Essays on supersymmetry. With contributions by M. Flato, C. Fronsdal, T. Hirai
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 second essay Fronsdal analyzes, using group theoretical methods, superfields (with any index structure) in \((3+2)\)-dimensional de Sitter space. The spectral decomposition of the Dirac operator (an invariant first order differential operator constructed from the covariant derivatives) provides a reduction of the different fields into their irreducible parts. Massless fields are analyzed in detail and the gauge and quantization of the massless limit of a vector multiplet are studied. In particular \(N=1\) and \(N=2\) supersymmetric theories in de Sitter space are discussed and the different spin fields are investigated according to whether they admit (or they do not) wave-type equations. The third essay (Flato and Fronsdal) discusses ''spontaneously generated'' massless field theories that admit one-particle ghost states with zero energy. The motivation comes from electrodynamics in which the photon representations of the Poincaré group have vanishing Casimir operators. Replacing the Poincaré group by the \((3+2)\)-de Sitter group and looking for theories that admit similar degenerate vacuum states the authors construct two families of extended super quantum electrodynamics-type theories, which include conformal and supersymmetric generalizations of electrodynamics and \(N=6\) extended supergravity but they do not include ordinary gravity and simple supergravity. The last essay (Fronsdal) contains an extensive study of the geometry and the representations of the orthosymplectic superalgebras \({\mathfrak osp}(2n| 1)\). This superalgebra is characterized by the fact that its most singular representation (the oscillator representation) when restricted to the conformal subalgebra contains each and every massless representation just once; thus, it describes all massless particles, with all possible spins, simultaneously and without redundancies. The essay starts with the construction of the oscillator representation in the context of classical mechanics, expresses it in terms of superfields, and identifies the simplest homogeneous space on which the construction works. Then it studies in detail the physics of a particular ten- dimensional homogeneous space (for which the theory predicts a family of massless particles of all spins) and proposes an alternative study of the four-dimensional conformal scalar field theory in which the space-time is identified with the U(2) group manifold. The essays are written very clearly and the subjects could be easily understood by a motivated reader without much prior knowledge on supersymmetry. The bibliography is rather extensive and the index quite complete. Since the material has not been published elsewhere, the book would be a valuable addition to anyone interest in the covered topics.
scientific article; zbMATH DE number 3955269