# How should I understand rigorously the definition of normal ordering of free fields

Let $$\phi(x)$$ be a free Hermitian scalar field in $$4D$$ Minkowski spacetime with the metric $$(1,-1,-1,-1)$$.

Then, though I wrote it as $$\phi(x)$$, it is in fact an operator-valued tempered distribution on a Bosonic Fock space, requiring a test function to be smeared.

Moreover, there is a decompositon $$\phi(x)=\phi^+(x)+\phi^-(x)$$ into the creation part $$\phi^+(x)$$ and the annihilation part $$\phi^-(x)$$.

Let $$D(x-y)=[\phi^-(x),\phi^+(y)]$$ where $$[,]$$ is the commutator so that $$D(x-y)$$ is a version of the propagator. It is known that $$D(x-y)$$ is a numerical valued tempered distribution.

Then, the normal ordering is defined in all literatures as follows: $$$$:\phi^2:(x)=\lim\limits_{y \to x} \Bigl(\phi(x)\phi(y)+iD(x-y)\Bigr)$$$$

However, I have great difficulty understanding the above formula rigorously, since every term is a distribution. How should I make sense of the limit $$\lim\limits_{y \to x}$$ in this case?

The singular part of the operator product $$\phi(x)\phi(y)$$ in the limit $$y\rightarrow x$$ is canceled by the singular part of the commutator $$D(x-y)$$, so that the normally ordered product is continuous when $$y\rightarrow x$$.
This is explained, and worked out with an example, in a Scholarpedia article. As explained there, the limit $$y\rightarrow x$$ should be understood as the limit of a correlation function containing the normally ordered product.