量子重力理論と宇宙論 (上巻) 共形場理論と重力の量子論 浜田賢二 高エネルギー加速器研究機構 (KEK) 素粒子原子核研究所 量子重力の世界は霧に包まれた距離感のない幽玄の世界にたとえること ができる 深い霧が晴れて時空が

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1 量子重力理論と宇宙論 (上巻) 共形場理論と重力の量子論 浜田賢二 高エネルギー加速器研究機構 (KEK) 素粒子原子核研究所 量子重力の世界は霧に包まれた距離感のない幽玄の世界にたとえること ができる 深い霧が晴れて時空が現れる 国宝松林図屏風 (長谷川等伯筆) 平成 0 年 11 月初版/平成 1 年 09 月改定/ 平成 5 年 09 月再改定 (上下巻に分離)

2 Planck Planck BRST Planck Λ QG Planck GeV Planck Λ QG Friedmann CMB

3 Minkowski Wightman Fourier Feynman Euclid Euclid (OPE) Conformal Blocks Casimir Conformal Blocks Conformal Bootstrap

4 4 4 R S R S Minkowski R S R S Wess-Zumino Riegert-Wess-Zumino Riegert BRST R S BRST

5 5 A 17 A A B 133 B.1 P µ1 µ l,ν 1 ν l C 135 C.1 Wightman Fourier D 137 D.1 M D. M E 143 E.1 S E. SU() SU()Clebsch-Gordan E.3 S E.4 Clebsch-Gordan WignerD F 151 F.1 (Fradkin-Palchik ) G 155 G.1 Weyl H 159

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7 7 1 Planck Compton Planck Planck NASA Wilkinson (Wilkinson Microwave Anisotropies Probe, WMAP) (cosmic microwave background, CMB) 1 1 D. Spergel et al., Astrophys. J. Suppl. 148 (003) 175.

8 Planck WMAP CMB 1 WMAP GeV 1. Einstein Ricci Newton : 1. R. Utiyama and B. DeWitt, J. Math. Phys. 3 (196) B. DeWitt, in Relativity, Groups and Topology, eds. B. DeWitt and C. DeWitt (Gordon and Breach, New York, 1964); Phys. Rev. 160 (1967) 1113; Phys. Rev. 16 (1967) 1195, G. t Hooft and M. Veltman, Ann. Inst. Henri Poincare XX (1974) 69; M. Veltman, in Methods in Field Theory, Les Houches S. Weinberg, in General Relativity, an Einstein Centenary Survay, eds. S. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1979). 5. S. Deser, Proceedings of the Coference on Gauge Theories and Modern Field Theories, edited by R. Arnowitt and P. Nath (MIT Press, Cambridge, 1975). 6. S. Weinberg, Proceedings of the XVIIth International Conference on High Energy Physics, edited by J. R. Smith (Rutherford Laboratory, Chilton, Didcot, 1974).

9 Einstein (conformal field theory, CFT) 3 : 1. K. Stelle, Phys. Rev. D16 (1977) 953; Gen. Rel. Grav. 9 (1978) E. Tomboulis, Phys. Lett. 70B (1977) 361; Phys. Lett. 97B (1980) E. Fradkin and A. Tseytlin, Nucl. Phys. B01 (198) 469; Phys. Lett. 104B (1981) E. Fradkin and A. Tseytlin, Phys. Rep. 119 (1985) 33 [Review]. 4 : 1. V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) J. Distler and H. Kawai, Nucl. Phys. B31 (1989) F. David, Mod. Phys. Lett. A 3 (1988) N. Seiberg, Prog. Theor. Phys. Suppl. 10 (1990) 319 [Review]. 5. J. Teschner, Class. Quant. Grav. 18 (001) R153 [Review].

10 Weyl Einstein Planck (QCD) Planck 1.3 QCD Λ QCD Λ QG 5 Λ QG 5

11 Riemann Weyl Wheeler-DeWitt S Minkowski 6 (graviton) Planck Planck Planck Friedmann 6 S Einstein

12 1 1 de Sitter (inflaton) Friedmann

13 13 Minkowski (conformal field theory, CFT) Minkowski Minkowski Hamilton Hermite Euclid Euclid Minkowski ( ) Euclid Minkowski Euclid D 4.1 x µ x µ η µν dx µ dx ν η µν dx µ dx ν = Ω(x)η µν dx µ dx ν (.1.1) Ω Minkowski η µν = ( 1, 1,, 1)

14 14 Minkowski x µ x ν η µν x λ x σ = Ω(x)η λσ Ω = 1 Poincaré η µν η µν x µ x µ = x µ + ζ µ ζ µ µ ζ ν + ν ζ µ D η µν λ ζ λ = 0 Killing ζ λ Killing Ω = 1 + D λζ λ (.1.) Killing (η µν +(D ) µ ν ) λ ζ λ = 0 ζ µ (D +1)(D +)/ D (translation) D(D 1)/ Lorentz 1 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 δ λ µ x µ x λ (.1.3) Killing µ ζ ν + ν ζ µ = 0 Poincaré

15 .. 15 dilatation x µ x µ = λx µ, x µ x µ = x µ + a µ x 1 + a µ x µ + a x (conformal inversion) x µ x µ = xµ x x µ xµ x xµ x + aµ x µ x + a µ ( x µ x + a µ ) = x µ + a µ x 1 + a µ x µ + a x. Lorentz dilatation P µ M µν D K µ 1 (D + 1)(D + )/ SO(D, ) [P µ, P ν ] = 0, [M µν, P λ ] = i (η µλ P ν η νλ P µ ), [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 ] = i (η µν D + M µν ) (..1) Lorentz SO(D 1, 1) Poincaré Hermite P µ = P µ, M µν = M µν, D = D, K µ = K µ 1 dilatation D

