Few Comments on the Pair Concept

In my previous post I mentioned that

• Complex number can be thought of as a pair of real numbers.
• Quaternion can be thought of as a pair of complex numbers
• Octonion can be thought of as a pair of quaternions

The concept of “pair” is very important in physics. There are many examples. I hope to write about twistors and spinors in the future. Twistors can be thought of as a pair of spinors, for example. The Dirac electron is a pair of spinors too.The spinor itself is a pair.

I also would like to remind you about the pair that I described in  Concise Formulation of Maxwell Equations.

In the concise formulation of the Maxwell equations the number of equations is reduced from 4 to 2 at the expense of the introduction of a complex vector field. This should not be seen as an expense because $(\mathbf{E}+i\mathbf{B})$ gives us new insights. The “pair” I am talking about is $(\mathbf{E}+i\mathbf{B})$.

The electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ form a pair. $\mathbf{E}$ and $\mathbf{B}$ have different units. This is the difference between physics and mathematics. In mathematics there is no “physical unit” concept. In physics we can form a complex valued number or function by combining two different physical quantities as pairs in the form of $(\mathbf{E}+i\mathbf{B})$ , for example.

We don’t do this randomly. We pair the physical quantities that are very closely related. In the case of electric and magnetic fields we know from a century of observations that they are different aspects of the same reality. This justifies the $(\mathbf{E}+i\mathbf{B})$ pair.

I talked about a different type of pair, namely  horizontal and vertical attribute pairs in New Perspective on Unification. In that article I suggested that all entities are mixtures of vertical and horizontal attributes. I defined the horizontal and vertical attributes as follows:

• Horizontal attributes are associated with collectivity and multiplicity. They are attributes of the group behavior. Horizontal attributes are about the cross-sectional universe.
• Vertical attributes are about individuality, individual histories and individual characteristics. Vertical attributes are about the idiosyncratic behavior.

Speaking of electric and magnetic fields, it is not possible to think of them as a horizontal/vertical pair because both of them are rather individual in character. The horizontal/vertical pair refers to collectivity/individuality dichotomy.

There is no dichomity between the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ … there is duality between them. $\mathbf{E}$ and $\mathbf{B}$ are dual.

The complex number notation of $(\mathbf{E}+i\mathbf{B})$ is perfect to represent dual quantities because complex algebra manipulates $(\mathbf{E}+i\mathbf{B})$ as a whole.

I need to find a new mathematical notation to represent the horizontal/vertical attribute pairs.