# nLab Haag's theorem

### Context

#### Quantum field theory

functorial quantum field theory

# Contents

## Idea

Haag’s theorem (Haag 55, Hall 57) 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.

## References

Haag’s theorem was first stated in

• Rudolf Haag, On quantum field theorie, Det Kongelige Danske Videnskabernes 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 field theory , Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 31, nr. 5: 1-41.(1957)

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)