G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2

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もえりえいさか
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1 6 Feynman (Green ) Feynman 6.1 Green generating functional Z[J] φ 4 L = 1 2 µφ µ φ m 2 φ2 λ 4! φ4 (1) ( 1 S[φ] = d 4 x 2 φkφ λ ) 4! φ4 (2) K = ( 2 + m 2 ) (3) n G (n) (x 1, x 2,..., x n ) = φ(x 1 )φ(x 2 ) φ(x n ) = 0 T (φ(x 1 )φ(x 2 ) φ(x n ) 0 (4) qft1-6-1

2 G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2,..., x n ) n! n=0 (ij(x 1 ))(ij(x 2 )) (ij(x n )) = 1 Dφ exp (is[φ] + ij φ) (7) N J φ d 4 xj(x)φ(x) (8) φ J (1/i)(δ/δJ(x)) φ(x) Z[J] qft1-6-2

3 Z[J] = 1 + Z[J] : J J J J J J λ Z[J] λ free S 0 [φ] = d 4 x 1 ( ( φ) 2 m 2 φ 2) 2 = d 4 x 1 2 φ( 2 m 2 )φ (9) S I [φ] = λ d 4 xφ(x) 4 (10) 4! ( 1/N ) generating functional Z 0 [J] Z 0 [J] = Dφ exp (is 0 [φ] + ij φ) (11) J J qft1-6-3

4 φ : 1 2 φkφ + J φ = 1 2 (φ + K 1 J)(φ + K 1 J) 1 2 JK 1 J (12) ( Z 0 [J] = exp i ) 2 JK 1 J (13) K 1 (x, y) = F (x y) Feynman propagator ( 2 + m 2 iɛ)k 1 (x, y) = δ 4 (x y) K 1 d 4 p (x y) = (p)e ip (x y) (2π) 4K 1 K 1 (p) = 1 p 2 m 2 + iɛ δ/δ(ij) φ Z[J] (14) qft1-6-4

5 [ ] δ Z[J] = G F [ij] (15) [ ] δ(ij) δ G = exp (is I [δ/δ(ij)]) (16) δ(ij) ( ) i F [ij] = exp 2 (ij) F (ij) = Z 0 [J] = free part (17) S I Feynman diagram [ ] [ ] δ δ G F [ij] = G F [ij]e ijφ (18) δ(ij) δ(ij) φ=0 [ ] [ ] [ ] δ G F [ij]e ij φ δ δ = G F e ij φ (19) δ(ij) δ(ij) δφ qft1-6-5

6 G F [ ] [ ] [ ] δ δ δ = F G e ij φ = F G[φ]e ij φ (20) δφ δ(ij) δφ [ ] δ G F [ij] = F δ(ij) [ ] δ G[φ]e ij φ (21) δφ φ=0 Z[J] ( ) ( 1 δ Z[J] = exp 2δφ δ ) exp (is I [φ] + ij φ) δφ (22) φ=0 = i F (23) qft1-6-6

7 δ δφ δ δφ = d 4 x 1 d 4 δ x 2 δφ(x 1 ) (x δ 1 x 2 ) δφ(x 2 ) = ij i j (24) δ δφ(x i ) = i (25) i : N J N 0 (vacuum bubble): ( ) J = 0 e is I is I [φ] = i λ 4! φ4 (26) φ = 0 exp( 1 2 ij i j ) 4 Taylor 2 qft1-6-7

8 ( ) ! 2 ij i j ( i λ4! ) φ4 = 1 2 ( 1 2 ) 2 ( i λ ) φ 4 i 4! (27) i φ j = δ ij 4 φ 4 i = 4δ 4iφ 3 i 4! ( ) ( i λ ) 2 2 4! 4! ii ii = 1 2 ( iλ) }{{} standard factor 1 4 }{{} symmetry factor ii ii 2-loop vacuum bubble diagram i (28) vacuum bubble diagrams Z[J = 0] Dφe is vacuum bubble qft1-6-8

9 Dφe is+ij φ Z[J] = (29) Dφe is 2 (propagator): O(λ) 2 S I JJ Taylor 3 (S I +J φ) 3 3S I JJ 3 i 3 ( ) 3! 3 S I[φ](J φ) 2 = i3 iλ 3! 3 φ 4 i 4! J jφ j J k φ k (30) 6 exp( 1 2 ij i j ) 3 i 3 ( ) ( ) iλ ! 3 4! 3! J j J k (φ 4 i 2 φ jφ k ) (31) Diagram qft1-6-9

