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2 QED QED

3 4 QED,. ab dx ax + b x)), abc dx x dy ax + by + c x y)) 3 Γn) δ x x n ) dx dx n a a a n a x + +a n x n ) n Γn) [ ] a a n dt dt dt dt n e a t + a n t n ) dt n δt t t n ) e a t + a n t n ). t i tx i dt dx dx dx dx n t n δ t x x n ) ) e a x + a n x n )t dx n δ x x n ) dx n Γn) δ x x n ) a x + +a n x n ) n. dt t n e a x + a n x n )t [ ] 3

4 . η) d η k π) η a k iɛ) iγc η) c 4π) η Γc)a c η) a >, c Z + ), k k k k ) k, ɛ +. [ ] k ike, k k E ) d η k E ) i π) η a +ke iɛ)c. ke k E ) + k E ke +i i a + ke iɛ ke) + a iɛ ke) ia + ɛ ) a a a + k E > ke 3. C C R ke +.: s c Γc) dt t c e st Re s > ) 4

5 d η k E ) i dt t c e a +k π) η E )t Γc) i dt t c e a t d η k π) η E e k E t Γc) i dt t c e a t π ) η π) η Γc) t iπ η dt t c η e at iγc η) π) η Γc) 4π) η Γc)a c η). [ ] k µ k µ b µ, a a +b d η k π) η a + b k k iɛ) c iγc η) 4π) η Γc) a +b ) c η. b µ d η k π) η k µ a + b k k iɛ) c iγc η) 4π) η Γc) b µ a +b ) c η. d η k k µ k ν π) η a + b k k iɛ) i Γc η)b µ b ν ) Γc η )gµν c 4π) η Γc) a +b ) c η a +b ) c η.3 4 x) 4 L ) m λ 4! 4 m λ L ) m λ 4! 4 5

6 m λ x) m, λ, + ζ, m + ζ m + ζ m, λ + ζ λ + ζ ) λ. ζ, ζ m, ζ λ L ) m + ζ ) ζ mm λ + ζ λ) 4 4! ζ i i, m, λ).:. a) p µ a) i ζ p ζ m m ). b) 4 b) iλ + ζ λ) 4! a) ζ ) ζ mm ζ ζ m m ) + ) µ 4 p µ µ ip µ ) x) m λ.4 n n ) 6

7 n Π n 4.3 n n Π n.3:.4: p) d 4 x < T x)) > e ip x i p m +iɛ 7

8 p).4 + Π + Π Π + Π ) n Π Π i n p m +iɛ iπ m x) p) p m iɛ i lim p m iɛ p m +iɛ) p) i lim p m Π p m. Π p m, p Π p m λ 4 Π 4 p p p 3 p 4 m iλ 4 p µ, pµ, pµ 3, pµ 4 p i m ) p i p j 4 p µ + pµ + pµ 3 + pµ 4 µ : p µ pµ pµ 3 pµ 4 Π 4 µ iλ m, ) ).5 Π n ) Π n Π ) n + Π ) n + Π ) n +. 8

9 Π ) n n n ζ i ζ m) Π ) ζ i ζ ) i + ζ ) i + i m m.5 ) Π ).5: Π ) i ζ) p ζ m ) m ) + 4! iλ ) d η k i 4! π) η k m + iɛ Π ) η d η k i Γ η) I π) η k m + iɛ 4π) η m ) η. 4 η ɛ ɛ ) I m 6π Γ + ɛ)4π)ɛ m ) ɛ Γ + ɛ) Γɛ) + ɛ Γɛ) + ɛ + Oɛ )) Γɛ) + + Oɛ)), 9

10 x ɛ e ɛ log x + ɛ log x + Oɛ ) I m Γɛ) + + log 4π ) 6π m + Oɛ) Π ) iζ ) p ζ ) m m ) + iλm 3π Γɛ) + + log 4π m ζ ), ζ m ) λ Γɛ) + + log 4π ) 3π m Π ) lim Γɛ) m ɛ 4 ) !) iλ ) d η k i i 4! π) η k p ) m +iɛ k+p 3 ) m +iɛ λ λ λ d η k π) η Π ) d η k π) η m k p ) iɛ dx d η k dx π) η λ iλ 3π iλ 3π m k+p 3 ) iɛ ζ ) m k p ) iɛ ) x+ m k+p 3 ) iɛ ) x)) m xp +k xp ) x)p 3 x)p 3 k iɛ iγ η) 4π) η dx Γɛ) 4π)ɛ Γɛ) + log 4π m I m x x)p +p 3 ) ) η dx m x x)p +p 3 ) ) ɛ p +p 3 ) m )). ) )

11 Is) dx log x x)s ). ) Π ) iλζ ) 4 λ + iλ 3π p +p 3 ) I Π ) m 3 ) I Γɛ)+log 4π p +p 4 ) m m ) I p +p ) ) ) 4 µ I) ζ ) λ λ 3 Γɛ)+log 4π ) ) I4) 3π m Π ) iλ p +p 3 ) ) p +p 4 ) ) I +I +I 4 3π m m m p +p ) m ) ) I4) Γɛ) 6 Π ), Π ) 6, 8 ) 5 : L ) m k 5 k ) k : 5

12 .6 4 λ ).7a) M ) iλ.7: λ.7b) g) c) d) Π ) p µ 4 3 pµ 4 M ) iλ p p 3 ) ) p p 4 ) ) p +p ) ) ) I +I +I I4) 3π m m m Π ) ±p. θ p p 3 ) p cos θ), p p 4 ) p +cos θ), p +p ) 4m + p ). Is) dx log x x)s ) r log r+ r s < ) r log +r + iπr r s 4).

