# Why is the $dx$ right next to the integral sign in QFT literature?

I've noticed that in QFT literature, integrals are usually written as $\int \!dx ~f(x)$ instead of $\int f(x) dx$. Why?

IMHO, the notation $\int_a^b\mathrm{d}x\,f(x)$ is much cleaner than $\int_a^b f(x)\,\mathrm{d}x$, because the integration variable ($x$) and its associated integral range $(\int_a^b$) are kept together. This is particularly important in lengthy and multi-dimensional integrals. Consider $$\Upsilon_{pq}(k)= \int_0^\infty\mathrm{d}x \int_0^{\beta(x)}\mathrm{d}y \int_0^1\mathrm{d}t \int_{-\infty}^0\mathrm{d}\eta \int_{-\infty}^\infty\mathrm{d}\kappa\;\; f(x,y,t)\;\mathrm{e}^{i[k_xx+k_yy]-\eta t}\;\Theta_{pq}(\kappa,\eta k)$$ as opposed to $$\Upsilon_{pq}(k)= \int_0^\infty \int_0^{\beta(x)} \int_0^1 \int_{-\infty}^0 \int_{-\infty}^\infty\; f(x,y,t)\;\mathrm{e}^{i[k_xx+k_yy]-\eta t}\;\Theta_{pq}(\kappa,\eta k)\;\; \mathrm{d}\kappa\, \mathrm{d}\eta\, \mathrm{d}t\, \mathrm{d}y\, \mathrm{d}x.$$

• Indeed particularly the second form is ambiguous as we could think the int...dy thing as a parenthesised expression or not, and then becomes unclear if the dx applies to the first or last integration sign. – arivero Aug 15 '15 at 0:52

It's not just QFT literature. Physicists, especially adult research physicists, find this notation sensible and popular – even though it may be more popular among particle physicists than elsewhere.

Formally, $dx\,f(x)$ is a product of two factors and $\int$ is a form of a sum. Because product is commutative, it doesn't hurt when the order is interchanged. The advantage of the $\int dx\,f(x)$ notation is that we immediately learn what we integrate over, what is the integration variable. Also, we're sure that we won't forget about the "simple thing", $dx$, at the end. The integrand may be pretty complicated in particle physics – but also elsewhere.

If this convention has something specific to do with particle physics, it's a cultural coincidence. It may be adopted – and it may be criticized – in any discipline of mathematics or natural science that uses integration.

• You mean child research physicists don't use this notation? – Craig Feinstein Aug 14 '15 at 21:46
• Yup, among other things. They usually don't - it's thanks to the way how they get to the expertise. One simple has to be careful about the units at the beginning, not to get completely lost, and use some units close enough to everyday life. At least that's what the school systems and textbooks on more basic physics disciplines assume. That's why I am convinced that this slogan - which I have from an instructor in my undergraduate studies - is correct. Experienced physicists are much more likely to pick natural units or this non-basic-textbook ordering of integrals and other things. – Luboš Motl Aug 15 '15 at 5:51

Besides the reasons listed in Lubos Motl's answer, here is another reason for the $\int \!dx ~f(x)$ notation: By writing the integral sign $\int_a^b$ and $dx$ next to each other in multiple nested integrations, it becomes more easy to trace which limits belong to which integration. This becomes particularly handy when changing the orders of integration.

Secondly, let us mention Grassmann-odd Berezin integration. In that case $d\theta$ carries odd Grassmann parity itself, so moving its position in the integrand may induce a sign factor, so the issue is no longer purely notational and innocent. Unfortunately, there exist different sign conventions in the literature, so seeing the position of $d\theta$ in the integrand is no guarantee for the absolute sign convention of the integral. Berezin choose a sign convention such that integration $$\int \!d\theta~f(\theta)~=~\frac{\partial_Lf(\theta)}{\partial \theta}$$ and differentiation from left is the same operation.