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

Generalizations of coordinate x-momentum p_x Uncertainty Principle, with Δ x and Δ p_x dependent terms (Δ denoting standard deviation), Δ x Δ p_x≥ iħ (1+αΔ p_x^2 +βΔ x^2) have provided rich dividends as a poor person's approach towards Quantum Gravity, because these can introduce coordinate and momentum scales (α,β ) 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 Uncertainty Principle has serious 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 no so well known), works by Judge and by Judge and Lewis, that explains in detail how the popularly known structure of Extended Uncertainty Principle is generated in the case of angle ϕ angular momentum L_z, ΔϕΔ L_z ≥ iħ (1 +νΔϕ^2) purely from a consistent implementation of periodic nature of the angle variable ϕ, without changing the ϕ, L_z canonical commutation relation. We directly apply this formalism to construction to formulate generalizations in Δ x Δ p_x Uncertainty Principle. We identify β with an observed length scale relevant in astrophysics context. We speculate about the α extension.