Essays on supersymmetry. With contributions by M. Flato, C. Fronsdal, T. Hirai
type: Item
instance of: scholarly article
title: Essays on supersymmetry. With contributions by M. Flato, C. Fronsdal, T. Hirai
zbMATH Open document ID: 0594.22010
published in: Mathematical Physics Studies
publication date: 1986-01-01T00:00:00Z
review text: The book is described well by its title: It contains four long essays on supersymmetry by C. Fronsdal, with collaborations from M. Flato and T. Hirai. The arena is always \((3+2)\)-dimensional de Sitter space and the aim is to understand the mathematical structure of Lie superalgebras, the associated local supergroups, their representations, and the relevance of the latter to physics. The central motiviation and the theme of the book is that ''physics has much to gain, and nothing to loose, by admitting that [the cosmological constant] can never be shown experimentally to be exactly zero''. The book starts with a short introduction in which the motivation for going to supersymmetry is explained (it attempts to unify all particles, bosons and fermions, in a single irreducible representation), it stresses the fundamental roles of supersymmetry, masslessness, and conformal invariance in the construction of finite field theories, and justifies the choice of the \((3+2)\)-de Sitter space as the fundamental arena from which, à la Kaluza-Klein, the correct four-dimensional theory should ultimately emerge. The first essay (Fronsdal and Hirai) addresses the question when there exist infinite dimensional integrable and unitarizable (i.e., constructed by unitary operators in a Hilbert space) representations of a given Lie superalgebra. The answer to this question depends crucially on the definition of the local supergroup obtained from the given Lie superalgebra. In the present essay the supergroup is defined to be the direct product of a Lie group by an invariant ''nilpotent'' subsupergroup. The irreducible representations are studied in more detail and it is found that every irreducible representation of a supergroup has a finite reduction on the even subgroup. A (non necessarily positive definite) invariant measure is determined and it is used for integration on the subsupergroups. And as an example, these ideas are applied to the algebras \({\mathfrak osp}(2n| 1).\) In the…
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81T60
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 22E05
Mathematics Subject Classification ID: 20G45
Mathematics Subject Classification ID: 22-06
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 00Bxx
Mathematics Subject Classification ID: 53-06
zbMATH DE Number: 3955269
zbMATH Keywords: Essays
zbMATH Keywords: Supersymmetry
zbMATH Keywords: integration on subsupergroups
zbMATH Keywords: representations of superalgebras
zbMATH Keywords: supersymmetry
zbMATH Keywords: de Sitter space
zbMATH Keywords: Lie superalgebras
zbMATH Keywords: local supergroups
zbMATH Keywords: conformal invariance
zbMATH Keywords: finite field theories
zbMATH Keywords: unitary operators
zbMATH Keywords: irreducible representations
zbMATH Keywords: spectral decomposition
zbMATH Keywords: Dirac operator
zbMATH Keywords: wave-type equations
zbMATH Keywords: singular representation
zbMATH Keywords: oscillator representation
zbMATH Keywords: superfields
zbMATH Keywords: homogeneous space
zbMATH Keywords: bibliography
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3955269