A note concerning the Dirac equation and its derivation

I will give here some comments and  clarifications about the derivation and developing of the Dirac equation from a historical point of view.

dirac001

The Dirac equation for a free particle is given by the following definitions and expressions:

  dirac-eq-01

In the above ψ is the wave function for the free particle , p is the momentum operator with three components , c is the speed of light ,  ħ  is the reduced Planck constant , i=√(-1) is of course the imaginary number unit , H is the Hamiltonian , and t is time.Dirac first elaborated and published his equation in 1928. The spatial coordinates have been written in different books and textbooks using different notations:

dirac-eq-02a

With the superscript notation , the Dirac equation takes the following condensed form:

dirac-eq-03

The α symbols in the equation are the original Dirac matrices :

dirac-matrices-00

These matrices take the more detailed form:

dirac-matrices

And the wave function ψ is expressed by:

psi-01

If we multiply , develop and expand the terms of the Dirac equation , we get the following set of four equations:

dirac-eq-04

Rearranging and reordering these equations we obtain:

dirac-eq-05

In his book about The Theory of Spinors  (originally published in 1938) , Elie Cartan obtained the following four Dirac equations:

cartan-dirac

The letter V represents the magnetic potential , which is mostly expressed by the letter A in modern notation. But since we’re considering the equation for a free particle we can neglect V and make it equal to zero.   Cartan noted that we pass from his notation to that of Dirac by making the following replacements or substitutions:

dirac-replace

  In fact  after replacing we get the following set of equations:

dirac-eq-06

Now if we combine ,add or subtract these equations two by two we get the Dirac equations.
Adding equations (2.1) and (2.4) and simplifying gives equation (1.2) above.
Similarly: Adding equations (2.2) and (2.3) and simplifying gives equation (1.3) above.
(2.1) – (2.4) give (1.4) , and (2.3) – (2.3) give (1.1) .

The Dirac equation presented a relativistic version of the Schrodinger equation for spin 1/2 particles and predicted the existence of antimatter. It is of importance  in many domains such as relativistic quantum mechanics and  Quantum Field Theory.

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