16 16 Minkowski SO(D, ) 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 = ( 1, 1,, 1, 1) a, b = 0, 1,,, D, D+ 1 µ, ν = 0, 1,, D 1 M µν = J µν, D = J D+1D, P µ = J µd+1 J µd, K µ = J µd+1 + J µd Lorentz dilatation (..1) l O µ1 µ l Hermite O µ 1 µ l (x) = O µ1 µ l (x) O µ 1 µ l (x ) = Ω(x) l x ν 1 x xνl O µ 1 x µ ν1 ν l l (x) (..) SO(D 1, 1) D µν Jacobian xν x µ = Ω(x) 1/ D ν µ (x) O j (x) R[D] jk D = 4 Lorentz SO(3, 1) (j, j) j = j = l/ O µ 1 µ l = σ µ 1 α 1 α 1 σ µ l α l α l O α 1 α l α 1 α l σ µ α α σ µβdotb ε αβ ε α β j j (1/, 0) (1/, 0) (1, 1/) (1/, 1) Rarita-Schwinger (1, 0) (0, 1)

17 .. 17 O j(x ) = Ω(x) / R[D(x)] k j O k (x) 0 O 1 (x 1 ) O n (x n ) 0 = 0 O 1(x 1 ) O n(x n ) 0 (..3) 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 )/ Ω (.1.) (..) δ ζ O µ1 µ l (x) = (ζ ν ν + ) D νζ ν O µ1 µ l (x) + 1 l ( ) µj ζ ν ν ζ µj Oµ1 µ j 1νµ j+1 µ l (x) j=1 δ ζ O µ1 µ l (x) = i [Q ζ, O µ1 µ l (x)] Q ζ Killing ζ µ (D + 1)(D + )/ Killing ζ λ T,L,D,S(.1.3) i [P µ, O λ1 λ l (x)] = µ O λ1 λ l (x), i [M µν, O λ1 λ l (x)] = (x µ ν x ν µ iσ µν ) O λ1 λ l (x), i [D, O λ1 λ l (x)] = (x µ µ + ) O λ1 λ l (x), i [K µ, O λ1 λ l (x)] = ( x µ x µ x ν ν x µ + ix ν Σ µν ) Oλ1 λ l (x) Σ µν O λ1 λ l = i l ( ) ηµλj δ σ ν η νλj δ σ µ Oλ1 λ i 1σλ i+1 λ l j=1 (..4)

18 18 Minkowski Σ µν O λ1 λ l = (Σ µν ) λ 1 λ l σ 1 σ l O σ1 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 1 σ l λ 1 λ l = δ σ 1 λ 1 j=1 δ σ j 1 λ j 1 (Σ µν ) σ j λ j δ σ j+1 λ j+1 δ σ l λ l O 1/ Σ µν ψ = i 1 4 [γ µ, γ ν ]ψ {γ µ, γ ν } = η µν Killing Q ζ = d D 1 xζ λ T λ0 Killing T µ µ = µ T µν = 0 η Q ζ = 0 ζ λ ζ λ T,L,D,S(.1.3) P µ = d D 1 xt µ0, M µν = d D 1 x (x µ T ν0 x ν T µ0 ), D = d D 1 xx λ T λ0, K µ = d D 1 x ( ) x T µ0 x µ x λ T λ0 (..5) D.1 0 Q ζ (= Q ζ ) Q ζ 0 = 0 Q ζ = 0 n O j (j = 1,, n) 0 [Q ζ, O 1 (x 1 ) O n (x n )] 0 = 0

19 .3. Wightman 19 n δ ζ 0 O 1 (x 1 ) O n (x n ) 0 = i 0 O 1 (x 1 ) [Q ζ, O j (x j )] O n (x n ) 0 = 0 j=1 (..3) O j j Q ζ D K µ (..4) ( ) n x µ j j=1 x µ + j 0 O 1 (x 1 ) O n (x n ) 0 = 0, j ( ) n x j j=1 x µ x jµ x ν j j x ν j x jµ 0 O 1 (x 1 ) O n (x n ) 0 = 0 j.3 Wightman l Wightman W µ1 µ l,ν 1 ν l (x y) = 0 O µ1 µ l (x)o ν1 ν l (y) 0 (.3.1) 1 W µ1 µ l,ν 1 ν l (x) = cp µ1 µ l,ν 1 ν l (x) (x ) x 0 x 0 iϵ ϵ UV P µ1 µ l,ν 1 ν l [ B ] Wightman O(x)O(0) 0 = c (x ) = c x 0 x 0 iϵ (x + iϵx 0 ) x 0 0 ϵ 1

20 0 Minkowski Wightman 1 0 O µ (x)o ν (0) 0 = ci µν (x ) x 0 x 0 iϵ 0 O µν (x)o λσ (0) 0 = c 1 ( I µλ I νσ + I µσ I νλ ) 1 D η µνη λσ (x ) x µ I µν I µν = η µν xµ x ν x x 0 x 0 iϵ I λ µ I λν = η µν I µ µ = D l P µ1 µ l,ν 1 ν l = 1 l! (I µ 1 ν 1 I µl ν l + perms) traces perms traces c ( ) c > 0 c = 1 Wightman (.3.1) f 1, (x) : (f 1, f ) = d D xd D yf µ 1 µ l 1 (x)w µ1 µ l,ν 1 ν l (x y)f ν 1 ν l (y). Wightman Fourier W µ1 µ l,ν 1 ν l (k) = d D xw µ1 µ l,ν 1 ν l (x)e ik µx µ (f 1, f ) = d D k (π) D f µ 1 µ l 1 (k)f ν 1 ν l (k)w µ1 µ l,ν 1 ν l (k) f 1, (k) Fourier k (f, f) > 0

21 .4. Fourier 1 Wightman s D 1 for s = 0, D + s for s 0 (.3.) (unitarity bound).4 Fourier D = 4 Wightman W (x) Fourier [ C ] π( 1) W (k) = (π) 4 1 Γ( ) θ(k0 )θ( k )( k ) (f, f) = d 4 k f(k) W (k)/(π) 4 1 = 1 lim 1 ( 1)θ( k ) = δ( k ) 1 (π) lim (f, f) = 1 d 4 k (π) 4 f(k) πθ( k 0 )δ( k ) = d 3 k 1 (π) 3 k f(k) A µ 0 A µ (x)a ν (0) 0 ( = η µν α x ) µx ν 1 x (x ) x 0 x 0 iϵ = 1 { α ( 1)( ) η µν α } 1 µ ν 1 (x ) 1 x 0 x 0 iϵ

