26 3 ( 29 5 ) D 3 1 (KEK) 1 Minkowski (conformal field theory, CFT) Minkowski Minkowski Hamilton Hermite Euclid

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1 26 3 ( 29 5 ) D 3 (KEK) Minkowski (conformal field theory, CFT) Minkowski Minkowski Hamilton Hermite Euclid Euclid Minkowski () Euclid Minkowski Euclid D 4 (, 206)

2 . x µ x µ η µν dx µ dx ν η µν dx µ dx ν = Ω(x)η µν dx µ dx ν (.) Ω Minkowski η µν = (,,, ) x µ x ν η µν x λ x σ = Ω(x)η λσ Ω = Poincaré η µν η µν x µ x µ = x µ + ζ µ ζ µ µ ζ ν + ν ζ µ 2 D η µν λ ζ λ = 0 Killing ζ λ Killing Ω = + 2 D λζ λ (.2) Killing (η µν 2 +(D 2) µ ν ) λ ζ λ = 0 ζ µ (D +)(D +2)/2 2

3 D (translation) D(D )/2 Lorentz dilatation D (special conformal transformation) ζ λ T,L,D,S (ζ λ T ) µ = δ λ µ, (ζ λ L) µν = x µ δ λ ν x ν δ λ µ, ζ λ D = x λ, (ζ λ S) µ = x 2 δ λ µ 2x µ x λ (.3) Killing µ ζ ν + ν ζ µ = 0 Poincaré dilatation x µ x µ = λx µ, x µ x µ = x µ + a µ x 2 + 2a µ x µ + a 2 x 2 (conformal inversion) x µ x µ = xµ x 2 x µ xµ x 2 xµ x 2 + aµ x µ x 2 + a µ ( x µ x 2 + a µ ) 2 = x µ + a µ x 2 + 2a µ x µ + a 2 x 2.2 Lorentz dilatation P µ M µν D K µ 2 (D + )(D + 2)/2 SO(D, 2) [P µ, P ν ] = 0, [M µν, P λ ] = i (η µλ P ν η νλ P µ ), 2 dilatation D 3

4 [M µν, M λσ ] = i (η µλ M νσ + η νσ M µλ η µσ M νλ η νλ M µσ ), [D, P µ ] = ip µ, [D, M µν ] = 0, [D, K µ ] = ik µ, [M µν, K λ ] = i (η µλ K ν η νλ K µ ), [K µ, K ν ] = 0, [K µ, P ν ] = 2i (η µν D + M µν ) (.4) Lorentz SO(D, ) Poincaré Hermite P µ = P µ, M µν = M µν, D = D, K µ = K µ SO(D, 2) J ab [J ab, J cd ] = i (η ac J bd + η bd J ac η bc J ad η ad J bc ) Hermite J ab = J ab J ab = J ba η ab = (,,,, ) a, b = 0,, 2,, D, D+ µ, ν = 0,,, D M µν = J µν, D = J D+D, P µ = J µd+ J µd, K µ = J µd+ + J µd Lorentz dilatation (.4) (primary) l O µ µ l 3 Hermite O µ µ l (x) = O µ µ l (x) 3 D = 4 Lorentz SO(3, ) (j, j) j = j = l/2 O µ µ l = (σ µ ) α α (σ µl ) α l α l O α α l α α l η µν (σ µ ) α α (σ ν ) β β ε αβ ε α β j j (/2, 0) (0, /2) (, /2) (/2, ) Rarita-Schwinger (, 0) (0, ) 4

5 O (x ) = Ω(x) /2 O(x) O µ µ l (x)dx µ dx µ l l O µ µ l (x ) = Ω(x) l 2 x ν x xνl O µ x µ ν ν l l (x) (.5) SO(D, ) D µν Jacobian xν x µ = Ω(x) /2 D ν µ (x) O j (x) R[D] jk O j(x ) = Ω(x) /2 R[D(x)] k j O k (x) 0 O (x ) O n (x n ) 0 = 0 O (x ) O n(x n ) 0 (.6) 0 x j x µ x µ = x µ + ζ µ x O j O j δ ζo j (x) = O j (x) O j(x) ζ µ O j(x ) = O j(x) + ζ µ µ O j (x) D µ ν = δ µ ν ( ν ζ µ µ ζ ν )/2 Ω (.2) (.5) δ ζ O µ µ l (x) = (ζ ν ν + D νζ ν ) O µ µ l (x) + l ( ) µj ζ ν ν ζ µj Oµ µ 2 j νµ j+ µ l (x) j= 5

6 δ ζ O µ µ l (x) = i [Q ζ, O µ µ l (x)] Q ζ Killing ζ µ (D + )(D + 2)/2 Killing ζ λ T,L,D,S(.3) i [P µ, O λ λ l (x)] = µ O λ λ l (x), i [M µν, O λ λ l (x)] = (x µ ν x ν µ iσ µν ) O λ λ l (x), i [D, O λ λ l (x)] = (x µ µ + ) O λ λ l (x), i [K µ, O λ λ l (x)] = ( x 2 µ 2x µ x ν ν 2 x µ + 2ix ν Σ µν ) Oλ λ l (x) Σ µν O λ λ l = i l ( ) ηµλj δ σ ν η νλj δ σ µ Oλ λ j σλ j+ λ l j= Σ µν O λ λ l = (Σ µν ) λ λ l σ σ l (.7) O σ σ l Lorentz M µν (Σ µν ) σ λ = i(η µλ δ σ ν (Σ µν ) η νλ δ σ µ ) l l σ σ l λ λ l = δ σ λ j= δ σ j λ j (Σ µν ) σ j λ j δ σ j+ λ j+ δ σ l λ l O /2 Σ µν ψ = i 4 [γ µ, γ ν ]ψ {γ µ, γ ν } = 2η µν Killing Q ζ = d D xζ λ T λ0 6

