nLab Haag's theorem



Quantum field theory


algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



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 irreducible 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…


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

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:

See also

Last revised on April 9, 2024 at 04:30:38. See the history of this page for a list of all contributions to it.