Duality Rotation

In recent posts ([1] [2] [3] [4] [5] [6] [7]) I have been introducing the $\mathbb{C}$ (Confinement) and $\mathbb{L}$ (Liberation) hypothesis . So far, I have discussed two modalities of the { $\mathbb{C} , \mathbb{L}$ } interplay: 1) orthogonality 2) coupling. Here, I will discuss another modality which can be summarized in one sentence. $\mathbb{C}$ and $\mathbb{L}$ are abstract and they are not tied to attributes.

One sentence summaries are good but it is useful to provide mathematical descriptions. That way we gain more insights.

$\mathbb{C}$ and $\mathbb{L}$ are abstract. $\mathbb{C}$ refers to the confinement process irrespective of the confining medium or agent. Similarly, $\mathbb{L}$ refers to the liberation process irrespective of the liberating medium or agent. $\mathbb{C}$ and $\mathbb{L}$ can switch their association with attributes and agents. The “switch” can be described as a rotation. I will refer to this type of rotation as “duality rotation” because it is so similar to the duality rotation of electromagnetism.

Before I go into the mathematical description, I would like to remind you about the CLA (circle, line, angle) symbolism introduced in [6] [7]. $\mathbb{C}$ is symbolized by a circle. $\mathbb{L}$ is symbolized by a straight line. The coupling between $\mathbb{C}$ and $\mathbb{L}$ is symbolized by a non-90-degree angle.

Duality rotation of Electromagnetism

The laws of electromagnetism are summarized in the form of Maxwell equations. In the absence of charges Maxwell equations preserve form when we switch the electric and the magnetic fields in this way: $\mathbf{E} \rightarrow \mathbf{B}$ and $\mathbf{B} \rightarrow - \mathbf{E}$ .

The “switch” can be described as a rotation. Remember the $(\mathbf{E} + i \mathbf{B})$ construct from the article titled “Concise Formulation of Maxwell Equations” ( $i=\sqrt{-1}$ is the “imaginary number” of complex numbers ).

Consider the continuous transformation known as the duality rotation of electromagnetism

$(\mathbf{E} + i \mathbf{B}) \rightarrow e^{i\phi} (\mathbf{E} + i \mathbf{B})$

where the phase angle $\phi$ can have any value. Maxwell equations preserve form under this transformation.

Mathematical form that makes orthogonality explicit

The mathematical form $(\mathbf{E} + i \mathbf{B})$ explicitly states that the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ of an electromagnetic wave are orthogonal. The geometrical meaning of $i=\sqrt{-1}$ is 90-degree rotation. The 90-degree angle represents orthogonality.

Consider $\mathbb{C} + i \; \mathbb{L}$

The mathematical form $\mathbb{C} + i \; \mathbb{L}$ states that $\mathbb{C}$ and $\mathbb{L}$ are fundamentally orthogonal.

The electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ of an electromagnetic wave are orthogonal in space. The orthogonality represented by $\mathbb{C} + i \; \mathbb{L}$ is more abstract and general. See [6] for more on orthogonality.

Abstract duality of $\mathbb{C}$ and $\mathbb{L}$

Consider the transformation

$(\mathbb{C} + i \mathbb{L}) \rightarrow e^{i\phi} (\mathbb{C} + i \mathbb{L})$

Using the Cotes-Euler identity

$e^{i \phi } = cos(\phi) + i sin(\phi)$

and taking $\phi = \pi/2$ we get

$(\mathbb{C} + i \mathbb{L}) \rightarrow (-\mathbb{L} + i \mathbb{C})$

which expresses the switch ( $\mathbb{L} \rightarrow \mathbb{C}$ , $\mathbb{C} \rightarrow -\mathbb{L}$ )

The “switch” implies that the “circle” characteristic is switched with the “line” characteristic.

An agent or medium may exhibit $\mathbb{C}$ (Confinement, “circle”) characteristic in one setting and $\mathbb{L}$ (Liberation, “line”) characteristic in another setting. $\mathbb{C}$ and $\mathbb{L}$ are abstract. $\mathbb{C}$ and $\mathbb{L}$ are not tied to attributes.

Circle and line characteristics in electromagnetism

We can speak of “circle” characteristic and “line” characteristic in the context of electromagnetism as well. If the source current (electric field) has “line” characteristic the resultant magnetic field has “circle” characteristic . In the reverse situation, if the source magnetic field has “line” characteristic the resultant current (electric field) has “circle” characteristic

More examples

Electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ are attributes. $\mathbb{C}$ and $\mathbb{L}$ can change their association with $\mathbf{E}$ and $\mathbf{B}$ .

In most cases magnetic field has the characteristics of $\mathbb{C}$ since the curving influence of the magnetic field is a confining action. But, we should remember that $\mathbb{C}$ cannot be tied to attributes or agents. We cannot claim that a magnetic field always displays the characteristics of $\mathbb{C}$ .

Magnetic field can be associated with $\mathbb{L}$ as well. For example, the uniform magnetic field inside a selonoid coil is an expression of $\mathbb{L}$ . The solenoid coil represents $\mathbb{C}$ because it forces the electrical current to flow in a helical path. In this case, the $\mathbb{L}$ reaction is the appearance of a uniform magnetic field inside the selenoid.

Now, let’s look at an example where $\mathbb{C}$ is associated with the magnetic field (usual case). In a structure known as the wiggler a series of magnets are oriented in such a way that the injected electrons follow a wiggling path and radiate photons as a result. When charged particles are forced to follow a curved path they radiate light known as the synchrotron radiation. The magnetic field inside the wiggler is clearly acting as the confiner ( $\mathbb{C}$ ). The emission of light is the liberation ( $\mathbb{L}$ ) reaction.

Seleonoid and wiggler are both man-made structures. In these cases $\mathbb{C}$ and $\mathbb{L}$ are acting through human agents.

Atoms are not man-made. The electric field created by the protons of the atomic nucleus captures electrons to form the atom. In this case the electric field is an expression of $\mathbb{C}$ . The $\mathbb{L}$ reaction manifests as quantum effects.

To repeat one more time, $\mathbb{C}$ and $\mathbb{L}$ are not tied to attributes.

Magnitude preserving transformation

Duality rotation of electromagnetism preserves the magnitude of the complex vector $(\mathbf{E} + i \mathbf{B})$ . This also means that the quantity $\mathbf{E}^2 + \mathbf{B}^2$ is preserved. The quantity $\mathbf{E}^2 + \mathbf{B}^2$ is proportional to the energy of the electromagnetic wave. Duality rotation of electromagnetism does not violate the principle of the conservation of energy.

Similarly, duality rotation of $\mathbb{C}$ and $\mathbb{L}$ implies the existence of a conservation law involving $\mathbb{C}^2 + \mathbb{L}^2$ . More on this later.

Fun with geometry

CLA (circle, line, angle) symbolism is useful. I take CLA symbolism seriously but how far can I stretch this particular symbolism?

$\mathbb{C}^2$ : what does it mean to take the square of a circle?

$\mathbb{L}^2$ : what does it mean to take the square of a line?

I have a post titled “Multiplication table of shapes” where I show sketches prepared by Tai-Danae Bradley. She explains that circle multiplied by circle is a torus; line segment multiplied by line segment is a square. This is fun.

References

[1] Ignoring Agency is Ignorance

[2] Confinement and Liberation

[3] Individuality and Collectivity

[4] Semiotic Closure

[5] Tropes in Physics

[6] Orthogonality

[7] Coupling