7 d D x Killing µ T µν = 0 η Q ζ = (/D) d D x λ ζ λ T µ µ ζ λ ζt,l,d,s(.3) λ P µ = d D xt µ0, M µν = d D x (x µ T ν0 x ν T µ0 ), D = d D xx λ T λ0, K µ = d D x ( ) x 2 T µ0 2x µ x λ T λ0 (.8) C 0 Q ζ (= Q ζ ) Q ζ 0 = 0, 0 Q ζ = 0 n O j (j =,, n) 0 [Q ζ, O (x ) O n (x n )] 0 = 0 n δ ζ 0 O (x ) O n (x n ) 0 = i 0 O (x ) [Q ζ, O j (x j )] O n (x n ) 0 = 0 j= (.6) O j j Q ζ D K µ (.7) ( ) n x µ j j= x µ + j 0 O (x ) O n (x n ) 0 = 0, j ( ) n x 2 j j= x µ 2x jµ x ν j 2 j x ν j x jµ 0 O (x ) O n (x n ) 0 = 0 j.3 Wightman l Wight- 7

8 man W µ µ l,ν ν l (x y) = 0 O µ µ l (x)o ν ν l (y) 0 (.9) W µ µ l,ν ν l (x) = cp µ µ l,ν ν l (x) (x 2 ) x 0 x 0 iϵ ϵ UV P µ µ l,ν ν l [ A ] 2 Wightman 0 O(x)O(0) 0 = c (x 2 ) = c x 0 x 0 iϵ (x 2 + 2iϵx 0 ) x 0 0 ϵ 2 2 Wightman 0 O µ (x)o ν (0) 0 = ci µν (x 2 ) x 0 x 0 iϵ 0 O µν (x)o λσ (0) 0 = c ( I µλ I νσ + I µσ I νλ 2 ) 2 D η µνη λσ (x 2 ) x µ I µν I µν = η µν 2 xµ x ν x 2 x 0 x 0 iϵ I λ µ I λν = η µν I µ µ = D 2 l P µ µ l,ν ν l = l! (I µ ν I µl ν l + perms) traces perms traces c ( ) c > 0 c = 8

9 Wightman (.9) f,2 (x) : (f, f 2 ) = d D xd D yf µ µ l (x)w µ µ l,ν ν l (x y)f ν ν l 2 (y). Wightman Fourier W µ µ l,ν ν l (k) = d D xw µ µ l,ν ν l (x)e ik µx µ d D k (f, f 2 ) = (2π) f µ µ l D (k)f ν ν l 2 (k)w µ µ l,ν ν l (k) f,2 (k) Fourier k 2 (f, f) > 0 Wightman s D for s = 0, 2 D 2 + s for s 0 (.0) (unitarity bound).4 Fourier D = 4 Wightman W (x) Fourier [ B ] 2 2π( ) W (k) = (2π) 4 Γ( ) 2 θ(k0 )θ( k 2 )( k 2 ) 2 9

10 (f, f) = d 4 k f(k) 2 W (k)/(2π) 4 = lim ( )θ( k 2 ) 2 = δ( k 2 ) (2π) lim (f, f) = 2 d 4 k (2π) 4 f(k) 2 2πθ(k 0 )δ( k 2 ) = d 3 k (2π) 3 2 k f(k) 2 A µ 2 0 A µ (x)a ν (0) 0 ( = η µν 2α x ) µx ν x 2 (x 2 ) x 0 x 0 iϵ = { α 2 2( )( 2) η µν 2 α } µ ν (x 2 ) x 0 x 0 iϵ Fourier Fourier W (α) µν W µν (α) (k) = (2π) 2 2π( ) 4 Γ( )Γ( + ) θ(k0 )θ( k 2 )( k 2 ) 2 { ( α)η µν 2α( 2) k } µk ν k 2 O µ α = W () µν = W µν α = A µ O µ O O = Wightman f µ f µ (k)f ν (k)w (α) µν (k) k µ = (K, 0, 0, 0) f µ f ν W (α) µν = Cθ(K)θ(K 2 ) { [(2 3)α ] f ( α) f j 2} K 2( 2) 0

11 C = 4(2π) 3 ( )/4 Γ( )Γ( + ) (2 3)α 0 α 0 3α 2α, α α = 3 4 = 3 µ O µ = 0 µ W µν (x) = 0 (x 0).5 O P µ µ ν O O (descendant) O 2 D = 4 O µ O 2 O O(x) 2 O(0) 0 = 6 2 ( + )( ) (x 2 ) +2 x 0 x 0 iϵ 4 F µν 2( )

12 2 O 2 > = 2 O = 0 { 0 µ O(x) ν O(0) 0 = 2 η µν 2( + ) x µx ν x 2 } (x 2 ) + x 0 x 0 iϵ Wightman O µ µ O µ 0 µ O µ (x) ν O ν (0) 0 = 4( )( 3) (x 2 ) + x 0 x 0 iϵ > 3 = 3 O µ µ O µ = 0 µ O µν 2 0 µ O µν (x) λ O λσ (0) 0 = ( 4)(4 7) { η νσ } x ν x σ x 2 (x 2 ) + x 0 x 0 iϵ µ ν O µν 0 µ ν O µν (x) λ σ O λσ (0) 0 = 24 ( )( 3)( 4) (x 2 ) +2 x 0 x 0 iϵ 4 Wightman = 4 O µν µ O µν = 0 2

13 .6 Feynman Feynman Wightman 0 T [O µ µ l (x)o ν ν l (0)] 0 = θ(x 0 ) 0 O µ µ l (x)o ν ν l (0) 0 + θ( x 0 ) 0 O ν ν l (0)O µ µ l (x) 0 Fourier 0 T [O µ µ l (x)o ν ν l (0)] 0 = d 4 k µxµ eik D (2π) 4 µ µ l,ν ν l (k) 0 T [O(x)O(0)] 0 = θ(x 0 ) (x 2 + 2iϵx 0 ) + θ( x0 ) (x 2 2iϵx 0 ) = (x 2 + iϵ) 2ϵ x 0 ϵ Fourier 2 Γ(2 ) D(k) = i(2π) 4 Γ( ) (k2 iϵ) 2 0 T [O µ (x)o ν (0)] 0 = { 2 2( 2) η µν 2 µ ν } (x 2 + iϵ) Fourier D µ,ν (k) = i (2π)2 Γ(2 ) 4 Γ( + ) { ( )ηµν k 2 2( 2)k µ k ν } (k 2 iϵ) 3 3