22 Minkowski Fourier Fourier W (α) µν W µν (α) (k) = (π) π( 1) 4 1 Γ( )Γ( + 1) θ(k0 )θ( k )( k ) { ( α)η µν α( ) k } µk ν k O µ α = 1 W (1) µν = W µν α = A µ O µ O O = 1 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 ) { [( 3)α ] f 0 + ( α) f j } K ( ) C = 4(π) 3 ( 1)/4 Γ( )Γ( + 1) ( 3)α 0 α 0 α 3 α 1 α = = 3 µ O µ = 0 µ W µν (x) = 0 (x 0) 3 1 F µν ( )

23 O P µ µ ν O O (descendant) O D = 4 O µ O O 0 O(x) O(0) 0 = 16 1 ( + 1)( 1) (x ) + x 0 x 0 iϵ O > 1 = 1 O = 0 { 0 µ O(x) ν O(0) 0 = η µν ( + 1) x } µx ν 1 x (x ) +1 x 0 x 0 iϵ Wightman 1 O µ µ O µ 0 µ O µ (x) ν 1 O ν (0) 0 = 4( 1)( 3) (x ) +1 x 0 x 0 iϵ > 3 = 3 O µ µ O µ = 0

24 4 Minkowski µ O µν 0 µ O µν (x) λ O λσ (0) 0 = ( 4)(4 7) { η νσ } x ν x σ x 1 (x ) +1 x 0 x 0 iϵ µ ν O µν 0 µ ν O µν (x) λ σ O λσ (0) 0 1 = 4 ( 1)( 3)( 4) (x ) + x 0 x 0 iϵ 4 Wightman = 4 O µν µ O µν = 0.6 Feynman Feynman Wightman 0 T [O µ1 µ l (x)o ν1 ν l (0)] 0 = θ(x 0 ) 0 O µ1 µ l (x)o ν1 ν l (0) 0 + θ( x 0 ) 0 O ν1 ν l (0)O µ1 µ l (x) 0 Fourier 0 T [O µ1 µ l (x)o ν1 ν l (0)] 0 = d 4 k µxµ eik D (π) 4 µ1 µ l,ν 1 ν l (k) 0 T [O(x)O(0)] 0 = θ(x 0 1 ) (x + iϵx 0 ) + 1 θ( x0 ) (x iϵx 0 ) = 1 (x + iϵ)

25 .6. Feynman 5 ϵ x 0 ϵ Fourier Γ( ) D(k) = i(π) 4 1 Γ( ) (k iϵ) 0 T [O µ (x)o ν (0)] 0 = 1 { 1 ( ) η µν 1 1 µ ν } 1 (x + iϵ) 1 Fourier D µ,ν (k) = i (π) Γ( ) 4 1 Γ( + 1) { ( 1)ηµν k ( )k µ k ν } (k iϵ) 3 O f I int = g d 4 x (fo + H.c.) S S = 1 + it f f i f T f = g = g d 4 xf (x) d 4 k (π) 4 f (k)f(k)d(k) d 4 yf(y) 0 T [O(x)O(y)] 0 S S = 1 Im(T ) = T 0 Im f T f = g d 4 k (π) 4 f(k) Im {id(k)} 0 (x+iϵ) λ (x iϵ) λ = i sin(πλ)θ( x)( x) λ sin(πλ) = π/γ(λ)γ(1 λ) π( 1) Im {id(k)} = (π) 4 1 Γ( ) θ( k )( k ) Wightman Fourier [θ(k 0 ) 1/ ] 1

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27 7 3 Euclid Euclid Minkowski Euclid 3.1 Ising D D Euclid 1 Euclid T T c T T c O(x)O(0) e x /ξ ξ T = T c ξ O(x)O(0) = 1 x S CFT 1 D Minkowski D 1

28 8 3 Euclid < D relevant O t = (T T c )/T c ( 1) S CFT S CFT + ta D d D xo(x) a ξ D ta D ξ at 1/(D ) O = ε relevant ε ν ξ at ν ν = 1/(D ε ) 3. Euclid Euclid R D SO(D + 1, 1) η µν δ µν M D (..1) (..4) P µ D Hermite P µ = K µ, D = D SO(D, ) J ab η ab = ( 1, 1,, 1, 1) D +

29 3.. Euclid 9 a, b = 0, 1,, D, D + 1 D Euclid µ, ν = 1,, D SO(D + 1, 1) 0 M µν = J µν, D = ij D+10, P µ = J µd+1 ij µ0, K µ = J µd+1 + ij µ0 J ab (..) Hermite Euclid Hermite l O µ1 µ l (x)o ν1 ν l (y) = cp µ1 µ l,ν 1 ν l 1 (x ) P µ1 µ l,ν 1 ν l x M D Euclid I µν I µν = δ µν x µx ν x P µ1 µ l,ν 1 ν l = 1 l! (I µ 1 ν 1 I µl ν l + perms) traces c c = 1 Hermite x µ Rx µ = x µ x O µ 1 µ l (x) = 1 (x ) I µ 1 ν 1 (x) I µl ν l (x)o ν1 ν l (Rx) (3..1) Hermite Hermite i[p µ, O(x)] = µ O(x)

30 30 3 Euclid Hermite i[p µ, O (x)] = µ O (x) Hermite y µ = x µ /x O (x) = (y ) O(y) Hermite P µ = K µ i(y ) [K µ, O(y)] = y ν { (y ) O(y) } x µ y ν = (y ) ( y µ y µ y ν ν y µ ) O(y) (y ) 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 ) I νλ [K µ, O λ (y)] = y ν { (y ) I νλ O λ (y) } x µ y ν ( ) = (y ) {I νλ y µ y µ y σ σ y µ Oλ (y) ( + δ µν y λ δ µλ y ν + 4 y ) } µy ν y λ O y λ (y) I µλ I λν = δ µν i[k µ, O λ (y)] = (y µ y µ y σ σ y µ +iy σ Σ µσ )O λ (y) iσ µσ O λ = δ µλ O σ + δ λσ O µ O µ1 µ l O µ 1 µ l Rx = 1/x O (x)o(0) = 1 (x ) O(Rx)O(0) = 1 x (c = 1 ) I µλ I λν = δ µν O µ(x)o ν (0) = δ µν, O µν(x)o λσ (0) = 1 (δ µλ δ νσ + δ µσ δ νλ ) D δ µνδ λσ

