What are Wightman fields/functions

Simple question: What are Wightman fields? What are Wightman functions? What are their uses? For example can I use them in operator product expansions? How about in scattering theory?

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1 Answer

In axiomatic approaches to quantum field theory, the basic field operators are usually realized as operator-valued distributions. That's what Wightman fields are: operator-valued distributions satisfying the Wightman Axioms.

Wightman functions are the correlation functions of Wightman fields, nothing more. There's a nice theorem that says if you have a bunch of functions that look like they're the Wightman functions of QFT, then you can actually reconstruct the Hilbert space and the algebra of Wightman fields from it.

Neither of these concepts is anything new physically. They're just more precise ways of speaking about things physicists already know. (Thinking of fields as operator-valued distributions instead of operator-valued functions lets you make precise what goes wrong when you multiply two fields at the same point.) You can talk about OPEs or scattering theory in this language, but it won't gain you anything, unless you're trying to publish in math journals.

If you're interested in the Wightman Axioms, they're explained nicely in the first of Kazhdan's lectures in the IAS QFT Year.

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Ok, so based on my reading, the field operators of a theory are understood to be distribution-valued, that is, to be integrated over a smooth function:  e.g.  $\hat\phi(f)\equiv\int dx f(x)\hat\phi(x)$.  But if the smooth function has support over such a large region in space-time that if I try to compute time-ordered products, I run into some ambiguity.  How do I define the time-ordering symbol for Wightman functions? – QuantumDot Nov 17 '12 at 22:23
@QuantumDot:  Why not just ask this as another question instead of trying to have a discussion here in the comments? – user1504 Nov 18 '12 at 0:13
– QuantumDot Nov 18 '12 at 0:18
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