Relativistic Quantum Mechanics Sheet 2

Relativistic Quantum MechanicsSheet 21. Show that the commutator of the Dirac Hamiltonian with the orbital angular momentumoperator ⃗ L = ⃗r × ⃗p is[H, ⃗ L] = −i⃗α × ⃗p.[Hint: Write ⃗ L = ɛ ijk x i p j ⃗e k , where ⃗e k (k = 1, 2, 3) are basis vectors.]Show that the commutator of the Dirac Hamiltonian with the quantity⃗Σ =⎛⎝ ⃗σ 00 ⃗σ⎞⎠ = −iα 1 α 2 α 3 ⃗αis[H, ⃗ Σ] = 2i⃗α × ⃗p.[Hint: Write α 1 α 2 α 3 = 1 3! ɛ ijkα i α j α k and use α 2 i = 1 to show Σ i = − i 2 ɛ ijkα j α k .]What is the significance of the quantity ⃗ L + 1 2 ⃗ Σ?2. Show that γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 in the Dirac representation is given byγ 5 =⎛⎝ 0 11 0⎞⎠ .Also show that{γ µ , γ 5 } = 0,in all representations of the γ-matrices.3. Prove that:γ µ γ µ = 4,γ µ γ α γ µ = −2γ α ,γ µ γ α γ β γ µ = 4g αβ ,γ µ γ α γ β γ ρ γ µ = −2γ ρ γ β γ α .1

4. If Γ represents a string of γ-matrices (not including γ 5 ) and Γ R is its reverse (i.e. thesame γ-matrices in reverse order), show that,[ū(k ′ )Γu(k)] † = ū(k)Γ R u(k ′ ).What is the analogous expression for [ū(k ′ )γ 5 Γu(k)] † ?5. Prove the following trace identities:Tr γ µ γ ν = 4g µν ,Tr γ µ γ ν γ ρ = 0,Tr γ µ γ ν γ ρ γ σ = 4 (g µν g ρσ − g µρ g νσ + g µσ g νρ ) .2