Noncommutative spacetime symmetries: Twist versus covariance

Abstract

We prove that the Moyal product is covariant under linear affine spacetime transformations. From the covariance law, by introducing an $(x,\Theta )$-space where the spacetime coordinates and the noncommutativity matrix components are on the same footing, we obtain a noncommutative representation of the affine algebra, its generators being differential operators in $(x,\Theta )$-space. As a particular case, the Weyl Lie algebra is studied and known results for Weyl invariant noncommutative field theories are rederived in a nutshell. We also show that this covariance cannot be extended to spacetime transformations generated by differential operators whose coefficients are polynomials of order larger than one. We compare our approach with the twist-deformed enveloping algebra description of spacetime transformations.

19 pages in revtex, references added

References in corpus (6)

• A Gravity Theory on Noncommutative Spaces
• Twisted Gauge Theories
• Twist Symmetry and Gauge Invariance
• Remarks on twisted noncommutative quantum field theory
• Twist to close
• Twisted Conformal Symmetry in Noncommutative Two-Dimensional Quantum Field Theory

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