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For the best explanation, see the section "How to stop worrying about Haag's theorem" in the book "The Conceptual Framework of Quantum Field Theory" by A. Duncan
In particular Haag's theorem, that one cannot consistently define the vacuum in interacting theories, is now considered to have been the wrong way to look at the problem, and largely disappears with considerations of RG flow.A Practical Introduction to Quantum Field Theory by Felix Flicker
According to something called Haagís theorem there appears to be no known consistent formalism within which interacting quantum field theory can be expressed.
p. 115 in "An interpretative introduction to quantum Field theory" by Teller (1995)