nLab
Haag's theorem

Idea

Haag’s theorem (due to Rudolf Haag, D. Hall, and Arthur Wightman) is a statement in the context of the fact that the Stone-von Neumann theorem fails for a non-finite number of dimensions/degrees of freedom: it states that in interacting quantum field theory (as opposed to quantum mechanics with finitely many degrees of freedom) there is in general no unitary equivalence between the free field CCR representation and that of the interacting fields.

References

Haag’s theorem was first stated in

• Rudolf Haag, On quantum fi…eld theorie, Det Kongelige Danske Viden- skabernes Selskab, Matematisk-fysiske Meddelelser 29, nr. 12: 1-37.(1955)

but the proof had some gaps. It was completed in

• D. Hall, Arthur Wightman A theorem on invariant analytic functions with applications to relativistic quantum fi…eld theory , Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 31, nr. 5: 1-41.(1957)

A brief statement in context is on pages 54-55 of

• Rudolf Haag, Local Quantum Physics -- Fields, Particles, Algebras

A thorough discussion of meaning and implications of Haag’s theorem (pointing out plenty of flaws on this point in the standard literature) is in

• John Earman, Doreer Fraser, Haag’s theorem and its implications for the foundations of quantum field theory (pdf)