K p k'k' p'p' probability amplitude locality,Lorentz inv.gauge inv. spinor vector A T electron quark scattering scattering cross section Feynman.

k p k'k' p'p' probability amplitude locality,Lorentz inv.gauge inv. spinor vector A T electron quark scattering scattering cross section Feynman rules photonfermionvertex loop fermion loop T matrix Feynman rules:draw graphs with& arrange the factors etc. photonfermionvertex loop fermion loop T matrix Feynman rules:draw graphs with& arrange the factors photonfermionvertex loop fermion loop T matrix Feynman rules:draw graphs with& arrange the factors photonfermionvertex loop fermion loop T matrix loop diagram divergent for q divergent for k photonfermionvertex loop fermion loop T matrix divergent for q photonfermionvertex loop fermion loop T matrix complete the square in the denominater with respect to q. use complete the square in the denominater with respect to q. use dimensional regularization: extend dimension n to non-integer I converges for non-integer n, and diverges as n 4. Finally we will renormalize the divergences. dimensional regularization: extend dimension n to non-integer I converges for non-integer n, and diverges as n 4. Finally we will renormalize the divergences. change the integration variable The parts odd in q' vanish. n n q 'q ' extend k 0 to complex k02k2Lik02k2Li Euclidian vector Gamma function (def.) Wick rotation extend k 0 to complex k02k2Lik02k2Li dnKdnK K2K2 ( ) n /2 1 Lt ( ) Lt L m ( t 1) m beta function : totally symmetric tensor : totally symmetric tensor : totally symmetric tensor Tr(odd matrices) = 0 change variables The parts odd in k' vanish. F e = #ext. fermion lines B e = #ext. photon lines L = #loops I diverges for primitive divergence fermion self energy part photon self energy part vertex part convergent owing to gauge invariance F i = #int. fermion lines B i = #int. photon lines V = #vertices for L = 1 proof of (1) holds. (1) (1) holds. inserting fermion lines inserting photon lines Incertions of photon lines & fermion lines do not change the l.h.s of (1) (1) always holds. proof of (1) (cont'd) fermion self energy part photon self energy part vertex part proper part(1 particle irreducible part) 1 particle reducible renormalization ( ) Consider a system with spinor & photon A Lagrangian Feynman rules internal lines photon fermion vertex particle anti-particle loop fermion loop T matrix external lines fermion self energy part photon self energy part proper vertex part primitive divergences the following items are added to the Feynman rules If we add to the Lagrangian which always appear in sum with the primitive divergences, and, hence, can be taken so as to cancel out all the divergences. the term (The gauge fixing term is redefined.) : renormalization constants All the divergences arising from these rules can be canceled out by choosing appropriately. Then We re-derive the Feynman rules for it. Relations among observables do not depend on the choices. : renormalized Take Thus, quantum electrodynamics is renormalizable. ( ) If the Lagrangian includes the terms with the mass dimension greater than 4, the theory is not renormalizable. It the theory includes interactions with coupling constants with negative mass dimensions, the theory is not renormalizable. ( 0 ) (Lagrangian 4 ) ( ) The theory, however, is not necessarily renormalizable, even if all the coupling constants have non-negative mass dimensions,