What's so not-quantum about tree-level diagrams?
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$\begingroup$ From the tag of the question, you are already assuming "FIELD" theory, are you? So the queston is how classical the fields are? $\endgroup$– user135Commented Dec 18, 2011 at 16:34
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2 Answers
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The reasons were given here. Essentially, at tree level you recover classical results. Loop corrections are proportional to powers of $\hbar$ and these are quantum terms.
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$\begingroup$ It is a bit more obscure for particles with spin, because then the classical result has still an $\hbar$. $\endgroup$– user135Commented Dec 18, 2011 at 16:33
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$\begingroup$ That is right, but classically spin does not exist and so, the result is consistent. $\endgroup$– JonCommented Dec 18, 2011 at 17:29
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Tree level diagrams are one-particle relativistic quantum mechanics, but not quantum field theories.
The point is obscured by two reasonable details in modern QFT books: they avoid to speak of "2nd quantisation", and they set h=1 everywhere (so for instance it is not so clear how different the h->0 limit is for fermions than for scalar fields, or how bosons can add to build a classical electromagnetic field but fermions can not). It is expected that you go across a relativistic quantum mechanics textbook before jumping to QFT, but sometimes the career path is different.
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$\begingroup$ There are relativistic quantum mechanics textbooks that are not quantum field theory texts? $\endgroup$ Commented Dec 20, 2011 at 18:29