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I understood that quantum field theory is essentially based on a problematic mathematical basis. Can someone please explain what is the fundamental problem to formulate QFT as a rigorous mathematical theory? It's clear that it has to do with renormalization but similar techniques are already exist in analytically number theory (and are considered to be a mathematical theory).

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Be aware that there are multiple QFT's, some of which aren't so problematic; see Which QFTs were rigorously constructed?. –  Chris White Nov 15 '13 at 1:00
    
Related: physics.stackexchange.com/q/27665/2451 and links therein. –  Qmechanic Nov 15 '13 at 2:16
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marked as duplicate by John Rennie, Qmechanic Nov 15 '13 at 11:51

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