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AIP conference proceedings

Heisenberg uncertainty in reduced power algebras

Elemér Elad Rosinger

Published Jan 1, 2012• 15 citations
Publisher

Abstract

The Heisenberg uncertainty relation is known to be obtainable by a \npurely mathematical argument. Based on that fact, here it is shown \nthat the Heisenberg uncertainty relation remains valid when Quantum \nMechanics is re-formulated within far wider frameworks of scalars, \nnamely, within one or the other of the infinitely many reduced power \nalgebras which can replace the usual real numbers R, or complex numbers \nC. A major advantage of such a re-formulation is, among others, \nthe disappearance of the well known and hard to deal with problem \nof the so called ”infinities in Physics”. The use of reduced power algebras \nalso opens up a foundational question about the role, and in \nfact, about the very meaning and existence, of fundamental constants \nin Physics, such as Planck’s constant h. A role, meaning, and existence \nwhich may, or on the contrary, may not be so objective as to \nbe independent of the scalars used, be they the usual real numbers \nR, complex numbers C, or scalars given by any of the infinitely many \nreduced power algebras, algebras which can so easily be constructed \nand used.

DOI
10.1063/1.4773168

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The Heisenberg uncertainty relation is known to be obtainable by a \npurely mathematical argument. Based on that fact, here it is shown \nthat the Heisenberg uncertainty relation remains valid when Quantum \nMechanics is re-formulated within far wider frameworks of scalars, \nnamely, within one or the other of the infinitely many reduced power \nalgebras which can replace the usual real numbers R, or complex numbers \nC. A major advantage of such a re-formulation is, among others, \nthe disappearance of the well known and hard to deal with problem \nof the so called ”infinities in Physics”. The use of reduced power algebras \nalso opens up a foundational question about the role, and in \nfact, about the very meaning and existence, of fundamental constants \nin Physics, such as Planck’s constant h. A role, meaning, and existence \nwhich may, or on the contrary, may not be so objective as to \nbe independent of the scalars used, be they the usual real numbers \nR, complex numbers C, or scalars given by any of the infinitely many \nreduced power algebras, algebras which can so easily be constructed \nand used.
The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of {\it scalars}, namely, within one or the other of the infinitely many {\it reduced power algebras} which can replace the usual real numbers R\mathbb{R}R, or complex numbers C\mathbb{C}C. A major advantage of such a re-formulation is, among others, the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics". The use of reduced power algebras also opens up a foundational question about the role, and in fact, about the very meaning and existence, of fundamental constants in Physics, such as Planck's constant hhh. A role, meaning, and existence which may, or on the contrary, may not be so objective as to be independent of the scalars used, be they the usual real numbers R\mathbb{R}R, complex numbers C\mathbb{C}C, or scalars given by any of the infinitely many reduced power algebras, algebras which can so easily be constructed and used.
The No-Cloning property in Quantum Computation is known not to depend on the unitarity of the operators involved, but only on their linearity. Based on that fact, here it is shown that the No-Cloning property remains valid when Quantum Mechanics is re-formulated within far wider frameworks of {\it scalars}, namely, one or the other of the infinitely many {\it reduced power algebras} which can replace the usual real numbers R\mathbb{R}R, or complex numbers C\mathbb{C}C.
Background: There has been altercation for and against Heisenberg uncertainty principles, whose earlier mathematical form has assumed other forms. This is despite the fact that the original form ought to be for the hydrogen atom only. Methods: the method is purely theoretical and computational, involving the use of spectroscopic data in the literature. Aim and objectives: The aim of the research was to show that half of reduced Planck’s constant (ℏ/2) cannot be restricted to a lower limit to the product of the uncertainty in the momentum and the uncertainty in the position. The objectives were: 1) to undertake the derivation of an equation unifying the Heisenberg uncertainty principle and atomic principles; and 2) to give evidence that half of the reduced Planck’s constant (ℏ/2) is strictly for hydrogen. Results and discussion: With hydrogen apart from other elements, fractions and multiples of ℏ/2, 0.0053–0.0264 corresponding to the radii ranging between a0/100 and a0/20 (a0 is the Bohr’s radius), and 5-26 corresponding to the radii ranging between 4a0 and 49a0 were demonstrated; the probability of locating the position and momentum increases with decreasing radii. Conclusion: The equations for the determination of the fractions and multiples of ℏ/2 were derived. Computation gave values less than and higher than ℏ/2; the latter was found to be restricted to hydrogen only. The possibility of linking the product of uncertainties in position and momentum to Pa may be reserved for a future study.
Uncertainty relations are commonly praised as one of the central pillars of quantum theory. Usually, they are taught in the first weeks of a beginners lecture, and introduced in the ?rst chapters of a textbook. However, their precise operational meaning and a formulation in a general context, i.e. beyond the example of position and momentum observables, are often left out. The reasoning for this is twofold: On one hand, an exact operational definition of uncertainty, indeterminacy and a corresponding uncertainty principle has been the content of many debates since the early days of quantum mechanics until today. From a modern perspective, we have the consent that there are at least the two notions of preparation and measurement uncertainty: the first notion prohibits the existence of dissipation free states and the latter one the existence of error free joint measurements. On the other hand, we have that, the mathematical tools, which are needed for comprehensive treatment of uncertainty relations in a general context, are still under development and usually go far beyond the mathematical level of an introductory course. In this thesis we will investigate these two notions of uncertainty, their corre- sponding uncertainty relations, such as their interplay. The aim of this thesis is to give answers to the central questions: (1.) Which quantities should be used to formulate uncertainty? (2.) How can we compute uncertainty relations for those? (3.) Are there connections between the two notions of uncertainty? We will do this, whenever possible, in a most general context and with a focus on relevant examples, otherwise. Therefore, we will consider constructions of mea- surement errors and deviation measures that quantify uncertainty, based on, so called, cost functions. Commonly used uncertainty measures like variances, en- tropies, and the Hamming distance are examples for these. We will investigate the structure of the corresponding uncertainty relations and provide several methods that enable us to compute them. The third question is addressed by a theorem that shows, for sharp observables, that measurement uncertainty relations can be lower bounded by preparation uncertainty relations, whenever the same cost function is used.
The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relation to be universally valid and how we experimentally quantify the error and the disturbance to refute the old relation and to confirm the new relation.