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

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 $\mathbb{K}^{m|n}(p)$ and para-projective spaces $\mathbb{KP}^{m|n}(p)$, with $\mathbb{K}$ being $\mathbb{R}$ and $\mathbb{C}$, 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.