New Uncertainty Principle for a particle on a Torus Knot
Madhushri Roy Chowdhury, Subir Ghosh
TL;DR
This work investigates uncertainty relations for a quantum particle constrained to move along a torus knot embedded on a torus, emphasizing the role of two intrinsic periodicities and knot geometry. It adapts periodic Cartesian coordinates within the Kennard–Robertson framework and, in the thin-torus limit, derives URs that explicitly involve knot- and torus-parameters (e.g., $\alpha=-\frac{q}{p}$ and $\gamma=\cosh\eta_0$), such as $\sigma_x^2\sigma_{L_z}^2 \ge \frac{\hbar^2}{4}\langle y+\frac{\alpha}{a}zx \rangle^2$. The authors show two distinct SD/UR patterns corresponding to knot-restrictions $|n-k|=p$ or $|n-k|=p+q$, and demonstrate that in the thin-torus limit the results reduce to the circle case, with the local geometry, not topology, driving the uncertainty relations. They further introduce a mean resultant length and a combined SD–$L_z$ uncertainty, discussing experimental prospects in knotted fibre systems and proposing extensions to noncommutative torus knots.
Abstract
The present work deals with quantum Uncertainty Relations (UR) subjected to the Standard Deviations (SD) of the relevant dynamical variables for a particle constrained to move on a torus knot. It is important to note that these variables have to obey the two distinct periodicities of the knotted paths embedded on the torus. We compute generalized forms of the SDs and the subsequent URs (following the Kennard-Robertson formalism). These quantities explicitly involve the torus parameters and the knot parameters where restrictions on the latter have to be taken into account. These induce restrictions on the possible form of wave functions that are used to calculate the SDs and URs and in our simple example, two distinct SDs and URs are possible. In a certain limit (thin torus limit), our results will reduce to the results for a particle moving in a circle. An interesting fact emerges that in the case of the SDs and URs, the local geometry of the knots plays the decisive role and not their topological properties.