# Can one classify irreducible unitary representations of the Weyl algebra?

I saw in this MO post: Is there a machinery describing all the irreducible representations ? that classifying irreducible representations of the Weyl algebra is essentially intractable. My question is if it becomes more reasonable if one restricts to unitary irreps.

To be completely precise: By the Weyl algebra I mean the complex algebra generated by $1$, $a_{+}$, and $a_{-}$ subject to the relations $[a_{+},a_{-}] = 1$. A unitary representation is a complex vector space $V$ with a Hermitian form, on which the Weyl algebra acts in such a way that $a_{+}$ and $a_{-}$ are adjoint pairs (and $1$ acts as $1$). There is no topological component: $V$ need not be complete, the operators must be everywhere defined, and irreducible means no subrepresentations (not just 'no closed subrepresentations').

I know that the Fock space (generated by a vector $v$ with $a_{-}v = 0$) is one such irrep, and one can conjugate by automorphisms to the algebra to obtain other irreps: $a_{+} \mapsto \alpha a_{+} + \beta a_{-}$; $a_{-} \mapsto \overline{\beta}a_{+} + \overline{\alpha} a_{-}$; where $|\alpha|^2 - |\beta|^2 = 1$. The automorphisms where $\beta = 0$ do not give distinct irreps, so one gets a two-dimensional space of irreps (unless there are other redundancies here).

Any futher explanations or references would be great, thanks!

EDIT: Since I think I was unclear, my question is specifically about how unitary irreps of a Lie algebra are different from unitary irreps of the corresponding Lie group. The corresponding Lie group here (exponentiating the anti-self-adjoint elements) is the Heisenberg group, which has a unique unitary irrep (fixing a nonzero central action). But it seems that the Weyl algebra has multiple nonequivalent unitary irreps.

• But those are only the ones that come from representations of the Heisenberg group (see, for example, "the Weyl form of the CCR" on that page). – Alex Zorn Nov 17 '14 at 22:26
• The Stone-von Neumann theorem referred above have been extended by Mackey to characterize the irreducible representations of locally compact groups. In quantum field theories however, the Weyl algebra is no longer the algebra of a locally compact group. I don't know if there are ways of extending the Mackey results to groups that are not locally compact (I am not an expert in the field), however I know (but maybe also you) that the existence of inequivalent irreducible representations of CCR with infinite degrees of freedom is central in the so-called... – yuggib Nov 18 '14 at 15:50
• ... Haag's Theorem of QFT. Anyways, a brief review of Mackey's work, with a (very sketchy) reference to the problems in QFT can be found in this pdf by Varadarajan. Don't know if this could be useful for you ;-) – yuggib Nov 18 '14 at 15:53
• @AlexZorn Sorry, I come from a physical background, and I misunderstood. Again because of my background, I find your requirements for a unitary irrep a bit confusing. One of them is that the operators must be everywhere defined. So the "Fock space" representation that you are considering is not what the physicists usually call a Fock space, with $a_+$ and $a_-$ being the annihilation and creation operators: with the "physical" definition, these operators are unbounded and thus only densely defined. – yuggib Nov 18 '14 at 22:22