10 combinatirics Combinatorics combinatorics 1 δ 2δφ δ δφ (φ 1φ 2 ) = 1 δ ij (δ j1 φ 2 + φ 1 δ j2 ) 2δφ i = 1 2 ij(δ j1 δ i2 + δ j2 δ i1 ) = 12 ( ij = ji ) 2n φ 1 φ 2 φ 2n ( 1 1 δ n! 2δφ δ ) n (φ 1 φ 2 φ 2n) = 1 ( δ δφ n!2 n δφ δ ) n (φ 1 φ 2 φ 2n) δφ δ/δφ 2n φ i 2n 1 φ i (2n)! qft1-6-10

11 1/2 n n! (2n)! (2n)!!(2n 1)!! = = (2n 1)!! 2 n n! 2 n n! propagators combinations φ i pair φ j 2n 1 propagator 2n 2 φ 2n 3 propagator choice (2n 1)!! n propagator 1 n! ( 1 δ 2δφ δ ) n (φ 1 φ 2 φ 2n) = i1 i δφ 2 i3 i 4 i2n 1 i 2n (32) distinct qft1-6-11

12 1 propagator φ i propagator Symmetry factor vertex 1 φ 4 : (4 1)!! = 3 diagrams (33) 3 i qft1-6-12

13 2 φ 1 φ 2 φ 4 3 : (6 1)!! = 15 2 (a) (b) (a) 12 vacuum bubble 3 (b) φ 1 φ 2 φ 4 3 contract 4 3 = symmetry factor qft1-6-13

14 6.2 (connected) W [J] diagram connected disconnected W [J] 1 iw [J] = d 4 x 1 d 4 x n G (n),c (x 1, x 2,..., x n ) n! = n=0 (ij(x 1 ))(ij(x 2 )) (ij(x n )) 1 n! G(n),c i 1 i 2...i n (ij i1 )(ij i2 ) (ij in ) (34) n=0 Z[J] W [J] Combinatoric Method Z[J] n W [J] n W [J] n qft1-6-14

15 W [J] n! W [J] i convention Z[J] = i n n! W [J]n = e iw [J] (35) W [J] Helmholtz F (V, T ) V J, T : rescale T coupling constant φ 4 φ = λ 1/2 χ S = 1 ( 1 d 4 x λ 2 µχ µ χ m 2 χ2 1 ) 4! χ4 (36) T λ qft1-6-15

16 φ i c = 1 Z δ Z = δiw δij i δij i = δw δj i (37) Z disconnected diagrams φ i φ j = 1 1 δ 1 δ Z Z i δij i i δij j W [J] : = δ2 iw δij i δij j + δiw δij i δiw δij j = φ i φ j c + φ i c φ j c (38) W [J] connected cluster decomposition N- cluster space-like qft1-6-16

17 connected disconnected Connected propagator damp disconnected damp connected graph source support qft1-6-17

18 Ω 1 Ω 2 J 1 (x) 0, J 2 (x) = 0 J 2 (x) 0, J 1 (x) = 0 J(x) = J 1 (x) + J 2 (x) S local interaction S[φ] + J φ = d n x(l + J 1 φ) + d n x(l + J 2 φ) Ω 1 Ω 2 + d n xl (J i ) + Ω i x / Ω 1,Ω 2 (39) qft1-6-18

19 generating functional factorize : Z[J] = Z 1 [J 1 ]Z 2 [J 2 ]Z 12 [J 1, J 2 ] (40) Z i [J i ] = (41) x Ω i Dφe i(s+j i φ) Z 12 = Ω i (42) linear scale up volume Z = e iw W [J] = W 1 [J 1 ] + W 2 [J 2 ] (43) W qft1-6-19

20 J iw [J] = 1 dx 1 dx 2 dx n ig (n) c (x 1,..., x n )(ij(x 1 )) (ij(x n )) n! = 1 dx 1 dx p dy p+1 dy n n! ig (n) c (x 1,..., x p, y p+1... y n )i n J 1 (x 1 ) J 1 (x p ) J 2 (y p+1 ) J 2 (y n ) x i Ω 1, y j Ω 2 iw 1 [J 1 ] + iw 2 [J 2 ] source G (n) c (x 1,..., x p, y p+1... y n ) 0 as min x i y j (44) connected disconnected diagrams qft1-6-20

21 6.3 1PI = W [J] Helmholtz Gibbs G(P, T ) Γ[Φ] V J P Φ i Legendre Φ i δiw = δw = φ i c in the presence of J i (45) δij i δj i ( iγ[φ]) (ij i )Φ i iw Γ[Φ] = W J i Φ i (46) (45) (46) Φ i δ( iγ) δφ i = ij i + δij j Φ j δiw δφ i δφ i = ij i + δij j Φ j δiw δij j (47) δφ i δij j δφ i (45) 2 3 δ( iγ) δφ i = ij i δγ δφ i = J i (48) qft1-6-21

