On group properties and reality conditions of UOSp(1|2) gauge transformations

TL;DR

This paper investigates the structure and reality conditions of UOSp(1|2) gauge transformations, focusing on Grassmann-odd parameters, their algebraic properties, and compatibility with hermitian conjugation in graded representations.

Contribution

It formulates a reality condition for Grassmann-odd parameters in UOSp(1|2) and analyzes their gauge transformation properties and compatibility with graded hermitian conjugation.

Findings

Grassmann-odd parameters are clarified using su(2)-spinors.

The graded hermitian conjugation is compatible with the Dirac adjoint.

The graded unitary condition aligns with the proposed reality condition.

Abstract

For osp(1|2;C) graded Lie algebra, which proper Lie subalgebra is su(2), we consider the Baker-Campbell-Hausdorff formula and formulate a reality condition for the Grassmann-odd transformation parameters that multiply the pair of odd generators of the graded Lie algebra. Utilization of su(2)-spinors clarifies the nature of Grassmann-odd transformation parameters and allow us an investigation of the corresponding infinitesimal gauge transformations. We also explore action of the corresponding group element of UOSp(1|2) on an appropriately graded representation space and find that the graded generalization of hermitian conjugation is compatible with the Dirac adjoint. Consistency of generalized (graded) unitary condition with the proposed reality condition is shown.