# Question #efb13

##### 1 Answer

There are many interesting aspects of the Dirac Equation, but I'd say the three most important points are:

- It's a relativistic wave equation
- It predicts the existence of antiparticles
- It predicts the existence of spin

#### Explanation:

The *Dirac Equation*

proposed by Paul Dirac in 1928, is considered by many to be one of the most important equations of Physics, since it symbolizes the union of Quantum Mechanics and Special Relativity. In summary, it describes the way a particle that obeys the laws of Quantum Mechanics and behaves as time passes by.

For this reason, in order to have a basic understanding, we need to be acquainted with the basic principles of these two theories. So we must have some basic knowledge of those fields to understand the points above.

To understand the first point, we need to know that a relativistic wave equation is an equation for a wave function (a quantum mechanical description of matter) that obeys the relativistic energy relation

The first wave equation developed to describe matter in the quantum mechanical level was the *Schrödinger equation*

where

One of the basic results of Quantum Mechanics is that a measurement of the momentum of a particle is related (via an eigenvalue equation) to the momentum operator:

Then, if we take a look at the Schrödinger equation, we see that it can be written as:

Remember that the expression for the kinectic energy in classical mechanics is

Taking one more look at the Schrödinger equation, we can rewrite it as:

Thus, it makes sense to define the energy operator as:

Now, we still need to get to the Dirac equation. For simplicity, we'll have to ignore the potential energy term, that is set *Klein Gordon equation*, wich is an adequate description for bosons, but does not account for some of the behaviour of fermions.

So Dirac derived a new form of the relativistic energy relation:

where

This is just the old relation written in a different way, but for the quantum version of the theory, it has a fundamental difference: it's more similar to the Schrödinger equation since it has an energy term that is linear, and not quadratic (the Klein-Gordon equation has a quadratic term on the energy operator and has some problems with probability densities due to this).

Now we can try to arrive at the Dirac equation. We just need to plug in the energy operator defined above in place of

Dirac's insight was to define matrices

We reach this equation:

It turns out there are some requirements that these matrices must abide and the smallest matrices that do are some

To make it easier on notation, we can define

or

There is even a beautiful notation (called the Feynman slash notation) that, combined with natural units, that is

It turns out that this vector quality of *spinor*).

If we take a look at the equation

Dirac's ideia was that all negative energy states were occupied in the vaccuum and to create positive energy states we'd need to supply energy to bump a negative state up to a positive energy level.

This ideia (known as the Dirac Sea) corresponds to pair production and we can relate 2 of the 4 components of

What's most remarkable about this is that antimater was first discovered experimentally in 1932 - 4 years after Dirac's prediction!

Finally, we take a look at spin. In Pauli's theory of spin, a wave function is actually a 2 component spinor, in wich the first component corresponds to the *up* state while the second one corresponds to the *down* state. Recall that we have 2 components of out 4 component spinor dedicated to particles and 2 dedicated to antiparticles.

As said before, the *are* higher dimensional versions of Pauli matrices with positive and negative signs organized in a way that particles and antiparticles have oposite spin relations!

So the Dirac equation not only links Special Relativity and Quantum Mechanics, it also explains a previously known effect (spin) and predicted correctly a previously unknown type of matter (antimatter), making it one of the greatest theoritical successes in Physics - and the reason Dirac won the 1938 Nobel Prize in Physics.