31 3.. Euclid 31 Euclid M µν {µ 1 µ l }, = (Σ µν ) ν1 ν l,µ 1 µ l {ν 1 ν l },, id {µ 1 µ l }, = {µ 1 µ l },, K µ {µ 1 µ l }, = 0 {µ 1 µ l }, = O µ1 µ l (0) 0 (3..) (state-operator correspondence) 0 P µ Hermite y µ = Rx µ I µν (x) = I µν (y) Hermite O µ 1 µ l (0) = lim x 0 (x ) I µ1 ν 1 I µl ν l O ν1 ν l (Rx) = lim y (y ) I µ1 ν 1 I µl ν l O ν1 ν l (y) (3..) {µ 1 µ l }, = 0 O µ 1 µ l (0) = lim x (x ) I µ1 ν 1 I µl ν l 0 O ν1 ν l (x) f µ1 µ l (f, f) = f µ 1 µ l f ν1 ν l {µ 1 µ l }, {ν 1 ν l }, = f µ1 µ l > 0 Minkowski = O(0) 0 M D O (x) = O(x) = 0 O (0)O(0) 0 = 0 O(0)O(0) 0

32 3 3 Euclid 3.3 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 (x ) O(x) Hermite (3..1) O(x)O(x ) = 1 (x ) O (Rx)O(x ) = 1 (x ) e ik µ(rx) µ e ip νx ν = O(0) 0 = 0 O( ) K µ P ν O(x)O(x ) = 1 ( x C (x ) n (x, x ) x n=0 C n ) n/ C n = 1 (n!) x µ1 x µn x ν 1 x ν n (x x ) n/ K µ1 K µn P ν1 P νn Gegenbauer nc n = ( + n 1)zC n 1 ( + n )C n z = x x / x x C n z Gegenbauer ( = 1/ Legendre ) 1 (1 zt + t ) = C n (z)t n n=0

33 3.4. (OPE) 33 z t = x /x O(x)O(x ) = 1/(x x ) 3.4 (OPE) ϕ (operator product expansion, OPE) ϕ ϕ I + T µν + l=0,,4, O µ1 µ l I T µν ( D ) O µ1 µ l l OPE ( ) ϕ l d ϕ(x 1 )ϕ(x ) = O µ1 µ l (x 1 )O ν1 ν l (x ) = 1 x 1, d [ ] 1 1 x 1 l! (I µ1ν1 I + perms) traces µlνl (x 1 ) µ = x 1µ x µ I µν = I µν (x 1 ) f,l ϕ(x 1 )ϕ(x )O µ1 µ l (x 3 ) = Z µ = (x 13) µ x 13 f,l x 1 d +l x 13 l x 3 l (Z µ 1 Z µl traces), (x 3) µ x 3 f,l OPE ϕ OPE ϕ(x)ϕ(y) = = 1 x y d + l=n 1 x y d + l=n f,l [ (x y)µ1 (x y) µl x y d +l O µ1 µ l (y) + f,l x y d C,l(x y, y )O,l (y) (3.4.1) ]

34 34 3 Euclid l O,l (y) C,l (x y, y ) l = 0 C,0 OPE O = O,0 ϕ(x)ϕ(y)o(z) = f,0 x y d C,0(x y, y ) O(y)O(z) 1 C,0 (x y, y ) y z = 1 x z y z Feynmann Γ( ) 1 [t(1 t)] 1 Γ( )Γ( ) dt 0 [t(x z) + (1 t)(y z) ] = 1 1 B(, ) dt[t(1 t)] 1 0 n=0 ( ) n n! [ t(1 t)(x y) ] n ([y z + t(x y)] ) +n (a) n = Γ(a + n)/γ(a) Pochhammer ( ) n 1 (x ) = 1 4n ( ) n ( + 1 D/) n (x ), +n 1 [(y + tx) ] = 1 etx y (y ) 1/ y z y C,0 (x y, y ) = 1 1 B(, ) dt[t(1 t)] 1 0 ( 1) n [t(1 t)a ] n ( 4 n n! ( + 1 D/) y) n e ta y n n=0 a=x y C,0 (x y, y ) = (x y) µ µ y + + 8( + 1) (x y) µ(x y) ν µ y ν y 16( + 1)( + 1 D/) (x y) y +

35 3.5. Conformal Blocks 35 (3.4.1) l Conformal Blocks j ϕ j ϕ 1 (x 1 )ϕ (x )ϕ 3 (x 3 )ϕ 4 (x 4 ) = ( ) 1 ( ) 34 x4 x14 G(u, v) x 14 x 13 x 1 1+ x ij = i j u v u = x 1x 34, v = x 14x 3 x 13x 4 x 13x 4 ϕ 1 ϕ OPE ϕ 1 ϕ 4 OPE (x, ) (x 4, 4 ) (x, ) (x 3, 3 ) (crossing symmetry) G(u, v) G(v, u) G(1/u, v/u) 1 = = 3 = 4 OPE G(u, v) = 1 +,l f,lg,l (u, v) d ϕ d ϕ d (x 1 )ϕ d (x )ϕ d (x 3 )ϕ d (x 4 ) = 1 [ 1 + f x 1 d x 34,lg d,l (u, v) ],l

