Moyal Quantization on Fuzzy Sphere

TL;DR

This paper explores Moyal quantization on the fuzzy sphere, constructing the $su(2)$ algebra from canonical coordinates and defining a novel vacuum state without using traditional creation and annihilation operators.

Contribution

It introduces a new approach to fuzzy sphere quantization based on Moyal methods, constructing the algebra and vacuum state without standard creation and annihilation operators.

Findings

Constructed $su(2)$ algebra from canonical coordinates.

Defined vacuum as powers of $a^*$, differing from flat space.

Demonstrated analogy of creation operator without traditional operators.

Abstract

We study the quantization of compact space on the basis of the Moyal quantization. We first construct the $su(2)$ algebra that are the functions of canonical coordinates $a$ and $a^*$. We make use of them to define the adjoint operators, which is used to define the fuzzy sphere and constitute the algebra. We show that the vacuum is constructed as the powers of $a^*$, in contrast to the flat case where the vacuum is defined by the exponential function of $a$ and $a^*$. We present how the analogy of the creation operator acting on the vacuum is obtaied. The construction does not resort to the ordinary creation and annihilation operators.