2017 II 1 Schwinger Yang-Mills 5. Higgs 1

2017 II 1 Schwinger 2 3 4. Yang-Mills 5. Higgs 1

1 Schwinger Schwinger φ 4 L J 1 2 µφ(x) µ φ(x) 1 2 m2 φ 2 (x) λφ 4 (x) + φ(x)j(x) (1.1) J(x) Schwinger source term) c J(x) x S φ d 4 xl J (1.2) φ(x) m 2 φ(x) 4λφ 3 (x) + J(x) 0 (1.3) ˆφ(x) m 2 ˆφ(x) 4λ ˆφ 3 (x) + J(x) 0 (1.4) ˆφ(x) c- J(x) 0, + ˆφ(x) m 2 ˆφ(x) 4λ ˆφ 3 (x) + J(x) 0, J 0 (1.5) 0, J J 0, + J 0, J J 0, + 0, + 0, J (1.6) Schwinger h i δ δj(x) 0, + 0, J 0, + ˆφ(x) 0, J (1.7) 2

H J d 3 x{π(t, x) φ(t, x) L J } d 3 x{ 1 2 [Π(t, x)]2 + 1 2 φ(t, x) φ(t, x) + 1 2 m2 φ(t, x) 2 φ(t, x)j(t, x)} 0, + 0, J 0, + exp[ ī h dth J] 0, J(x) δj(x) δj(y) δ(x y) d 4 y δj(x) δj(y) δj(y) d 4 yδ(x y)δj(y) δj(x) ( h δ δ i )2 δj(x) δj(y) 0, + 0, J 0, + T ˆφ(x) ˆφ(y) 0, J (1.8) T T ˆφ(x) ˆφ(y) ˆφ(x) ˆφ(y) for x 0 > y 0, ˆφ(y) ˆφ(x) for y 0 > x 0 (1.9) (1.8) x 0 > y 0 h i δ δj(x) 0, + 0, J 0, + ˆφ(x) 0, J (1.10) n 0, + ˆφ(x) n, x 0 n, x 0 0, J δ δj(y) h i n, x 0 0, J Schwinger 3

( h δ δ i )2 δj(x) δj(y) 0, + 0, J 0, + ˆφ(x) n, x 0 h δ n i δj(y) n, x0 0, J 0, + ˆφ(x) n, x 0 n, x 0 ˆφ(y) 0, J n 0, + ˆφ(x) ˆφ(y) 0, J (1.11) x 0 < y 0 ˆφ(x) ˆφ(y) n ( h δ i )n δj(x 1 )... δ δj(x n ) 0, + 0, J 0, + T ˆφ(x 1 )... ˆφ(x n ) 0, J Green (1.12) ig(x 1,..., x n ) 0, + T ˆφ(x 1 )... ˆφ(x n ) 0, J0 (1.13) Green LSZ S- Green T T 0 T ˆφ(x) ˆφ(y) 0 (1.14) T ( ) T ˆφ(x) ˆφ(y) ˆφ(x) ˆφ(y) for x 0 > y 0, x 0 y 0 x 0 y 0 T ˆφ(y) ˆφ(x) for y 0 > x 0 (1.15) T ˆφ(x) ˆφ(y) ˆφ(x) ˆφ(y)θ(x 0 y 0 ) + ˆφ(y) ˆφ(x)θ(y 0 x 0 ) (1.16) 4

x 0θ(x0 y 0 ) δ(x 0 y 0 ), θ(x 0 y 0 ) 1 for x 0 > y 0, 1 2 for x0 y 0 0 for y 0 > x 0 (1.17) x 0θ(y0 x 0 ) δ(x 0 y 0 ) (1.18) f(x 0 ) f(± ) 0 ( y 0 ))f(x 0 )dx 0 x 0θ(x0 y 0 )f(x 0 ))dx 0 x 0(θ(x0 θ(x0 y 0 ) )dx 0 x 0f(x0 y 0 )f(x 0 ))dx 0 )dx 0 x 0(θ(x0 y 0 +0 x 0f(x0 θ(x 0 y 0 )f(x 0 )) f(x 0 ) y 0 +0 f(y 0 + 0) f(y 0 ) θ(x 0 y 0 ) 1 2 x0 y 0 T x 0 y 0 T x 0T ˆφ(x) ˆφ(y) T ˆφ(x) ˆφ(y) x0 x0t ˆφ(x) ˆφ(y) T x ˆφ(x) ˆφ(y) + [ 0 ˆφ(x0, x), ˆφ(y 0, y)]δ(x 0 y 0 ) T T T Bjorken- Johnson-Low 5

1.1 Feynman Green 0, + 0, J 0, + 0, J Schwinger [ h δ i δj(x) m2 h δ i δj(x) 4( h δ 3 i )3 δj(x) + J(x)] 0, + 0, 3 J 0, + ˆφ(x) m 2 ˆφ(x) 4λ ˆφ 3 (x) + J(x) 0, J 0 (1.19) J(x) 0, + 0, J Lagrangian [ h i 0, + 0, J Dφ exp{ ī h d 4 xl J }. (1.20) δ δj(x) m2 h δ i δj(x) 4( h δ 3 i )3 δj(x) + J(x)] 0, + 0, 3 J Dφ[ φ(x) m 2 φ(x) 4λφ 3 (x) + J(x)] exp{ ī d 4 yl J } h δ Dφ( d 4 yl J ) exp{ ī d 4 yl J } δφ(x) h Dφ h δ i δφ(x) exp{ ī d 4 yl J } (1.21) h ( y ϵ(x) D(φ + ϵ) Dφ (1.22) Dφ h i δ δφ(x) exp{ ī h d 4 yl J } 0 (1.23) Dφ exp{ ī h d 4 yl J (φ)} 6

