Deriving Maxwell's Equations From Heisenberg and Einstein

What is the node that connects Heisenberg's uncertainty principle, Einstein's field equations, and Maxwell's equations? To find out, let's start with Heisenberg's uncertainty principle and see what we can derive:

We derive a distance (r). We can do the same if we start with Einstein's field equations:

We can now equate the right side of equation 6) with equation 11). If we do this, we can discover the above-mentioned node.

Equation 15) reveals the "node" to be energy (E). Equation 16) shows that energy not only connects the uncertainty principle and field equations, it shows the electric field is linked as well. From 16) we can derive the integral form of Gauss's law for electric fields:

Equation 20) basically says electric charge produces an electric field. The field flux passing through a closed surface is proportionate to the charge contained within the surface.

If we take equation 20's integral and divide it by a volume (V), we can derive the differential form of Gauss's law for electric fields:

Equations 22) and 23) are just different ways of saying the electric field produced by electric charge diverges from a positive charge, or it converges upon a negative charge.

Going back to energy (E), we can begin again and derive Gauss's law for magnetic fields:

Equation 35) is the integral form. It says the magnetic flux passing through a closed surface is zero. According to equation 36), the magnetic field's divergence at any point is zero.

Given what we have derived so far, we can also derive the integral form of Faraday's law:

Equation 44) describes the electric generator. As the magnetic flux through a surface changes, an electric field is induced.

The differential form of Faraday's law is derived as follows:

According to equation 51), a magnetic field that changes with time produces a circulating electric field.

Next, we shall derive the integral form of the Ampere-Maxwell law:

Equation 63) says an electric current (I) produces a circulating magnetic field. The differential form is derived as follows:

According to equation 67), an electric field that changes with time produces a circulating magnetic field.

To put a cherry on top of the work we've done so far, we use Heisenberg's uncertainty principal and Einstein's field equations to derive the electromagnetic tensor (of equation 82) below):

Notice in equation 82) the Schur product was used to multiply the matrices to get the final result: the electromagnetic tensor.

To learn more about Maxwell's equations, there is an excellent book entitled "A Student's Guide to Maxwell's Equations" by Daniel Fleisch who explains every detail, symbol, and nuance. There's also this video which is also excellent: