Neeb's classification of the irreducible semibounded unitary representations of hermitian Lie groups (Banach–Lie groups attached to infinite-dimensional hermitian Hilbert symmetric spaces) leaves open the precise relation between the semiboundedness condition and the physically motivated positive energy condition. Semiboundedness implies positive energy (up to passing to the dual representation), but the converse is not known: it is an open problem whether every irreducible positive energy representation of a full hermitian Lie group is semibounded. The obstruction is a missing Peter–Weyl-type decomposition theory for representations of Hilbert–Lie groups such as the full unitary group U(H)U(\mathcal H) on Fréchet spaces of smooth vectors.

Background

For a (possibly infinite-dimensional) Lie group GG, a continuous unitary representation π\pi on a Hilbert space H\mathcal H yields, on the Fréchet space H∞\mathcal H^\infty of smooth vectors (vectors vv for which g↦π(g)vg\mapsto\pi(g)v is a smooth map), a derived representation dπ\mathrm{d}\pi of the Lie algebra g\mathfrak g by (essentially) skew-adjoint operators, so that i dπ(x)i\,\mathrm{d}\pi(x) is self-adjoint. The representation is called semibounded if the spectra of the operators i dπ(x)i\,\mathrm{d}\pi(x) are uniformly bounded from above on some non-empty open subset of g\mathfrak g, and it is of positive energy with respect to a distinguished element d∈gd\in\mathfrak g if sup⁡Spec⁡(i dπ(d))<∞\sup\operatorname{Spec}(i\,\mathrm{d}\pi(d))<\infty (in physics −i dπ(d)-i\,\mathrm{d}\pi(d) is a Hamiltonian bounded below). Semiboundedness is thus a stable version of the positive energy condition. A hermitian Lie group, in the sense of the source paper, is a triple (G,θ,d)(G,\theta,d) consisting of a connected Banach–Lie group GG, an involutive automorphism θ\theta with K=(Gθ)0K=(G^\theta)_0 the identity component of the fixed-point group, and an element dd in the center of k=Lie⁡(K)\mathfrak k=\operatorname{Lie}(K) such that, for p≅g/k\mathfrak p\cong\mathfrak g/\mathfrak k (identified with a closed complement), the space p\mathfrak p is a complex Hilbert space, k\mathfrak k acts on it by bounded skew-hermitian operators, [d,x]=ix[d,x]=ix for x∈px\in\mathfrak p, and the kernel of this action is central; the quotient M=G/KM=G/K then carries the structure of a hermitian symmetric space modeled on a Hilbert space, and gC=p+⊕kC⊕p−\mathfrak g^{\mathbb C}=\mathfrak p^+\oplus\mathfrak k^{\mathbb C}\oplus\mathfrak p^- is a 3-grading. The triple is called full (and GG a full hermitian Lie group) if p\mathfrak p is a simple so-called JH*-triple (a complex Hilbert space carrying a continuous Jordan triple product {x,y,z}=12[[x,y‾],z]\{x,y,z\}=\tfrac{1}{2}[[x,\overline{y}],z]) and the action of k\mathfrak k on p\mathfrak p realizes the full Lie algebra of triple automorphisms. The main examples are central extensions of the unitary group U(H)U(\mathcal H) built on the Hilbert–Schmidt Lie algebra, and the restricted unitary group Ures(H)U_{\mathrm{res}}(\mathcal H) attached to a polarization H=H+⊕H−\mathcal H=\mathcal H^+\oplus\mathcal H^- (off-diagonal blocks Hilbert–Schmidt), i.e. the groups acting on symmetric Hilbert domains and on the restricted Grassmannian. Neeb classified the irreducible semibounded unitary representations of these groups for infinite-dimensional irreducible symmetric spaces: under a separability assumption they are obtained by holomorphic induction from bounded representations of KK (realizing H\mathcal H as a Hilbert space of holomorphic vector-valued functions on the symmetric space M=G/KM=G/K determined by the bounded KK-representation). Theorem 5.2(iv) of that paper shows the converse direction: an irreducible semibounded representation is of positive energy for a suitable element, or its dual is. What is missing is the reverse implication. Concretely, for an irreducible positive energy representation (π,H)(\pi,\mathcal H), the space V=(H∞)p−‾V=\overline{(\mathcal H^\infty)^{\mathfrak p^-}} of smooth vectors annihilated by p−\mathfrak p^- (a lowest-energy space) generates H\mathcal H as a GG-module, and holomorphic induction applies as soon as the KK-representation on VV is bounded; in the separable case one can find a bounded irreducible KK-subrepresentation W⊆VW\subseteq V, but it is not known whether WW can be chosen so that W∩H∞W\cap\mathcal H^\infty is dense in WW. The source identifies the missing tool as a Peter–Weyl-type decomposition theory (the classical Peter–Weyl theorem achieves such a decomposition into irreducibles for compact groups) for representations of Hilbert–Lie groups such as U(H)U(\mathcal H) on Fréchet spaces of smooth vectors.

Problem Statement

Is every irreducible positive energy unitary representation of a full hermitian Lie group semibounded? Precisely: let (G,θ,d)(G,\theta,d) be a full hermitian Lie group in the sense of Neeb (Definitions 1.1 and 2.4 of the source paper; a connected Banach–Lie group with involution θ\theta and grading element dd such that p\mathfrak p is a simple JH*-triple, k\mathfrak k acts on it by the full automorphism Lie algebra, p\mathfrak p is a Hilbert space, and [d,x]=ix[d,x]=ix for x∈px\in\mathfrak p), and let (π,H)(\pi,\mathcal H) be an irreducible continuous unitary representation with sup⁡Spec⁡(i dπ(d))<∞\sup\operatorname{Spec}(i\,\mathrm{d}\pi(d))<\infty (positive energy). Does it follow that π\pi is semibounded, i.e. that sup⁡x∈Usup⁡Spec⁡(i dπ(x))<∞\sup_{x\in U}\sup\operatorname{Spec}(i\,\mathrm{d}\pi(x))<\infty for some non-empty open subset UU of the Lie algebra g\mathfrak g? The converse implication is known (Theorem 5.2(iv) of the source). For all full hermitian Lie groups, and without separability restrictions, does the positive energy condition pick out the same irreducible representations as semiboundedness?

References

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