+ D(φ + ϵ) exp{ ī h Dφ exp{ ī h d 4 yl J (φ + ϵ)} d 4 yl J (φ + ϵ)} Dφ exp{ ī d 4 yl J (φ)} h Dφ ī d 4 δ xϵ(x)( d 4 yl J ) exp{ ī h δφ(x) h d 4 yl J (φ)} (1.24) ϵ(x) ϵ(x) x Dφ h i δ δφ(x) exp{ ī h 0, + 0, J d 4 yl J } 0 (1.25) Dφ exp{ ī h d 4 xl J } (1.26) Feynman (1.22) Dφ x dφ(x) (1.27) (1.22) Grassmann Dψ x δ δψ(x) (1.28) (1.22) Schwinger Grassmann ψ(x)ψ(y) + ψ(y)ψ(x) 0 (1.29) 7

ψ(x)ψ(x) 0 (1.30) (1.22) 2 Feynman φ 4 L J 1 2 µφ(x) µ φ(x) 1 2 m2 φ 2 (x) λ 4! φ4 (x) + φ(x)j(x) (2.1) Feynman T 1 2 µφ(x) µ φ(x) V 1 2 m2 φ 2 (x) + λ 4! φ4 (x) 1 2 m2 φ 2 (x) 2 λ 4! φ4 (x) λ 4! φ4 (x) (2.2) 0, + 0, J Dφ exp{ ī h d 4 xl J } (2.3) Green 8

Z(J) Schwinger L (0) J 1 2 µφ(x) µ φ(x) 1 2 m2 φ 2 (x) + φ(x)j(x) (2.4) Z(J) n0 Dφ exp{ ī h Dφ exp{ ī h d 4 xl J } d 4 x[l (0) J Dφ 1 iλ ( n! 4! h λ 4! φ4 (x)]} d 4 xφ 4 (x)) n exp{ ī h d 4 xl (0) J } (2.5) λ Feynman h i δ δj(y) exp{ ī h d 4 xl (0) J } φ(y) exp{ ī h d 4 xl (0) J } (2.6) δ δj(y) J(x) δ4 (x y) (2.7) δ 4 (x y) Dirac δj(x) d 4 y Z(J) n0 n0 Dφ 1 iλ ( n! 4! h 1 iλ [ n! 4! h d 4 y( h i Dφ exp{ ī h δ δj(y) J(x) δj(y) d 4 yφ 4 (y)) n exp{ ī h δ δj(y) )4 ] n 9 Dφ exp{ ī h d 4 xl (0) J } d 4 xl (0) J } (2.8) d 4 xl (0) J } (2.9)

Feynman h 1 ig F (x y) i h m 2 + iϵ δ4 (x y) i h d 4 k m 2 + iϵ (2π) 4eik(x y) d 4 k i h (2π) 4 k 2 m 2 + iϵ eik(x y) (2.10) ϵ Dirac δ 4 (x y) d 4 xl (0) J d 4 k (2π) 4eik(x y) (2.11) d 4 x{ 1 2 µφ(x) µ φ(x) 1 2 m2 φ 2 (x) + φ(x)j(x)} d 4 x{ 1 2 φ(x)[ m2 ]φ(x) + J(x)φ(x)} d 4 x 1 2 φ (x)[ m 2 ]φ (x) + 1 i 2 h φ (x) φ(x) + d 4 x d 4 yj(x)ig F (x y)j(y) 1 m 2 + iϵ δ4 (x y)j(y)d 4 y (2.13) [ m 2 1 ] m 2 + iϵ δ4 (x y)j(y)d 4 y J(x) (2.14) Dφ exp{ ī h d 4 xl (0) J } 10 (2.12)

Dφ exp{ ī d 4 x 1 h 2 φ (x)[ m 2 ]φ (x) + 1 2 ( ī h )2 d 4 x d 4 yj(x)ig F (x y)j(y)} Dφ exp{ ī d 4 x 1 h 2 φ (x)[ m 2 ]φ (x) + 1 2 ( ī h )2 d 4 x d 4 yj(x)ig F (x y)j(y)} constant exp{ 1 2 ( ī h )2 d 4 x d 4 yj(x)ig F (x y)j(y)} (2.15) Dφ D(φ + ϵ) Dφ Dφ Feynman J 1 Dφ exp{ ī d 4 xl J } Z(J 0) h Feynman W (J) e iw (J) Z(J) iw (J) ln Z(J) Z(J) n0 Dφ exp{ ī d 4 xl J } h 1 iλ [ d 4 y( h δ n! 4! h i δj(y) )4 ] n exp{ 1 2 ( ī h )2 d 4 x d 4 yj(x)ig F (x y)j(y)} (2.16) Z(J) 2 J 0 (2.16) Z(J) 2 Green ( h δ δ i )2 δj(x 1 ) δj(x 2 ) Z(J) J0 T φ(x 1 )φ(x 2 ) (2.17) (2.16) 2 n 0 ( h δ δ i )2 δj(x 1 ) δj(x 2 ) exp{1 2 ( ī h )2 d 4 x d 4 yj(x)ig F (x y)j(y)} J0 11

