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Someone mentioned to me in passing that it had been proven that the Wightman axioms are over-restrictive in four dimensions and provably always result in trivial correlators (or maybe a trivial S-matrix, I don't remember). He didn't know any more specifics off hand, but I've looked around and I can't find anything approaching that strong of a statement. The only known problems with the Wightman axioms (and the constructive field theory program in general) I can find are:

  • The axioms as stated are only formalized for a single scalar field, but it's generally considered that there isn't a fundamental impediment to writing them down for other matter content (even though naively there might be for gauge theories there are many proposals for how to extend the axioms to Wilson loops).
  • The axioms are only valid in flat spacetime, and rely on global symmetries in the formalism (but this is only a problem for the Wightman axioms in particular).
  • Two of the assumptions (uniqueness and cyclicity of the vacuum) are now known to be wrong in many quantum field theories (i.e. those with physically inequivalent vacua and those that exhibit phenomena like instantons and monopoles), but many of the results that were proven do not rely on these in an essential way.
  • It's suspected that there are no non-trivial pure scalar field theories in 4D.

But the rest of the axioms seem extremely benign. Essentially you're assuming that a quantum field theory (a) is a quantum mechanical system in the sense of having a hilbert space realization (b) is relativistically invariant and (c) exhibits locality and causality properly. The formalization of fields as operator valued distributions, while seemingly complicated, is really a very conservative stance as in any reasonable field operators ought to be able to be smeared.

One other problem that he mentioned (and this might be a large enough topic to warrant a second question) is that in some supersymmetric quantum field theories the partition function vanishes, so it's unclear how to define expectation values and therefore it's unclear how to think of the theory as an ordinary quantum mechanical system.

So ultimately my questions are: Does anyone known of a result that resembles what my friend mentioned to me? And are there any known problems with the Wightman axioms in 4D that are more serious than the ones outlined above?

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    $\begingroup$ Triviality of scalar field theories in dimension $d> 4$ is proved in this articles by Aizenman and Fröhlich. It is worth noting that such triviality is proved only in taking the limit of scalar QFT on the lattice; so in principle with other methods one may obtain non-trivial theories. In addition, these results do not hold for $d=4$. $\endgroup$ – yuggib Aug 22 '15 at 18:40
  • $\begingroup$ There is however concrete hope that (even if the scalar theory turn out to be trivial) the Wightman axioms may be satisfied by non-abelian gauge theories such as Yang-Mills. Sadly, there is not yet a rigorous proof of an interacting system in $d=4$ satisfying the Wightman axioms. $\endgroup$ – yuggib Aug 22 '15 at 18:43
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    $\begingroup$ I'm pretty sure your acquaintance was simply mistaken. $\endgroup$ – Danu Aug 22 '15 at 19:17
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    $\begingroup$ I could easily see him having misremembered $d>4$ as $d=4$. Thank you. $\endgroup$ – James Hanson Aug 22 '15 at 19:43
  • $\begingroup$ Well, there are (qualified) people that think $(\varphi^4)_4$ is trivial; however it is not known. $(\varphi^4)_2$, $(\varphi^4)_3$ (for small coupling constants) are not trivial; $(\varphi^4)_{> 4}$ are trivial...You may think what you want about $d=4$ ;-) $\endgroup$ – yuggib Aug 22 '15 at 20:29

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