36 36 3 Euclid g,l (u, v) conformal block x x 4 v d G(u, v) = u d G(v, u) conformal block u d v d =,l f,l [ v d g,l (u, v) u d g,l (v, u) ] (3.5.1) Conformal block g,l OPE (l = 0) OPE g,0 (u, v) = x 1 x 34 1 C,0 (x 1, )C,0 (x 34, 4 ) x 4 1 C,0 (x 1, )C,0 (x 34, 4 ) x 4 = 1 B(, dtds[t(1 t)s(1 s)] 1 ) 0 ( 1) n+m ( ) n+m ( ) n+m [t(1 t)x 1] n [s(1 s)x 34] m n!m! ( ) n ( ) m [(x 4 + tx 1 sx 34 ) ] +n+m n,m=0 = + 1 D/ A = t(1 t)x 1 B = s(1 s)x 34 (x 4 + tx 1 sx 34 ) = Λ A B, Λ = tsx 13 + t(1 s)x 14 + s(1 t)x 3 + (1 t)(1 s)x 4 1 B(, ) dtds [t(1 t)s(1 s)] 1 (Λ A B ) F 4(, ;, ; X, Y ) X = A /(Λ A B ) Y = B /(Λ A B ) F 4 Appell (double series) F 4 (a, b;, c, d; x, y) = n,m=0 1 n!m! (a) n+m (b) n+m x n y m (c) n (d) m

37 3.5. Conformal Blocks 37 Gauss F 1 ( a F 4 (a, b; c, d; x, y) = (1 x y) a F 1, a + 1 ) ; b; 4xy (1 x y) 1 B(, ) 1 0 dtds [t(1 t)s(1 s)] 1 ( a F (Λ ) 1, a + 1 ; b; 4A B ) Λ t s t a 1 (1 t) b 1 dt [tα + (1 t)β] = 1 B(a, b), a+b α a βb s a 1 (1 s) b 1 ds (1 sα) c (1 sβ) = B(a, b)f 1(a, c, d; a + b; α, β) d F 1 F 1 (a, b, c; d; x, y) = n,m=0 1 n!m! (a) n+m (b) n (c) m x n y m (d) n+m Gauss ( F 1 (a, b, c, b + c; x, y) = (1 y) a F 1 a, b; b + c; x y ) 1 y 1 x 13 x 4 v n=0 u n n! ( ) 4 n ( ) n ( ) n F 1 ( + n, + n; + n; 1 v ) u = u/v v = 1/v ( ) n 4 n ( ) n( +1 ) n = ( ) n 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) n+m x n y m g,0 ( + n) m = ( ) n+m/( ) n ( + n) m = ( ) n+m /( ) n x 3 x 4

38 38 3 Euclid g,l (u, v) = g,l (u, v ) g,0 u v u v ( g,0 (u, v) = u G,, + 1 D ), ; u, 1 v conformal block g,l Gauss u = z z, v = (1 z)(1 z) G(a, b, c 1, c; u, 1 v) = 1 [ z F 1 (a, b; c; z) F 1 (a 1, b 1; c ; z) z z z F 1 (a, b; c; z) F 1 (a 1, b 1; c ; z) ] Gauss k β (x) = x β F 1 ( β, β, β; x ) (3.5.) D = 4 g,0 (u, v) D=4 = z z z z [k (z)k ( z) (z z)] l 1 OPE l conformal blocks l D = 4 g,l (u, v) D=4 = ( 1)l l z z z z [k +l(z)k l ( z) (z z)] (3.5.3) g,l (u, v) D= = ( 1)l l [k +l (z)k l ( z) + (z z)] (3.5.4) D = 3 l z = z

39 3.6. Casimir Conformal Blocks Casimir Conformal Blocks conformal block Casimir SO(D + 1, 1) J ab Casimir C = 1 J ab J ab C = 1 M µνm µν D 1 (K µp µ + P µ K µ ) l C, l = C,l, l, C,l = ( D) + l(l + D ) ϕ 1 (x 1 )ϕ (x )ϕ 3 (x 3 )ϕ 4 (x 4 ) =,l 0 ϕ 1 (x 1 )ϕ (x ), l, l ϕ 3 (x 3 )ϕ 4 (x 4 ) [ J ab, [ J ab, ϕ 1 (x 1 )ϕ (x ) ]], l = 0 ϕ 1 (x 1 )ϕ (x )C, l = C,l 0 ϕ 1 (x 1 )ϕ (x ), l { (x 1 1 µ µ (x 1 ) µ (x 1 ) ν 1 µ ν 1 (x 1 ) µ µ + (x 1 ) µ 1 µ + ( 1 + ) ( 1 + D) } 0 ϕ 1 (x 1 )ϕ (x ), l O,l conformal block g,l 0 ϕ 1 (x 1 )ϕ (x ), l, l ϕ 3 (x 3 )ϕ 4 (x 4 ) 0 ( ) x 1 / ( = 4 x ) 34 / 14 f,lg,l (u, v) (x 1) ( 1+ )/ (x 34) ( 3+ 4 )/ x 14 x 13

40 40 3 Euclid conformal block Dg,l (u, v) = 1 C,lg,l (u, v), D = (1 u + v)u ( u ) + [ (1 v) u(1 + v) ] ( v ) u u v v (1 + u v)uv u v Du u + 1 [ ( ( 1 34 ) (1 + u v) u u + v ) (1 u v) ] v v (1 + u v) z z D = z (1 z) z + z (1 z) z + 1 ( ( 1 34 ) z z + ) z z (z + z) + (D ) z z ( (1 z) z z z (1 z) ) z D = 4, (3.5.3) (3.5.4) (3.5.) ( k β (x) = x β β F 1 1, β + ) 34 ; β; x 3.7 (.3.) D = 4 µ,, µ ν, = δ δ µν

41 µ; ν, = P µ ν, µ; λ, ν; σ, = λ, [K µ, P ν ] σ, = λ, i (Dδ µν + M µν ) σ, = δ ( δ µν δ λσ δ µλ δ νσ + δ νλ δ µσ ) 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) a b 3 ( 3) ( 1) ( + 1) 3 SO(4) {r} {r}, M µν {r}, = (Σ µν ) {r },{r} {r },, id {r}, = {r},, K µ {r}, = 0 SO(4) SU() SU() SU() j 1, j {r} (j 1, j ) (j 1 + 1)(j + 1) l O µ1 µ l j 1 = j = l/ P µ n n µ 1 µ n ; {r}, = P µ1 P µn {r}, {r }, {r}, = δ {r }{r}δ