ig F (x 1 x 2 ) (2.18) λ ( h δ δ iλ i )2 [ δj(x 1 ) δj(x 2 ) 4! h exp{ 1 2 ( ī h )2 d 4 x iλ 2! h d 4 y( h i δ δj(y) )4 ] d 4 yj(x)ig F (x y)j(y)} J0 d 4 yig F (x 1 y)ig F (y y)ig F (y x 2 ) (2.19) λ 2 Green T φ(x 1 )φ(x 2 ) ig F (x 1 x 2 ) (2.20) iλ d 4 yig F (x 1 y)ig F (y y)ig F (y x 2 ) 2! h Feynman Green ( h δ δ δ δ i )4 δj(x 1 ) δj(x 2 ) δj(x 3 ) δj(x 4 ) Z(J) J0 T φ(x 1 )φ(x 2 )φ(x 3 )φ(x 4 ) (2.21) ( h δ δ δ δ iλ i )4 [ δj(x 1 ) δj(x 2 ) δj(x 3 ) δj(x 4 ) 4! h exp{ 1 2 ( ī h )2 d 4 x iλ h d 4 y( h i δ δj(y) )4 ] d 4 yj(x)ig F (x y)j(y)} J0 (2.22) d 4 yig F (x 1 y)ig F (x 2 y)ig F (x 3 y)ig F (x 1 y) Green iλ h d 4 y (2.23) 12

ig F (x 1 x 2 ) (2.24) Feynman ϵ T φ(x)φ(y) ig F (x y) d 4 k i (2π) 4 k 2 m 2 + iϵ eik(x y) (2.25) d 4 k i (2π) 4 k 2 m 2 + iϵ eik(x y) d 3 k i (2π) 4 dk0 (k 0 ) 2 ( k 2 + m 2 iϵ) eik0 (x0 y0) i k( x y) d 3 k i (2π) 4 dk0 (x 0 y 0 ) i k( x y) (2.26) [k 0 (E iϵ)][k 0 ( E + iϵ)] eik0 E E k2 + m 2 > 0 (2.27) k 0 k 0 d 3 k 2π (2π) 4[ 2E θ(x0 y 0 )e ie(x0 y 0 ) i k( x y) + 2π 2E θ(y0 x 0 )e ie(y0 x 0 ) i k( x y) d 3 k y 0 )e ie(x0 y 0 ) i k( x y) + θ(y 0 x 0 )e ie(y0 x 0 ) i k( x y) ]. 2E(2π) 3[θ(x0 ig F (x y) (2.28) d 3 k y 0 )e ie(x0 y 0 ) i k( x y) + θ(y 0 x 0 )e i( E)(x0 y 0 ) i k( x y) ] 2E(2π) 3[θ(x0 13

ig F (x y) y x 1 E > 0 y 0 x 0 (x 0 > y 0 ) 2 E < 0 y 0 x 0 (y 0 > x 0 ) Feynman m iϵ 3 Maxwell A µ (x) A µ (t, x), µ 0, 1, 2, 3 L 1 4 µν ( µ A ν (x) n A µ (x))( µ A ν (x) n A µ (x)) 1 4 F µνf µν 1 2 [ E 2 B 2 ] (3.1) g µν (1, 1, 1, 1) F µν F νµ E B E (F 01, F 02, F 03 ), B ( F23, F 31, F 12 ). (3.2) F µν ( ) A µ (x) A µ(x) A µ (x) + µ ω(x) (3.3) 14

F µν F µν ω(x) ω(t, x) ω(x) A µ (x) ω(x) A µ (x) Coulomb A µ (x) k A k (x) 3 A k(x) 0 (3.4) k k1 Landau( Lorentz) Lorentz µ A µ (x) 0 (3.5) Weyl A 0 (x) 0 (3.6) 3.1 Maxwell Lagrangian S Maxwell d 4 xl d 4 x 1 4 F µνf µν (3.7) DA µ exp[ ī h S Maxwell] (3.8) DA µ x A 0 (x) x A 1 (x) x A 2 (x) x A 3 (x) (3.9) 4 4 15

Maxwell S Maxwell A µ (x) A ω µ(x) A µ (x) + µ ω(x) (3.10) ω(x) S Maxwell 4 (A µ ) A 0 (x) ω(x) DA µ DA ω µ DA µ (3.11) dµ dµ exp[ ī h S Maxwell] [DA µ /Dω] exp[ ī h S Maxwell] (3.12) DA µ Dω 3.2 Coulomb Coulomb k A k (x) 0 (3.13) [DA µ /Dω] DA ω µδ( k A ω k )det δ k A ω k (y) (3.14) δω(x) δ( k A ω k ) x δ( k A ω k (x)) (3.15) k A k (x) 0 16

k A ω k (x) 0 B(x) δ( k A ω k ) DB exp[i 3.14) d 4 xb(x) k A ω k (x)] (3.16) det δ k A ω k (y) (3.17) δω(x) 4 y x ω(x) k A ω k (y) 3.14) DA µ [DA µ /Dω]Dω DA ω µδ( k A ω k )det δ k A ω k (y) Dω δω(x) DA ω µδ( k A ω k )D( k A ω k ) DA µ δ( k A ω k )D( k A ω k ) DA µ (3.18) k A ω k (x) k A k (x) + k k ω(x) k A k (x) ω(x) (3.19) det δ k A ω k (y) det δ(y x) (3.20) δω(x) 3 DA µ δ( k A k )det δ(y x) exp[ ī h 17 d 4 x 1 4 F µνf µν ]

DA µ δ( k A k )det δ(y x) exp{ ī h DA µ δ( k A k )det δ(y x) exp{ ī h DA µ δ( k A k )det δ(y x) exp{ ī h d 4 x 1 2 [A ν A ν + ν A ν µ A µ ]} d 4 x 1 2 [A ν A ν + 0 A 0 0 A 0 ]} d 4 x 1 2 [A k A k A 0 A 0 ]} (3.21) ω Maxwell A ω µ A µ k A k 0 DA 0 exp{ ī h d 4 x 1 2 [ A 0 A 0 ]} 1 det δ(y x) (3.22) A 0 Ã0 δã0/δa 0 det δ(y x) DA 0 exp{ ī d 4 x 1 h 2 [ A 0 A 0 ]} 1 DÃ0 exp{ ī d 4 x 1 det δ(y x) h 2 [ Ã0Ã0 ]} 1 (3.23) det δ(y x) O(x) ϕ(x) Dϕ exp{ ī h ϕ(x)o(x)ϕ(x)d 4 x} det[o(x)δ(x y)] 1/2 DA k δ( k A k ) det δ(y x) exp{ ī h d 4 x 1 2 [A k A k ]} (3.24) 2 1 (3.25) det δ(y x) 18