14 O f I int = g d 4 x (fo + H.c.) S S = + it f f i f T f = g 2 = g 2 d 4 xf (x) d 4 k (2π) 4 f (k)f(k)d(k) d 4 yf(y) 0 T [O(x)O(y)] 0 S S = 2Im(T ) = T 2 0 Im f T f = g 2 d 4 k (2π) 4 f(k) 2 Im {id(k)} 0 (x+iϵ) λ (x iϵ) λ = 2i sin(πλ)θ( x)( x) λ sin(πλ) = π/γ(λ)γ( λ) 2 π( ) Im {id(k)} = (2π) 4 Γ( ) 2 θ( k2 )( k 2 ) 2 Wightman Fourier [θ(k 0 ) /2 ] 4

15 2 Euclid Euclid Minkowski Euclid δ µν 2. Ising D D Euclid 5 Euclid T T c T T c O(x)O(0) e x /ξ ξ T = T c ξ O(x)O(0) = x 2 S CFT < D relevant 5 D Minkowski D 5

16 O t ( ) S CFT S CFT ta D d D xo(x) a ξ D ta D ξ at /(D ) 6 O ε t = (T T c )/T c ε ν ξ at ν ν = /(D ε ) D 2.2 Euclid Euclid R D SO(D +, ) η µν δ µν M D (.4) (.7) P µ D Hermite P µ = K µ, D = D SO(D, 2) J ab η ab = (,,,, ) D ξ a dξ/da = 0 t β = adt/da = (D )t 6

17 a, b = 0,,, D, D + D Euclid µ, ν =,, D SO(D +, ) 0 M µν = J µν, D = ij D+0, P µ = J µd+ ij µ0, K µ = J µd+ + ij µ0 J ab (.5) Hermite Euclid Hermite l O µ µ l (x)o ν ν l (0) = cp µ µ l,ν ν l (x 2 ) P µ µ l,ν ν l x M D Euclid I µν I µν = δ µν 2 x µx ν x 2 P µ µ l,ν ν l = l! (I µ ν I µl ν l + perms) traces c c = Hermite x µ Rx µ = x µ x 2 O µ µ l (x) = (x 2 ) I µ ν (x) I µl ν l (x)o ν ν l (Rx) (2.) Hermite Hermite i[p µ, O(x)] = µ O(x) 7

18 Hermite i[p µ, O (x)] = µ O (x) Hermite y µ = x µ /x 2 O (x) = (y 2 ) O(y) Hermite P µ = K µ i(y 2 ) [K µ, O(y)] = y ν { (y 2 ) O(y) } x µ y ν = (y 2 ) ( y 2 µ 2y µ y ν ν 2 y µ ) O(y) (y 2 ) O i[d, O(x)] = (x µ µ + )O(x) Hermite Hermite D = D I µν (x) = I µν (y) i[p µ, O ν (x)] = µ O ν (x) Hermite i(y 2 ) I νλ [K µ, O λ (y)] = y ν { (y 2 ) I νλ O λ (y) } x µ y ν ( ) = (y 2 ) {I νλ y 2 µ 2y µ y σ σ 2 y µ Oλ (y) ( + 2δ µν y λ 2δ µλ y ν + 4 y ) } µy ν y λ O y 2 λ (y) I µλ I λν = δ µν i[k µ, O λ (y)] = (y 2 µ 2y µ y σ σ 2 y µ +2iy σ Σ µσ )O λ (y) iσ µσ O λ = δ µλ O σ + δ λσ O µ O µ µ l O µ µ l Rx 2 = /x 2 O (x)o(0) = (x 2 ) O(Rx)O(0) = x (c = ) I µλ I λν = δ µν O µ(x)o ν (0) = δ µν, O µν(x)o λσ (0) = (δ µλ δ νσ + δ µσ δ νλ 2 ) 2 D δ µνδ λσ 8

19 Euclid M µν {µ µ l }, = (Σ µν ) ν ν l,µ µ l {ν ν l },, id {µ µ l }, = {µ µ l },, K µ {µ µ l }, = 0 7 {µ µ l }, = O µ µ l (0) 0 (2.2) (state-operator correspondence) 0 P µ Hermite y µ = Rx µ I µν (x) = I µν (y) Hermite O µ µ l (0) = lim x 2 0 (x2 ) I µ ν I µl ν l O ν ν l (Rx) = lim y 2 (y2 ) I µ ν I µl ν l O ν ν l (y) (2.2) {µ µ l }, = 0 O µ µ l (0) = lim x 2 (x2 ) I µ ν I µl ν l 0 O ν ν l (x) f µ µ l (f, f) = f µ µ l f ν ν l {µ µ l }, {ν ν l }, = f µ µ l 2 > 0 7 Minkowski = O(0) 0 M D O (x) = O(x) = 0 O (0)O(0) 0 = 0 O(0)O(0) 0 9

20 2.3 Hermite R D O(x) = e ip µx µ O(0)e ip µx µ Hermite P µ = K µ O (x) = e ik µx µ O( )e ik µx µ O( ) = O (0) = lim x 2 (x 2 ) O(x) Hermite (2.) O(x)O(x ) = (x 2 ) O (Rx)O(x ) = (x 2 ) e ikµ(rx)µ e ipνx ν = O(0) 0 = 0 O( ) K µ P ν O(x)O(x ) = ( x C (x 2 ) n (x, x 2 ) x 2 n=0 C n ) n/2 C n = (n!) 2 x µ x µn x ν x ν n (x 2 x 2 ) n/2 K µ K µn P ν P νn Gegenbauer nc n = 2( + n )zc n (2 + n 2)C n 2 z = x x / x 2 x 2 C n z Gegenbauer ( = /2 Legendre ) ( 2zt + t 2 ) = C n (z)t n z t = n=0 x 2 /x 2 O(x)O(x ) = /(x x ) 2 20

