Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

• invariant overlap from conserved current:∫∫Q = d 3 ⃗x j 0 (x) = d 3 ⃗x ¯ψ 1 (x)γ 0 ψ 2 (x) =∫d 3 ⃗x ψ † 1 (x)ψ 2(x)x 0 =tx 0 =tx 0 =t∫(d 3 ⃗p 1=ψ (+)†(2π) 3 1(⃗p)ψ (+)2(⃗p) + ψ (−)†1(−⃗p)ψ (+)2(⃗p) · e −2ip 0t2p 0 2p 0)+ ψ (+)†1(⃗p)ψ (−)2(−⃗p) · e 2ip0t + ψ (−)†1(−⃗p)ψ (−)2(−⃗p)∫d 3 ⃗p=(2π) 3 2p 02∑ ()b † 1,k (p)b 2,k(⃗p) + d † 1,k (p)d 2,k(⃗p)k=1(69)∴ normalization positive everywhere!Problem 9. Show the Partial Conservation of the Axial Current (PCAC) for solutions ψ 1 (x)und ψ 2 (x) of the Dirac equation∂ µ [ ¯ψ 1 (x)γ µ γ 5 ψ 2 (x) ] = 2im ¯ψ 1 (x)γ 5 ψ 2 (x) . (70)Problem 10. Compute σ klGamma matrices.= i 2 [γ k, γ l ] −for k, l = 1, 2, 3 in the Dirac realization of the∴ the matrices {σ 23 , σ 31 , σ 12 } can take over the rôle of the Pauli matrices {σ 1 , σ 2 , σ 3 }and distinguish spin up from spin down in the rest frame12 σ 12u 1 ( ⃗ 0) = + 1 2 u 1( ⃗ 0) ,12 σ 12v 1 ( ⃗ 0) = + 1 2 v 1( ⃗ 0) ,12 σ 12u 2 ( ⃗ 0) = − 1 2 u 2( ⃗ 0) (71a)12 σ 12v 2 ( ⃗ 0) = − 1 2 v 2( ⃗ 0) (71b)• just as before in the case of scalar particles, we can interpret solutions on thenegative energy mass shell for spin-1/2 particles as anti particles that move inthe opposite direction in space time:– u k (p) amplitude for a particles in the initial state– ū k (p) amplitude for a particle in the final state– v k (p) amplitude for an anti particle in the final state– ¯v k (p) amplitude for an anti particle in the initial state• in addition to the other quantum numbers, we must flip the spins so that theoverall balance is maintainedQ, ⃗p, s−Q, −⃗p, −s13