DA k DB det δ(y x) exp{ ī d 4 x[ 1 h 2 A k A k + B k A k ]} 1 DB det δ(y x) exp{ ī det δ(y x) 3/2 h 1 1 det δ(y x) det δ(y x) 3/2 det δ(y x) 1/2 1 det δ(y x) d 4 x[ 1 2 k B 1 kb]} (3.26) Feynman Schwinger source DA k DB det δ(y x) exp{i DA k DB det δ(y x) exp{i d 4 x[ 1 2 ( A k 1 ( kb + J k ) 1 2 ( kb + J k ) 1 ( k B + J k )]} d 4 x[ 1 2 A k A k + B k A k J k A k ]} ) ( A k 1 ) ( k B + J k ) 1 DB det δ(y x) det δ(y x) 3/2 exp{i d 4 x[ 1 2 ( k B + J k ) 1 ( kb + J k )]} det δ(y x) DB det δ(y x) 3/2 exp{i d 4 x[ 1 2 B k k B 1 2 J 1 k J k + 1 ( B 1 ) 2 k J k + k 1 J k B ]} det δ(y x) DB det δ(y x) 3/2 exp{i d 4 x[ 1 2 (B kj k 1 det δ(y x) exp{i ) (B kj k ) + 1 2 ( kj k 1 d 4 x[ 1 2 J k [ g kl + k l ]J l ]} 19 kj k 1 2 J 1 k J k ]} (3.27)

k k ( 1 i )2 J k 0, g kl δ kl δ δj l (x) δ δj m (y) 0 T A l (x)a m (y) 0 1 i [δ lm 1 l m]δ(x y) d 4 k (2π) 4i[δ lm k l k m / k 2 ] e ik(x y) k 2 + iϵ d 3 k (2π) 3 2ω(k) [δ lm k l k m /( k) 2 ]e ik(x y) θ(x 0 y 0 ) d 3 k + (2π) 3 2ω(k) [δ lm k l k m /( k) 2 ]e ik(y x) θ(y 0 x 0 ) (3.28) k 0 ω(k) ck 0 > 0 Coulomb 3.3 Landau Landau µ A µ (x) 0 (3.29) µ A ω µ(x) µ A µ (x) + µ µ ω(x) µ A µ (x) + ω(x) (3.30) det δ µ A ω µ(y) det δ(y x) (3.31) δω(x) DA µ δ( µ A µ )det δ(y x) exp[i DA µ DBdet δ(y x) exp{i DA µ DBD cdc exp{i d 4 x 1 4 F µνf µν ] d 4 x[ 1 4 F µνf µν + B(x) µ A µ (x)]} d 4 x[ 1 4 F µνf µν + B(x) µ A µ (x) i c(x) c(x)]} 20 (3.32)

δ δ( µ A µ ) DB exp{i d 4 xb(x) µ A µ (x)} (3.33) Grassmann c(x) c(x) Faddeev-Popov det δ(y x) D cdc exp{i d 4 x[ i c(x) c(x)]} (3.34) 3.4 BRST(Becchi-Rouet-Stora-Tyutin) Faddeev-Popov Faddeev-Popov, c(x) c(x), λ BRST A µ (x) A µ (x) + iλ µ c(x), c(x) c(x), c(x) c(x) + λb(x), B(x) B(x), (3.35) c(x) c(x) λ c(x) c(x), c(x) c(x), λ λ. (3.36) BRST (A µ (x) + iλ µ c(x)) A µ (x) i µ c(x)λ A µ (x) + iλ µ c(x) (3.37) 21

µ c(x)λ λ µ c(x) (3.38) BRST ω(x) iλc(x) Maxwell Lorentz Landau µ A µ (x) 0 (3.39) L 1 4 F µνf µν + B(x) µ A µ (x) i c(x) µ µ c(x) (3.40) 2 ( c(x) + λb(x)) µ (A µ (x) + iλ µ c(x)) c(x) µ A µ (x) + λ[b(x) µ A µ (x) i c(x) µ µ c(x)] (3.41) 2 λ c(x) µ A µ (x) BRST BRST BRST (3.40) BRST DA µ DBD cdc exp{i d 4 x[ 1 4 F µνf µν + B(x) µ A µ (x) i c(x) c(x)]} (3.42) BRST BRST D(A µ + iλ µ c)dbd( c + λb)dc DA µ DBD cdc (3.43) Faddeev-Popov (3.32) BRST Slavnov-Taylor T c(x)a µ (y) T ( c(x) + λb(x))(a µ (y) + iλ µ c(y)) (3.44) 22

BRST Grassmann λ T λb(x)a µ (y) + T c(x)iλ µ c(y) 0 (3.45) Grassmann λ c(x) c(x)λ λ c(x) T B(x)A µ (y) T c(x)i µ c(y) (3.46) Slavnov-Taylor 3.5 Grassmann Grassmann Faddeev-Popov D cdc exp{i d 4 x[ i c(x) c(x)]} (3.47) Grassmann c(x) c(x) Grassmann c(x)c(y) + c(y)c(x) 0, c(x)c(y) + c(y) c(x) 0, c(x) c(y) + c(y) c(x) 0 (3.48) c(x)c(x) + c(x)c(x) 2c(x) 2 0, (3.49) c(x) 2 0 D cdc x δ δ δ c(x) y δc(y) (3.50) δ δc(y) d 4 x[ i c(x) c(x)] 23 δ δc(y) d 4 x[i c(x) c(x)]

