Related papers: Multiplication in Newton's Principia
In a mathematical context in which one can multiply distributions the "`formal"' nonperturbative canonical Hamiltonian formalism in Quantum Field Theory makes sense mathematically, which can be understood a priori from the fact the so…
Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this…
We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For…
Application of Newton's ideas from "Principia" gives many new results in mechanics. Here we explore the question ``What form of extra force will maintain the magnitude of a vector constant of the motion while changing its direction?''
A new formulation of quantum mechanics is proposed based on a new principle that can be considered a generalization of the Born rule. The principle is composed of a mathematical expression and an associated interpretation, and establishes a…
A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with…
This work develops a conceptual framework for the foundations of quantum physics, linking two main approaches: the algebraic formulation and quantum probability. Rather than proposing new axioms or theories, the text reorganizes and…
Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…
We define the notion of mutual quantum measurements of two macroscopic objects and investigate the effect of these measurements on the velocities of the objects. We show that multiple mutual quantum measurements can lead to an effective…
The additivity of classical probabilities is only the first in a hierarchy of possible sum-rules, each of which implies its successor. The first and most restrictive sum-rule of the hierarchy yields measure-theory in the Kolmogorov sense,…
The centuries-long practice of the teaching turned mechanics into an academic construct detached from its underlying science, the physics of macroscopic bodies. In particular, the regularities that delineate the scope of validity of…
This article is an attempt for a new vision of the basics of Physics, and of Relativity, in particular. A new generalized principle of inertia is proposed, as an universal principle, based on universality of the conservation laws, not…
Despite its enormous empirical success, the formalism of quantum theory still raises fundamental questions: why is nature described in terms of complex Hilbert spaces, and what modifications of it could we reasonably expect to find in some…
Newton's "superb theorem" for the gravitational inverse-square-law force states that a spherically symmetric mass distribution attracts a body outside as if the entire mass were concentrated at the center. This theorem is crucial for…
A new formulation of quantum mechanics is developed which does not require the concept of the wave-particle duality. Rather than assigning probabilities to outcomes, probabilities are instead assigned to entire fine-grained histories. The…
We consider a quantum test particle in the background of a Newtonian gravitational field in the framework of Cartan's formulation of nonrelativistic spacetimes. We have proposed a novel quantization of a point particle which amounts to…
The principle of mass additivity states that the mass of a composite object is the sum of the masses of its elementary components. Mass additivity is true in Newtonian mechanics but false in special relativity. Physicists have explained why…
Newton introduced the concept of mass in his {\it Principia} and gave an intuitive explanation for what it meant. Centuries have passed and physicists as well as philosophers still argue over its meaning. Three types of mass are generally…
Applications of quantum mechanics have led to many successful predictions and explanations of puzzling phenomena, and we now apply quantum mechanics to gain, process, and communicate information in novel ways. We can understand quantum…
The concept of quantum commutativity with respect to an action or coaction of a given Hopf algebra is used for the algebraic description of a system of particles and their interaction with certain quantum field. Graded commutativity and…