scholarly journals The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 659
Author(s):  
Gerrit Coddens
Keyword(s):  
Quantum Mechanics ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Uniform Motion ◽  
Statistical Interpretation ◽  
Geometrical Meaning ◽  
New Approach ◽  
Minkowski Space Time ◽  
Homogeneous Lorentz Group

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART I: THRESHOLD OF COMPLEXITY

2002 ◽  
Vol 12 (12) ◽  
pp. 2655-2766 ◽  
Author(s):  
LEON O. CHUA ◽  
SOOK YOON ◽  
RADU DOGARU
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Dynamical System ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Local Rule ◽  
Boolean Cube ◽  
New Approach ◽  
Nonlinear Dynamical ◽  
Complexity Index ◽  
Science Part

This tutorial provides a nonlinear dynamics perspective to Wolfram's monumental work on A New Kind of Science. By mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram's empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenomena is that it is based on concepts from nonlinear dynamics and attractors where many fuzzy concepts introduced by Wolfram via brute force observations can be defined and justified via mathematical analysis. The main result of Part I is the introduction of a fundamental concept called linear separability and a complexity index κ for each local Rule which characterizes the intrinsic geometrical structure of an induced "Boolean cube" in three-dimensional Euclidean space. In particular, Wolfram's seductive idea of a "threshold of complexity" is identified with the class of local Rules having a complexity index equal to 2.

Complex potential equations II. An application to general relativity

1976 ◽  
Vol 80 (2) ◽  
pp. 349-355 ◽  
Author(s):  
C. B. Collins
Keyword(s):  
General Relativity ◽  
Complex Potential ◽  
Three Dimensional ◽  
Arbitrary Real Number ◽  
Dimensional Euclidean Space ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Minkowski Space Time ◽  
Potential Formalism ◽  
Image Position

AbstractThe results of a previous paper are applied to a study of the class of Kerr–Schild metrics in general relativity. These metrics have the formwhere η is the flat (Minkowski) space-time metric, m is an arbitrary real number, and 1 is a null covector. It is already known that for a certain restricted subclass of these metrics, the vacuum Einstein field equations, viz.can be written in the formwhere γ is a complex potential. Using the methods developed in a previous paper, such space-times are characterized by means of a special family of complex surfaces in three-dimensional Euclidean space, and the exact solutions for the metric g are consequently recovered. It is also shown that the field equations for a much wider class of Kerr–Schild metrics can be expressed in terms of a potential formalism, not only in the vacuum case, but also for many electrovacuum solutions.

Mechanics, classical

2018 ◽  
Author(s):  
Mark Wilson
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
Dimensional Euclidean Space ◽  
Absolute Space ◽  
Theory Of Relativity ◽  
Isaac Newton ◽  
The World ◽  
Physical Phenomena

Understood at its most general, ‘classical mechanics’ covers the approach to physical phenomena that dominated science from roughly the time of Galileo until the early decades of the twentieth century. The approach is usually characterized by the assumption that bodies carry an inherent mass and well-defined positions and velocities. Bodies subsist within a three-dimensional absolute space and influence one another through reciprocal forces. These objects obey the three laws of motion articulated by Isaac Newton in 1686 in a deterministic manner: once a mechanical system is assembled, its future behaviour is rigidly fixed. Such ‘classical’ assumptions were eventually rejected by Einstein’s theory of relativity, where the assumption of a three-dimensional Euclidean space is abandoned, and by quantum mechanics, where determinism and well-defined positions and velocity are eschewed. Classical mechanics is frequently characterized as ‘billiard ball mechanics’ or ‘the theory of mechanism’ on the grounds that the science treats its materials in the manner of colliding particles, or clockwork. Such stereotypes should be approached with caution because the basic framework of classical mechanics has long been subject to divergent interpretations that unpack the content of Newton’s ‘three laws’ in remarkably different ways. These differing interpretations provide incompatible catalogues of the basic objects that are supposed to comprise the ‘classical world’ – are they point masses, rigid bodies or flexible substances? Or, as many writers have suggested, should mechanics not be regarded as ‘about’ the world at all, but merely as a source of useful but fictitious idealizations of reality? These foundational disagreements explain why classical mechanics has often found itself entangled in metaphysics. Much of modern philosophy of science is characterized by attitudes that were originally articulated during the nineteenth century’s attempts to clarify the grounds of classical mechanics.

