arXiv physics/97100381997-10-27math-phmath.MP

Angular Momentum, Quaternion, Octonion, and Lie-Super Algebra osp(1,2)

Susumu Okubo

备注：: 17 pages, TEX
DOI：: 10.1142/9789812816566_0014

英文摘要：

We will derive both quaternion and octonion algebras as the Clebsch-Gordan algebras based upon the su(2) Lie algebra by considering angular momentum spaces of spin one and three. If we consider both spin 1 and 1/2 states, then the same method will lead to the Lie-super algebra osp(1,2). Also, the quantum generalization of the method is discussed on the basis of the quantum group $su_q(2)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Octonions

In this paper we review the topological model for the quaternions based upon the Dirac string trick. We then extend this model, to create a model for the octonions - the non-associative generalization of the quaternions.

2010-10-15hep-thmath-phmath.MP

Notes on Spinors and Polyforms II: Quaternions and Octonions

Pauli matrices are 2x2 tracefree matrices with a real diagonal and complex (complex-conjugate) off-diagonal entries. They generate the Clifford algebra Cl(3). They can be generalised by replacing the off-diagonal complex number by one taking values in either quaternions or octonions (or their split versions). These quaternionic and octonionic generalisations generate well-known models of Cl(5) and Cl(9) respectively. The main aim of the paper is to explicitly relate these models to the models arising via the creation/annihilation operator construction. We describe in details the models related…

2022-05-10hep-thmath-phmath.DG

Volumetric (CUBIC) Octonion Sigma-Matrices and their Properties

This work rests upon the certainty that only fields of real and complex numbers, quaternions and octonions have algebras of all four arithmetical operations. Also quaternions are good to represent 3-dimensional Euclid space and 4-dimensional Minkowski space. Moreover algebra of Pauli classical sigma-matrices isomorphic to a split quaternion algebra. In this article author supposes that the hyperspace can be described with octonions and if 4-dimensional Minkowski space is a projection of the hyperspace, consequently equations for 4-dimensional space must be projections of hyperspace equations. …

2011-06-02math-phmath.MP

Octonions in random matrix theory

The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random by symmetry considerations. Only for $N=2$ is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for $N=3$. We then proceed to consider the matrix structure $X^\dagger X$, when $X$ has random octonion entri…

2016-10-25math-phmath.MP

Non-associativity, supersymmetry and hidden variables

It is shown that the supersymmetric quantum mechanics has an octonionic generalization. The generalization is based on the inclusion of quaternions into octonions. The elements from the coset octonions/quaternions are unobservables bacause they cannot be considered as quantum operators as a consequence of their non-associative properties. The idea that the octonionic generalization of the supersymmetric quantum mechanics describes an observable particle formed with unobservable ``particles'' is presented.

2008-04-22hep-phhep-thmath-ph

Octonions, E6, and Particle Physics

In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3x3 Hermitian matrices over the octonions. We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E6, which preserve…

2010-02-25math-phmath.MPmath.RA

更多相关论文收起更多论文

Clifford algebra-parametrized octonions and generalizations

Introducing products between multivectors of Cl(0,7) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphere, and the XY-product. We also present the formalism necessary to construct Clifford algebra-parametrized octonions. Finally we introduce a method to construct generalized octonionic algebras, where their octonionic units are parametrized by arbitrary Clifford multivec…

2006-03-21math-phmath.MPmath.QA

A series of algebras generalizing the octonions and Hurwitz-Radon identity

We study non-associative twisted group algebras over $(\Z_2)^n$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: $(a_1^2+.…

2011-02-11math-phmath.MPmath.RA

Nonassociative quantum theory, emergent probability, and coquasigroup symmetry

This paper follows recent steps towards a nonassociative quantum theory and points out the mathematical structure behind the proposed modifications to conventional quantum theory. An N=1 supersymmetry model and a strong force glueball ansatz is highlighted. Using nonassociative complex octonion algebra, it is shown how the Lorentz Lie algebra can be understood as a four dimensional generalization of the algebra of spin-1/2 operators in physics. Probability is speculated to become an emergent phenomenon from some nonassociative geometry in which to better understand the fluxes involved. A proto…

2011-04-22math-phmath.MP

A perspective on the Magic Square and the 'special unitary' realizations of simple Lie algebras

This article contains the last part of the mini-course `Spaces: a perspective view' delivered at the IFWGP2012. Here I deal with the part of the mini-course which centers on the classification questions associated to the simple real Lie groups. I review the original introduction of the Magic Square `a la Freudenthal', putting the emphasis in the role played in this construction by the four normed division algebras ${\mathbb R}, {\mathbb C}, {\mathbb H}, {\mathbb O}$. I then explore the possibility of understanding some simple real Lie algebras as `special unitary' over some algebras ${\mathbb …

2022-08-31math-phmath.MP