Integration on minimal Z22 -superspace and emergence of space

Abstract

We investigate the possibilities of integration on the minimal Z2 2-superspace. Two definitions are taken from the works by Poncin and Schouten and we examine their generalizations. It is shown that these definitions impose some restrictions on the integrable functions. We then introduce a new definition of integral, which is inspired by our previous work, and show that the definition does not impose restrictions on the integrable functions. An interesting feature of this definition is the emergence of a spatial coordinate which means that the integral is defined on R<sup>2 despite the fact that the (0,0) part of the minimal Z2 2-superspace is R

Orthosymplectic Z2×Z2Z2×Z2 -graded Lie superalgebras and parastatistics

“…Since the recognition of a Z 2 × Z 2 -graded Lie superalgebra underlying the symmetries of Lévy-Leblond equations [11,12], these Z 2 × Z 2 -graded algebras have experienced a revival in mathematical physics. They appeared in graded (quantum) mechanics and quantization [13][14][15][16][17][18], in Z 2 × Z 2graded two-dimensional models [19][20][21], in Z 2 × Z 2 -graded superspace formulations [22][23][24][25] and in particular in parastatistics [4,26] and in the description and application of other types of parabosons and parafermions [27,28].…”