Spherically Symmetric Noncommutative Spacetime via Exotic Atomic Transitions

In discussing non-commutative spacetime, the generally studied $\theta$-Poincare model is inconsistent with bound states. In this Letter, we develop the formalism and study the phenomenology of another model $\mathcal{B}_{\chi \hat{n}}$ by the twisted permutation algebra and extend the Pauli Exclusion Principle(PEP) into non-commutative spacetime. The model also implies time quantization and can avoid UV/IR mixing. Applying it to atomic systems, we show that the model with newly induced phase factors can cause exotic transitions consisting of three electrons in the 1S orbit of atoms. The transition rate is derived, and the upper bound of non-commutative parameter $\chi$ is thus set by utilizing data from the low-energy and low-background experiments, where strongest constraint $\chi\leq4.05\times10^{-30}$ eV$^{-1}$ at 90\% C.L. is given by XENONnT, with the time quanta $\Delta t\sim 2.67\times 10^{-45} s$, equivalent to twenty times smaller than the Planck time.