When canonically quantising a scalar field without normal ordering, one comes across the following expression:
$$\int_\mathbb{R}E(x)\delta(x-x)dx.$$ Here, $E(x)$ is a smooth unbound function.
I have read that this is infinity because, 'the value of the Dirac function at zero is infinity.'
For obvious reasons, this explanation is very unsatisfying - $\delta$ is not a function but a distribution. And the first question therefore is: How does the expression
$$\int\varphi(x)\delta(x-x)dx$$
mathematically make sense? And for this, let's first assume that $\varphi$ is a Schwartz-function, unlike $E(x)$. I know that $\int\varphi(x)\delta(x-a)dx=\varphi(a)$. But here $a$ is a constant!
The second question is then how does the above equation make sense, if you swap $\varphi$ for $E$.