Monopoles - 3 - Consequence of Duality transformation

The major consequence of this duality transformation is that, you can never say it for sure that whether any charged particle in Nature has only electric charge or magnetic charge or both. 
For example, let us say there are 3 planets separated very far from each other  with 3 different kind of aliens, where the laws of physics are the same and so the Maxwell's equations are equally applicable anywhere in these 3 planets. 

They will predict exactly the same results for any Electromagnetic phenomena in their universe using the common Maxwell's equations. But they needn't to have the same form. If they vary their definition of Electric and magnetic fields according to duality transformation, there is no way finding which one is true. (It is not correct use the word "true" - after all their definition are different but they will conclude the same results).


It is a possibility for the first one (let us say humans are the first type of aliens) to describe any EM phenomena with our usual definition of Electric and Magnetic field where $q^m = 0$ magnetic monopole charge is zero, and the second planet is our inverse where they define only the magnetic charge with no electric charge by choosing $q^e = 0$
Unlike it so happens that, the third Planet define their electron with both electric and magnetic charge!

So, let us leave the first two planets and go to the third planet, where we will try to get some intrinsic physical understanding of their definitions.  

In this planet, we will first take two positive electric charges with respect to our conventional Maxwell's equations $Q_1$ and $Q_2$. 
The electrostatic repulsion force is given by coulomb's law as,
$$\vec{F_{21}}=\frac{Q_1Q_2}{r^2}\hat{r_{12}}$$
But, if the aliens define it in a such a way that it has both the electric and magnetic charge as, $$ Q_1 = q_1^e + q_1^m $$ and $$ Q_2 = q_2^e + q_2^m $$
Then the two charges as we study in Electrostatics and Magneto statics (no changing Electric or Magnetic fields), the force on charge 2 due to charge 1 is given by, $$ F_{Q_2Q_1} = q_2^e \left[\vec{E} + \frac{(\vec{v}\times\vec{B})}{c}\right] + q_2^m \left[ \vec{B} - \frac{(\vec{v}\times\vec{E})}{c}\right] $$

Since we are working with static conditions, $\vec{v} = 0$.
So, $$F_{Q_2Q_1} = q_2^e\vec{E} + q_2^m\vec{B}$$ where E and B due to $Q_1$ is given by, 
$$ \vec{E} = \frac{q_1^e}{r^2}\hat{r_{12}}$$ and $$ \vec{B} = \frac{q_1^m}{r^2} \hat{r_{12}}$$ because, now we just consider the problem as the combination of Electric and Magnetic charge placed closed together at the same point.

The Net force, $$ F_{Q_2Q_1} = q_2^e\frac{q_1^e}{r^2}\hat{r_{12}} + q_2^m \frac{q_1^m}{r^2}\hat{r_{12}}$$
or simply, $$ F_{21} = \frac{(q_2^eq_1^e+q_2^mq_1^m)}{r^2}\hat{r_{12}} $$ Since, they both direct along the same direction, we will just see some Net force acting as a repulsion force. 

So, you will always see the same result independent of the assigned electric or magnetic charge to the charged particle. Physically observable results are invariant under different definitions. 

Then, how do we decide the truth? what we really mean by a magnetic monopole?

The definition Magnetic monopole is,
Given the usual conventional Maxwell's equations, where there is only Electric charge, we haven't found any particle in Nature with pure magnetic charge i.e.the particle that transformed with $\frac{\pi}{2}$ angle in duality transformation equations.  

But, still it doesn't answer whether I have an Electromagnetic charge or pure electric charge in my hand!! 

If I say, I have pure electric charge and expect to find in Nature a new particle with pure magnetic charge, then I can also expect for another particle with both Electric and Magnetic charge (i.e. Electromagnetic charge).

After all there is no any kind of specification about the quantization or anything about the charge in Maxwell's equations. It can just assume any arbitrary value of unlike the reality where it can have only discrete values (also in energy, angular momentum, etc.).

Thus, the role of Quantum Mechanics is inevitable when you talk about any subatomic particle in reality. It is the reason why, subsequent development about Magnetic monopoles were first made by Paul Dirac with his new concept of Dirac string.