**The Dirac Sea**

So we have managed to understand that we have left movers and right moving wave, and in order for Dirac to stop real particles loosing energy and simply falling back into the vacuum, he required what he called a sea of negative particles, both contained with an equal proportion of left movers and right movers so it is symmetric with respect to left and right so there is no longer such a huge momentum in one particular directionality. The spin states would cancel out particles quite simply, and whenever we create a particle from the vacuum, this leaves a hole - the hole is the antiparticle of the electron, the positron.

So last time we more or less ended with our standard matrix notation expressed in column vectors:

φ_{R}

φ_{L} = ψ

This kind of notation is very handy to say the least; if one was to calulate this, we would find a matrix we will introduce soon. Usually we would place a dot above the expressions φ_{R} and φ_{L} to express that we are taking the time derivative of the quantities.

Such an equation would be read as:

(∂/∂dt) ψ = α (∂ψ/∂x)

We have a newcomer, and his name is alpha α and he is a matrix. The importance of alpha will become clear as we proceed.

If we take the square root of the equation

ω^{2}=m^{2}+k^{2}

we naturally have ω=√m^{2}+k^{2}

What we would like to do now, is rewrite this equation (∂/∂dt) ψ = α (∂ψ/∂x) in a more compact form. When you take a wave and hit it with ∂/∂dt it may be remembered that this pulls down -iω. We then look at ∂ψ/∂x and this pulls down a -ik. The equation now has the more simplified form of:

-iω=-α-ik

Cancelling out the minus signs and the i's we finish with:

ω=αk

The alpha here acts like an instruction to express k either positively or negatively. Because of our relativistic relationship ω=√m^{2}+k^{2} we would like to consider a mass and a new matrix which is called beta:

ω=αk+mβ

**Anticommutation in a Clifford Algebra**

A requirement on our matrices is that whatever α and β are, they must satisfy that the square of ω is equal to m^{2}+k^{2}. So let's square it:

ω^{2}=α^{2}k^{2}+m^{2}β^{2}

Decomposing this equation, we find a usual notation:

ω^{2}=(αk+mβ)(αk+mβ)

we are now going to expand our terms - we get:

(αk+mβ)(αk+mβ)=α^{2}k^{2}+m^{2}β^{2}+αβKm+βαkm

we can rewrite the superlfuous quantities αβKm+βαkm here, knowing that k is simply momentum and m is the mass as:

(αβ+βα)km

why are they superfluous? Remember what I said, the matrices require that omega squared is equal to k^{2}+m^{2} so looking at our terms, we can see that the first lot of them ie. α^{2}k^{2}+m^{2}β^{2} already satisfy the omega squared part. Because of this, (αβ+βα)km can be seen as an unecessery left over.

Whatever alpha is (we know its a matrix) but whatever it would have been to the imagination before this, we would know that through matrix calculations, α^{2}=1. Likewise, whatever beta would be also requires β^{2}=1. Because (αβ+βα)km has no place in our formula, this means that we need α and β to satisfy an algebraic property called anticommutation in that (αβ+βα)=0. This is actually a clifford algebra. With a little matrix calculation on the side, we must have alpha obide by the algebraic properties of

10

0-1 = α

so α^{2} is

10

01 = α^{2}

β cannot be the same as α because if it where, αβ+βα would just be twice alpha squared, so that doesn't work. So:

01

10 = β

If you are any good at metrix algebra, you can check that by squaring it.

Once we know alpha and beta, we can explicitely rewrite everything to produce the famous Dirac Equation.

**The Dirac Equation in a Finalized Form**

After some mathematics expressed in matrix notation, one can arrive at the equation:

iψ_{R}'=-i∂_{x}ψ_{R}+mψ_{L} which are right moving waves

iψ_{L}'=+i∂_{x}ψ_{L}+mψ_{R} which are left moving waves

Where the dash A' of an object here just means taking the mathematical object by its time derivative, as an alternative to upper dot notation. And ∂_{x} is standard partial derivative notation is just ∂/∂x. Because beta interchanges the sign of the wave from psi-left and psi-right, the equation:

iψ'=-iα∂_{x}ψ+mβ

becomes a coupled equation which just means our left movers and right movers become coupled and this is what a mass term does acting as a scale factor of β. Spin enters the Dirac equation when Pauli Matrices are taken into consideration.

The Dirac Equation explicitely describes fermions with an intrinsic spin, and if you wanted to persue an equation which is void of spin, the Klein-Gorden Equation would satisfy.

In a compact notation, the theory of spin would arise from two specific matrices:

0I

I0 = β

Where ''I'' is the unit matrix

σ_{j}0

0σ_{j} = α

Each entry here is a 2X2 matrix and σ_{j} is the presence of the pauli matrix where j=1,2,3.