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I understand that Wightman axioms were defined for scalar quantum field theories. However, what prevents axiomatization of non-scalar quantum field theories, such as gauge theories, even for free field?

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Well, nothing prevents a generalization of Wightman's axioms in case of free gauge theories. This, however, requires in turn a generalization of Hilbert spaces, known as Krein Spaces. The only lucid and thorough presentation I know of is Chapter #10 of BLOT. I hope this boook is available to you.

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As @DanielC Pointed out, Krein spaces are the main ingredient to generalise Wightman axioms to free gauge theories. There is a framework, which you might be looking for: the Strocchi Wightman framework. You can find a detailed description in chapter 7 of the following notes:

https://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/WojciechDybalski/Notes-QFT41.pdf

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