The "definition" of the normal ordering of a single operator in CFT is understandable looks a bit vague to mebut it is not clear as to how products of normal ordered operators or normal order of product of operators are defined? .
Some more elaboration on what about normal ordering I am concerned about.
The problem is that I can't these books give an honest definition of what it means to "normal order" operators in CFT. Like there is a very clean definition in rest of QFT whose relation to time-ordering is given by the Wick's Theorem. Here in CFT one is supposed to understand that while normal ordering a string of operators inserted at different points on the space-time one is subtracting away from the product every possible way in which one or more pairs of insertion points can coincide and produce a singularity
Like if A,B,C,D are 4 different Bosonic operators say inserted at 4 different space-time points. Then one would define normal ordering as,
$:ABCD: = ABCD - (AB):CD: - (AC):BD: - (AD):BC:-(BC):AD:-(BD):AC:$$$-(CD):AB:-(AB)(CD)-(AC)(BD)-(AD)(BC)$$
where () denotes the correlation function of the operators.
Now the point is whether one is supposed to take the above kind of equations as being just well-motivated definition or is there is anything more fundamental from which it is derivable?
There is definitely an issue about defining difference of two divergent expressions here.