R-deformed Heisenberg algebra

TL;DR

This paper explores the R-deformed Heisenberg algebra, revealing its finite-dimensional representations, connections to generalized statistics, and applications in supersymmetry and fractional spin fields in 2+1 dimensions.

Contribution

It introduces finite-dimensional representations of the R-deformed Heisenberg algebra and demonstrates their relevance to supersymmetry and fractional spin field descriptions.

Findings

Finite-dimensional representations equivalent to paragrassmann algebra.

Relation to Guon-like algebra and generalized statistics.

Applications in (2+1)-dimensional supersymmetry and fractional spin fields.

Abstract

It is shown that the deformed Heisenberg algebra involving the reflection operator R (R-deformed Heisenberg algebra) has finite-dimensional representations which are equivalent to representations of paragrassmann algebra with a special differentiation operator. Guon-like form of the algebra, related to the generalized statistics, is found. Some applications of revealed representations of the R-deformed Heisenberg algebra are discussed in the context of OSp(2|2) supersymmetry. It is shown that these representations can be employed for realizing (2+1)-dimensional supersymmetry. They give also a possibility to construct a universal spinor set of linear differential equations describing either fractional spin fields (anyons) or ordinary integer and half-integer spin fields in 2+1 dimensions.