δ δ c(y) d 4 x[ i c(x) c(x)] d 4 x[i x δ(x y) c(x)] i y c(y), d 4 x[ iδ(x y) x c(x)] i y c(y) (3.51) c(x) ϵ(x) Grassmann D(c + ϵ) δ y δ(c(y) + ϵ(y)) δc(y) δ y δ(c(y) + ϵ(y)) δc(y) δ δc(y) Dc (3.52) D( c + ϵ) D c (3.53) D cdc exp{i D( c + ϵ)dc exp{i D cdc exp{i d 4 x[ i c(x) c(x)]} d 4 x[ i( c(x) + ϵ(x)) c(x)]} d 4 x[ i c(x) c(x) iϵ(x) c(x)]} (3.54) ϵ(x) D cdc( d 4 y iϵ(y) c(y)) exp{i d 4 x[ i c(x) c(x)]} 0 (3.55) ϵ(y) ĉ(y) D cdc( c(y)) exp{i d 4 x[ i c(x) c(x)]} 0 (3.56) 24

ĉ(y) 0 Z D cdc exp{i d 4 x[ i c(x) c(x)]} det[ δ(x y)] (3.57) c (x) c(x) Grassmann Jacobian Dc det[ δ(x y)] 1 Dc (3.58) D cdc exp{i d 4 x[ i c(x)c (x)]} det[ δ(x y)] 1 D cdc exp{i d 4 x[ i c(x) c(x)]} (3.59) D cdc exp{i d 4 x[ i c(x)c (x)]} (3.57) Faddeev-Popov (3.32) (3.42) 3.6 Feynman Lorentz Landau DA µ DBD cdc exp{i d 4 x[ 1 4 F µνf µν + B(x) µ A µ (x) i c(x) c(x)]} (3.60) Feynman S d 4 x[ 1 2 Aµ ( g µν µ ν )A ν + B(x) µ A µ (x) i c(x) c(x)] (3.61) 25

Schwinger source S J d 4 x[ A µ (x)j µ (x) + J B (x)b(x) + J c (x) c(x) + J c (x)c(x)] (3.62) J c (x) J c (x) Grassmann A µ (x) a µ (x) + 1 (g µν µ ν )J ν (x) 1 µj B (x), B(x) b(x) 1 µj µ (x), c(x) c (x) i J c(x) c(x) c (x) + i J c(x), (3.63) S + S J d 4 x[ 1 2 aµ ( g µν µ ν )a ν + b(x) µ a µ (x) i c (x) c (x)] + d 4 x[ 1 2 J µ (x) 1 (g µν µ ν )J ν (x) J B (x) 1 µ J µ (x) +J c (x) i J c(x)] (3.64) Z J DA µ DBD cdc Da µ DbD c Dc (3.65) Da µ DbD c Dc exp{i(s + S J )} (3.66) Z J exp{i d 4 x[ 1 2 J µ (x) 1 (g µν µ ν )J ν (x) J B (x) 1 µ J µ (x) +J c (x) i J c(x)]} (3.67) 26

Feynman T A µ (x)a ν (y) ( 1 δ δ i )2 δj µ (x) δj ν (y) Z J J0 i (g µν µ ν )δ(x y) d 4 k (g µν kµkν ( i) k ) 2 e ik(x y), (2π) 4 k 2 + iϵ T B(x)A ν (y) ( 1 δ δ i )2 δj B (x) δj ν (y) Z J J0 i δ(x y) yν d 4 k k ν (2π) 4 k 2 + iϵ eik(x y), T c(x)c(y) ( 1 δ δ i )2 δj c (x) δj c (y) Z J J0 4 Dirac 1 δ(x y) d 4 k 1 (2π) 4 k 2 + iϵ eik(x y), (3.68) Dirac Fermi Ld 3 xdt ψ(t, x)[iγ µ µ (mc/ h)]ψ(t, x)d 3 xdt (4.1) 4 4 Dirac γ µ, γ µ γ ν + γ ν γ µ 2g µν, (γ 0 ) γ 0, 2 2 Pauli τ 1 0 1 1 0 µ 0, 1, 2, 3 (γ k ) γ k, k 1, 2, 3 (4.2), τ 2 0 i i 0 27, τ 3 1 0 0 1 (4.3)

4 4 γ 0 1 0 0 1, γ k 0 τ k τ k 0, k 1, 2, 3 (4.4) γ 0 1 2 2 Dirac γ 5 γ 5 iγ 0 γ 1 γ 2 γ 3, {γ 5, γ µ } + 0 (4.5) γ γ 5 0 1 1 0, (γ 5 ) 2 1 0 0 1 (4.6) ψ(t, x) α, α 1 4 4 4 4 Dirac ψ(t, x) ψ(t, x) γ 0 (4.7) 4.1 Dirac ψ(t, x) ψ(t, x) Grassmann D ψdψ exp{ ī h ψ(x)[iγ µ µ (mc/ h)]ψ(x)d 4 x} (4.8) D ψdψ x δ δ δ ψ(x) y δψ(y) (4.9) D( ψ + ϵ)d(ψ + ϵ) D ψdψ (4.10) 28

