The Uncertainty Principle

The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$ , ( $w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω$ . 3) The time weight scale $a$ , ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$ , ${w_2}_b(ω)=w_2(ωb^{-1})$ . "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights $w_1(x)$ and $w_2(ω)$ , we introduced a three dimensional set in $R^3$ that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$ , $k\in N$ , up to a constant.The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$, ($w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω$. 3) The time weight scale $a$, ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$, ${w_2}_b(ω)=w_2(ωb^{-1})$. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights $w_1(x)$ and $w_2(ω)$, we introduced a three dimensional set in $R^3$ that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, $w_1(x)=w_2(x)=x^k$, $k\in N$, up to a constant.

arXiv:0807.2190Jul 14, 20084 citations

Giulio Armando Ottonello

Springer geochemistryJan 1, 202415 citations

Research on Generalized Uncertainty Principles

Xiaotong Wang, Guanlei Xu, Lijia Zhou +2 more

The uncertainty principle plays an important role in both physics and mathematics, and it also plays an important role in information science. This paper reviewed the Heisenberg’s uncertainty principles, windowed uncertainty principles, logarithmic uncertainty principles, entropic uncertainty principles and so on in the fractional Fourier transform domains and linear canonical transform domains in great details. The development and applications of generalized uncertainty principles were analyzed, and the trend and direction were given as well.The uncertainty principle plays an important role in both physics and mathematics, and it also plays an important role in information science. This paper reviewed the Heisenberg’s uncertainty principles, windowed uncertainty principles, logarithmic uncertainty principles, entropic uncertainty principles and so on in the fractional Fourier transform domains and linear canonical transform domains in great details. The development and applications of generalized uncertainty principles were analyzed, and the trend and direction were given as well.

Jan 1, 20163 citations

The Uncertainty Principle

University of Pittsburgh Press eBooksSep 7, 20173 citations

The Uncertainty Principle

Mark Kraushaar

Nov 8, 20110 citations

Expanding Uncertainty Principle To Certainty-Uncertainty Principles With Neutrosophy And Quad-Stage Method

Fu Yuhua

The most famous contribution of Heisenberg is uncertainty principle. But the original uncertainty principle is improper. Considering all the possible situations (including the case that people can create laws) and applying Neutrosophy and Quad-stage Method, this paper presents "certainty-uncertainty principles" with general form and variable dimension fractal form. According to the classification of Neutrosophy, "certainty-uncertainty principles" can be divided into three principles in different conditions: "certainty principle", namely a particle’s position and momentum can be known simultaneously; "uncertainty principle", namely a particle’s position and momentum cannot be known simultaneously; and neutral (fuzzy) "indeterminacy principle", namely whether or not a particle’s position and momentum can be known simultaneously is undetermined.The most famous contribution of Heisenberg is uncertainty principle. But the original uncertainty principle is improper. Considering all the possible situations (including the case that people can create laws) and applying Neutrosophy and Quad-stage Method, this paper presents "certainty-uncertainty principles" with general form and variable dimension fractal form. According to the classification of Neutrosophy, "certainty-uncertainty principles" can be divided into three principles in different conditions: "certainty principle", namely a particle’s position and momentum can be known simultaneously; "uncertainty principle", namely a particle’s position and momentum cannot be known simultaneously; and neutral (fuzzy) "indeterminacy principle", namely whether or not a particle’s position and momentum can be known simultaneously is undetermined.

Neutrosophic sets and systemsAug 3, 20150 citations

\int_{-\infty}^\infty |f(x)|^2w_1(x)dx

, (

w_1(x)

is a non-negative function). 2) The weighted frequency spreading

\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω

. 3) The time weight scale

a

{w_1}_a(x)=w_1(xa^{-1})

and 4) The frequency weight scale

b

{w_2}_b(ω)=w_2(ωb^{-1})

. "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights

w_1(x)

and

w_2(ω)

, we introduced a three dimensional set in

R^3

that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights,

w_1(x)=w_2(x)=x^k

k\in N

, up to a constant.