algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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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
but the proof had some gaps. It was completed in
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
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.