Fermions and the Dirac Equation

In 1928 Dirac proposed the following form for the electron wave equation:

The four µ matrices form a Lorentz 4-vector, with components, µ. That is, they transform like a 4-vector under Lorentz transformations between moving

frames. Each µ is a 4x4 matrix.

4x4 matrix 4x4 unit matrix

4-row column matrix

Dirac wanted an equation with only first order derivatives:

But the above must satisfy the Klein=Gordon equation:

This must =

Klein-Gordon equation

10 conditions must be satisfies

Pauli spinors:

=

1 =

2 =

3 =

They must be 4x4 matrices!

The 0 and 1,2,3 matrices anti-commute

The 1,2,3, matrices anti-commute with each other

The square of the 1,2,3, matrices equal minus the unit matrix

The square of the 0 matrixequals the unit matrix

4x4 unit matrix

All of the above can be summarized in the following expression:

Here gµ is not a matrix, it is a component

of the inverse metric tensor.

4x4 matrices

With the above properties for the matrices one can show that if satisfies the Dirac equation, it also satisfies the Klein Gordon equation. It takes some work.

4 component column matrix

4x4 unit matrix

Klein-Gordon equation:

Dirac derived the properties of the matrices by requiring that the solution tothe Dirac equation also be a solution to the Klein-Gordon equation. In the process

it became clear that the matrices had dimension 4x4 and that the was a column matrix with 4 rows.

The Dirac equation in full matrix form

0 1 2

3

spin dependence

space-timedependence

p0

After taking partial derivatives … note 2x2 blocks.

Writing the above as a 2x2 (each block of which is also 2x2)

Incorporate the p and E into the matrices …

The exponential (non-zero) can be cancelled out.

Finally, we have the relationships between the upper and lower spinor components

p along z and spin = +1/2

p along z and spin = -1/2

Spinors for the particle with p along z direction

p along z and spin = +1/2

p along z and spin = -1/2

Spinors for the anti-particle with p along z direction

Field operator for the spin ½ fermion

Spinor for antiparticlewith momentum p and spin s

Creates antiparticle with momentum p and spin s

Note: pµ pµ = m2 c2

Creates particle with momentum p and spin s

r, s = 1/2

Lagrangian Density for Spin 1/2 Fermions

1. This Lagrangian density is used for all the quarks and leptons – only the masses will be different!

2. The Euler Lagrange equations, when applied to this Lagrangian density, give the Dirac Equation!

Comments:

3. Note that L is a Lorentz scalar.

Dirac equation

The Euler – Lagrange equations applied to L:

[ A B ]†= B † A †

Lagrangian Density for Spin 1/2 Quarks and Leptons

Now we are ready to talk about the gauge invariance that leads tothe Standard Model and all its interactions. Remember a “gaugeinvariance” is the invariance of the above Lagrangian under

transformations like e i . The physics is in the -- which can be a matrix operator and depend on x,y, z and t.