21 2.4 (OPE) ϕ (operator product expansion, OPE) ϕ ϕ I + T µν + l=0,2,4, O µ µ l I T µν ( 2 D ) O µ µ l l OPE ( ) ϕ l d 2 ϕ(x )ϕ(x 2 ) = O µ µ l (x )O ν ν l (x 2 ) = x 2, 2d [ ] x 2 2 l! (I µν I + perms) traces µlνl (x 2 ) µ = x µ x 2µ I µν = I µν (x 2 ) f,l ϕ(x )ϕ(x 2 )O µ µ l (x 3 ) = Z µ = (x 3) µ x 2 3 f,l x 2 2d +l x 3 l x 23 l (Z µ Z µl traces), (x 23) µ x 2 23 f,l OPE ϕ OPE ϕ(x)ϕ(y) = = x y 2d +,l(=2n) x y 2d +,l(=2n) (2.3) f,l [ (x y)µ (x y) µl x y 2d +l O µ µ l (y) + f,l x y 2d C,l(x y, y )O,l (y) l O,l (y) C,l (x y, y ) 2 ]

22 l = 0 C,0 OPE O = O,0 ϕ(x)ϕ(y)o(z) = f,0 x y 2d C,0(x y, y ) O(y)O(z) C,0 (x y, y ) y z = 2 x z y z Feynmann Γ( ) [t( t)] 2 Γ( )Γ( ) dt 0 [t(x z) ( t)(y z) 2 ] = B(, ) dt[t( t)] n=0 ( ) n n! [ t( t)(x y) 2 ] n ([y z + t(x y)] 2 ) +n (a) n = Γ(a + n)/γ(a) Pochhammer ( 2 ) n (x 2 ) = 4n ( ) n ( + D/2) n (x 2 ), +n [(y + tx) 2 ] = etx y (y 2 ) / y z 2 y C,0 (x y, y ) = B(, ) dt[t( t)] ( ) n [t( t)a 2 ] n ( 2 4 n n! ( + D/2) y) n e ta y n n=0 a=x y C,0 (x y, y ) = + 2 (x y) µ µ y ( + ) (x y) µ(x y) ν µ y ν y 6( + )( + D/2) (x y)2 y 2 + (2.3) l 0 22

23 2.5 Conformal Blocks j ϕ j ϕ (x )ϕ 2 (x 2 )ϕ 3 (x 3 )ϕ 4 (x 4 ) = ( ) 2 ( ) 34 x24 x4 G(u, v) x 4 x 3 x x ij = i j u v u = x2 2x 2 34, v = x2 4x 2 23 x 2 3x 2 24 x 2 3x 2 24 ϕ ϕ 2 OPE ϕ ϕ 4 OPE (x 2, 2 ) (x 4, 4 ) (x 2, 2 ) (x 3, 3 ) (crossing symmetry) G(u, v) G(v, u) G(/u, v/u) = 2 = 3 = 4 OPE G(u, v) = +,l f 2,lg,l (u, v) d ϕ d ϕ d (x )ϕ d (x 2 )ϕ d (x 3 )ϕ d (x 4 ) = [ + f x 2 2d x 34,lg 2 2d,l (u, v) ] g,l (u, v) conformal block x 2 x 4 v d G(u, v) = u d G(v, u) conformal block u d v d =,l f 2,l,l [ v d g,l (u, v) u d g,l (v, u) ] (2.4) 23

24 Conformal block g,l OPE (l = 0) OPE g,0 (u, v) = x 2 x 34 C,0 (x 2, 2 )C,0 (x 34, 4 ) x 24 2 C,0 (x 2, 2 )C,0 (x 34, 4 ) x 24 = 2 B(, dtds[t( t)s( s)] )2 0 ( ) n+m ( ) n+m ( ) n+m [t( t)x 2 2] n [s( s)x 2 34] m n!m! ( ) n ( ) m [(x 24 + tx 2 sx 34 ) 2 ] +n+m n,m=0 = + D/2 A 2 = t( t)x 2 2 B2 = s( s)x 2 34 (x 24 + tx 2 sx 34 ) 2 = Λ 2 A 2 B 2, Λ 2 = tsx t( s)x s( t)x ( t)( s)x 2 24 B( 2, 2 )2 0 dtds [t( t)s( s)] 2 (Λ 2 A 2 B 2 ) F 4(, ;, ; X, Y ) X = A 2 /(Λ 2 A 2 B 2 ) Y = B 2 /(Λ 2 A 2 B 2 ) F 4 Appell (double series) F 4 (a, b;, c, d; x, y) = n,m=0 n!m! (a) n+m (b) n+m x n y m (c) n (d) m Gauss 2 F ( a F 4 (a, b; b, b; x, y) = ( x y) a 2F 2, a + ) 2 ; b; 4xy ( x y) 2 B( 2, 2 )2 0 dtds [t( t)s( s)] 2 (Λ 2 ) 2F 24 ( 2, + ; 2 ; 4A2 B 2 ) Λ 4

25 t s 0 0 t a ( t) b dt [tα + ( t)β] = B(a, b), a+b α a βb s a ( s) b ds ( sα) c ( sβ) = B(a, b)f (a, c, d; a + b; α, β) d F F (a, b, c; d; x, y) = n,m=0 n!m! (a) n+m (b) n (c) m x n y m (d) n+m Gauss ( F (a, b, c, b + c; x, y) = ( y) a 2F a, b; b + c; x y ) y ( ) 4 u n ( x 3 x 24 v 2 2 n n! ( ) 2n ( ) 2F n 2 + n, ) 2 + n; + 2n; v n=0 u = u/v v = /v ( ) 2n 4 n ( 2 ) n( + 2 ) n = ( ) 2n G(a, b, c, d; x, y) = n,m=0 (d a) n (d b) n n! (c) n (a) n+m (b) n+m m! (d) 2n+m x n y m g,0 ( 2 + n) m = ( 2 ) n+m/( 2 ) n ( + 2n) m = ( ) 2n+m /( ) 2n x 3 x 4 g,l (u, v) = g,l (u, v ) g,0 u v u v g,0 (u, v) = u 2 G ( 2, 2, + D 2, ; u, v ) conformal block g,l Gauss u = z z, v = ( z)( z) 25

