Operator Ordering Problem of the Nonrelativistic Chern-Simons Theory

TL;DR

This paper demonstrates that by imposing Galilei covariance, the operator ordering ambiguity in nonrelativistic Abelian Chern-Simons theory can be resolved at the quantum level, ensuring consistent equations of motion regardless of regularization choices.

Contribution

It shows how Galilei covariance constrains operator ordering in nonrelativistic Chern-Simons theory, resolving quantization ambiguities and maintaining consistent equations of motion.

Findings

Galilei covariance fixes operator ordering ambiguities

Regularization prescriptions do not alter the physical equations

Unique ordering is determined for point sources

Abstract

The operator ordering problem due to the quantization or regularization ambiguity in the Chern-Simons theory exists. However, we show that this can be avoided if we require Galilei covariance of the nonrelativistic Abelian Chern-Simons theory even at the quantum level for the extended sources. The covariance can be recovered only by choosing some particular operator orderings for the generators of the Galilei group depending on the quantization ambiguities of the $gauge-matter$ commutation relation. We show that the desired ordering for the unusual prescription is not the same as the well-known normal ordering but still satisfies all the necessary conditions. Furthermore, we show that the equations of motion can be expressed in a similar form regardless of the regularization ambiguity. This suggests that the different regularization prescriptions do not change the physics. On the other hand, for the case of point sources the regularization prescription is uniquely determined, and only the orderings, which are equivalent to the usual one, are allowed.