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.

https://www.mdpi.com/2227-7390/3/3/781/pdf

ℏ derived from cosmology and origin of special relativity and QM.

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

A charged particle immersed in the random fluctuating zeropoint field is considered as an oscillator with oscillations at random directions and the stochastic average of all such oscillations may be considered as local complex rotation in complex vector space. The average internal oscillations or rotations of the charged particle reveal the particle extended structure with separated centre of charge and centre of mass. The aim of this short review article is to give an account of the extended particle structure in complex vector space and to study the origin of relativistic effects due to a charged particle motion in the presence of zeropoint field.

background

https://medcrave.org.medcraveonline.com/index.php/OAJMTP/article/download/21/pdf

The relativistic effects of charged particle with complex structure in zeropoint field

The Boltzmann equation with electron-electron (e-e) interactions has been reduced to a Fokker-Planck equation 
(e-e FP) in a previous paper. In steady-state conditions, its solution q 0 ( v ) (where v is the electron speed)
depends on the square of the acceleration a = e E / m . If we introduce the nonrenormalized zero-point field (ZPF)
of QED, i.e., the one considered in stochastic electrodynamics, so that 〈a 2 〉 = 〈( a D.C. + a ZPF ) 2 〉 s a 2 ZPF , then
 q 0 ( v ) becomes similar to the Fermi-Dirac equation, and the two collision frequencies ν 1 ( v ) and ν 2 ( v ) appearing
in the e-e FP become both proportional to 1/ v in a small d v interval. The condition υ 1 ( v ) ∝ υ 2 ( v ) ∝ 1/ v is
at the threshold of the runaways. In the same d v range, the time-dependent solution q 0 ( v , t) of the e - e FP
decays no longer exponentially but according to a power law ∝ t -e where 0.004 < e < 0.006, until t → ∞.
That extremely long memory of a fluctuation implies the same dependence τ -e for the conductance correlation
function, hence a corresponding power-spectral noise S ( f )∝ f e-1 where f is the frequency. That behaviour is
maintained even for a small sample because the back diffusion velocity of the electrons in the effective range d v ,
where they are in runaway conditions, is much larger than the drift velocity.

Very long decay time for electron velocity distribution in semiconductors, and consequent 1/f noise

FRANKLY, we find the title of Prof. Newman's new book a little puzzling. He sets out to deal with recent advances in non-atomic physics, and, knowing his flair for exposition, we looked forward to a readable account, with due emphasis given to the experimental side, of recent developments in what we might call macroscopic or molar physics. His account is readable enough, and interesting experimental detail bulks largely therein, but the choice of topics bears singularly small relation to the title. Wave mechanics; statistical mechanics; electromagnetic radiation, including a discussion of infra-red spectra; γ-rays, cosmic rays, and the scattering and diffraction of X-rays; ninety pages on modern magnetics, and, under the heading electricity, a discussion, inter alia, of electrolytic conduction and of super-conductivity. These be strange themes to meet in a work where we had hoped a comfortable certainty would reign supreme and electron waves would cease to surge and roar. Recent Advances in Physics (Non-Atomic). By Prof. F. H. Newman. Pp. ix + 378. (London: J. and A. Churchill, 1932.) 15s.

/pdf/physics-in-the-making-7eryqi3e89.pdf

ℏ derived from cosmology and origin of special relativity and QM.

Citations

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

The relativistic effects of charged particle with complex structure in zeropoint field

Very long decay time for electron velocity distribution in semiconductors, and consequent 1/f noise

References

Physics in the Making

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