3. Solving the Dirac Equation Flashcards
What is the first step to solving the Dirac equation?
Look for positive energy plane wave solutions
- Start by looking at a plane wave
Describe the Helicity operator?
An operator that distinguishes between the two linearly independent solutions of the Dirac equation
- It involves a projection of spin onto the particle direction
Should the Helicity operator commute with the Hamiltonian, and why?
Yes as the two linearly independent solutions have the same energy
What is the condition from Heisenberg’s equation of motion?
That any operator that commutes with the Hamiltonian represents a constant of motion
Is the Helicity operator h(p) Lorentz invariant, and why?
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
What is the spin operator, and is it Lorentz invariant and why?
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
What is the chirality operator?
The gamma 5 matrix
Which hand spinors does the weak interaction only operate on?
Left handed spinors
Is the chirality operator, and the handed operators Lorentz invariant?
Yes
Why can chirality and the helicity operator be confused?
In the relativistic limit where (E-m) ~ (E+M) ~ |p|or in the massless limit
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?
When considering phase space, the electron is more likely as it has more available momentum states. However, the muon is more likely
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?
The helicity is positive when the spin points in the direction of motion, and vv
What is the handedness of a neutrino and anti neutrino?
Neutrino is Left handed
Anti neutrino is Right handed
What is the handedness of the muon produced in the pion decay, and why?
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
What is the helicity in the massless limit?
Helicity + is RH
Helicity - is LH
If a particle has mass, how is the LH chiral state related to the helicity states?
The LH chiral state is a mixture of the two helicity states
- Split into a component that promote helicity + and another than promotes helicity -
Describe the Dirac Sea
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
Describe what happens if a photon with sufficient energy hits an electron in the negative energy in the Dirac sea
It would promote an electron from the E < 0 to E > 0 state
- Leaves behind a hole (a positron) in the Sea
What is Feyman’s interpretation of the neagtive energy solutions?
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
What are the 4 types of discrete transformations?
Charge conjugation
Parity
Time reversal
Supersymmetry
Describe what the operations of the 4 discrete transformations are
Charge: Flip the sign of all charges
Parity: x -> -x
Time: t -> -t
Supersymmetry: Fermions flip to bosons and vv
What happens under parity to the dirac current?
The probability density is unchanged, but the current flips sign
