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This paper explores the consequences of discretization of space in quantum mechanics. Specifically, I show that under normal conditions in space where the wave function is uniform across a quantum of space, Heisenberg’s uncertainty principle is identically satisfied, where the “normal conditions” are defined as p x Δ x / ħ ≪ 1 with Δ x characterizing the size of a quantum of position space or equivalently Δ p x x / ħ ≪ 1 with Δ p x characterizing the size of a quantum of momentum space (this limiting behavior is also at the core of Nyquist sampling theorem; see Jerri (1977); Vaidyanathan (IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(9), 1094–1109, 2001); Ozaktas et al. (2001)). Moreover, I derive an equation similar to Heisenberg’s uncertainty principle which imposes a constraint on the minimum values of “precision” of space (i.e. the maximum values for Δ x and Δ p x ). Additionally, I show that in discretized space the simultaneous minimum position and minimum momentum uncertainties are unphysical. [ABSTRACT FROM AUTHOR]

In the present paper, we review some recent results about commutative positive operator valued measures (POVMs) and single out some open problems. We introduce a conjecture about the extension of some recent results and prove some important consequences of such conjecture. In particular, we prove that it implies the universal character of some of the mathematical objects we introduce, i.e., the fact that they do not depend on the POV measure we are considering. We analyze the relevance of this result. Finally, we point out that some of the results we present admit a constructive proof and we show the relevance of this fact to the theory of commutative POV measures. [ABSTRACT FROM AUTHOR]

The rules of union and intersection of probabilistic fuzzy sets guided us to construct a related operator algebra. In a Hilbert space, where each fuzzy set is represented by an orthonormal vector, the union and the intersection operators generate a well-defined algebra with a unique representation. [ABSTRACT FROM AUTHOR]