nLab Haag's theorem

Idea

Haag’s theorem [Haag (1955), Hall & Whightman (1957)] says that for a non-finite number of generators the canonical commutation relations do not have a unique (up to isomorphism) irreducible unitary representation.

This is in contrast to the Stone-von Neumann theorem which says that for a finite number of generators the Schrödinger representation is, up to isomorphism, the unique irrducible unitary representation of the canonical commutation relations.

While canonical commutation relations with a finite number of generators appear in quantum mechanics, those appearing in quantum field theory (see e.g. Wick algebras for free fields) generically have infinitely many generators and hence are subject to Haag’s theorem.

More pertinently, the theorem goes on to say that, therefore, the interaction picture in interacting quantum field theory does not behave as in quantum mechanics…

References

Haag’s theorem was first stated in

• Rudolf Haag, On quantum field theories, Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 29 12 (1955) 1-37 [cds:212242, pdf]

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 field theory, Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 31 5 (1957) 1-41 [pdf]

A brief statement in context is in

• Rudolf Haag, pages 54-55 of 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)

Textbook account in a context of non-perturbative quantum field theory:

• Franco Strocchi, Thm. 3.1 in: An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press (2013) [doi:10.1093/acprof:oso/9780199671571.001.0001]

See also

• Wikipedia, Haag’s theorem