Explanation of the generalizations of uncertainty principle from coordinate and momentum space periodicity

Article Type

Research Article

Publication Title

European Physical Journal Plus

Abstract

Generalizations of coordinate x-momentum px uncertainty principle, with Δx- and Δpx-dependent terms (Δ denoting standard deviation), (Formula presented.) have provided rich dividends as a poor person’s approach toward 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 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 ϕ—angular momentum Lz, (Formula presented.) purely from a consistent implementation of periodic nature of the angle variable ϕ, without changing the ϕ,Lz canonical commutation relation. Structurally this is identical to the well-known extended uncertainty principle. We directly apply this formalism to formulate the ΔxΔpx extended uncertainty principle. We identify β with an observed length scale relevant in astrophysics context. We speculate about the α extension.

DOI

10.1140/epjp/s13360-024-05366-x

Publication Date

7-1-2024

Comments

Open Access; Green Open Access

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