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

Abstract

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.

Compressible Navier-Stokes Equations in Cylindrical Passages and General Dynamics of Surfaces—(I)-Flow Structures and (II)-Analyzing Biomembranes under Static and Dynamic Conditions

“…Multiplication of Equation (14) by ρ and Equation (15) below by L 1 , addition of the resulting equations [29], and using the ordinary product rule of differential multivariable calculus a form as in Equation (16) is obtained whereby a is given by a = ρ L 2 .…”

“…An imaginary vector is defined as a sum of two real vectors one of which is multiplied by the complex number i. If we take the complex number i and replace it by a pseudo-scalar I in the context of Geometric Algebra it can be shown that for compressible flow the density appearing in the Navier-Stokes equations can be written in terms of a "complex" exponential function in time or, even better, an exponential of a multivector which can be proven to be well defined [14]. The form of a complex vector has been used as a bivector to study polarization of electric fields [15,16].…”

“…This will be in the form of a sum of a vector in the usual sense of vector in three dimensional space added to a bivector in the GA sense. Multiplication of complex vectors by rotors can rotate the vector by an angle θ in the plane of the vector and bivector [14]. The fundamental lemma of Geometric-Algebra is that the geometric product can be decomposed into an inner product and outer product of two complex vectors.…”

Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space

“…The complexification process, in general, provides a pathway for generating higher dimensional geometric algebra from lower dimensions. This is explicitly illustrated through examples in the work of Muralidhar [7] (2)…”

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

“…Recently, the role of spin and the internal charged particle structure in complex vector formalism have been studied by the author. [23][24][25] In the presence of zeropoint field, an elementary charged particle oscillates in accordance with the oscillations of the random zeropoint field and an average of all such oscillations may be considered as complex rotations and the imaginary part of such complex rotation gives the classical origin of particle spin. 26,27 It has been shown that the mass of the particle may be interpreted to the zeropoint field energy associated with the local complex rotation or oscillation confined in a region of space of the order of Compton wavelength.…”