D ψdψ exp{ ī h D( ψ + ϵ)dψ exp{ ī h D ψdψ exp{ ī h ψ(x)[iγ µ µ (mc/ h)]ψ(x)d 4 x} ϵ ( ψ(x) + ϵ)[iγ µ µ (mc/ h)]ψ(x)d 4 x} ( ψ(x) + ϵ)[iγ µ µ (mc/ h)]ψ(x)d 4 x} (4.11) D ψdψ{ ī ϵ(y)[iγ µ µ (mc/ h)]ψ(y)d 4 y} h exp{ ī ψ(x)[iγ µ µ (mc/ h)]ψ(x)d 4 x} 0 (4.12) h ϵ(y) [iγ µ µ (mc/ h)] ˆψ(y) D ψdψ{[iγ µ µ (mc/ h)]ψ(y)} exp{ ī ψ(x)[iγ µ µ (mc/ h)]ψ(x)d 4 x} h 0 (4.13) Feynman Grassmann Schwinger η(x) η(x) Z J D ψdψ exp{ ī h ( ψ(x)[iγ µ µ (mc/ h)]ψ(x) + η(x)ψ(x) + ψ(x)η(x) ) d 4 x} (4.14) ψ(x) ψ 1 (x) iγ µ µ (mc/ h) η(x), ψ(x) ψ 1 (x) η(x) iγ µ µ (mc/ h) (4.15) Z J D ψ Dψ exp{ ī ( ψ (x)[iγ µ µ (mc/ h)]ψ (x) ) d 4 x} h 29

exp{ ī 1 η(x) h iγ µ µ (mc/ h) η(x) d 4 x} (4.16) D ψdψ D ψ Dψ (4.17) Z J exp{ ī 1 η(x) h iγ µ µ (mc/ h) η(x) d 4 x} Schwinger Feynman T ψ(x) ψ(y) ( h δ δ i )2 δ η(x) δη(y) Z J J0 i h iγ µ µ (mc/ h) δ4 (x y) 5 5.1 (4.18) i h d 4 k iγ µ µ (mc/ h) (2π) 4e ik(x y) d 4 k i h (2π) 4 γ µ k µ (mc/ h) + iϵ e ik(x y) (4.19) Maxwell Dirac d 4 xl d 4 { ψ(t, x)[iγ µ ( µ iea µ ) m]ψ(t, x) 1 4 ( µa ν ν A µ ) 2 } (5.1) Dirac γ 0 γ k g µν (1, 1, 1, 1) γ µ γ ν + γ ν γ µ 2g µν (5.2) 30

γ 5 iγ 0 γ 1 γ 2 γ 3 γ 5, (γ 5 ) 2 1 (5.3) γ 5 A µ ψ D µ µ iea µ (x) (5.4) ω(x) Maxwell A µ (x) A µ(x) A µ (x) + 1 e µω(x), ψ(x) ψ(x) U(x)ψ(x) exp[iω(x)]ψ(x), ψ(x) ψ(x) ψ(x)u(x) ψ(x) exp[ iω(x)] (5.5) ( ) D µ D µ U(x)D µ U(x) µ ie(a µ (x) + 1 ω(x)) (5.6) e ψ ψ(x) U(x)ψ(x) ( ) () U 1 (x)u 2 (x) exp[iω 1 (x)] exp[iω 2 (x) exp[i(ω 1 (x) + ω 2 (x))] (5.7) Abel U(1) A µ Abel Abel Yang-Mills 31

D µ ψ(x) D µ ψ(x) U(x)D µ U(x) U(x)ψ(x) U(x)D µ ψ(x) (5.8) ψ(x) [D µ, D ν ] D µ D n D ν D µ ie( µ A ν ν A µ ) ief µν (5.9) [UD µ U, UD ν U ] U[D µ, D ν ]U ief µν (5.10) Jacobi [D µ, [D ν, D α ]] + [D ν, [D α, D µ ]] + [D α, [D µ, D ν ]] 0 (5.11) Maxwell 4 2 µ F να + ν F αµ + α F µν 0 (5.12) Maxwell d 4 xl maxwell d 4 x{ 1 4 F µνf µν } (5.13) Pauli gf µν ψ(x)[γ µ, γ ν ]ψ(x) Pauli (minimal coupling) 0, + 0, J exp{i D ψdψda µ DBDcD c (5.14) d 4 x{ ψ(t, x)[iγ µ ( µ iea µ ) m]ψ(t, x) 1 4 ( µa ν ν A µ ) 2 +B(x) µ A µ (x) + i µ c µ c + ξ 2 B2 A µ (x)j µ (x) + η(x)ψ(x) + ψ(x)η(x)}} 32

ξ ( Abel Faddeev-Popov ) Dirac Grassmann ξ 2 B2 B(x) BRST BRST Feynman Feynman Schwinger c (source) J µ (x) Dirac Grassmann η(x) η(x) dµ D ψdψda µ DBDcD c (5.15) Bose Fermi ( 0 ) 5.2 h c 1 ( ) L J ψ(x)[iγ µ ( µ iea µ ) m]ψ(x) 1 4 ( µa ν ν A µ ) 2 +B(x) µ A µ (x) + i µ c µ c + ξ 2 B2 A µ (x)j µ (x) + η(x)ψ(x) + ψ(x)η(x) L 0 J + L int (5.16) L (0) J L int ea µ (x) ψ(x)γ µ ψ(x) (5.17) 0, + 0, J (5.18) dµ exp[i d 4 xl (0) J (x)] 1 n0 n! (i d 4 xl int (x)) n exp[i d 4 iδ xl int { δj µ (x), iδ δη(x), iδ δ η(x) }] dµ exp[i d 4 xl (0) J (x)] exp[ie d 4 iδ iδ iδ x{ δj µ γµ (x) δη(x) δ η(x) }] dµ exp[i d 4 xl (0) J (x)] 33

