What is the meaning of $\mathrm{d}^4k$ in this integral? From Gerardus 't Hooft's Nobel Lecture, December 8, 1999, he states the following equation (2.1):
$$
\int \mathrm{d}^4k \frac{\operatorname{Pol}(k_{\mu})}{(k^2+m^2)\bigl((k+q)^2+m^2\bigr)}  =  \infty
$$
in relation to weak interactions theory, where $\operatorname{Pol}(k_{\mu})$ stands for some polynomial in the integration variables $k_{\mu}$, and then goes on to say that physically it must be a nonsense.
Why is it a nonsense?
What sort of integral is this, and how does one interpret it?
Is the $\mathrm{d}^4k$ shorthand for 4th degree integration?
At what stage and subject of a physics course does one learn about it (A pre-fresher is asking)?
 A: The equation is a term in the calculation of a scattering probability. Obviously a scattering probability must be between zero and one, like any other type of probability. So when the calculation of a scattering probability returns a value of $\infty$ that isn't physically possible, and it shows that the method we are using to calculate the probability is incorrect.
That is what 't Hooft means by nonsense - it means the method of doing the calculation is wrong. His Nobel prize was earned showing us the correct way to do the calculation.
The parameter $k$ is a wave vector, or more precisely the special relativistic form of a wave vector. This is a 4D vector so it has four independant components normally written as $k^0$, $k^1$, $k^2$ and $k^3$. Note that the superscript is a label and doesn't mean you're raising $k$ to a power. The integration is over all possible values of each of the four components, so it's really four integrations:
$$ \int \int \int \int\,dk^0 \,dk^1 \,dk^2 \,dk^3 $$
Writing $d^4k$ is a common shorthand for this.
You are unlikely to study quantum field theory in any depth unless you do a postgraduate course in physics, though I guess some universities may offer it as a final year option.
