Understanding Dimensional Regularization

Question

I am diving into the potential minefield that is learning regularization and renormalization, and I am currently lost on dimensional regularization. I understand the intuitive idea using dimension as a complex parameter, and finding open sets where relevant integrals are defined, and then analytically continuing them to the real line, with poles emerging along the integers, explaining various divergences. However, I am not happy with that explanation, since it does not make mathematical sense, so far as I know, to integrate over some "2.5-dimensional" space, except if one wants to talk about fractals and Hausdorff measure integrations.

However, I have found two, or possibly three, more mathematically rigorous explanations, and I am lost on if they are the same explanations, or if they're distinct.

The first is in John Collins' Renormalization: An Introduction. It is done by working over an infinite-dimensional space, and finding relevant finite-dimensional subspaces to integrate over. This makes enough sense to me, especially since some work goes into showing the independence of results from the choice of subspaces.

However, I only found that after first finding Quantum Fields and Strings: A Course for Mathematicians and, in particular, Eqn. (9):

$$I_{D}^{E}(f) = \int_{S^{2} E^{*}} \rho_{D}^{E}(A) f(A) \mathrm{d} A $$

where $S^{2} E^{*}$ is the space of symmetric bilinear forms on a vector space $E$ with dimension $N$, $f \in \mathcal{S}\big( S^{2} E^{*} \big)$ is a complex-valued Schwartz function on $S^{2} E^{*}$, $D$ is such that $\Re(D) > N - 1$, and $\rho_{D}^{E}$ is a function on $S^{2} E^{*}$ defined as:

$$\rho_{D}^{E}(A) = \pi^{N D/2} \Gamma_{N} (D/2)^{-1} (\det(A))^{(D - N - 1)/2}$$

This is given as the definition of the $D$-dimensional integral, followed by Proposition 5 on page 599, reiterated here:

Proposition 5: Let $D \geq 0$ be a nonnegative integer, and $V$ be a vector space of dimension $D$, with a positive definite symmetric bilinear form $\beta$. Then for any $f ∈ \mathcal{S}\big( S^{2} E^{*} \big)$ one has

$$I_{D}^{E}(f) = \int_{\mathrm{Hom}(E, V)} f\big( x^{*}(β)\big) \mathrm{d} x$$

where $x^{*}(β)$ denotes the inverse image of $\beta$ under $x$ (a nonnegative symmetric bilinear form on $E$).

It is not clear to me if this is a more explicit formulation of Collins' basic idea.

However, one additional wrinkle is thrown in when considering Connes-Kreimer Renormalization theory. This post claims that Connes-Kreimer Renormalization theory 'makes mathematical sense out of dimensional regularization." However, looking at the arXiv article, and especially the top of page 25, it only seems to use dimensional regularization as formulated as Collins, not place it on a more foundational footing.

Thus, with all that context out of the way, let me state my questions.

1. Is John Collins' formulation formulation of dimensional regularization what people usually mean by it, even if only implicitly, since most probably don't try to dive to the rigorous heart of the idea.

2. Is the second formulation given in terms of bilinear forms the same formulation, an equivalent one, or something different one entirely?

3. Does Connes-Kreimer Renormalization provide yet another formulation, or does it just use Collins' work?

Answer 1

First of all, in general you just cannot define a QFT in non-integer dimension. Just tell me this: how many elements does the Clifford algebra have in, says, 3.8-dimensional spacetime? That doesn't even begin to make sense. So there is no spinor for you. There is no sensible definition of the Poincare algebra, either, and you can't do any gauge (vector) field. Even the plain old scalar runs into trouble if there is an underlying crystal lattice and... just what is a crystal point group in 2.8 dimensional space, anyway?

The safest point of view is just to regard dim. reg. as a mathematical procedure that allows for regularization of logarithmically divergent integrals. The QFT itself is defined in, says, $d = 4$, and you keep working in the fixed integer dimension until you write down a divergent integral. And only then do you apply dim. reg. to the integral to isolate the divergence.

One can wrap the narrative in fancy mathematical languages, but I just don't see how the non-integer dimension can be incorporated into the basic definition of a QFT beyond just scalar fields in isotropic flat space.

From the physics point of view, we usually assume that the UV completion of a QFT exists, and it gives a meaning to those divergent integrals, but we generally have no idea what that is or how that works. But if you take the difference of two similarly-regulated divergent integrals, the (unknown) detail of the regulator cancels out anyway. That's why we can get away with using any weird trick for regularization, as long as we do it consistently.

-edit-

Response to OP's question in the comment.

Dim. reg. defines a family of integral $I(d)$ in the so-called d-dimensions. Say the integral in question is logarithmically divergent in $d = 4$, and then we expect a finite window, like $2<d<4$, where $I(d)$ is convergent. It doesn't mater how you define dim. reg., it must reproduces the convergent integral. And then the pole of $I(d)$ at $d=4$ is uniquely determined and cannot depend on how dim. reg. is defined, either.

Also, since we don't know anything about the UV theory, regularization is no more than a mathematically trick. All physical results should be independent of the regularization scheme; if they are not, you did something wrong.

In this regard, it matters little how you regularize the divergence. The important thing is what you do with the regularized integral, i.e. how you handle the subtraction.

Take the following example: $$ I(d) = \int d^d k \, \frac{\mu^{4-d}}{k^2 + m^2} $$ where $\mu$ is the renormalization scale. Dim. reg. and MS-bar (modified minimal subtraction) is in fact equivalent to $$ I(d) \rightarrow \mu^{4-d} \int \left(\frac{1}{k^2 + m^2} - \frac{1}{k^2} - \frac{\mu^2}{k^2(k^2-\mu^2)}\right) $$ (Thanks J.G. for pointing out the typo.)

The subtracted $I(d)$ is fully convergent. You can evaluate it however you want; either you get the unique correct answer or you make a mistake. Dim. reg. just allows me to do the integral quickly.

Answer 2

You might find the nlab article on renormalisation interesting. They say:

The construction of a perturbative QFT from a given local Lagrangian density ... involves ambiguities associated with the detailed nature of the quantum processes at point interactions. What is called renormalization is making a choice of fixing these ambiguities to produce a perturbative QFT ...

...the renormalization ambiguities are understood as the freedom of extending the time-ordered product of operator-valued distributions to the interaction points, in the sense of extension of distributions. The notorious “infinities that plague quantum field theory” arise only if this extension is not handled correctly and has to be fixed.