Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

• we can verify the anti commutation relations by explicit calculation.NB: block matrices are multiplied just like ordinary matrices with non commutingmatrix elements( ) ( ) ( ) ( )γ0 2 1 0 1 0 1 0= = = 1 (42a)0 −1 0 −1 0 1( ) ( ) ( ) ( )[γ 0 , γ i] 1 0 0 σ = + γ0 γ i + γ i γ 0 i 0 σi 1 0=0 −1 −σ i +0 −σ i 0 0 −1() ()0 1 · σ=i 0 (−σ(−1) · (−σ i +i ) · 1) 0 σ i (42b)· (−1) 0( ) ( )0 σi 0 −σi=σ i +0 −σ i = 0 (42c)0Problem 3. Verify the remaining (k, l = 1, 2, 3) anti commutation relations (34):• in the Dirac realization (41) we have obviously• however, since (γ 0 ) 2 = 1 and (γ i ) 2 = − 1∴ γ 0 must have only real eigenvalues, and∴ all γ i must have only imaginary eigenvalues∴ this must be true in all realizations.[γ k , γ l ] + = −2δ kl · 1 . (43)γ † 0 = γ 0 , γ † i = − γ i (44)• Another useful and ubiquitous notation is therefore the Dirac adjoint for matricesA = γ 0 A † γ 0 , γ µ = γ 0 γ † µγ 0 = γ µ (45)• NB: on the next page we will meet the related, but different Dirac adjoint forcolumn vectorsv = v † γ 0 (46)• Don’t mix them up!2.6 Free Spin-1/2 Particles• Ansatz:∫ψ(x) = ˜dp ( ψ (+) (p)e −ipx + ψ (−) (p)e ipx) (47)(i/∂ − m) ψ(x) = 0 ⇔{(/p − m) ψ (+) (p) = 0(/p + m) ψ (−) (p) = 0(48)9