Quantum Field Theory. Note 5

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Parity transformation

Parity transformation of the coordinates

P ( ^t_x ) = ( ^t_-x )

Transformation of the generators A and B of the Lorentz group

PAP^-1 = B ; PBP^-1 = A

Therefore under the parity transformation the representations (j₁j₂) and (j₂j₁) of the Lorentz group transform into each other

d^(j₁j₂) ↔ d^(j₂j₁)

Irreducible representation D(j₁,j₂) of the general Lorentz group can then be made as a direct sum (j₁j₂)⊕(j₂j₁)

D(j₁,j₂) = [

d^(j₁j₂)	0
0	d^(j₂j₁)

]

For the important spin ½ field the Lorentz transformation is

( ^φ_χ )' = [ ^d₀ ⁰_(d⁺)^-1 ] ( ^φ_χ )

Bilinear forms of bispinors. Dirac matrices

The Dirac matrices reduce a direct product ψ^*⊗ψ into irreducible objects

Scalar, S= ψψ ≡ ψ⁺γ₀ψ , γ₀=[⁰₁ ¹₀]

Pseudoscalar, P= ψγ₅ψ , γ₅=i[¹₀ ⁰_-1]

Four-vector, V_i= ψγ_iψ , γ=[⁰_σ ^-σ₀]

Pseudo-vector, A_i= ψγ_iγ₅ψ ,

Antisymmetric tensor T_ij = ψ½(γ_iγ_j-γ_jγ_i)ψ

Exercises

(Mulders, Excercise 3.5) Show that if the spinor φ transforms under (½0) then the spinor φ^* transforms under (0½). Hint: show that there exists a unitary transformation U⁺σU=-σ^*