26 G(a, b, c, c; u, v) = [ z 2 F (a, b; c; z) 2 F (a, b ; c 2; z) z z z 2 F (a, b; c; z) 2 F (a, b ; c 2; z) ] Gauss k β (x) = x β2 2F ( β 2, β 2, β; x ) (2.5) D = 4 g,0 (u, v) D=4 = z z z z [k (z)k 2 ( z) (z z)] l OPE l conformal blocks l D = 4 g,l (u, v) D=4 = ( )l 2 l z z z z [k +l(z)k l 2 ( z) (z z)] (2.6) 2 g,l (u, v) D=2 = ( )l 2 l [k +l (z)k l ( z) + (z z)] (2.7) D = 3 l z = z 2.6 Casimir Conformal Blocks conformal block Casimir SO(D +, ) J ab Casimir C 2 = J ab J 2 ab C 2 = 2 M µνm µν D 2 2 (K µp µ + P µ K µ ) 26

27 { (x 2 2 µ 2 µ 2(x 2 ) µ (x 2 ) ν µ 2 ν 2 (x 2 ) µ 2 µ (x 2 ) µ µ l C 2, l = C,l, l, C,l = ( D) + l(l + D 2), l n n;, l ϕ (x )ϕ 2 (x 2 )ϕ 3 (x 3 )ϕ 4 (x 4 ) =,l,n 0 ϕ (x )ϕ 2 (x 2 ) n;, l n;, l ϕ 3 (x 3 )ϕ 4 (x 4 ) [ J ab, [ J ab, ϕ (x )ϕ 2 (x 2 ) ]] n;, l = 0 ϕ (x )ϕ 2 (x 2 )C 2 n;, l = C,l 0 ϕ (x )ϕ 2 (x 2 ) n;, l C 2 P µ n + ( + 2 ) ( + 2 D) } 0 ϕ (x )ϕ 2 (x 2 ) n;, l O,l conformal block g,l 0 ϕ (x )ϕ 2 (x 2 ) n;, l n;, l ϕ 3 (x 3 )ϕ 4 (x 4 ) 0 n ( ) x 2 2 /2 ( = 24 x 2 ) 34 /2 4 f,lg 2,l (u, v) x 2 4 x 2 3 (x 2 2) ( + 2 )/2 (x 2 34) ( 3+ 4 )/2 conformal block Dg,l (u, v) = 2 C,lg,l (u, v), 27

28 D = ( u + v)u ( u ) + [ ( v) 2 u( + v) ] ( v ) u u v v 2( + u v)uv 2 u v Du u + [ ( 2 ( 2 34 ) ( + u v) u u + v ) ( u v) ] v v ( + u v) z z D = z 2 ( z) 2 z + 2 z2 ( z) 2 z + ( 2 2 ( ) z 2 z + ) z2 z (z + z) + (D 2) z z ( ( z) z z z ( z) ) z D = 4, 2 (2.6) (2.7) (2.5) ( k β (x) = x β2 β 2F 2 2 2, β 2 + ) 34 2 ; β; x 2.7 (.0) D = 4 µ,, µ ν, = δ δ µν µ; ν, = P µ ν, µ; λ, ν; σ, = λ, [K µ, P ν ] σ, = λ, 2i (Dδ µν + M µν ) σ, = 2δ ( δ µν δ λσ δ µλ δ νσ + δ νλ δ µσ ) 28

29 Hermite P µ = K µ K µ ν, = ν, P µ = 0 λ, M µν σ, = (Σ µν ) λσ a = (µ, λ) b = (ν, σ) 6 6 a b 3 2( 3) 2( ) 2( + ) 3 SO(4) {r} {r}, M µν {r}, = (Σ µν ) {r },{r} {r },, id {r}, = {r},, K µ {r}, = 0 SO(4) SU(2) SU(2) SU(2) j, j 2 {r} (j, j 2 ) (2j + )(2j 2 + ) l O µ µ l j = j 2 = l/2 P µ n n µ µ n ; {r}, = P µ P µn {r}, {r }, {r}, = δ {r }{r}δ µ; {r}, = P µ {r}, ( µ; {r }, ν; {r}, = δ 2 δ{r }{r} + 2 {r }, im µν {r}, ) (2.8) 29

30 Lorentz im µν = i 2 (δ µαδ νβ δ µβ δ να ) M αβ = 2 (Σ αβ) µν M αβ Σ αβ µ µ M {v} αβ ν = (Σ αβ) µν 2 {r }, im µν {r}, = µ {r }, M {v} αβ M {r} αβ {r}, ν (2.9) M {R} αβ M {v} αβ = M {v} αβ + M {r} αβ M {r} αβ = 2 M {R} αβ M {R} αβ 2 M {v} αβ M {v} αβ 2 M {r} αβ M {r} αβ = c 2 ({R}) c 2 ({v}) c 2 ({r}) c 2 SO(4) Casimir {r} SU(2) SU(2) (j, j 2 ) {v} (/2, /2) {R} (J, J 2 ) J,2 j,2 ± /2 (2.9) Casimir 2J (J +)+2J 2 (J 2 +) 3 2j (j + ) 2j 2 (j 2 + ) (2.8) (2.8) j, j 2 0 (2.8) J = j /2, J 2 = j 2 /2 2 2(j + j 2 + 2) j + j for j, j 2 0 j = j 2 = l/2 l l + 2 j = 0, j 2 0 J = /2, J 2 = j 2 /2 30

31 2 2(j 2 + ) j 2 + for j = 0, j 2 0 j j 2 j = j 2 = P µ P µ K µ K µ P ν P ν = 32 ( )δ for j = j 2 = Conformal Bootstrap 3.5 (2.4) conformal block g,l p,l F d,,l (z, z) =,,l F d,,l (z, z) = vd g,l (u, v) u d g,l (v, u) u d v d (2.0) p,l = f,l 2 (d = ) l p,l = δ,l+2 δ l,2n 2 l+ (l!) 2 /(2l)! OPE f,l p,l 0 3

32 OPE D = 4 d OPEϕ d ϕ d + O (d ) /2 + 2.(d ) (d ) 3/2 + o((d ) 2 )(2.) D = 2 Ising ϕ d σ O ε d = σ = /8 = ε = d = /8 OPE = D = 3 Ising z = /2 + X + iy X Y N Λ[F ] = m,n=even 2 m+n N λ m,n m X n Y F X=Y =0 X = Y = 0 (z = z = /2) (2.0) p,l Λ [F d,,l ] = 0 (2.2),l 32

