Monopoles - 7 - Dirac Monopoles in Quantum Mechanics - Part - 3

From our previous post we found that, the gauge invariance allows to reconsider our ordinary wave function with the phase factor as the wavefunction of an electron in the Electromagnetic potentials given by, $$ \vec{A} = \frac{\hbar{c}}{e}\vec{k}$$ and $$ V = \frac{-\hbar}{e}k_0$$ 
So, the Magnetic and Electric fields are given by, $$ \nabla\times\vec{A} = \vec{B} $$$$\rightarrow\,\,\,\,\,\,\nabla \times\vec{k} = \frac{e}{\hbar{c}} \vec{B}$$ and $$ E = -\nabla{V} - \frac{1}{c}\frac{\partial{A}}{\partial{t}} = \frac{\hbar}{e}\nabla{k_0} - \frac{\hbar}{e} \frac{\partial{\vec{k}}}{\partial{t}} $$ $$ \rightarrow\,\,\,\,\,\,\nabla{k_0} - \frac{\partial{\vec{k}}}{\partial{t}} = \frac{e}{\hbar}\vec{E}$$ 

Looking at these relations we see that, a new physical meaning is given to "k" in terms of the potentials of the Electromagnetic field. So, whatever the mathematical manipulation we do in "k" will give rise to a new physical meaning in terms of Electric and Magnetic fields. That makes our problem physically more significant as it is in Mathematics.

So, we need to go back and take a re look at our initial definitions of phase factor and its non-integrability and to check whether we could impose some more conditions, so that its applicability can be generalized.  

We have seen in the beginning that the change in phase (the change in $\beta$) of the wave function around a closed curve is same for all the wave functions. But even at the single point, there is some arbitrary freedom in choosing the phase of the wavefunction, so that we can add any integer multiples of phase $2\pi{n}$ to the wave function and we will still get the same result. 
$$ e^{i\beta} = e^{i (\beta + 2\pi)} = e^{i(\beta+4\pi)} = e^{i(\beta+2n\pi)}$$
So, it is always the phase and the change in phase is undetermined up to the addition of multiples of $2\pi$. 

From this we can conclude, the change in phase of any wave function around a closed loop should be equal the Electromagnetic flux penetrating through the area enclosed by the loop and with the addition of arbitrary multiples of $2\pi$.  

$$\oint_{(4d)}\,d\beta = 2n\pi + \oint_{(4d)}\vec{K_{(4d)}} \,dl_{(4d)} \\~\\= 2n\pi + \int (\nabla\times\vec{K}) \vec{da_{(6d)}}$$
Using Stokes' theorem for 4-dimensional vectors. Finally, we get the change in phase, $$ \Delta\beta = 2n\pi + \frac{e}{\hbar{c}}  \int \vec{B}\, \vec{da}$$
Only, magnetic flux comes into play, as we are considering the region where the monopole is enclosed. 
In a special case, if you take a very small closed curved where the functions are smooth, then the change in phase cannot be in terms of the integer multiples of $2\pi$. Only, the flux term will play the role, which by itself is unique and cannot vary as we take the small curve. This wouldn't be the case, if there is any singularity in the function that is in the potentials. 

It can be concluded that, for a very small closed curve with smooth functions (potential), the change in phase for different wave functions cannot arbitrarily vary in multiples of $ 2\pi$ but is determined by the flux penetrating through the surface. But the same wave function, with singularity will give the change as, $$ \Delta\beta = 2n\pi + \int \vec{B} \,\vec{da} $$ For a given monopole the second term cannot change for various wave functions. And so, the change in phase of various wave functions should depend on the various values of the first term "n" which itself depends on the singularities enclosed within the surface. 

Now, if make my closed curve very small approaching near to zero, then the change in phase should be reduced to give zero (from the simple logic that the wave function at a single point should have definite phase - and so definite probability). 

Remember, it was not considered an empty space, but the space with singularity passing through our point. 

From the above discussion we get, $$ \Delta\beta = 0 $$ So, $$2n\pi = \frac{-e}{\hbar{c}} \int \vec{B}\cdot\vec{da} $$
where $$\int\vec{B}\cdot\vec{da}$$ in three dimensions gives the net magnetic flux penetrating through the closed surface, which is equal to $ 4\pi{q_m}$. The positive and negative sign depends on the nature of singularity. 
On equating, we finally get the quantization condition for the charges as, $$ 4\pi{q_m} = \frac{2n\pi\hbar{c}}{e}$$ and $$ q_m = \frac{n\hbar{c}}{2e}$$

Thus, Dirac proved that even if one magnetic monopole exist in Nature, it would explain why all the electric charges in the Universe are Quantized.  

Let us consider next, the problem worked out by Dirac with a little more explanation on its solution.