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Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics

Physics Department, National Defence Academy, Khadakwasla, Pune-411 023, India
Academic Editor: Palle E.T. Jorgensen
Mathematics 2015, 3(3), 781-842;
Received: 14 March 2015 / Revised: 28 July 2015 / Accepted: 14 August 2015 / Published: 20 August 2015
(This article belongs to the Special Issue Mathematical physics)
PDF [523 KB, uploaded 26 August 2015]


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. View Full-Text
Keywords: geometric algebra; complex vectors; particle spin; zeropoint field geometric algebra; complex vectors; particle spin; zeropoint field
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Muralidhar, K. Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics. Mathematics 2015, 3, 781-842.

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