Explanation of the Generalizations of Uncertainty Principle from Coordinate and Momentum Space Periodicity

Abstract: Generalizations of coordinate $x$-momentum $p_x$ Uncertainty Principle, with $\Delta x$ and $\Delta p_x$ dependent terms ($\Delta$ denoting standard deviation), $$\Delta x \Delta p_x\geq i\hbar (1+\alpha\Delta p_x^2 +\beta \Delta x^2)$$ have provided rich dividends as a poor person's approach towards Quantum Gravity, because these can introduce coordinate and momentum scales ($\alpha,\beta$ ) that are appealing conceptually. However, these extensions of Uncertainty Principle are purely phenomenological in nature. Apart from the inherent ambiguity in their explicit structures, the introduction of generalized commutations relations compatible with the the uncertainty relations has some drawbacks. In the present paper we reveal that these generalized Uncertainty Principles can appear in a perfectly natural way, in canonical quantum mechanics, if one assumes a periodic nature in coordinate or momentum space, as the case may be. We bring in to light quite old, (but not so well known), works by Judge and by Judge and Lewis, that explain in detail how a consistent and generalized Uncertainty Principle is induced in the case of angle $\phi$ - angular momentum $L_z$, $$\Delta \phi \Delta L_z \geq i\hbar (1 +\nu \Delta \phi^2)$$ purely from a consistent implementation of {\it{periodic}} nature of the angle variable $\phi $, without changing the $\phi, L_z$ canonical commutation relation. {\it{Structurally this is identical to the well known Extended Uncertainty Principle.}} We directly apply this formalism to formulate the $\Delta x \Delta p_x $ Extended Uncertainty Principle. We identify $\beta$ with an observed length scale relevant in astrophysics context. We speculate about the $\alpha$ extension.