What exactly is regularization in QFT?

Question

The question.

Does there exist a mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory? If so, what is it?

Motivation and background.

As pointed out by user drake in his nice answer to my previous question Regulator-scheme-independence in QFT , it is often said that in a renormalizable quantum field theory, results for physical quantities (such as scattering amplitudes) when written in terms of other physical quantities (like physical masses, physical couplings, etc.) do not depend on the regularization procedure one chooses to use. In fact, user drake takes this desirable property as part of the definition of the term "renormalizable."

In my mind, in order for such a statement or definition to be meaningful and useful, it helps to have a precise, mathematically explicit notion of what constitutes a regularization procedure. That way, when one wants to compute something physical, one can use any procedure one wishes provided it satisfies some general properties.

My current understanding in a (small) nutshell.

When we perturbatively compute, say, correlation functions for some theory pre-regularization/renormalization, we obtain formal power series' in the bare parameters that characterize the theory. Such power series' contain expressions for loop integrals that generically diverge, often due to high-momentum (UV) effects, so we "regularize", namely we implement some procedure by which these integrals are made to depend on some parameter, call it $\Lambda$, and are rendered finite provided $\Lambda$ doesn't take on a certain limiting value $\Lambda_*$ (that could be $\infty$) corresponding to the physical regime (like the UV) that led to the original divergence. We then renormalize and find (in renormalizable theories) that $\Lambda$ drops out of physical results.

What sort of an answer am I looking for?

I am looking for something like this.

A regularization procedure is a prescription by which all divergent integrals encountered in perturbation theory are made to depend on a parameter $\Lambda$ and which satisfies the following properties: (1) All divergent integrals are rendered finite for all but a certain value of $\Lambda$. (2)...

I know that there are other properties, but I don't know what constitutes a sufficiently complete list of such properties such that if you were to show me some procedure, I could say "oh yes, that counts as a valid regularization procedure, good job!" Surely, for example, the putative regularization procedure cannot be too destructive; I cannot, for instance, simply replace every loop integral with $3\Lambda$ and call it a day because that would completely destroy all information about how many loops the corresponding diagrams contained. How much of the "formal structure" of the integrals does the procedure need to preserve?

As far as I can tell, there is no discussion of this in any of the standard QFT texts who simply adopt tried-and-true procedures like a hard UV cutoff, dim reg, Pauli-Villars, etc. without commenting on general conditions sufficient to guarantee that these particular procedures count as "good" ones. There is, of course, a lot of discussion of whether certain regulators preserve certain symmetries, but that's a distinct issue.

Edit. (January 8, 2014)

Discussions with fellow graduate students have led me to believe that the appropriate definition proceeds by appealing to the idea of effective field theory. In particular, if we view our theory as an effective low-energy description of some more complete theory that works at higher energy scales, then imposing a high momentum cutoff has a conceptually privileged position among regulators; it is the natural way of encoding the idea that the putative theory only works below a certain scale.

This can then be used to define a regularization procedure that, in some sense, reproduces the same structure encoded in regularizing with a cutoff. Unfortunately, I'm still not entirely sure if this is the correct way to think about this, and I'm also not sure how to formalize the notion of preserving the structure the comes out of cutoff regularization. My inclination is that the most important structure to preserve is the singular behavior of regularized integrals as the cutoff $\Lambda$ is taken to infinity.

Answer 1

The definite answer to your question is: There is no mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory.

Instead, there are various regularization schemes with their advantages and disatvantages.

Maybe you'll find Chapter B5: Divergences and renormalization of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html illuminating. There I try to abstract the common features and explain in general terms what is needed to make renormalization work. The general belief is that the details of the regularization scheme don't matter, though in fact it is known that sometimes some regularization schemes give apparently incorrect results.

This is to be expected since the unregularized theory is ill-defined, and can be made well-defined in different ways, just as a divergent infinite series can be given infinitely many different meanings depending how you group the terms to sum them up.

If at any time in the future there will be a positive answer to your question, it will be most likely only when someone found a logically sound nonperturbative definition of the class of renormalizable quantum field theories.

On the other hand, if you want to have a mathematically rigorous treatment of some particular regularization schemes for some particular theories, you should read the books by (i) Salmhofer, Renormalization: an introduction, Springer 1999, and (ii) Scharf, Finite quantum electrodynamics: the causal approach, Springer 1995.

Answer 2

I'm going to give a silly answer, but I think this is the best we can do. A regulator is any prescription for defining the path integral such that after adding a sum of local counterterms to the action and allowing the physical couplings to depend on the renormalization scale $\mu$, the correlation functions are equal to those obtained by taking a continuum limit of a lattice theory. This is similar in motivation to your most recent edit, except that we really need to use a lattice instead of a Euclidean momentum space cutoff, because the latter breaks gauge invariance.

I'm pessimistic about the existence of a better answer, just because some regulators satisfy some nice properties (i.e. unitarity, symmetries, etc.), while others don't.

