DiracEquation
The Schrodinger equation in non-relativistic quantum mechanics (QM) expresses the non-relativistic energy-momentum relation, {$ E=p^2 / 2 m$}
In the standard way, {$E$} and {$p$} are turned into differential operators, {$E=-i \hbar \frac{\partial}{\partial t}$} and {$p=i \hbar \frac{\partial}{\partial x}$} for one dimension {$x$} or in general {$\vec{p}=i \hbar \nabla$}
The Klein-Gordon equation describes scalar particles using the relativistic energy relation {$E^2=p^2 c^2 + m^2 c^4$}. I believe Schrodinger had originally conjectured this form of the equation. It has the form of a wave equation.
Dirac sought an equation expressing the relativistic energy relation yet linear in FIRST derivatives. The Klein-Gordon equation allowed negative energy solutions because it is based on an energy equation for energy squared. Dirac's equation eventually also displayed this behavior, leading to the prediction of antiparticles.
The story goes that he conjectured his equation while staring into a fireplace. The requirement of linear in first derivatives, reducing to the relativistic energy relation, can only be met if the wave function is now a "Dirac spinor", with 4 complex components in a column vector, multiplied by 4 complex matrices, 4x4, called the Dirac Gamma matrices.
The Dirac equation can be made to express different physical interpretations depending on the choice of Gamma matrices. Any choices are possible as long as they satisfy the Clifford Algebra, that is, that they square to 1, and different {$\Gamma$} anticommute.
Dirac representation: The four components of the Dirac spinor describe a spin up and down fermion and spin up and down anti-fermion.
Weyl representation: The four components of the Dirac spinor describe spin up and down and Left and Right handed fermions
The spinor wave function now no longer expresses just a probability but a probability current.
The Dirac Equation describes single-particle relativistic quantum physics, though it has both fermion and antifermion solutions. The form of the Lagrangian for the Dirac equation can also be used in its quantum field theory.
Educational resources on the Dirac Equation:
- William O. Straub's description of spinors and the Dirac equation (archived from his WeylMann blog)
- Gamma matrices are essential to understanding the Dirac equation. They can be built up from Pauli matrices (Wikipedia links)
- The Dirac Gamma matrices obey a Clifford Algebra (Wikipedia link)
- Bernd Thaller's book on the Dirac Equation which somehow I have a digital copy of
NEW
An annotated letter from Richard Feynman from 1947 in which he struggles with finding a deeper explanation for the Dirac equation and a path integral formulation