Algebra of complex vectors and applications in electromagnetic theory and quantum mechanics
type: Item
instance of: scholarly article
title: Algebra of complex vectors and applications in electromagnetic theory and quantum mechanics
zbMATH Open document ID: 1417.15037
author: K. Muralidhar
published in: Mathematics
publication date: 2016-01-12T00:00:00Z
review text: Summary: A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher \(n\)-dimensional geometric algebra from \((n-1)\)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 6528849
zbMATH Keywords: geometric algebra
zbMATH Keywords: complex vectors
zbMATH Keywords: particle spin
zbMATH Keywords: zeropoint field
MaRDI profile type: Publication
full work available at URL: math3030781
OpenAlex ID: W1930582205
cites work: New Foundations in Mathematics
cites work: Q4435147
cites work: Unitary geometric algebra
cites work: Complex vector formalism of harmonic oscillator in geometric algebra: particle mass, spin and dynamics in complex vector space
cites work: Quantum/classical interface: classical geometric origin of fermion spin
cites work: Geometric matrix algebra
cites work: An introduction to geometric algebra with an application in rigid body mechanics
cites work: Time as a geometric concept involving angular relations in classical mechanics and quantum mechanics
cites work: An introduction to geometric calculus and its application to electrodynamics
cites work: Revisiting Zitterbewegung
cites work: Q3467328
cites work: Zitterbewegung in quantum mechanics
cites work: Random electrodynamics
cites work: The Emerging Quantum
cites work: On the Quantum Correction For Thermodynamic Equilibrium
cites work: Classical aspects of the Pauli-Schrödinger equation
cites work: The vacuum electromagnetic fields and the Schrödinger equation
cites work: Quantum mechanics in the geometry of space-time. Elementary theory.
DOI: 10.3390/MATH3030781
description: scientific article; zbMATH DE number 6528849