British physicist, Paul A. M. Dirac created an equation which at the time revolutionized physics and it's ability to describe Fermion particles. Fermion particles are spin-1/2 particles. Spin is a quantum property of all particles in the universe with maybe the exception of one... the Higgs Boson. Spin comes in either integer or half-integer values and is denoted by the angular momentum (the quantum action) ħ.
Dirac's equation is called a wave equation - it is also relativistic making it a classical wave equation. Classical equations don't take into account, quantum uncertainty (Heisenberg's Uncertainty Principle). Dirac started off with the relativistic energy-momentum formula discovered by Einstein
E² = p²c² + M²c²²
(When you see something raised to ²² that is equivalent to stating it is raised to the 4th power). We will work in natural units by setting the speed of light equal to 1 (c=1). This makes the relativistic equation
E² = p² + M²
We will now consider the energy as E = ħω where ω is the angular frequency and we shall replace the momentum with the wave number k, but we shall also set ħ=1 which means we have
ω² = M² + k²
We will concentrate now on a function
ψ = e^(i(kx - ωt))
here, ''i'' is an imaginary number, t is time and x is position. We may pull down the derivatives with respect to x and k which gives us an equation
iω - ik = 0
Keep these things in mind, as we shall now move onto the concept of ''right moving waves'' and ''left moving waves.'' Right movers and left movers are another way of saying particles and antiparticles. A wave equation with right movers can be given as
∂ψ_(R)/∂x + ∂ψ_(R)/∂t = 0
A wave equation with left movers (antiparticles) may be described as thus
-∂ψ_(L)/∂x + ∂ψ_(L)/∂t = 0
Notice we have a sign difference. We may write ∂ψ/∂x as the time derivative of ψ but we shall sneakily add a coefficient α - jumping ahead slightly, I wish to explain very briefly what it is... α is a matrix - it is one of two matrices that appear in the Dirac Equation. α has the form
The square of this matrix is the identity matrix
Which is an important property for the Dirac Equation as we will see in the Clifford algebra part we will be coming too soon. Now, before we race ahead too much, we shall take a few steps back and write the time derivative we spoke about with our matrix coefficient
∂ψ/∂t = α(∂ψ/∂x)
How did we get this? Remember, we had
-∂ψ_(L)/∂x + ∂ψ_(L)/∂t = 0
Ignoring the left moving subscript descriptions, we simply rearrange that equation to find the one above. When you hit ∂/∂t with a wave ψ this gives iω. When you hit ∂/∂x with ψ this gives you an ik, so what we now have is (after removing the imaginary numbers)
-ω² = -α - k²
Cancelling the minus signs we have
ω² = αk²
Now, we will add a mass term to this, but we shall introduce a new matrix... the beta matrix β
ω² = αk² + βM²
β has the form
It also has the property that when squared will give you unity, or better said, the (unity matrix) which is often written as I
α² = β² = I
This means the matrices are Hermitian - Hermitian matrices represent ''observables'' in physics. If it is not Hermitian, then it is regarded as not being an observable. These matrices are also traceless and have eigenvalues equal to either +1 or -1. Some words in here I cannot explain all, but it is a just a quick summery over some other related mathematical concepts.
Now, let us square the matrices in ω² = αk² + βM² and doing so we may be able to factorize the equation
ω² = α²k² + β²M² = (αk + βM)(αk + βM)
Working it out it gives
= α²k² + M²β² + αkβM + αβkM
Now this tells us something interesting, we know that alpha matrix squared and beta matrix squared is equal to 1, which means that whatever αkβM + αβkM is, it is not required because the first two terms k² + M² is known to equal ω². Therefore, there is a unique property involved with the matrices, we find that property as
αβ + βα = 0
This is called a Clifford Algebra. There are many kinds of Clifford Algebra in the world of mathematics and Dirac discovered this one by accident. Ok, the Dirac Equation has now taken shape, we may describe left movers and right movers as
i(∂ψ_(R)/∂t) = -iα∂_(x)ψ_(R) + βMψ_(L)
i(∂ψ_(L)/∂t) = +iα∂_(x)ψ_(L) + βMψ_(R)
What a mass term does in the Dirac equation (I will simply tell you but not vigorously work it out as it requires writing down more matrices) is that it mixes left movers and right movers together and you may have noticed this with the terms of both right movers and left movers on the RHS (Right hand side) of both equations above.
And that is the Dirac Equation in a nutshell. For a particle at rest you just throw away the space variation.
i(∂ψ_(R)/∂t) = βMψ_(L)
i(∂ψ_(L)/∂t) = βMψ_(R)
This part ∂_(x) is partial differential notation, it basically means ∂/∂x. You can also work out what is called the Dirac Langrangian by variating the Dirac Equation with psi-star ψ* which actually produces psi-dash ψ' (usually a psi-bar but the ability to write certain notation here is crippling).
The Dirac Equation helps describe particles like electrons and protons with incredible accuracy.