2 L (0) J L (0) J Schwinger Dirac ψ 0, + [iγ µ µ m] ˆψ(x) + η(x) 0, (0) dµ{[iγ µ µ m]ψ(x) + η(x)} exp[is (0) J ] dµ iδ exp[is(0) J ] 0 (5.19) δ ψ(x) Dyson S- Wick Feynman D (0) S (0) F (x y) 1 iγ µ µ m + iϵ δ4 (x y), F (x y) µν g µν + (ξ 1) µ ν / µ µ δ 4 (x y) (5.20) µ µ iϵ µ x µ A µ (x) A µ(x) + ψ(x) ψ (x) ψ(x) ψ (x) D (0) F (x y) µν J ν (y)d 4 y, S (0) F (x y)η(y)d 4 y, η(y)s (0) F (x y)d 4 y (5.21) Dψ Dψ 0, + 0, (0) J dµ exp{i 34 d 4 xl (0) J } J

dµ exp[i d 4 xl (0) J0] (5.22) exp{ i [ 1 2 J µ (x)d (0) F (x y) µν J ν (y) + η(x)s (0) F (x y)η(y)]d 4 xd 4 y} 0, + 0, (0) J0 1 0, + 0, (0) J (5.23) exp{ i [ 1 2 J µ (x)d (0) F (x y) µν J ν (y) + η(x)s (0) F (x y)η(y)]d 4 xd 4 y} Schwinger iδ iδ 0, + 0, (0) J J0 δ η(x) δη(y) 0, + T ψ(x) ψ(y) 0, (0) J0 is (0) F i 1 (x y) i iγ µ µ m + iϵ δ4 (x y) d 4 p 1 (2π) 4 p m + iϵ e ip(x y) d 3 p (2π) 3 2p 0 [( p + m) αβ e ip(x y) θ(x 0 y 0 ) ( p m) αβ e ip(x y) θ(y 0 x 0 )] (5.24) p 0 > δ δ 4 (x y) d 4 p exp[ ip(x y)] (2π) 4 1 p m + iϵ p + m p 2 m 2 + iϵ p + m p 2 m 2 + iϵ iϵ p 0 Feynman iϵ Dirac 35

( ) iδ iδ 0, + 0, δj µ (x) δj ν (0) J J0 (y) 0, + T A µ (x)a ν (y) 0, (0) J0 id (0) F (x y) µν i g µν + (ξ 1) µ ν / µ µ δ 4 (x y) µ µ iϵ d 4 p g µν + (ξ 1)p µ p ν /p µ p µ i e ip(x y) (2π) 4 p µ p µ + iϵ d 3 p ( g (2π) 3 µν )[e ip(x y) θ(x 0 y 0 ) + e ip(x y) θ(y 0 x 0 )](5.25) 2p 0 ξ 1 Feynman ξ ( ) iϵ Maxwell ( ) 0, + 0, J exp[ie d 4 iδ iδ iδ x{ δj µ γµ }] (5.26) (x) δη(x) δ η(x) exp{ i [ 1 2 J µ (x)d (0) F (x y) µν J ν (y) + η(x)s (0) F (x y)η(y)]d 4 xd 4 y} e 0 Green 6 Yang-Mills Abel Abel Abel T a [T a, T b ] T a T b T b T a if abc T c (6.1) 36

a, b, c f abc structure constant) trt a T b 1 2 δ ab (6.2) SU(2) 3 Pauli T 1 1 2 0 1 1 0, T 2 1 2 0 i i 0, T 3 1 2 1 0 0 1 (6.3) [T a, T b ] iϵ abc T c (6.4) ϵ abc 3 ϵ 123 1 D µ D µ µ iga a µ(x)t a a µ iga µ (x) (6.5) Abel T a Abel D µ µ iea µ (x) (6.6) Abel 2 A µ A a µt a F a µν E B [D µ, D ν ] ig( µ A a ν ν A a µ + gf abc A b µa c ν)t a igf µν (6.7) L Y M Abel Yang-Mills L Y M 1 2 trf µνf µν 1 4 FµνF a aµν µ,ν,a 1 4 ( µa a ν ν A a µ + gf abc A b µa c ν) 2 (6.8) 37

Maxwell SU(2) SU(2) ω a (x) g(x) exp[i 3 ω a (x)t a ] SU(2) (6.9) a1 Pauli SU(2) T a Abel A µ (x) a A µ (x) a T a A µ(x) g(x)a µ (x)g (x) + 1 ig ( µg(x))g (x) (6.10) A µ 2 D µ D µ µ iga µ(x) g(x)( µ iga µ (x))g (x) g(x)d µ g (x) (6.11) [D µ, D ν] igf µν g(x)[d µ, D ν ]g (x) g(x) ( igf µν ) g (x) (6.12) g(x)g (x) 1 Lagrangian L 1 4 F a µνf aµν 1 2 trf µνf µν 1 2 trf µνf µν (6.13) trt a T b (1/2)δ ab g(x) 38

ω a (x) g(x) exp[i 3 ω a (x)t a ] a1 1 + i 3 ω a (x)t a a1 1 + iω(x) (6.14) Abel A µ (x) a A µ (x) a T a A µ(x) A µ (x) + i[ω(x), A µ (x)] + 1 g µω(x) (6.15) A µ (x) a A a µ (x) A µ (x) a + f abc A µ (x) b ω(x) c + 1 g µω(x) a (6.16) dµ exp[ ī h Lagrangian L eff d 4 xl eff ] (6.17) L eff 1 4 F a µνf aµν + B a (x) µ A a µ i c a (x) µ [ µ c a (x) + gf abc A b µ(x)c c (x)] (6.18) dµ DA a µdb a D c a Dc a (6.19) Lorentz Landau) µ A a µ(x) 0 (6.20) B a (x) c a (x) Faddeev-Popov 39

c a (x) c a (x) c a (x) Grassmann Yang-Mills () BRST λ Grassmann A a µ(x) A a µ(x) + iλ( µ c a (x) + gf abc A b µ(x)c c (x)) c a (x) c a (x) iλ g 2 f abc c b (x)c c (x) c a (x) c a (x) + λb a (x) B a (x) B a (x) (6.21) L eff BRST 6.1 QCD Abel Abel QCD Quantum Chromodynamics ) Lagrangian L ψ(i D m)ψ + L Y M (6.22) Dirac γ 0 γ k g µν (1, 1, 1, 1) γ µ γ ν + γ ν γ µ 2g µν (6.23) γ 5 iγ 0 γ 1 γ 2 γ 3 γ 5, (γ 5 ) 2 1 (6.24) γ 5 D µ D γ µ D µ γ µ ( µ a iga a µt a ) γ µ ( µ iga µ ) (6.25) Abel T a Abel 2 40

