New Quantum Spin Perspective and Geometrical Operators of Quantum Geometry

In this paper, we propose a new perspective of quantum spin (angular momentum) in which the Boltzmann constant $k_{\beta}$, Planck temperature $T_{P}$, Planck mass $m_{P}$ and Planck area $l_{P}^{2}$ are the integral part of the total angular momentum $J$. With the aid of this new perspective, we modify the equation of the area and volume operator. In the quantum geometry, for $SO(3)$ group, the angular momentum operators $J^{k}$ is the $k$th Lie group generator $T^{k}$; hence, $T^{k} \equiv J^{k}$. Therefore, new perspective of quantum spin can be directly applicable to quantum geometry. From data, the value of the area operator $\hat{A}_{S}$ increases with $n^{2}$ in discrete way that suggests discrete spectrum of the area operator similar to the actual formula of the area operator. This perspective provides an auto-correct or auto-balance mechanism within the equation of these geometrical operators. At the quantum gravity scale, it means that the mutual small change in $T_{P}$, $m_{P}$, and $l_{P}^{2}$ occur in such a way that $\hbar$, $l_{P}$ and $\hat{A}_{S}$ and $\hat{V}_{S}$ remain invariant for a value of $j_{i}$. The constancy of the reduced Planck constant $\hbar$ in the geometrical operators can provide a way through which smooth transition of the Planck scale to the nuclear or the atomic scale can be understood.