Asymptotic commutativity of quantized spaces: the case of $\mathbb{CP}^{p,q}$

TL;DR

This paper develops a method to quantize complex projective spaces P^{p,q} and constructs star products that preserve their isometry algebra, showing that quantum corrections vanish asymptotically, making the spaces approach classical geometry.

Contribution

It introduces a unique quantization procedure for P^{p,q} that maintains full isometry algebra and extends previous results on Euclidean AdS_2 to these spaces.

Findings

Quantum corrections to Killing vectors vanish asymptotically.

Star product trivializes to pointwise product in the asymptotic limit.

Quantized spaces approach classical P^{p,q} in the limit.

Abstract

We present a procedure for quantizing complex projective spaces $\mathbb{CP}^{p,q}$, $q\ge 1$, as well as construct relevant star products on these spaces. The quantization is made unique with the demand that it preserves the full isometry algebra of the metric. Although the isometry algebra, namely $su(p+1,q)$, is preserved by the quantization, the Killing vectors generating these isometries pick up quantum corrections. The quantization procedure is an extension of one applied recently to Euclidean $AdS_2$, where it was found that all quantum corrections to the Killing vectors vanish in the asymptotic limit, in addition to the result that the star product trivializes to pointwise product in the limit. In other words, the space is asymptotically anti-de Sitter making it a possible candidate for the $AdS/CFT$ correspondence principle. In this article, we find indications that the results for quantized Euclidean $AdS_2$ can be extended to quantized $\mathbb{CP}^{p,q}$, i.e., noncommutativity is restricted to a limited neighborhood of some origin, and these quantum spaces approach $\mathbb{CP}^{p,q}$ in the asymptotic limit.