Preprint Article Version 1 This version not peer reviewed

A Paradox of Unity

Version 1 : Received: 29 December 2017 / Approved: 2 January 2018 / Online: 2 January 2018 (06:47:57 CET)

How to cite: Fischer, J.V. A Paradox of Unity. Preprints 2018, 2018010003 (doi: 10.20944/preprints201801.0003.v1). Fischer, J.V. A Paradox of Unity. Preprints 2018, 2018010003 (doi: 10.20944/preprints201801.0003.v1).

Abstract

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.

Subject Areas

Dirac impulse; generalized functions; tempered distributions; paradox of unity; Dirac's delta squared; quantum mechanics; Born's principle of reciprocity; scalar self-reciprocity

Readers' Comments and Ratings (2)

Comment 1
Received: 12 January 2018
Commenter: Jens V. Fischer
The commenter has declared there is no conflict of interests.
Comment: I admit that I obviously overloaded this paper. Despite the circumstance that no major mistake has been found so far, a reviewed version of this paper will not appear. Smaller, more manageable portions of it will be submitted instead. Please feel free to send me your comments.
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Comment 2
Received: 14 January 2018
Commenter: Jens V. Fischer
The commenter has declared there is no conflict of interests.
Comment: More precisely, the term "discrete function" in Figure 2 refers to all generalized functions in S' which are not smooth (infinitely differentiable) in the ordinary functions sense. In this way, the sum of the function that is constantly one and a Dirac delta would be a “fully discrete function”. The concatenation of periodization and discretization (or discretization and periodization) hence becomes an injection into the subspace of “fully discrete functions”.
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