33 l Λ[F d,,l ] 0 p,l 0 OPE ϕ d ϕ d + f O + l>0 l=even O,l D 2+l O f l > 0 d f f (l = 0) D 2 + l (l > 0) Λ[F d,,l ] 0 λ m,n p,l 0,l p,l Λ [F d,,l ] 0 (2.2) d f d f d f f c (d) D/2 f c (d) l l (linear programing method) λ m,n (2.) OPE O f ( f c ) = f c Ising σ = 0.582(3) ε =.43() 33

34 OPE ( ) 2.9 Wilson-Fisher Wilson-Fisher 8 D = 4 ϵ 4 S = [ ] d D x 2 ( ϕ)2 + λϕ 4 β λ = ϵλ + 9λ2 2π 2 ϵ 0 λ = ϵ2π 2 /9 ϕ : ϕ 2 : γ = 3λ 2 /6π 4 δ = 3λ/2π 2 9 ϕ = D λ2 ( 6π = ϵ ) + ϵ ϕ 2 = D 2 + 3λ 2π 2 = (2 ϵ) + ϵ 3 Ising OPEσ σ ε ϕ ϕ ϕ 2 σ = ϕ ε = ϕ 2 Ising ϵ σ = 0.5 ε = J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press. 9 S. Hathrell, Ann. Phys. 39 (982)36 [ L. Brown, Ann. Phys. 26 (980) 35 ] 0 o(ϵ 5 ) σ = ε = 34

35 3 3. Wess-Zumino Weyl δg µν = 2ωg µν Γ δ ω Γ = d 4 x gω { ac 2 µνλσ + bg 4 + cr 2 + d 2 R + ef 2 µν } C µνλσ 2 Weyl G 4 Euler C 2 µνλσ = R 2 µνλσ 2R 2 µν + 3 R2, G 4 = R 2 µνλσ 4R 2 µν + R 2 (3.) Fµν 2 Weyl Wess-Zumino (Wess- Zumino integrability condition)[δ ω, δ ω2 ] = 0 2 (3.) [δ ω, δ ω2 ] Γ = 24c d 4 x gr ( ω ˆ 2 ω 2 ω 2 ˆ 2 ω ).402 S. Rychkov and Z. Tan, J. Phys. A48 (205) 29FT0 R. Gopkumar et al, Phys. Rev. Lett. 8 (207) 0860 M. Duff, Nucl. Phys. B25 (977) L. Bonora, P. Cotta-Ramusino and C. Reina, Phys. Lett. B26 (983)

36 c = 0 R Riegert-Wess-Zumino g µν = e 2ϕ ḡ µν ϕ Euler Euler E 4 = G R Euler E 4 ge 4 = ḡ(4 4 ϕ+ Ē 4 ) S RWZ (ϕ, ḡ) = b (4π) 2 = b (4π) 2 ϕ d 4 x dϕ ge 4 0 d 4 x ḡ ( 2ϕ 4 ϕ + Ē4ϕ ) (3.2) 3 g 4 (self-adjoint) d 4 x ga 4 B = d 4 x g( 4 A)B 4 4 = 4 + 2R µν µ ν 2 3 R µ R µ 3 R. Riegert, Phys. Lett. 34B (984) 56 36

37 Riegert-Wess-Zumino ϕ Riegert b N X N W Weyl N A b = (N X + ) N W + 62N A f I Riegert ϕ I(f, g) = I(f, ḡ)( f ϕ ) Riegert ϕ g µν (= e 2ϕ ḡ µν ) ḡ µν Jacobian [df] g = [df]ḡe is(ϕ,ḡ) e iγ(g) = [df] g e ii(f,g) = e is(ϕ,ḡ) [df]ḡe ii(f,ḡ) = e is(ϕ,ḡ) e iγ(ḡ) g µν ϕ ϕ ω, ḡ µν e 2ω ḡ µν (3.3) e is(ϕ ω,e2ω ḡ) e iγ(e2ω ḡ) = e is(ϕ ω,e2ω ḡ) e is(ω,ḡ) e iγ(ḡ) e iγ(g) S S(ϕ ω, e 2ω ĝ) + S(ω, ĝ) = S(ϕ, ĝ) Wess-Zumino S RWZ (3.2) [0, ϕ] [0, ω] [ω, ϕ] 37

38 (3.3) Riegert-Wess-Zumino Riegert-Wess-Zumino S RWZ Riegert ϕ (3.3) Γ(ḡ) Γ Riegert (g) = S RWZ (ϕ, ḡ) + Γ Riegert (ḡ) Γ Riegert (g) = b d 4 x ge (4π) 2 4 E 4 4 Riegert-Wess-Zumino S RWZ Wess-Zumino Riegert 3.3 Weyl QED U() Weyl U() F 2 µν Wess-Zumino S QED(ϕ, ḡ) = aϕ gf 2 µν/4 ϕ QED Riegert ϕ ϕ S QED ϕ Γ QED (ḡ) Γ QED (ḡ) = { ( )} e2 r k 2 gf 4 2π log 2 2 µ 2 µν 38

39 µ k 2 = ḡ µν k µ k ν ḡ µν ḡ µν Minkowski QED β e = e 3 r/2π 2 e r Wess-Zumino (3.3) k 2 e 2ω k 2 Wess-Zumino a = e 2 r/6π 4 { + e2 r 6π ϕ e2 r 2 2π log 2 ( k 2 µ 2 )} gf 2 µν g µν (= e 2ϕ ḡ µν ) p 2 = k2 e 2ϕ (3.4) Γ QED (g) = 4 { e2 r 2π 2 log ( p 2 µ 2 )} gf 2 µν µ p k (comoving momentum) Weyl t Weyl (/t 2 ) d 4 x gc 2 µνλσ ḡ µν Weyl { ( ) } k 2 gc 2 Γ Weyl (g) = + β 0 log 2β 0 ϕ µνλσ (3.5) t 2 r ḡ µν h µν ḡ µν Minkowski 39 µ 2

40 ḡ µν = η µν + th µν I int = t d 4 xh µν T µν /2 T µν h µν 2 β 0 4 β 0 = 240(4π) 2 (N X + 3N W + 2N A ) t β t = β 0 t 3 r (3.5) Wess-Zumino (3.3) p t 2 r(p) = β 0 log(p 2 /Λ 2 QG) (3.6) Γ Weyl (g) = gc 2 t 2 µνλσ (3.7) r(p) Λ QG = µ exp( /2β 0 t 2 r) 5 t (Riegert ) 4 2 β 0 5 Adler-Bardeen 40

