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:
requiring that it satisfies the Maxwell system of equations in free space, which in geometrical algebra terms is one equation:
where and multiplications are the geometrical algebra ones.
Element in (1) is a constant element of geometric algebra and is unit value bivector of a plane S in three dimensions, that is a generalization of the imaginary unit  ,  . The exponent in (1) is unit value element of  :
Solution of (2) should be sum of a vector (electric field E) and bivector (magnetic field ):
with some initial conditions:
In the magnetic field the item 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.
2. Solution in the Geometric Algebra Terms
The derivative by time gives
The geometric algebra product is:
depending on do we write or . The result should be the same since is a scalar.
Commutativity is true only if . The following agreement takes place between orientation of , orientation of and direction of vector k  . The vector is orthogonal to the plane of and its direction is defined by orientations of and . Rotation of right/left hand screw defined by orientation of gives movement of right/left hand screw. This is the direction of the vector . That means that the matching between and should be or 1.
Assuming that orientation is , the Maxwell equation becomes:
Left hand side is sum of vector and bivector, while right hand side is scalar plus bivector , plus pseudoscalar , plus vector . It follows that both E and H lie on the plane of and then:
Thus, and we get equation from which particularly follows and .
The result for the case is that the solution of (2) is
where and are arbitrary mutually orthogonal vectors of equal length, lying on the plane S. Vector k should be normal to that plane, and .
In the above result the sense of the
orientation and the direction of
For a plane S in three dimensions Maxwell equation (2) has two solutions
・ , with , , and the triple is right hand screw oriented, that’s rotation of to by π/2 gives movement of right hand screw in the direction of .
・ , with , , and the triple is left hand screw oriented, that’s rotation of to by π/2 gives movement of left hand screw in the direction of or, equivalently, movement of right hand screw in the opposite direction, .
・ E0 and H0, initial values of E and H, are arbitrary mutually orthogonal vectors of equal length, lying on the plane S. Vectors are normal to that plane. The length of the wave vectors is equal to angular frequency w.
Maxwell Equation (2) is a linear one. Then any linear combination of and saving the structure of (1) will also be a solution.
Then for arbitrary scalars λ and μ:
is solution of (2). The item in second parenthesis is weighted linear combination 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.
One another option is:
which is just rotation, along with possible change of length, of electric and magnetic initial vectors in their plane.
3. Transformations of Polarization States
Polarizations, in our approach, exponents in the solution of (3), have the form of states  , that’s elements of : , distributed in space. They are operators than can act on observables, also elements of , particularly other polarizations. Such states can be depicted in the current geometric algebra formalism using a triple of basis bivectors in three dimensions (Figure 1):
The basis bivectors satisfy multiplication rules (in the righth and screw orientation of ):
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  ,  .
The difference between units of information in classical computational scheme, quantum mechanical conventional computations (qubits) and geometric algebra
Figure 1. Basis of bivectors and unit value pseudoscalar.
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 on unit sphere . It can be subjected to a Clifford translation
executing displacement at point along intersection of with the unit bivector plane .
Let’s make notations more like conventional quantum mechanical ones. I will write:
and use Hamiltonian like form of the Clifford translation bivector.
Figure 2. Differences between bits, qubits and g-qubits.
with removed not important scalar γ, has the lift in  :
Then the associated Clifford translation plane bivector is . By normalizing the bivector to unit value we get generalization of imaginary unit
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 is integral2
along the curve on unit sphere composed of infinitesimal displacements by
Let’s calculate the result of the right-hand side of (5) in general case when the plane of differs from .
To calculate the geometric algebra product of the two exponents in Clifford translation with not coinciding exponent planes, , , let’s first expand in original basis to get formulas for generators of Clifford translation. If then a part of geometrical product is:
(see Figure 3)
where and are vectors dual to bivectors and .
Thus, the full product is:
Figure 3. Two bivector geometrical product.
4. Transformations of Circular Polarized Electromagnetic Fields
Now we have everything to retrieve action of Clifford translation generated by a Hamiltonian on general solution (4):
To make expressions simpler I will use notations
. Then we get (see Sections 1.3 and 1.6 in
Let’s take popular case of (plane orthogonal to axis) and (or , does not matter.) The above formula becomes:
It makes simpler if and are equally weighted, say both λ and μ are equal to one:
5. 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 , 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.
and taking a bivector operand (observable) we get the result of measurement, action of the state on observable (see  ,  for details):
One interesting remark. If the observable belongs only to the plane, that’s , the result of measurement has only components in and , projections of the value due to rotation with angular velocity 2ω around the axis.
6. Polarization States Acting on Multiple Observables
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. The core of quantum computing scheme should be in manipulation and transferring of sets of states as operators decomposed in geometrical algebra variant of qubits (g-qubits), or four-dimensional unit sphere points, if you prefer. Such operators can act on observables, particularly through measurements. From the recent calculation we realize that the action of state, which depends on , on an observable can be done only if observable is defined at the same point where the state is defined  . In this way quantum computer is an analog computer keeping information in sets of objects with infinite number of degrees of freedom, contrary to the two value bits or two-dimensional Hilbert space elements, qubits.
Thus, if we have a state
as in the case of polarization defined states, it becomes a state acting on a set of observables if the latter are defined at some given points:
Then the state transforms into multi-observable one:
Figure 4. Decoding of g-qubit message.
This formula for bears clear physical and geometrical sense, contrary to conventional quantum mechanics definition following formally from tensor product which does not have good physical interpretation but is the root of entanglement-based quantum computing.
The formula also prompts how quantum encryption decoding can be effectively implemented with the bivector value security key (see Figure 4).
The formula can also be applied to challenging area of anyons in three dimensions.
Two seminal ideas―variable and explicitly defined complex plane in three dimensions, and the states5 as operators acting on observables―allow to put forth comprehensive and much more detailed formalism appropriate for quantum mechanics in general and particularly for quantum computing schemes. The approach may be thought about, for example, as a far going geometric algebra generalization of some proposals for quantum computing formulated in terms of light beam time bins, see  ,  , but giving much more strength and flexibility in practical implementation.
1For any vector we write .
2In the case of constant plane of Hamiltonian, it easily follows the Schrodinger equation of conventional quantum mechanics with clearly defined imaginary unit.
3In the case we trivially have rotation of by angle .
4Easy to see that the left-hand side is unit value element of .
5Good to remember that “state” and “wave function” are (at least should be) synonyms in conventional quantum mechanics.
 Soiguine, A. (2015) Geometric Algebra, Qubits, Geometric Evolution, and All That.
 Soiguine, A. (2016) Anyons in Three Dimensions with Geometric Algebra.