13 r 4/s) M ) iλ r 3π log r + r + r log r + r 4 + r 3 log +r ) 3 + iπr 3 r 3 m r + m r p + cos θ), p +cos θ), r 3 p m + p. λ.7 QED QED QED L ψi/ m)ψ 4 F µνf µν α A) qa µ ψγ µ ψ ψ + ζ ψ ψ, A µ + ζ A A µ, α + ζ A )α, m + ζ m + ζ q m, q + ζ ψ + ζ ψ ) q + ζ A ) L ψi/ m)ψ 4 F µνf µν α A) + ζ ψ ψi/ ψ ζm m ψψ 4 ζ AF µν F µν q + ζ q )A µ ψγ µ ψ ψ ψ iζ ψ /p ζ m m), iζ A 4 F µνf µν iζ A Aµ µ ν + g µν )A ν + ) A µ A ν iζ A /)p µ p ν g µν p ). ψ ψa µ iq + ζ q )γ µ 3

14 α ) α ) QED 97).8 QED QED.8 3 ψ ψ iζ) ψ /p ζ) m m), d η ) k π) iqγ µ) igµν i/k + m) )tr iqγ η ν ) iɛ k m + iɛ Π ) /k trγ ν ) 3 d η k π) iqγ i/p /k + m) µ) η p k) m + iɛ iqγ ν) igµν k + iɛ d q η k /k /p + m dx π) η m p )x + xp k k iɛ ) iq Γɛ)/p 4m) + O) 6π O) ɛ ɛ ζ ) ψ q Γɛ) + O), 6π ζ) Π ) ψ m q Γɛ) + O) 4π ψ Π ) A µ A ν A p µp ν g µν p ), ) iqγ µ ) d η k )tr π) η 4q d η k π) η iζ ) iq π Γɛ)p µp ν g µν p ) + O) i/k + m) k m + iɛ iqγ ν) i/k /p + m) k p) m + iɛ dx k µk ν g µν k p µ k ν p ν k µ +g µν p k+g µν m m xp + xp k k iɛ ) 4

15 .8: QED p µ p ν g µν p ) Π ) ζ ) A q Γɛ) + O) π A µ A ν ζ ) A ) Π ) ψ ψa iqζ ) µ q γ µ, L µ p µ, pµ, pµ 3 pµ + pµ + pµ 3 L µ p µ pµ L µ L µ p µ pµ + L µ pν p ν + p ν L µ p µ pµ p ν +. p µ pµ L µ 4 d η k π) iqγ i/k + m) ρ) η k m + iɛ iqγ i/k + m) µ) k m + iɛ iqγ σ) igρσ k + iɛ iq3 6π Γɛ)γ µ + O) 5

16 ) Π ) ψ ψa µ ζ q ) q Γɛ) + O) 6π Π ) Π ) A µ A ν A λ A µ A ν A ρ A σ Π ) A µ A ν A λ.9 ) A µ A ν A ρ A 3! 6 σ ) 5 Π ).9: * ) d η k k µ k ν π) η k + iɛ)k m + iɛ) iγɛ)gµν 64π + O) I µν I µν Cg µν d I µ µ η k ηc π) η k m + iɛ) iγɛ) 6π + O) C iγɛ) + O) 64π.9 QED n n j µ x) ψx)γ µ ψx) C j µ x)) C Cj µ x)c j µ x) C > > n < T j µ x )j µ x ) j µ n x n ) > ) 6

17 C C n n ) < T ψx)γ µ ψx) > trγ µ )). x x ) < T ψx) ψx ) > d 4 k i/k + m) e ik x x ) π) 4 k m + iɛ ψ ψ n 3 ) Π ) < T ψx )γ µ ψx ) ψx )γ µ ψx ) ψx 3 )γ µ 3 ψx 3 ) > n tr γ µ x x )γ µ x x 3 )γ µ 3 x 3 x ) ) +tr γ µ x x 3 )γ µ 3 x 3 x )γ µ x x ) ) A µ A µ A 3, µ ) Π ) n 5, n 7,. ψx) S d 4 x ψ x) i t + ) m ψx). m ) i t + ) m ψx), [ ψ s x), ψs x ) ] tt δ ss δ 3 r r ) 7

18 ) d 3 k ψ s x) π) c sk) e ik r iɛk)t, ɛk) k 3 m [ cs k), c s k ) ] π) 3 δ ss δ 3 k k ). ) < T ψ s x)ψs x ) > θt t ) < ψ s x)ψs x ) > dω d 3 k πi ω iɛ eiωt t ) δ ss eik r r ) iɛk)t t ) π) 3 dωd 3 k iδ ss eik r r ) iɛk) ω)t t ) π) 4 ω+iɛ d 4 k iδ ss e ik x x ) π) 4 k ɛk)+iɛ k ɛk) ω 4 k, k) iδ ss /k ɛk) + iɛ) S I V d 4 x ψ x)ψx) d 4 xd 4 x Gx x )ψ x)ψx)ψ x )ψx ). V Gx) q δt) 4π r d 4 k π) 4 q k e ik x q. a) iv δ s s, b) iq k δ s s δ s3 s 4 k ) V V V ) + V ) + Π ) ψψ. iv ). iq d 4 k i π) 4 k ɛk)+iɛ iq d 3 k π) 3. 8

19 .:.: 3 p d 4 k i iq π) 4 k ɛk)+iɛ k p iq d 3 k π) 3 k : { dx +iπ a > ) x ia iπ a < ) k 3 V ) Π ) ψψ 3 4 p, p) 9

20 4 ψψ ψψ dk k + A + iɛ k + B + iɛ ) dk B A k + A + iɛ k + B + iɛ Π ). QED + )

21

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