Unitarity of The Realization of Conformal Symmetry in The Quantum Hall Effect

TL;DR

This paper investigates the realization of conformal symmetry in the quantum Hall effect, highlighting its classical unitarity, quantum non-unitarity, and how proper measure adjustments can restore unitarity, with implications for Chern-Simons theory.

Contribution

It demonstrates how conformal symmetry in QHE can be made unitary through measure modifications and explores the connection between unitarity, anomalies, and geometric transformations.

Findings

Classical conformal symmetry in QHE is unitary.

Quantum conformal symmetry is initially non-unitary.

Proper measure adjustments restore unitarity in the quantum case.

Abstract

We study the realization of conformal symmetry in the QHE as part of the $W_\infty$ algebra. Conformal symmetry can be realized already at the classical level and implies the complexification of coordinate space. Its quantum version is not unitary. Nevertheless, it can be rendered unitary by a suitable modification of its definition which amounts to taking proper care of the quantum measure. The consequences of unitarity for the Chern-Simons theory of the QHE are also studied, showing the connection of non-unitarity with anomalies. Finally, we discuss the geometrical paradox of realizing conformal transformations as area preserving diffeomorphisms.