A comment on the magical realizations of W_3

TL;DR

This paper investigates the quantization of magical realizations of W_3 related to Jordan algebras, concluding that none of these special realizations can be quantized, thus resolving an open problem in the field.

Contribution

It proves that all four sporadic magical realizations of W_3 cannot be quantized, completing the analysis of their quantization properties.

Findings

All magical realizations fail to survive quantization.

The generic realizations are quantizable, unlike the magical ones.

The problem of quantizing magical realizations is definitively closed.

Abstract

In the process of investigating classical realizations of W_3 in terms of free bosons, Romans unveiled a relation to finite-dimensional Jordan algebras with a cubic norm. These algebras have been classified and consist of an infinite series (yielding the ``generic'' realizations) and four sporadic algebras associated to the real division algebras (which yield the ``magical'' realizations). The generic realizations were shown by Romans to quantize, who left the problem of the quantization of the magical realizations open. In later work, Mohammedi showed that the first two magical realizations did not survive quantization. In this note we close the problem by showing that neither do the other two magical realizations.