Para-spaces, their differential analysis and an application to Green's quantisation

2023/12/07 by R. B. Zhang, Zhang, Ruibin · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2312.04250

openalex publication_date 2023/12/07 · openalex created_date 2023/12/09 · openalex updated_date 2026/07/28

Abstract

We introduce a class of non-commutative geometries, loosely referred to as para-spaces, which are manifolds equipped with sheaves of non-commutative algebras called para-algebras. A differential analysis on para-spaces is investigated, which is reminiscent of that on super manifolds and can be readily applied to model physical problems, for example, by using para-space analogues of differential equations. Two families of examples, the affine para-spaces \mathbbK^m|n(p) and para-projective spaces \mathbbKP^m|n(p), with \mathbbK being ℝ and ℂ, are treated in detail for all positive integers p. As an application of such non-commutative geometries, we interpret Green's theory of parafermions in terms of para-spaces on a point. Other potential applications in quantum field theory are also commented upon.

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