Unitary Representations of Super Lie Groups and Applications to the Classification and Multiplet Structure of Super Particles

Abstract

It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincaré groups in arbitrary dimension. Finally we compare our results with those in the physical literature on the structure and classification of super multiplets.

George Mackey and his work on representation theory and foundations of physics

“…One can then prove the imprimitivity theorem for a super homogeneous space G/H where G is a super Lie group and H is a closed super Lie subgroup, at least when X = G/H is purely even, i.e., a classical manifold. This leads as in the classical theory to a complete description of irreducible unitary representations of super semi direct products (not just the super Poincaré groups) A× ′ H (A is a super abelian group) which are regular at the classical level (when all odd coordinates are put to 0) [CCTV1]. In turn this leads to the classification of super particles and the elucidation of the concept of super multiplets.…”

“…It was also under the influence of his lectures that I wrote my paper on the logic of quantum mechanics [Va3] and subsequently the book [Va2] on the geometry of quantum theory. Years later, when I returned again to quantum foundations, my work was still animated by the themes he had introduced [CTV] [ CCTV1].…”

Differentiable vectors and unitary representations of Fréchet–Lie supergroups

“…We begin by recalling the definition of Lie supergroups and their unitary representations. See [CCTV06] and [MNS11] for further details. By a locally convex Lie superalgebra we mean a Lie superalgebra g = g 0 ⊕ g 1 over R or C with the following two properties.…”

“…In Section 6 we apply all this to unitary representations of Lie supergroups (G, g), which we consider as a pair consisting of a Lie superalgebra g = g 0 ⊕g 1 and a Lie group G whose Lie algebra is the even part g 0 of g (see [CCTV06]). A crucial difficulty in dealing with unitary representations of Lie supergroups is the specification of the common domain of the operators corresponding to the odd part g 1 (see [CCTV06] and [MNS11] for a detailed discussion).…”

Unitary Representations of Nilpotent Super Lie Groups

“…1. One can combine ρ π and π ∞ to obtain a representation of g in H ∞ where an element [CCTV,Proposition 1] it follows that for every X ∈ g 0 , Y ∈ g 1 , and v ∈ H ∞ we have…”