scholarly journals A Paradox of Unity

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Summation Formula ◽  
Generalized Functions ◽  
The Other ◽  
Complex Numbers ◽  
Dirac Delta ◽  
Max Born ◽  
Open Question

In previous studies we found that generalized functions can be smooth, discrete, periodic or discrete periodic and they can either be local or global and they are regular or generalized functions. We also saw that these properties were related to Poisson’s summation formula on one hand and to Heisenberg’s uncertainty principle on the other. In this paper, we interlink these studies and show that scalars (real or complex numbers) considered as trivial functions are discrete and periodic, local and global as well as regular and generalized, simultaneously. However, this is also a paradox because it means that Dirac’s δ and 1 (its Fourier transform) coincide. They both are unity. We show that δ and 1 coincide in the sense of scalars (real or complex numbers) but they differ in the sense of (generalized) functions. This result can moreover be related to Max Born’s principle of reciprocity. It also answers an open question in present-day quantum mechanics because it means that the Dirac delta squared is simply delta.

scholarly journals On the Duality of Regular and Local Functions

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Summation Formula ◽  
Generalized Functions ◽  
Tempered Distributions ◽  
Regular Functions ◽  
Local Functions ◽  
The Fourier Transform

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

scholarly journals On the Duality of Regular and Local Functions

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Summation Formula ◽  
Generalized Functions ◽  
Tempered Distributions ◽  
Regular Functions ◽  
Local Functions ◽  
The Fourier Transform

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

CHANGES OF TOPOLOGY AND CHANGES OF SIGNATURE

1994 ◽  
Vol 03 (01) ◽  
pp. 61-70 ◽  
Author(s):  
G W GIBBONS
Keyword(s):  
Quantum Mechanics ◽  
The Other ◽  
Complex Numbers ◽  
Physical Processes ◽  
Closed Timelike Curves ◽  
Important Distinction ◽  
Evolution Of The Universe ◽  
Lorentzian Signature ◽  
The Universe

Some recent ideas about topology and signature changing spacetimes are described. If spacetime is everywhere Lorentzian but non-orientable, one can sometimes avoid closed timelike curves, but one must must consider pinors rather than spinors. One finds that there is now an important distinction between signature (+ + + −) and (− − − +). In some cases one signature may be excluded and the other allowed. Topology changing spacetimes with domains of non-Lorentzian signature are considered. These domains may be Riemannian or Kleinian (+ + − −). It is argued that our present signature, together with the idea of time must have arisen as the consequence of physical processes. This emergence of the idea of time is also connected with the origin of the complex numbers in Quantum Mechanics which should also be regarded as the consequence of the evolution of the universe.

Paradoxes of Nature

2020 ◽  
pp. 87-98
Author(s):  
Demetris Nicolaides
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Apparent Motion ◽  
Space Time ◽  
The Other ◽  
Even Start ◽  
Other Hand ◽  
Logical Paradoxes

Parmenides’s insinuation of an unchanging universe is assertively supported by Zeno with various logical paradoxes that question the very nature of plurality, space, time, and the reality of apparent motion. The dichotomy is his most famous paradox. To begin a trip, say, from here to the door, a traveler must travel the first half of it, but before she does that she must travel half of the first half, and in fact half of that, ad infinitum. Since there will always exist a smaller first half to be traveled first, Zeno questions whether a traveler can ever even start a trip. Zeno’s analysis is logical; on the other hand, things everywhere appear to be moving. Hence, either Zeno’s reasoning is wrong or appearances are deceptive. Empowered by the uncertainty principle of quantum mechanics, it will be argued that, at best, the phenomenon of motion is experimentally unverifiable!

