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But those axioms don't work for 3+1 QFT, at least that's what I understand from Witten's and Jaffa's Millenium prize proposal.
We don't...
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You should look at the 4 Wightman axioms W0-W3 and their consequences, as summarized for example here...
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How about this one
"Quantum Field Theory: A Tourist Guide for Mathematicians" - Gerald Folland
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I am trying to teach myself QFT. So, that includes understanding the mathematical definition of a Quantum Field. My education is in...