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When reading the definition of the Haag-Lopuszanski-Sohnius theorem, it mentions a 'consistent 4-dimensional quantum field theory':

the Haag–Lopuszanski–Sohnius theorem shows that the possible symmetries of a consistent 4-dimensional quantum field theory do not only consist of internal symmetries and Poincaré symmetry, but can also include supersymmetry as a nontrivial extension of the Poincaré algebra.

What is the exact mathematical definition of consistent quantum field theory in this context?

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The paper referenced, http://www.sciencedirect.com/science/article/pii/0550321375902795 states the assumptions on the genrator of a symmetry or supersymmetry in Section 2: (i) it commutes with the S-matrix; (ii) it acts additively on the states of several incoming particles.

Consistent refers to interpreting these assumptions in the frame of Haag's local quantum field theory (as is said explicitly in the third paragraph of the section). The latter is known to be mathematically consistent - there are many interacting examples in 2 and 3 space-time dimensions (though, due to technical difficulties, so far none in four).

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  • $\begingroup$ Sir, could you please write those part of the link that is concerned here? Otherwise the answer would become invalid once the link becomes dead. $\endgroup$
    – user36790
    Nov 8, 2015 at 18:53
  • $\begingroup$ @user36790: My link is the official publication link and is likely to be valid longer than the link in the PO's question. - Apart from that I wrote already the essential part of the definition - namely Haag's local quantum theory. $\endgroup$
    – Arnold Neumaier
    Nov 8, 2015 at 18:59
  • $\begingroup$ I only have access to the abstract, cannot read section 2 $\endgroup$
    – diffeomorphism
    Nov 8, 2015 at 20:53
  • $\begingroup$ @diffeomorphism: I added a bit more detail; but to understand the full argument you should read the paper. $\endgroup$
    – Arnold Neumaier
    Nov 9, 2015 at 9:01

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