scholarly journals On Free Products of Cyclic Rotation Groups

1959 ◽  
Vol 11 ◽  
pp. 67-69 ◽  
Author(s):  
TH. J. Dekker
Keyword(s):  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Third Order ◽  
Free Products ◽  
Cyclic Groups ◽  
Rotation Axes ◽  
Abelian Subgroups ◽  
Rotation Groups ◽  
Positive Determinant

We consider the group of rotations in three-dimensional Euclidean space, leaving the origin fixed. These rotations are represented by real orthogonal third-order matrices with positive determinant. It is known that this rotation group contains free non-abelian subgroups of continuous rank (see 1).In this paper we shall prove the following conjectures of J. de Groot (1, pp. 261-262):Theorem 1. Two rotations with equal rotation angles a and with arbitrary but different rotation axes are free generators of a free group, if cos α is transcendental.Theorem 2. A free product of at most continuously many cyclic groups can be isomorphically represented by a rotation group.

scholarly journals Topological correlators and surface defects from equivariant cohomology

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Rodolfo Panerai ◽  
Antonio Pittelli ◽  
Konstantina Polydorou
Keyword(s):  
Quantum Mechanics ◽  
Equivariant Cohomology ◽  
Surface Defects ◽  
Star Product ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional System ◽  
The Novel ◽  
One Dimensional ◽  
Dual Quantum

Abstract We find a one-dimensional protected subsector of $$ \mathcal{N} $$ N = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S3. Then, we apply it to the novel case of S2× S1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncom- mutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general $$ \mathcal{N} $$ N = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

scholarly journals Exploration of collagen cavitation based on peptide electrolysis

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Rui Zhai ◽  
Hui Chen ◽  
Zhihua Shan
Keyword(s):  
Structural Changes ◽  
Preparation Method ◽  
Supporting Electrolyte ◽  
Three Dimensional ◽  
Anode Region ◽  
New Approach ◽  
New Material ◽  
Animal Skin ◽  
Exchange Membrane ◽  
Material Preparation

AbstractElectrochemical modification of animal skin is a new material preparation method and new direction of research exploration. In this study, under the action of the electric field using NaCl as the supporting electrolyte, the effect of electrolysis on Glycyl-glycine(GlyGl), gelatin(Gel) and Three-dimensional rawhide collagen(3DC) were determined. The amino group of GlyGl is quickly eliminated within the anode region by electrolysis isolated by an anion exchange membrane. Using the same method, it was found that the molecular weight of Gel and the isoelectric point of the Gel decreased, and the viscosity and transparency of the Gel solution obviously changed. The electrolytic dissolution and structural changes of 3DC were further investigated. The results of TOC and TN showed that the organic matter in 3DC was dissolved by electrolysis, and the tissue cavitation was obvious. A new approach for the preparation of collagen-based multi-pore biomaterials by electrochemical method was explored.

scholarly journals A NEW APPROACH TO DYNAMIC FINITE-SIZE SCALING

2003 ◽  
Vol 14 (07) ◽  
pp. 945-954 ◽  
Author(s):  
MEHMET DİLAVER ◽  
SEMRA GÜNDÜÇ ◽  
MERAL AYDIN ◽  
YİĞİT GÜNDÜÇ
Keyword(s):  
Three Dimensional ◽  
Finite Size ◽  
Taylor Series Expansion ◽  
Scaling Behavior ◽  
Dynamic Scaling ◽  
Finite Size Scaling ◽  
New Approach ◽  
Size Scaling ◽  
Dynamic Exponent ◽  
Good Agreement

In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behavior of two- and three-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For the three-dimensional Ising model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

Sign in / Sign up

Export Citation Format

Share Document