Friday, January 15, 2016

soft question - 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?



Answer



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 $$ \Upsilon_{pq}(k)= \int_0^\infty \int_0^{\beta(x)} \int_0^1 \int_{-\infty}^0 \int_{-\infty}^\infty\; f(x,y,t)\;\mathrm{e}^{i[k_xx+k_yy]-\eta t}\;\Theta_{pq}(\kappa,\eta k)\;\; \mathrm{d}\kappa\, \mathrm{d}\eta\, \mathrm{d}t\, \mathrm{d}y\, \mathrm{d}x. $$


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