Representations of the Poincare group on relativistic phase space

2008/02/01 by Yaakov Friedman, Friedman, Yaakov · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.0802.0070

arxiv created 2008/02/01 · openalex publication_date 2008/02/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a complex relativistic phase space as the space ℂ⁴ equipped with the Minkowski metric and with a geometric tri-product on it. The geometric tri-product is similar to the triple product of the bounded symmetric domain of type IV in Cartan's classification, called the spin domain. We define a spin 1 representations of the Lie algebra of the Poincaré group by natural operators of this tri-product on the complex relativistic phase space. This representation is connected with the electromagnetic tensor. A spin 1/2 representation on the complex relativistic phase space is constructed be use of the complex Faraday electromagnetic tensor. We show that the Newman-Penrose basis for the phase space determines the Dirac bi-spinors under this representation. Quite remarkable that the tri-product representation admits only spin 1 and spin 1/2 representations which correspond to most particles of nature.

Cited by

• The scalar complex potential of the electromagnetic field

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• The relativistic phase space and Newman-Penrose basis
• The Majorana spinor representation of the Poincare group
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• Complex hidden symmetries in real spacetime and their algebraic structures