A µ A a µt a F a µν E B [D µ, D ν ] ig( µ A a ν ν A a µ + gf abc A b µa c ν)t a igf µν (6.26) L Y M Abel Yang-Mills L Y M 1 4 FµνF a aµν µ,ν,a 1 4 ( µa a ν ν A a µ + gf abc A b µa c ν) 2 (6.27) Maxwell Abel Abel Abel T a [T a, T b ] T a T b T b T a if abc T c (6.28) a, b, c f abc structure constant) trt a T b 1 2 δ ab (6.29) SU(2) 3 Pauli T 1 1 2 0 1 1 0, T 2 1 2 0 i i 0, T 3 1 2 1 0 0 1 (6.30) [T a, T b ] iϵ abc T c (6.31) ϵ abc 3 ϵ 123 1 SU(3) Gell-Mann Pauli 3 3 8 {λ a, a 1 8} T a 1 2 λa, a 1 8 (6.32) 41

SU(n) n 2 1 n 2 1 SU(n) Dirac SU(2) ψ(x) ψ 1(x) ψ 2 (x), (6.33) 4 Dirac ψ 1 (x), ψ 2 (x) Dirac 4 4 γ 2 8 8 QCD SU(3) 3 Dirac SU(2) SU(2) ω a (x) g(x) exp[i 3 ω a (x)t a ] SU(2) (6.34) a1 Pauli SU(2) T a Abel ψ(x) ψ(x) g(x)ψ(x), ψ(x) ψ ψ(x)g(x) (6.35) A µ (x) A µ (x) a T a A µ(x) g(x)a µ (x)g (x) + 1 a ig ( µg(x))g (x) ψ(x) g(x) A µ 2 D µ D µ µ iga µ(x) g(x)( µ iga µ (x))g (x) g(x)d µ g (x) (6.36) D µ ψ(x) D µψ (x) ( µ iga µ(x))ψ (x) g(x)( µ iga µ (x))ψ(x) g(x)d µ ψ(x) (6.37) 42

ψ(x) D µ (covariant derivative) [D µ, D ν] igf µν g(x)[d µ, D ν ]g (x) g(x) ( igf µν ) g (x) (6.38) g(x)g (x) 1 Lagrangian L ψ(i D m)ψ 1 4 F a µνf aµν ψ(i D m)ψ 1 2 trf µνf µν ψ (i D m)ψ 1 2 trf µνf µν (6.39) trt a T b (1/2)δ ab ψmψ g(x) dµ exp[ ī h Lagrangian L eff d 4 xl eff ] (6.40) L eff ψ(i D m)ψ 1 4 F a µνf aµν + B a (x) µ A a µ i c a (x) µ [ µ c a (x) + gf abc A b µ(x)c c (x)] (6.41) dµ D ψdψda a µdb a D c a Dc a (6.42) Lorentz Landau) µ A a µ(x) 0 (6.43) B a (x) c a (x) Faddeev-Popov c a (x) 43

6.2 Weinberg-Salam SU(2) L SU(2) TL a T a ( 1 γ 5 ) (6.44) 2 γ 5 Dirac ( 1 γ 5 2 ) (γ 5 ) 2 1, {γ 5, γ µ } + γ 5 γ µ + γ µ γ 5 0 (6.45) ( 1 γ 5 )( 1 γ 5 ) ( 1 γ 5 ) (6.46) 2 2 2 (6.44) TL a [TL, a TL] b iϵ abc TL c (6.47) T a L SU(2) L Dirac L ψi D( 1 γ 5 )ψ 1 2 4 F µνf a aµν ψiγ µ ( µ iga a µtl)( a 1 γ 5 )ψ 1 2 4 F µνf a aµν ψ L iγ µ ( µ iga a µt a L)ψ L 1 4 F a µνf aµν (6.48) Lagrangain g L (x) exp[iω a (x)t a L] (6.49) ψ L (x) ψ L (x) g L (x)ψ L (x), ψl (x) ψ L ψ L (x)g R (x) (6.50) A µ (x) A µ (x) a TL a A µ(x) g L (x)a µ (x)g L(x) + 1 a ig ( µg L (x))g L(x) D L γ µ ( µ iga µ (x))( 1 γ 5 ) D L g R (x) D L g 2 L(x) (6.51) 44

Dirac ψ L (x) ( 1 γ 5 )ψ(x) 2 ψ L (x) ψ L (x) γ 0 ψ (x)( 1 γ 5 )γ 0 2 ψ(x)( 1 + γ 5 ) (6.52) 2 g R (x) exp[ iω a (x)t a ( 1 + γ 5 )], 2 g L (x) γ 0 γ 0 g R (x), g R (x) γ µ γ µ g L (x) (6.53) L ψ L i D L ψ L 1 4 F a µνf aµν ψ L i D L ψ L 1 4 trf µνf µν ψ Li D Lψ L 1 4 trf µνf µν (6.54) F µν F a µνt a L (6.55) tr Dirac T rt a T b 1 2 δ a,b trt a LT b L δ a,b ψ(x)mψ(x) ψ R (x)mψ L (x) + ψ L (x)mψ R (x) (6.56) ψ R (x) ( 1 + γ 5 )ψ(x) (6.57) 2 45