Twisted Quantum Fields on Moyal and Wick-Voros Planes are Inequivalent

TL;DR

This paper demonstrates that quantum field theories on Moyal and Wick-Voros non-commutative planes are inequivalent due to differences in their twisted Poincaré symmetries, despite the planes being *-isomorphic.

Contribution

It shows that the *-isomorphism between Moyal and Wick-Voros planes does not preserve the twist of the Poincaré group, leading to inequivalent quantum field theories.

Findings

Wick-Voros plane exhibits non-trivial dependence on non-commutative parameter in self-energy diagrams.

Moyal plane's quantum field theory shows no such dependence in the absence of gauge fields.

The inequivalence arises from differences in the twisted Poincaré-Hopf symmetries.

Abstract

The Moyal and Wick-Voros planes A^{M,V}_{\theta} are *-isomorphic. On each of these planes the Poincar\'e group acts as a Hopf algebra symmetry if its coproducts are deformed by twist factors. We show that the *-isomorphism T: A^M_{\theta} to A^V_{\theta} does not also map the corresponding twists of the Poincar\'e group algebra. The quantum field theories on these planes with twisted Poincar\'e-Hopf symmetries are thus inequivalent. We explicitly verify this result by showing that a non-trivial dependence on the non-commutative parameter is present for the Wick-Voros plane in a self-energy diagram whereas it is known to be absent on the Moyal plane (in the absence of gauge fields). Our results differ from these of (arXiv:0810.2095 [hep-th]) because of differences in the treatments of quantum field theories.