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  k02k2Lik02k2Li

Euclidian vector Gamma function (def.) Wick rotation extend k 0 to complex k02k2Lik02k2Li ∟ 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 ( くりこみ ) 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. ( このように、量子電気力学はくりこみ可能である。 ) 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. ( 結合定数の質量次元が正か 0 であってもくりこみ可能 とは限らない。 ) (Lagrangian に、演算子部分の質量次元が 4 より大の相互 作用項を含む理論はくりこみ不可能である。 ) ( 結合定数の質量次元が負の相互作用項を含む理論はく りこみ不可能である。 ) The theory, however, is not necessarily renormalizable, even if all the coupling constants have non-negative mass dimensions,