22 : iγ, iw, ij i Γ[Φ] : Γ[Φ] 2 : (48) ij i δ ij = δ2 ( iγ) δij i δφ j = δφ k δij i δ 2 ( iγ) δφ k δφ j = φ i φ k δ2 ( iγ) δφ k δφ j (49) δ 2 ( iγ) δφ k δφ j = φ k φ j 1 (50) inverse propagator wave operator qft1-6-22

23 3 : (49) ij l 0 = φ i φ k φ l φ k φ j 1 δ 3 ( iγ) + φ i φ k φ l φ m δφ m δφ k δφ j iδ 3 Γ φ i φ j φ k = φ i φ i φ j φ j φ k φ k (51) δφ i δφ j δφ k propagator amputated 3-point function ij l φ i φ j φ k φ l = δ iδ 3 Γ ( φ i φ i φ j φ j φ k φ k ) δij l δφ i δφ j δφ k iδ 4 Γ + φ i φ i φ j φ j φ k φ k φ l φ l δφ i δφ j δφ k δφ l qft1-6-23

24 1 3 connected 3-pt functions (51) Γ (3) i(j, k) j(k, i) l k(i, j) + Γ 1 propagator 1 1 (1-particle-irreducible=1PI) 1PI n-point function (n 3) amputated Green s function i Γ Γ i1 i 2...i n qft1-6-24

25 Γ = i 2 Φ ig 1 ij Φ j i 1 n! Γ i 1 i 2...i n Φ i1 Φ i2 Φ in (52) 2 Γ ij Tree level : = G 1 ij (53) Tree level Γ scalar ( ) Γ (0) = 1 2 Φ i( 1 F ) ijφ j 1 n! λ i 1 i 2...i n Φ i1 Φ i2 Φ in = i 2 Φ i 1 ij Φ j 1 n! λ i 1 i 2...i n Φ i1 Φ i2 Φ in (54) qft1-6-25

26 Γ : Γ (0) i 1 i 2...i n = iλ i1 i 2...i n (55) δγ/δφ i = J i source S : 6.4 Γ[Φ] 1PI diagram 1PI diagram 1 line diagram connected effective action Γ[Φ] diagrams ( Φ φ ) qft1-6-26

27 Propagator trick: modify S ɛ = 1 dxdyφ(x)φ(y) [K(x, y) + ɛ] + V (φ) (56) 2 K(x, y) = wave operator, ɛ = small parameter (57) Modified propagator ɛ dz ɛ (x, z) [K(z, y) + ɛ] = δ(x y) (58) ɛ ɛ ɛ = + ɛ (1) + (59) = K 1 = propagator (60) (58) ɛ [ ] ɛ dz (x, z) + (1) (x, z) 1 (z, y) = 0 (61) (y, w) y dz (x, z) dy (y, w) + (1) (x, w) = 0 (62) qft1-6-27

28 η(x) dz (x, z) (1) (x, y) = η(x)η(y) (63) ɛ (x, y) = (x, y) ɛη(x)η(y) + O(ɛ 2 ) (64) ɛ factorize insert Feynman diagram line ɛ Γ ɛ [φ] O(ɛ) diagrams connected qft1-6-28

29 Γ ɛ [φ] : Z ɛ [J] O(ɛ) ɛ ( Z ɛ [J] e iwɛ[j] = Dφ 1 + i ɛ ) dxdyφ(x)φ(y) e i(s+j φ) + O(ɛ 2 ) ( 2 = 1 + i ɛ dxdy 1 ) δ 1 δ e iw [J] + ( 2 i δj(x) i δj(y) = 1 + i ɛ [ ( dx δw ) ]) δ 2 W dxdy e iw [J] + 2 δj(x) i δj(x)δj(y) W ɛ [J] W [J] + ɛ 2 [ ( dx δw ) δj(x) i ] δ 2 W dxdy δj(x)δj(y) (65) O(ɛ) disconnect Γ Legendre W [J] global parameter ɛ qft1-6-29

30 Legendre Γ[φ] = W [J] dxj(x)φ(x) (66) φ(x) = δw δj(x) = ɛ (67) Γ[φ] ɛ chain rule LHS = Γ ɛ + dx φ(x) δγ (68) ɛ δφ(x) RHS = W ɛ dxj(x) φ(x) (69) ɛ δγ/δφ(x) ɛ = 0 J(x) Γ ɛ = W ɛ=0 ɛ (70) ɛ=0 ɛ Γ W (65) Γ ɛ [φ] = Γ[φ] + ɛ dxdyφ(x)φ(y) + ɛ δ 2 W dxdy 2 2i δj(x)δj(y) + (71) qft1-6-30

31 2 actin 3 source connected propagator cut connected φ φ J φ φ φ φ J φ φ x y φ φ J φ φ φ φ J φ φ φ φ J φ φ Γ[φ] 1PI diagram generating function qft1-6-31

φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)

φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x