42 4 3 Euclid µ; {r}, = P µ {r}, ( µ; {r }, ν; {r}, = δ δ{r }{r} + {r }, im µν {r}, ) (3.7.1) Lorentz im µν = i 1 (δ µαδ νβ δ µβ δ να ) M αβ = 1 (Σ αβ) µν M αβ Σ αβ µ µ M {v} αβ ν = (Σ αβ) µν {r }, im µν {r}, = µ {r }, M {v} αβ M {r} αβ {r}, ν (3.7.) M {R} αβ M {v} αβ = M {v} αβ + M {r} αβ M {r} αβ = 1 M {R} αβ M {R} αβ 1 M {v} αβ M {v} αβ 1 M {r} αβ M {r} αβ = c ({R}) c ({v}) c ({r}) c SO(4) Casimir {r} SU() SU() (j 1, j ) {v} (1/, 1/) {R} (J 1, J ) J 1, j 1, ± 1/ (3.7.) Casimir J 1 (J 1 + 1) + J (J + 1) 3 j 1 (j 1 + 1) j (j + 1) (3.7.1) (3.7.1) j 1, j 0 (3.7.1) J 1 = j 1 1/, J = j 1/ (j 1 + j + ) j 1 + j + for j 1, j 0

43 3.8. Conformal Bootstrap 43 j 1 = j = l/ l l + j 1 = 0, j 0 J 1 = 1/, J = j 1/ (j + 1) j + 1 for j 1 = 0, j 0 j 1 j j 1 = j = 0 0 P µ P µ K µ K µ P ν P ν = 3 ( 1)δ 1 for j 1 = j = Conformal Bootstrap 3.5 (3.5.1) conformal block g,l p,l F d,,l (z, z) = 1,,l F d,,l (z, z) = vd g,l (u, v) u d g,l (v, u) u d v d (3.8.1) p,l = f,l (d = 1) l p,l = δ,l+ δ l,n l+1 (l!) /(l)!

44 44 3 Euclid OPE f,l p,l 0 OPE D = 4 d OPEϕ d ϕ d 1 + O (d 1) 1/ +.1(d 1) (d 1) 3/ + o((d 1) )(3.8.) D = Ising ϕ d σ O ε d = σ = 1/8 = ε = 1 d = 1/8 OPE 1 = 1 D = 3 Ising z = 1/ + X + iy X Y N Λ[F ] = m,n=even m+n N λ m,n m X n Y F X=Y =0

45 3.8. Conformal Bootstrap 45 X = Y = 0 (z = z = 1/) (3.8.1) p,l Λ [F d,,l ] = 0 (3.8.3),l l Λ[F d,,l ] 0 p,l 0 OPE ϕ d ϕ d 1 + f O + l>0 l=even O,l D +l O f l > 0 d f f (l = 0) D + l (l > 0) Λ[F d,,l ] 0 λ m,n p,l 0,l p,l Λ [F d,,l ] 0 (3.8.3) d f d f d f f c (d) D/ 1 f c (d) l l (linear programing method) λ m,n (3.8.) OPE O

46 46 3 Euclid f ( f c ) = f c Ising σ = 0.518(3) ε = 1.413(1) OPE ( ) 3.9 Wilson-Fisher 3 D = 4 ϵ 4 S = [ ] 1 d D x ( ϕ) + λϕ 4 β λ = ϵλ + 9λ π ϵ 0 λ = ϵπ /9 ϕ : ϕ : γ = 3λ /16π 4 δ = 3λ/π ϕ = D + 3λ (1 16π = ϵ ) + ϵ ϕ = D + 3λ π = ( ϵ) + ϵ 3 Ising OPEσ σ ε ϕ ϕ ϕ σ = ϕ ε = ϕ Press. 3 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ.

47 Ising ϵ 1 σ = 0.51 ε = o(ϵ 5 ) σ = ε = 1.410

48

49 49 4 R S 3 R S 3 R S 3 Euclid Minkowski 4.1 R S 3 R 4 R S 3 R 4 x µ r x µ x µ = r S 3 X µ X µ = 1 X µ = x µ /r R 4 ds R = dx 4 µ dx µ ds R = 4 dr + r dx µ dx µ = e ( ) τ dτ + dx µ dx µ = e τ ds R S 3 τ = log r R 4 R S 3 ds R S 3 R 4 R S 3 x µ (r, X µ ) O(x) = O(r, X) (r, X µ ) (τ, X µ ) O(x) = e τ O(τ, X) i [P µ, O(τ, X)] = e τ i [P µ, O(x)] = e τ µ O(x)

50 50 4 R S 3 ( τ = e τ x µ τ + X ) ν e τ O(τ, X) x µ X ν = e τ {X µ τ + (δ µν X µ X ν ) / X ν X µ } O(τ, X) i [K µ, O(τ, X)] = e τ ( x µ x µ x ν ν x µ ) O(x) (4.1.1) = e τ { X µ τ + (δ µν X µ X ν ) / X ν X µ } O(τ, X) Dilatation Lorentz (4.1.) i [D, O(τ, X)] = τ O(τ, X), ( ) i [M µν, O(τ, X)] = X µ X ν O(τ, X) (4.1.3) X ν X µ R S 3 Dilatation r = e τ τ = 0 O(0, X) O (0, X) = O(0, X) Hermite Minkowski τ O(τ, X) = e iτd O(0, X)e iτd D = D Hermite O (τ, X) = e iτd O(0, X)e iτd = O( τ, X) S 3 Euler ˆx j = (α, β, γ) [0, π], [0, π], [0, 4π] S 3 dx µ dx µ = ˆγ ij dˆx i dˆx j = 1 4 X µ Euler ( dα + dβ + dγ + cos βdαdγ ) X 0 = cos β cos 1 (α + γ), X 1 = sin β sin 1 (α γ), X = sin β cos 1 (α γ), X 3 = cos β sin 1 (α + γ)

