Homework 6

Question 1:
Verify explicity that the Dirac spinors (as defined in the lectures) satisfy u⁺(k,s') u(k,s) = 2E δ_ss' and
u—(k,s') u(k,s) = 2M δ_ss' , where u—= u⁺ γ⁰. (The superscript "+" here denotes the Hermitian conjugate.)

Question 2:
Summing over the spins s = +1/2 and -1/2, show that Σ_s u(p,s) u—(p,s) = ( γ^μ p_μ + M ) .

Question 3:
Verify explicitly the Gordon identity.

Question 4:
Show that the Dirac matrices satisfy γ^μ+ = γ⁰ γ^μ γ⁰ .

Question 5:
Verify for each component of μ and ν (= 0 or i) that Tr [ γ^μ γ^ν ] = 4 g^{μ ν}, where g^{μ ν} = diag(1,-1,-1,-1)
is the metric tensor. Show that Tr [ γ^μ γ^ν γ^λ γ^ρ ] = 4 ( g^{μ ν} g^{λ ρ} - g^{μ λ} g^{ν ρ} + g^{μ ρ} g^{λ ν} ) .

Question 6:
Defining γ⁵ = i γ⁰ γ¹ γ² γ³ , show that Tr [ γ⁵ γ^μ γ^ν ] = 0.