Polarizations as states and their evolution in geometric algebra terms with variable complex plane

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.


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.
Let's take the electromagnetic field in the form: ( ) (1) requiring that it satisfies the Maxwell system of equations in free space, which in geometrical algebra terms is one equation: Element 0 F in (1) is a constant element of geometric algebra 3 G and S I is unit value bivector of a plane S in three dimensions, that is a generalization of the imaginary unit [2], [3].The exponent in (1) is unit value element of 3 G + [3]: Solution of (2) should be sum of a vector (electric field E) and bivector (magnetic field 3 I H ): with some initial conditions:

Solution in the Geometric Algebra Terms
The derivative by time gives   The result for the case 3 ˆS where 0 E and 0 H are arbitrary mutually orthogonal vectors of equal length, lying on the plane S. Vector k should be normal to that plane, For a plane S in three dimensions Maxwell equation ( 2) has two solutions • E 0 and H 0 , initial values of E and H, are arbitrary mutually orthogonal vectors of equal length, lying on the plane S. Vectors to that plane.The length of the wave vectors ± k is equal to angular fre- quency ω.
Maxwell Equation ( 2) is a linear one.Then any linear combination of F + and F − saving the structure of (1) will also be a solution.
Let's write: Then for arbitrary scalars λ and μ: is solution of (2).The item in second parenthesis is weighted linear combination One another option is: ( ) which is just rotation, along with possible change of length, of electric and magnetic initial vectors in their plane.

Transformations of Polarization States
Polarizations, in our approach, exponents in the solution of (3), have the form of states [3] G , particularly other polarizations.Such states can be depicted in the current geometric algebra formalism using a triple of basis bivectors in three dimensions { } , , B B B (Figure 1): The basis bivectors satisfy multiplication rules (in the righth and screw orientation of 3 I ): One can identify basis bivectors with usual coordinate planes: Any one of these three bivectors can be taken as explicitly identifying imaginary unit, though any unit value bivector in three dimensions can take the role [2], [4].

Thus:
( ) The difference between units of information in classical computational scheme, quantum mechanical conventional computations (qubits) and geometric algebra scheme (g-qubits) with variable explicitly defined complex plane is seen from Figure 2.
Circular polarizations received as solutions of Maxwell Equation ( 2) is an excellent choice to have such g-qubits in a lab.
Commonly accepted idea to use systems of qubits to tremendously increase speed of computations is based on assumption of entanglement -roughly speaking when touching one qubit all the other in the system react instantly, in no time.A bit strange, though you should not care about that because our paradigm is very different.
Assume we have some general state: ( ) The state can be identified as a point ( ) , , , α β β β on unit sphere 3

Conventional Hamiltonian
Figure 2. Differences between bits, qubits and g-qubits.
with removed not important scalar γ, has the lift in 3 G + [3]: ( ) that is critical for the whole approach.Therefore, for some Δt, Clifford translation for a given Hamiltonian is: For an arbitrary sequence of infinitesimal Clifford translations, the final state along the curve on unit sphere 3 S composed of infinitesimal displacements by ( ) ( ) ( ) Let's calculate the result of the right-hand side of (5) in general case when the plane of ( ) ( ) To calculate the geometric algebra product of the two exponents in Clifford translation with not coinciding exponent planes, (see Figure 3) where γ and β are vectors dual to bivectors  ( ) ( )

Transformations of Circular Polarized Electromagnetic Fields
Now we have everything to retrieve action of Clifford translation generated by a Hamiltonian on general solution (4): In the case It makes simpler if F + and F − are weighted, say both λ and μ are equal to one: )

Action of Polarization States on Observables
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 3 G , the following is detailed description of the case when the element in parenthesis of the (6) expression acts on some bivector.Such operation is generalization of the Hopf fibration and rotates the bivector in three dimensions.
Denoting4 : ( In the magnetic field 3 I H the item 3 I is unit pseudoscalar in three dimen- sions 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.
the plane of S I and its direction is defined by orientations of 3 I and S I .Rotation of right/left hand screw defined by orientation of S I gives movement of right/left hand screw.This is the direction of the vector the above result the sense of the S I orientation and the direction of  �⃗ were assumed to agree with 3 ˆS + is right hand screw oriented, that's rotation of Ê to Ĥ by π/2 gives movement of right hand screw in the direction of − is left hand screw oriented, that's rotation of Ê to Ĥ by π/2 gives movement of left hand screw in the direction of , movement of right hand screw in the opposite direction, − −k .
.4236/***.2018.***** of two states with the same phase in the same plane but opposite sense of orientation.The states are strictly coupled, entangled if you prefer, because bivector plane should be the same for both, does not matter what happens with it.

Figure 1 .
Figure 1.Basis of bivectors and unit value pseudoscalar.
α β β β along intersection of 3 S with the unit bivector plane Cl I .Let's make notations more like conventional quantum mechanical ones.I will write: like form of the Clifford translation bivector.
By nor- malizing the bivector to unit value we get generalization of imaginary unit ( ) ( )

1 SI
in original basis to get formulas for generators of Clifford transla- full product is: A. Soiguine DOI: 10.4236/***.2018.*****710 Journal of Applied Mathematics and Physics