51 4.1. R S 3 51 S 3 ( ) ˆγ ij X µ ˆγ ij = X µ ˆx i X ν ˆx j δ µν (4.1.4) R S 3 S 3 Y JM 3 = ˆ j ˆ j S 3 Laplace 3 Y JM = J(J + )Y JM S 3 SO(4) = SU() SU() (J, J) M = (m, m ) m, m = J, J 1, J Wigner D Y JM = J + 1 V 3 D J mm dω 3 Y J 1 M 1 Y J M = δ J1 J δ M1 M dω 3 = ˆγd 3ˆx = sin βdαdβdγ/8 S 3 V 3 = dω 3 = π Y JM ϵ M = ( 1) m m Y JM = ϵ M Y J M M δ MN = δ mm δ nn J = 1/ Wigner D J = 1/ X µ D 1 mm = X0 + ix 3 X + ix 1 X + ix 1 X 0 ix 3 = (T µ ) M X µ T µ (T µ) M = ϵ M (T µ ) M (T µ) M (T µ ) N = δ MN, (Tµ) M (T ν ) M = δ µν M

52 5 4 R S 3 1 J = 1/ V3 Y 1 M = (T µ) M X µ : V 3 Y 1 4 MY 1 M = 1, V 3 ˆ i Y M 4 ˆ 1 M j Y 1 M = ˆγ ij, M V 3 4 ˆ i Y ˆ 1 M j Y 1 N = δ MN V 3 4 Y 1 MY 1 N. (4.1.5) M Y 1/M ˆ j Y 1/M = 0 X µ X µ = 1 (4.1.4) ˆγ ij X µ ˆx i X ν ˆx j = δ µν X µ X ν (4.1.6) X µ dx µ = 0 T µ T ν (4.1.5) (4.1.6) ˆγ ij X µ ˆx j = (δ µν X µ X ν ) ˆxi X ν (4.1.7) V3 ˆγij ˆx j Y 1 M = (T µ) M (δ µν X µ X ν ) ˆxi X ν (T µ ) M H = id, R MN = i(t µ) M (T ν ) N M µν, Q M = i(t µ) M K µ, Q M = i(t µ ) M P µ 1 T µ I Pauli σ i (T 0 ) M = 1 (I) M (T j ) M = i (σ j ) M Tµ Hermite M = (m, m )

53 4.1. R S 3 53 H Hermite Q M Q M Hermite S 3 R MN = R NM R MN = ϵ M ϵ N R N M [ QM, Q ] N = δ MN H + R MN, [H, Q M ] = Q M, [H, R MN ] = 0, [Q M, Q N ] = 0, [Q M, R NL ] = δ ML Q N ϵ N ϵ L δ M N Q L, [R MN, R LK ] = δ MK R LN ϵ M ϵ N δ NK R L M δ NL R MK + ϵ M ϵ N δ ML R NK (4.1.8) R MN Hamilton H Q M 1( 1) SU() SU() 4 M = {( 1, 1), ( 1, 1), ( 1, 1), ( 1, 1)} {1,, 3, 4} A + = R 31 A = R 31 A 3 = 1(R 11+R ) B + = R 1 B = R 1 B 3 = 1(R 11 R ) R MN SU() SU() [A +, A ] = A 3, [A 3, A ± ] = ±A ±, [B +, B ] = B 3, [B 3, B ± ] = ±B ± A ±,3 B ±,3 Dilatation Lorentz (4.1.3) H R MN i [H, O(τ, ˆx)] = i τ O(τ, ˆx), i [R MN, O(τ, ˆx)] = ρ µ MN ˆ µ O(τ, ˆx) ˆ µ = ( τ, ˆ j ) ĝ µν = (1, ˆγ ij ) R S 3 Killing υ µ = (i, 0, 0, 0) ρ µ MN = (0, ρ j MN) ρ j MN = i V 3 4 ( Y 1 M ˆ j Y 1 N Y 1 N ˆ j Y 1 M ) (4.1.9)

54 54 4 R S 3 ρ j MN S3 Killing ˆ i ρ j MN + ˆ i ρ j MN = 0 Killing M, N ρ j MN = ρ j NM ρj MN = ϵ M ϵ N ρ j N M Q M Hermite Q M K µ P µ (4.1.) (4.1.1) i [Q M, O(τ, ˆx)] = ρ µ M ˆ µ O(τ, ˆx) + 4 ˆ µ ρ µ MO(τ, ˆx), i [ Q M, O(τ, ˆx) ] = ρ µ M ˆ µ O(τ, ˆx) + 4 ˆ µ ρ µ M O(τ, ˆx) ρ µ M ρ µ M = ( ( ) ρ 0 M, ρ j ) V3 M = i eτ Y V3 1 M, i eτ ˆ j Y 1 M EuclidR S 3 Killing ˆ µ ρ ν M + ˆ ν ρ µ M = ĝ µν ˆ λ ρ λ M/ Killing ρ µ M Killing ρ0 M(τ, ˆx) = ρ 0 M( τ, ˆx) ρ j M(τ, ˆx) = ρ j M( τ, ˆx) Dilatation H S 3 R MN Q M Q M Killing υ µ ρ µ MN ρµ M ρµ M R 4 dx µ dx µ dr + ˆγ ij d x i d x j O µ (x)dx µ = O 0 (r, x)dr+ O j (r, x)d x j d x j O µ (x)dx µ = rdˆx j τ = log r = e τ {O 0 (r, x)dτ + O j (r, x)dˆx j } (r, x j ) ds R S 3 (τ, ˆx j ) O µ (r, x) = e τ O µ (τ, ˆx) { O µ (x) = e τ e τ O 0 (τ, ˆx) τ } + O j (τ, ˆx) ˆxj x µ x µ { = e τ X µ O 0 (τ, ˆx) + ˆγ jk X } µ ˆx O j(τ, ˆx) k (4.1.10)

