3. Solving the Dirac Equation

Question 1

What is the first step to solving the Dirac equation?

Answer 1

Look for positive energy plane wave solutions
- Start by looking at a plane wave

Question 2

Describe the Helicity operator?

Answer 2

An operator that distinguishes between the two linearly independent solutions of the Dirac equation
- It involves a projection of spin onto the particle direction

Question 3

Should the Helicity operator commute with the Hamiltonian, and why?

Answer 3

Yes as the two linearly independent solutions have the same energy

Question 4

What is the condition from Heisenberg’s equation of motion?

Answer 4

That any operator that commutes with the Hamiltonian represents a constant of motion

Question 5

Is the Helicity operator h(p) Lorentz invariant, and why?

Answer 5

No - Can boost to a frame for a massive particle with a velocity larger than the particle speed
- Flips the helicity
- It is Lorentz invariant for massless particles

Question 6

What is the spin operator, and is it Lorentz invariant and why?

Answer 6

Spin operator is a capital sigma vector which has the pauli matricies on the diagonal and is a 2x2 matrix
- This is Lorentz invariant, but not a constnat of motion

Question 7

What is the chirality operator?

Answer 7

The gamma 5 matrix

Question 8

Which hand spinors does the weak interaction only operate on?

Answer 8

Left handed spinors

Question 9

Is the chirality operator, and the handed operators Lorentz invariant?

Answer 9

Yes

Question 10

Why can chirality and the helicity operator be confused?

Answer 10

In the relativistic limit where (E-m) ~ (E+M) ~ |p|or in the massless limit

Question 11

When considering the pion decay into a muon and anti-muon neutrino, or an electron and anti-electron neutrino, which is more likely due to phase space?

Answer 11

When considering phase space, the electron is more likely as it has more available momentum states. However, the muon is more likely

Question 12

When considering the helicity of the pion decay in its rest frame with a muon and muon neutrino coming radially outwards, what defines the + and - of the helicity?

Answer 12

The helicity is positive when the spin points in the direction of motion, and vv

Question 13

What is the handedness of a neutrino and anti neutrino?

Answer 13

Neutrino is Left handed
Anti neutrino is Right handed

Question 14

What is the handedness of the muon produced in the pion decay, and why?

Answer 14

Muon is left handed as it is the weak interaction
- If the muon was massless, this decay into a muon and anti-muon neutrino would be forbidden

Question 15

What is the helicity in the massless limit?

Answer 15

Helicity + is RH
Helicity - is LH

Question 16

If a particle has mass, how is the LH chiral state related to the helicity states?

Answer 16

The LH chiral state is a mixture of the two helicity states
- Split into a component that promote helicity + and another than promotes helicity -

Question 17

Describe the Dirac Sea

Answer 17

We know negative energy solutions exist, but these are not observed
- Assume the negative energy solution states are already filled
- PEP will stop transitions from E > 0 to E < 0

Question 18

Describe what happens if a photon with sufficient energy hits an electron in the negative energy in the Dirac sea

Answer 18

It would promote an electron from the E < 0 to E > 0 state
- Leaves behind a hole (a positron) in the Sea

Question 19

What is Feyman’s interpretation of the neagtive energy solutions?

Answer 19

The negative energy solutions represent E>0 particles which are travelling backwards in time
- Or antiparticles travelling forwards in time
- Solves the problem of bosons vs fermions as Dirac said PEP was involved

Question 20

What are the 4 types of discrete transformations?

Answer 20

Charge conjugation
Parity
Time reversal
Supersymmetry

Question 21

Describe what the operations of the 4 discrete transformations are

Answer 21

Charge: Flip the sign of all charges
Parity: x -> -x
Time: t -> -t
Supersymmetry: Fermions flip to bosons and vv

Question 22

What happens under parity to the dirac current?

Answer 22

The probability density is unchanged, but the current flips sign