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I have one slight confusion with something that Blumenhagen & Plauschinn state in the book "Introduction to Conformal Field Theory: With Applications to String Theory". In page 24 regarding radial ordering they state:

Note that there is some ambiguity in Eq. (2.35) because we have to decide whether $w$ and $\bar w$ are inside or outside the contour ${\cal C}$. However, from quantum field theory we know that correlation functions are only defined as a time ordered product. Considering the change of coordinates (2.23), in a CFT the time ordering becomes a radial ordering and thus the product $A(z)B(w)$ does only make sense for $|z|>|w|$. To this end we define the radial ordering of two operators as $$R(A(z)B(w)):=\begin{cases}A(z)B(w),\quad \text{for $|z|>|w|$},\\ B(w)A(z),\quad\text{for $|w|>|z|$}.\end{cases}$$

I'm puzzled by their claim that "from quantum field theory we know that correlation functions are only defined as a time ordered product".

I mean, I do understand that what we really need are the time ordered correlation functions. In particular they are what naturally appear in the LSZ reduction formula which allow us to obtain the ${\cal S}$-matrix from correlation functions. Moreover, I do know that in the functional approach what we naturally get are time ordered correlation functions.

But for me, even though the non time-ordered ones are not useful and don't come out naturally in the path integral approach, that does not mean that they are ill-defined. Observe that the authors' claim is not that "we only care about time-ordered ones", he is claiming very clearly that only the time ordered ones are defined. He further confirms that when he says that, after translating to the plane, $A(z)B(w)$ only makes sense for $|z|>|w|$.

I'm clearly missing something very basic here. Why only time ordered correlation functions are defined? Why a product of two fields which is not time ordered does not make sense? In canonical quantization, for example, what would stop us from writing expressions like $\phi(x)\phi(y)$ for $x^0 < y^0$? It appears a valid product of operators in a Hilbert space to me.

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I'm not sure about this (so please watch for other answers), but I suspect that the "only" wording is a mild case of careless writing. The context suggests that the authors are thinking about QFT in euclidean signature. Wick-rotating from euclidean to lorentzian signature automatically gives time-ordered correlation functions, but after we have all of the QFT's time-ordered correlation functions, we can reconstruct an operators-on-a-Hilbert-space formulation which allows non-time-ordered products.

However, the meaning of "time-ordered" depends on which coordinate we decide to use as time, such as the radial coordinate. Different choices lead to different operators-on-a-Hilbert-space formulations. So maybe the authors meant something like this: non-time-ordered correlation functions can't be defined until after we decide which coordinate we want to use as time. The euclidean CFT doesn't tell us which coordinate we should use.

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    $\begingroup$ I have to say, theoretical physics textbooks are full of little potholes like this, and I think Physics.SE's greatest contribution is supplying ways around them. If it's in the cards, you should consider writing a textbook someday and fixing this, given the clarity of your answers! $\endgroup$
    – knzhou
    Jul 24 at 23:16

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