Minimal representations and reductive dual pairs in conformal field theory

A minimal representation of a simple non-compact Lie group is obtained by ``quantizing'' the minimal nilpotent coadjoint orbit of its Lie algebra. It provides context for Roger Howe's notion of a reductive dual pair encountered recently in the description of global gauge symmetry of a (4-dimensional) conformal observable algebra. We give a pedagogical introduction to these notions and point out that physicists have been using both minimal representations and dual pairs without naming them and hence stand a chance to understand their theory and to profit from it.Comment: 21 page

"Quantization is a mystery"

Expository notes which combine a historical survey of the development of quantum physics with a review of selected mathematical topics in quantization theory (addressed to students that are not complete novices in quantum mechanics). After recalling in the introduction the early stages of the quantum revolution, and recapitulating in Sect. 2.1 some basic notions of symplectic geometry, we survey in Sect. 2.2 the so called prequantization thus preparing the ground for an outline of geometric quantization (Sect. 2.3). In Sect. 3 we apply the general theory to the study of basic examples of quantization of Kaehler manifolds. In Sect. 4 we review the Weyl and Wigner maps and the work of Groenewold and Moyal that laid the foundations of quantum mechanics in phase space, ending with a brief survey of the modern development of deformation quantization. Sect. 5 provides a review of second quantization and its mathematical interpretation. We point out that the treatment of (nonrelativistic) bound states requires going beyond the neat mathematical formalization of the concept of second quantization. An appendix is devoted to Pascual Jordan, the least known among the creators of quantum mechanics and the chief architect of the "theory of quantized matter waves".Comment: lecture notes, 51 page

Exceptional quantum geometry and particle physics II

We continue the study undertaken in [13] of the relevance of the exceptional Jordan algebra $J^8_3$ of hermitian $3\times 3$ octonionic matrices for the description of the internal space of the fundamental fermions of the Standard Model with 3 generations. By using the suggestion of [30] (properly justified here) that the Jordan algebra $J^8_2$ of hermitian $2\times 2$ octonionic matrices is relevant for the description of the internal space of the fundamental fermions of one generation, we show that, based on the same principles and the same framework as in [13], there is a way to describe the internal space of the 3 generations which avoids the introduction of new fundamental fermions and where there is no problem with respect to the electroweak symmetry.Comment: 18 page