Free Will and Quantum Mechanics

The Monist ◽  
2020 ◽  
Vol 103 (4) ◽  
pp. 415-426
Author(s):  
Mario De Caro ◽  
Hilary Putnam
Keyword(s):  
Quantum Mechanics ◽  
Free Will ◽  
Uncertainty Principle ◽  
The Other ◽  
Quantum Mechanical ◽  
Quantum Jumps ◽  
Seminal Work ◽  
Single Atoms ◽  
The Way

Abstract In the last few decades, the relevance of quantum mechanics to the free-will debate has been discussed at length, especially in relation to the prospects of libertarianism (the view according to which humans enjoy an indeterministic kind of free will). Basing his interpretation on Anscombe’s seminal work, Putnam argued in 1979 that, given that quantum mechanical indeterminacy is holistic at the macrolevel—i.e., it is not traceable to atomistic events such as quantum jumps of single atoms—it can provide libertarians with the kind of freedom they seek. As shown in this article, however, Putnam ultimately reached the conclusion—together with the other author of this article—that his argument was wrong due to problems with the way it appealed to the Uncertainty Principle.**

scholarly journals The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1685
Author(s):  
Rui Jing ◽  
Bei Liu ◽  
Rui Li ◽  
Rui Liu
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Transformation Operator ◽  
Local Analysis ◽  
Transform Domain ◽  
Uncertainty Principles ◽  
The Fourier Transform ◽  
Heisenberg's Uncertainty Principle ◽  
Fourier Transform Domain

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.

scholarly journals Generalized uncertainty principle and its implications on geometric phases in quantum mechanics

2021 ◽  
Vol 136 (2) ◽  
Author(s):  
Giuseppe Gaetano Luciano ◽  
Luciano Petruzziello
Keyword(s):  
Quantum Mechanics ◽  
Phase Shift ◽  
Uncertainty Principle ◽  
Generalized Uncertainty Principle ◽  
Upper Bounds ◽  
The Other ◽  
Quantum Mechanical ◽  
Geometric Phases ◽  
Deformation Parameters ◽  
Aharonov Bohm

AbstractWe study the implications of the generalized uncertainty principle (GUP) with a minimal measurable length on some quantum mechanical interferometry phenomena, such as the Aharonov–Bohm, Aharonov–Casher, COW and Sagnac effects. By resorting to a modified Schrödinger equation, we evaluate the lowest-order correction to the phase shift of the interference pattern within two different GUP frameworks: the first one is characterized by the redefinition of the physical momentum only, and the other is a Lorentz covariant GUP which also predicts non-commutativity of spacetime. The obtained results allow us to fix upper bounds on the GUP deformation parameters which may be tested through future high-precision interferometry experiments.

Quantum Theory

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Principle ◽  
Quantum Spin ◽  
Quantum States ◽  
Wave Functions ◽  
Symbolic Representations ◽  
Quantum Numbers ◽  
The Subject ◽  
Heisenberg's Uncertainty Principle

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.

Incarnation as Awakening: Katsumi Takizawa Reading Karl Barth

Exchange ◽  
2007 ◽  
Vol 36 (2) ◽  
pp. 144-155
Author(s):  
Susanne Hennecke
Keyword(s):  
Karl Barth ◽  
The Other ◽  
Historical Jesus ◽  
Kyoto School ◽  
Dialectical Theology ◽  
Christian Theologian ◽  
The One ◽  
Open Question

AbstractThis contribution deals with the thinking of the Buddhist philosopher and Christian theologian Katsumi Takizawa (1909-1984) on incarnation. Firstly, it gives a short biographical and theological introduction to Takizawa, who was influenced not only by the "father" of the so-called dialectical theology, Karl Barth, but also by one of the famous figures of the Kyoto-school, the philosopher Kitaro Nishida.This contribution concentrates, secondly, on Takizawa's the-anthropological re-interpretation of the incarnation. It is argued that for Takizawa incarnation has to be seen as an awakening of the historical Jesus (or other historical phenomena) to what he calls the original fact: the eternal relationship between God and man.Thirdly, this contribution discusses the the-anthropological thinking of Takizawa about incarnation in five short points. Apart from the positive challenges of Takizawa's thinking especially for the theology of Karl Barth, it marks clearly the most thrilling point between Takizawa's thinking on the one side and that of scholars in Barthian theology on the other side. The open question that comes up is if incarnation really can be thought without a historical mediation or mediator, as Takizawa seems to claim.

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