41 t t = 0 4

42 A P µ µ l,ν ν l Euclid Minkowski η µν x 0 x 0 iϵ x µ = (Rx) µ = x µ x 2 Ω(x) = /x 4 R 2 = I x µ = (Rx ) µ O (x ) = Ω(x) /2 O(x) = x 2 O(x) 6 x O (x) = (/x 2 ) O(Rx) O x O (x )O (y ) = O(x )O(y ) (.6) (x 2 y 2 ) O(x)O(y) = O(Rx)O(Ry) (Rx Ry) 2 = x2 y 2 (x y) 2 O(x)O(y) = (x y) 2 O µ(x ) = Ω(x) ( )/2 x ν O x ν (x) = x 2 I µν (x)o ν (x) µ 6 Euclid O Hermite O 42

43 I µν (x) = δ µν 2x µ x ν /x 2 O µ(x )O ν(y ) = O µ (x )O ν (y ) (x 2 y 2 ) I µλ (x)i νσ (y) O λ (x)o σ (y) = O µ (Rx)O ν (Ry) I µλ (x)i νσ (y)i λσ (x y) = I µν (x y) + 2 x2 y 2 ( xµ x ν (x y) 2 x 2 = I µν (Rx Ry) y ) µy ν y 2 O µ (x)o ν (y) = I µν(x y) (x y) 2 P µ,ν = I µν B Wightman Fourier D Euclid Wightman2 O(x)O(0) Fourier (x 2 ) = (2π) D 2 Γ( D ) 2 4 D 4 Γ( ) d D k ( eik x k 2) (2π) D D 2 (B.) Minkowski Wightman Fourier Euclid k x k x+k D x D k x Minkowski D x D = ix 0 +ϵ (B.) Minkowski Wightman 0 O(x)O(0) 0 { (x 0 iϵ) 2 + x 2 } = (2π) D 2 Γ( D ) 2 d D k eik x 4 D 4 Γ( ) (2π) D dk D { (x 0 iϵ) 2π e kd k 2 + (k D ) 2} D 2 (B.2) 43

44 k D e iϵkd k D k D = ±i k k D = i k i k D = i k i < k D < ( = D/2 ) k D = ik 0 = i dk D { 2π e kd (x0 iϵ) k 2 + (k D ) 2} D 2 dk 0 { [k 0 2π e ik0 x0 ϵk0 ) 2 (k 0 io) 2] D 2 [ k 2 (k 0 + io) 2] } D 2 o (x + io) λ (x io) λ = 0 for x > 0 2i x λ sin πλ for x < 0 = 2i( x) λ θ( x) sin πλ [k 2 (k 0 io) 2 ] D/2 [k 2 (k 0 + io) 2 ] D/2 = 2i( k 2 ) D/2 θ( k 2 ) sin π( D/2) k 2 = k 2 (k 0 ) 2 (B.2) [ ( 2 sin π D )] D (2π) 2 Γ( D ) 2 d D k eik x 2 4 D 4 Γ( ) (2π) D dk 0 x0 e ik0 ( k 2 ) D 2 θ( k 2 ) 0 2π (2π) D 2 + d D k = eik x 4 D 4 Γ( )Γ( D + ) θ(k 0 )θ( k 2 )( k 2 ) (2π) D 2 Γ(λ)Γ( λ) = π/ sin πλ Γ(λ+) = λγ(λ) Fourier W (k) 44 D 2

45 C M 4 I = d 4 x ĝ (ĝ µν µ X ν X ˆRX ) 2 Minkowski ĝ µν = η µν P X = η X [X(η, x), P X (η, x )] = iδ 3 (x x ) X = X < + X > X > = X < X < (x) = d 3 k φ(k)e ik µx µ (2π) 3/2 2ω [φ(k), φ (k )] = δ 3 (k k ) 2 (Wightman ) 0 X(x)X(0) 0 = [X < (x), X > (0)] 0 X(x)X(0) 0 = d 3 k e i k (η iϵ)+ik x = (2π) 3 2 k 4π 2 (η iϵ) 2 + x 2 ϵ UV ĝ µν T µν = (2/ ĝ) δi/δĝ µν Minkowski T µν = 2 3 µx ν X 3 X µ ν X 6 η µν λ X λ X P 0 = H = d 3 xa, P j = d 3 xb j, M 0j = d 3 x ( ηb j x j A), M ij = d 3 x (x i B j x j B i ), 45

46 D = K 0 = K j = d 3 x ( ηa + x k B k + :P X X : ), { (η d 3 x 2 + x 2) A + 2ηx k B k + 2η :P X X : + } 2 :X2 :, d 3 x {( η 2 + x 2) B j 2x j x k B k 2ηx j A 2x j :P X X : } A B j A = 2 :P2 X : 2 :X 2 X :, B j =:P X j X : 2 = j j 2 0 X(x)X(x ) 0 = 4π 2 (x x ) 2 + ϵ 2, 0 X(x)P X (x ) 0 = i 2π 2 ϵ [(x x ) 2 + ϵ 2 ] 2, 0 P X (x)p X (x ) 0 = 2π 2 (x x ) 2 3ϵ 2 [(x x ) 2 + ϵ 2 ] 3 X P X [X(η, x), P X (η, x )] = 0 X(η, x)p X (η, x ) 0 0 X(η, x)p X (η, x ) 0 = i π 2 ϵ [(x x ) 2 + ϵ 2 ] 2 X P X 3 δ d 3 k δ 3 (x) = eik x ϵω = ϵ (2π) 3 π 2 (x 2 + ϵ 2 ) 2 OPE [:AB(x):, :CD(y):] = [A(x), C(y)] :B(x)D(y): + [A(x), D(y)] :B(x)C(y): + [B(x), C(y)] :A(x)D(y): + [B(x), D(y)] :A(x)C(y): +QC(x y) 46