Answer 3

As you said regularization is necessary to even begin making sense of the diagrams which appear in perturbation theory. The word "perturbation" already contains a hint towards the answer you may be looking for. If you are trying to make sense of the interacting theory $d\nu$ by perturbation, this implies that there is somebody being perturbed, namely a free theory $d\mu$. I think regularization is a feature of the pair $d\nu,d\mu$ rather than $d\nu$ alone. For instance take the functional measure $d\mu$ corresponding to the usual free field $$ d\mu(\phi)=\frac{1}{Z}e^{-\frac{1}{2}\int \{(\nabla\phi)^2+m^2\phi^2\}} D\phi $$ and let $d\nu(\phi)=\frac{1}{Z'}e^{-V(\phi)}d\mu(\phi)$ where $V$ is your favorite interaction potential, e.g., $$ V(\phi)=g\int \phi^4\ . $$ Let $\rho(x)$ be a mollifier (not sure how one should spell that), i.e., a smooth function with rapid decay or even compact support around the origin with $\int \rho=1$. Let $\rho_r(x)=2^{-dr}\rho(2^{-r} x)$ where $d$ is the dimension (of Euclidean space-time) and $r\rightarrow -\infty$ is the UV cut-off. Then if one denotes by $d\mu_r$ the law of $\rho_r\ast\phi$ with $\phi$ sampled according to the free probability measure $d\mu$, one has the weak convergence of $d\mu_r$ to $d\mu$ when $r\rightarrow-\infty$. In that sense, the modification we are doing is a sensible regularization of the measure $d\mu$. Now on closer inspection $e^{-V(\phi)}d\mu(\phi)$ does not make any sense. However $e^{-V(\phi)}d\mu_r(\phi)$ is perfectly fine. So the game becomes trying to obtain $d\nu$ as the limit of measures $d\nu_r(\phi)=\frac{1}{Z_r}e^{-V_r(\phi)}d\mu_r(\phi)$ when $r\rightarrow -\infty$. Notice that I changed $V$ into $V_r$ because renormalization theory tells us we have to let couplings like $g$ etc. depend on the cut-off $r$.

If one succeeds in this construction of the interacting theory $d\nu$, independence of the regularization can be understood as showing that the result is independent of the choice of the mollifier $\rho$ (heat kernel reularization or Pauli-Villars are particular cases).

In the related context of singular SPDEs, recent progress was made by Hairer with his theory of regularity structures (see also this pedagogical review). An important feature of this theory is that it allows one to prove similar statements about independence with respect to the choice of regularization procedure. In the article "A Second-Quantized Kolmogorov-Chentsov Theorem" I also give a general procedure for forming products of random Schwartz distributions thanks to Wilson's operator product expansion. There too one needs to introduce a regularization and then control its removal. Moreover, the end result is independent of the choice of regularization.

Note that there is no reason to restrict the above considerations to perturbed theories $d\mu$ which are Gaussian. In some sense, conformal perturbation theory is about similar perturbations around other nontrivial QFTs.

Of course, in Fourier space, the propagator for $d\mu_r$ is $$ \frac{|\widehat{\rho}(2^r p)|^2}{p^2+m^2} $$ but I prefer the definition in $x$ space as a convolution, since this amounts to taking local averages and is a continuous analogue of Kadanoff's block-spinning procedure.

BTW, the notion of regularization is not the exclusive property of QFT, it is used a lot in the mathematical theory of distribution as a tool for proving various theorems and dealing with nonlinear operations. In the French literature on the subject by Laurent Schwartz and many others the catchphrase is "troncature et regularisation", i.e., the use of IR and UV cutoffs respectively.

Answer 4

Regularization is a rewriting of your integral so that you can deal its divergences using other tricks.

For example in QFT you calculate some amplitude to a certain order in perturbation theory. The integrals that represent loop diagrams diverge. The most common regularization procedure is called dimensional regularization where you parametrize the dimension of your loop integral to, for example, d=4-c.

It turns out that your integral was divergent for large momentum, therefore having a branch cut. Now after dimensional regularization the branch cut divergence became a simple pole when c=0. As you know, it is easier to deal with a simple pole than a branch cut.

After making your integral more manageable you can renormalize and then separate the divergent and finite parts of your result. You eventually use other tricks to remove the divergent part, for example add counterterms to your lagrangian.

Answer 5

You have a theory, then you calculate some physical quantity, then you get some divergence . The idea of "regularization " is to define another theory depend on some parameter, when let the parameter send to some value ( say d->4, Λ->infinity, lattice length to zero...), That another theory will tend to original theory. This procedure will guarantee you get a finite result, yet when the parameter send to some value, recover the original theory.

The following step is "renormalization". We use the regularized theory to calculate some basic physical quantity, such as mass, scattering amplitude. All other physical quantity can be expressed by this basic quantity, the miracle of "renormalizable theory "is these expression is irrelevant of the parameter.