundamental idea of renormalization enters the game, namely the renormalization scale. So when learning about renormalization one should look not only at the quite abstract dimensional-regularization and MS-renormalization technique but also at other schemes like cutoff, Pauli-Villars, heat-kernel methods or, in my opinion the most transparent one, the BPHZ renormalization scheme, which does not use the intermediate step of a regularization at all but just substracts the renormalization parts on the level of the integrands occurring in the divergent integrals, using clearly defined renormalization conditions defining a renormalization scheme. Here, the renormalization scale enters in the most natural way, namely as scale, at which the parameters of the theory are defined.

 

[ChrisVer]

Here, the renormalization scale enters in the most natural way, namely as scale, at which the parameters of the theory are defined.

Well the fact that the scale is scheme-dependent also seems unphysical to me. As a result of calculations I was able to see that in class, but as a result of physical intuition it confused me more than normal. Eg when I'd say that \Lambda_{QCD} \sim 150~MeV I would have to say in which scheme this holds , cause if I changed the scheme, the scale would also change.

PS- I really liked your text notes as far as I looked at it. If I knew of them 1 semester ago, I would give my QFT2 class a second chance (for examination rather than general knowledge)