Deformed conformal and super-Poincar\'e symmetries in the non-(anti)commutative spaces

TL;DR

This paper constructs representations of super-Poincaré and conformal symmetries in non-(anti)commutative superspaces, showing that their algebras remain unchanged despite deformations, facilitating systematic field theory development.

Contribution

It introduces a method to represent symmetry generators in deformed superspaces using higher-derivative operators, extending previous work and maintaining algebraic structure.

Findings

Algebras of symmetries remain unchanged despite deformation.

Deformation affects comultiplication rules but not the algebraic structure.

Field transformations under deformed symmetries are characterized.

Abstract

Generators of the super-Poincar\'e algebra in the non-(anti)commutative superspace are represented using appropriate higher-derivative operators defined in this quantum superspace. Also discussed are the analogous representations of the conformal and superconformal symmetry generators in the deformed spaces. This construction is obtained by generalizing the recent work of Wess et al on the Poincar\'e generators in the $\theta$-deformed Minkowski space, or by using the substitution rules we derived on the basis of the phase-space structures of non-(anti)commutative-space variables. Even with the nonzero deformation parameters the algebras remain unchanged although the comultiplication rules are deformed. The transformation of the fields under deformed symmetry is also discussed. Our construction can be used for systematic developments of field theories in the deformed spaces.