Exercise 2: Clifford algebra

Quantum Field Theory

Set 12

Exercise 1: Transformation properties of spinor bilinears

Consider a Lorentz transformation parametrized by~θand ~η (respectively for rotation and boosts) and recall the representation on left-handed and right-handed spinors:

Λ_L(~η, ~θ)≡D₍1

2,0)(~η, ~θ) =e^{−12(~η+i~θ)·~σ}

ΛR(~η, ~θ)≡D_(0,1

2)(~η, ~θ) =e^{12(~η−i~θ)·~σ}

• Show that the spinor bilinearψ_R^†σ^µψR transforms as a 4-vector

• Consider thetensor defined in the lecture,=

0 1

−1 0

, which satisfies^T =⁻¹=−,⁻¹σⁱ=−(σⁱ)^∗=

−(σⁱ)^T.

Express⁻¹ΛLin terms of ΛR.

• Use the previous two results to determine the transformation property ofψ_L^†σ¯^µψL. Remember the definitions: σ^µ= (1, σⁱ), σ¯^µ= (1,−σⁱ)

Exercise 2: Clifford algebra

• Show thatγ^µ transforms as a vector:

Λ⁻¹_D γ^µΛD= Λ^µ_νγ^ν. where:

γ^µ=

0 σ^µ

¯ σ^µ 0

, (1)

ΛD=

ΛL 0

0 ΛR

• Compute the anticommutator of two Dirac matrices: {γ^µ, γ^ν}

• Define the matrixγ⁵≡iγ⁰γ¹γ²γ³. Using only the previous result show that{γ^µ, γ⁵}= 0

• Prove that PL ≡ ¹₂(1 +γ⁵), PR ≡ ¹₂(1−γ⁵) define two orthogonal projectors: PL+PR = 1, P_L² = PL, P_R²=PR,PLPR= 0.