This is the third in the trilogy of prequels of Quantum Integer Programming (QuIP), the previous two being:
Pythagoras Theorem and Time Dilation, covering Special Relativity
Newton v Schrodinger (and Heisenberg), covering (non-relativistic) Quantum Mechanics
Early naïve attempts (by Schrodinger, Klein, Gordon among others) to marry Special Relativity with Quantum Mechanics did not lead anywhere.
The time derivative in the resulting wave equation is of second order, while in Schrodinger Equation it is first order.
The stumbling block was that a square root of a second order differential operator is needed.
What does that even mean? Should you take the positive or the negative square root?
Dirac handled the issue deftly.
As you can see from the top of the post, instead of worrying about the square root of the entire expression, he looked to find the square root of just the first term.
What is remarkable is this. Dirac was not aware of Quaternions and Clifford Algebra, although they had been developed decades (half a century!) earlier.
He re-constructed the square root on his own, and so rediscovered a special case of the earlier results.
Then he did what I absolutely love (and have previously discussed in my post Imaginary Numbers and Electron Spin as a general methodology):
He factored the expression using imaginary numbers.
Then, he wrote it in a form consistent with Schrodinger Equation.
This had four terms, different from the known operators for position and momentum.
What did these new terms represent physically? Something new. New degree of freedom:
There is more. Spin accounts only for two of the four independent components. What else is the equation hiding?
Wait for it:
Some guys have all the luck. 😏