Polarizations as States and Their Evolution in Geometric Algebra Terms with Variable Complex Plane

Alexander Soiguine

doi:10.4236/jamp.2018.64063

Abstract

Recently suggested scheme [1] of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.

Highlights

The circular polarized electromagnetic waves are the only type of waves following from the solution of Maxwell equations in free space done in geometric algebra terms
Since a state in the described formalism is operator that gives the result of measurement when acting on observable, which can be any element of geometric algebra G3, the following is detailed description of the case when the element in parenthesis of the (6) expression acts on some bivector
The core of quantum computing should not be in entanglement as it understood in conventional quantum mechanics, which only formally follows from representation of the many particle states as tensor products of individual particle states and not supported by really operating physical devices

Summary

Introduction

The circular polarized electromagnetic waves are the only type of waves following from the solution of Maxwell equations in free space done in geometric algebra terms. Requiring that it satisfies the Maxwell system of equations in free space, which in geometrical algebra terms is one equation:. Solution of (2) should be sum of a vector (electric field E) and bivector (magnetic field I3H ): F= E + I3H with some initial conditions: E + I3 H =t 0,=r 0 = F0 = E=t 0,=r 0 + I3 H =t 0,=r 0 = E0 + I3 H0. In the magnetic field I3H the item I3 is unit pseudoscalar in three dimensions assumed to be the right-hand screw oriented volume, relative to an ordered triple of orthonormal vectors. Substitution of (1) into the Maxwell’s (2) will exactly show us what the solution looks like

Solution in the Geometric Algebra Terms

Transformations of Polarization States

Transformations of Circular Polarized Electromagnetic Fields

Action of Polarization States on Observables

Polarization States Acting on Multiple Observables

Conclusion