47 QC(x y) = A(x)C(y) B(x)D(y) + A(x)D(y) B(x)C(y) H.c. H.c. Hermite A B j [A(x), A(y)] = 2 i 2 xδ 3 (x y) (:P X (x)x(y): :X(x)P X (y):), [B j (x), B k (y)] = i x k δ 3 (x y) : j X(x)P X (y): +i x j δ 3 (x y) :P X (x) k X(y):, [A(x), B j (y)] = i x j δ 3 (x y) :P X (x)p X (y): 2 iδ 3(x y) : 2 X j X(y): 2 i 2 xδ 3 (x y) :X(x) j X(y): i 2 π 2 f j(x y) ( [A(x), :P X X(y):] = iδ 3 (x x) :P 2 X(y): + ) 2 :X 2 X(y): 2 i 2 xδ 3 (x y) :X(x)X(y): +i 0 f(x y), π2 [B j (x), :P X X(y):] = iδ 3 (x x)b j (y) + i j x δ(x y) :P X (x)x(y): f j f f j (x) = ϵx j (x 2 ϵ 2 ) f(x) = ϵ(5x 2 3ϵ 2 ) π 2 (x 2 + ϵ 2 ) 6 40π 2 (x 2 + ϵ 2 ) 5 f j (x) = j f(x) ϵ d 3 xf j (x) = 0, d 3 xf(x) = 0, d 3 xx j f(x) = 0 d 3 xx 2 f(x) = /60ϵ 2 ϵ 0 7 (.4) 7 f δ f(x) = ( /320) 2 ( δ 3 (x)/x 2) δ π 2 δ 3 (x) = 4ϵ 3 /(x 2 + ϵ 2 ) 3 47

48 :X n : [A(x), :X n (y):] = iδ 3 (x y) η :X n (y):, [B j (x), :X n (y):] = iδ 3 (x y) j :X n (y): +i 2π n(n )g j(x y) :X n 2 (y):, 2 [:P X X(x):, :X n (y):] = inδ 3 (x y) :X n (y): +i 3 2π n(n )g(x y) 2 :Xn 2 (y): g j g g j (x) = π 2 ϵx j g(x) = ϵ (x 2 + ϵ 2 ) 4 6π 2 (x 2 + ϵ 2 ) 3 g j (x) = j g(x) :X n : i [P µ, :X n (x):] = µ :X n (x):, i [M µν, :X n (x):] = (x µ ν x ν µ ) :X n (x):, i [D, :X n (x):] = (x µ µ + n) :X n (x):, i [K µ, :X n (x):] = ( x 2 µ 2x µ x ν ν 2x µ n ) :X n (x): :X n : n D S CFT relevant ( < D ) ε 48

49 t = (T T c )/T c S t = S CFT ta ε D d D xε(x) ε a 8 ξ ta ε D ξ ε D ξ at ν, ν = D ε (D.) relevant ε < D ν ξ t 0 t O O t = Oe S t O t = n=0 ( n! O ta ε D d D xε(x)) n O = Oe S CFT CFT ξ UV a = ε σ Ising OPE σ σ I + ε, σ ε ε, ε ε I 8 t a a 49

50 f C = 2 f/ t 2 = ε(0) t / t ξ (t ) CFT ε C = ε(0) d D xε(x) = d D x x ξ x ξ x 2 ε ε t ξ x ξ ξ C ξ D 2 ε + const. ξ 9 α C t α ξ t ν t α = ν(d 2 ε ) CFT σ ε OPE σε M = σ(0) t = t2 d D x d D y σ(0)ε(x)ε(y) 2! x ξ y ξ ξ (t 0) M t 2 ξ 2D σ 2 ε t ν σ 9 D = 2 Ising 50

51 σ M t β β = ν σ h S t,h = S CFT ta ε D d D xε(x) ha σ D d D xσ(x) χ = h σ(0) t,h = d D x σ(0)σ(x) ξ D 2 σ t ν(d 2 σ) h=0 x ξ χ t γ γ = ν(d 2 σ ) t = 0 h M h /δ h S h = S CFT ha σ D d D xσ(x) M = σ(0) h = h d D x σ(0)σ(x) x ξ ξ σ D ha σ D ξ (h 0) M h ξ D 2 σ h σ/(d σ ) δ = D σ σ 5

52 η 2 σ = D 2 + η (D.) σ ε α = 2 νd [Josephson slaw], β = ν(d 2 + η), 2 γ = ν(2 η) [Fisher slaw], δ = D + 2 η D 2 + η α + 2β + γ = 2 γ = β(δ ) [Rushbrooke slaw], [Widom slaw] D = 2 Ising ε = σ = /8 ν = η = /4 α = 0, β = 8, γ = 7 4, δ = 5 52

53 E E. Fradkin and M. Palchik, Conformal Quantum Field Theory in D-dimensions (Kluwer Academic Publishers, Dordrecht, 996) P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory (Springer, New York, 997). (unitarity bound) G. Mack, All Unitary Ray Representations of the Conformal Group SU(2, 2) with Positive Energy, Commun. Math. Phys. 55 (977). S. Minwalla, Restrictions imposed by Superconformal Invariance on Quantum Field Theories, Adv. Theor. Math. Phys. 2 (998) 78. B. Grinstein, K. Intriligator and I. Rothstein, Comments on Unparticle, Phys. Lett. B662 (2008) 367. D. Dorigoni and S. Rychkov, Scale Invariance + Unitarity Conformal Invariance?, arxiv Conformal Bootstrap A. Polyakov, Zh. Eksp. Teor. Fiz. 66 (974) 23. S. Ferrara, A. Grillo and R. Gatto, Ann. Phys. 76 (973) 6. G. Mack, Nucl. Phys. B8 (977)

54 Conformal Blocks F. Dolan and H. Osborn, Conformal Four Point Functions and the Operator Product Expansion, Nucl. Phys. B599 (200) 459. F. Dolan and H. Osborn, Conformal Partial Wave and the Operator Product Expansion, Nucl. Phys. B678 (2004) 49. F. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arxiv: Conformal Bootstrap R. Rattazzi, V. Rychkov, E. Tonni and A. Vichi, Bounding Scalar Operator Dimensions in 4D CFT, JHEP 082 (2008) 03. V. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D80 (2009) S. El-Showk, M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D86 (202) OPE D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE Convergence in Conformal Field Theory, Phys. Rev. D86 (202) M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, arxiv: