411 research outputs found

### 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

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