55 4.1. R S 3 55 dr = X µ dx µ (4.1.7) τ/ x µ = e τ X µ ˆx j = X ν ˆx j = e τ (δ µν X µ X ν ) ˆxj = ˆγ jk X µ x µ x µ X ν X ν ˆx k Hermite (3..1) O µ(τ, ˆx) = ( O 0 ( τ, ˆx), O j ( τ, ˆx)) EuclidR S 3 O λ1 λ l (τ, ˆx) 15 Killing {υ µ, ρ µ MN, ρ µ M, ρ µ M} ρ µ {H, R MN, Q M, Q M} Q ρ i [Q ρ, O λ1 λ l ] = (ρ λ ˆ λ + 4 ˆ ) λ ρ λ O λ1 λ l + 1 l ( ˆ λi ρ λ ˆ ) λ ρ λi Oλ1 λ i 1λλ i+1 λ l (4.1.11) i=1 h MN Ê MÊν µ N = ĝ µν, M,N ( Ê µ M = (T µ ) M X µ, γ jk X ) µ = ˆx k Ê µ MʵN = h MN ( ) V3 Y V3 1 M, ˆ j Y 1 M h MN = ϵ M δ MN Ê µ M = ϵ M Ê µ M = N h MN Ê λ N M ʵ M Êν N = µν ĝ ʵ M ʵN = δ MN O M1 M l = Êλ 1 M 1 Êλ l M l O λ1 λ l Hermite (3..1) O M 1 M l (τ, ˆx) = I M1 N 1 I Ml N l ϵ N1 ϵ Nl O N1 N l ( τ, ˆx) N 1, N l (4.1.10) O M (τ, ˆx) = ʵ M O µ(τ, ˆx) = e τ (T µ ) M O µ (x) O µ (x) O M (τ, ˆx)

56 56 4 R S 3 I MN = (T µ) M (T ν ) N (δ µν X µ X ν ) = δ MN V 3 4 Y 1 MY 1 N N I MN I NL = δ ML 3 O M1 M l (τ, ˆx) 4 i [H, O M1 M l ] = i τ O M1 M l, i [R MN, O M1 M l ] = (ρ µ MN µ + iσ MN ) O M1 M ( l i [Q M, O M1 M l ] = ρ µ M µ + 4 ˆ µ ρ µ M + 1 ) ˆ µ ρ µ NΣ MN N i [ Q ] ( M, O M1 M l = ρ µ M µ + 4 ˆ ) µ ρ µ M O M1 M l O M1 M l µ = ( τ, / ˆx j ) τ Euler l Σ MN O M1 M l = (Σ MN ) Ni M i O M1 M i 1 N i M i+1 M l i=1 N i (Σ MN ) KL = δ ML δ NK ϵ K ϵ L δ M K δ N L Σ MN R MN R MN M,N h MN A M B N (= M,N h MN A MB N) M,N h MN A MB N h MN 3 I MN = I NM N I MN j I NL = iρ jmn L,K I MKI LN ρ jlk = ρ jmn 4 R MN Êλ M ˆ 0 O λ = τ O M Êλ M ˆ j O λ = j O M + i N ρ jnm O N iρ j MN ρ jkl + Êj ˆ L j ρ k MN Ê kk = i(σ MN ) KL iρ jnm (4.1.1) Q M Q M ÊN λ ( ˆ λ ρ σ M ˆ σ ρ λm )Ê σl = iρj M ρ jln Êλ N ( ˆ λ ρ σ M ˆ σ ρ λm )Ê σl = i ρj M ρ jln iρ j M ρ jln = δ MN ρ 0 L ϵ Lδ M L ϵ N ρ 0 N i ρj M ρ jln = δ ML ρ 0 N ϵ N δ M N ϵ L ρ 0 L ρ 0 M = ˆ µ ρ µ M /4 ρ0 M = ˆ µ ρ µ M /4

57 4.. Minkowski R S 3 57 R S 3 ˆω µmn = Êλ M ˆ µ Ê λn = (0, iρ jnm ) (4.1.1) ˆ µ O M = µ O M + N ˆω µmn O N Q M Q M i [Q M, O M1 M l ] = (ρ µm ˆ µ + 4 ˆ µ ρ µm ˆ µ ρ µnσ ) MN O M1 M l i [ Q ] M, O M1 M l = ( ρ µm ˆ µ + 4 ˆ µ ρ µm ˆ µ ρ µnσ ) MN O M1 M l (3..) {µ 1 µ l }; = lim τ e τ O µ1 µ l (τ, ˆx) 0 (4.1.13) {M 1 M l } H, {r} =, {r}, R MN, {r} = {r } (Σ MN ) {r },{r}, {r }, Q M, {r} = 0 (4.1.14) {r} SU() SU() Σ MN, {r} Q M 4. Minkowski R S 3 MinkowskiR S 3 M 4 Euclid Wick

58 58 4 R S 3 MinkowskiR S 3 Killing ˆ µ ζ ν + ˆ ν ζ µ ĝ µν ˆ λ ζ λ / = 0 15 Killing Killing 3 η ζ 0 + ψ = 0, η ζ i + ˆ i ζ 0 = 0, ˆ i ζ j + ˆ j ζ i 3 ˆγ ijψ = 0 (4..1) ψ = ˆ i ζ i ψ ( 3 +3)ψ = 0 ( η + 1)ψ = 0 (4..1) ˆ j ˆ i Killing ψ = 0 ψ e ±iη Y 1 M ψ = 0 η ζ 0 = 3 ζ 0 = 0 S 3 Killing ˆ i ζ j + ˆ j ζ i = 0 η µ = (1, 0, 0, 0) ζ 0 = 0 η ζ i = 0 S 3 Killing ζ µ MN = ρ µ MN = (0, ρ j MN) S 3 Killing ρ j MN (4.1.9) ψ 0 Killing ζ µ M = ( ( ) ζm, 0 ζ j ) V3 M = eiη Y V3 1 M, i eiη ˆ j Y 1 M ζ µ M Euclid R S 3 Killing ρ µ τ = iη Wick Dilatation i[h, O λ1 λ l ] = η O λ1 λ l H MinkowskiR S 3 Hamilton Killing ζ µ Q ζ = dω 3 ζ µ T µ0 (4..) S 3

59 4.3. R S 3 59 (4..1) ˆ µ T µ0 = η T 00 + ˆ i T i0 = 0 η Q ζ = dω 3 ψt λ λ/3 = 0 15 Killing ζ µ = {η µ, ζ µ MN, ζ µ M, ζ µ M } Q ζ = {H, R MN, Q M.Q M} Hamilton S 3 H = dω 3 T 00, S 3 R MN = dω 3 ζmnt j j0 S 3 Killing Q M = V 3 P (+) dω 3 Y S 3 1 MT 00 (4..3) P (+) = e iη (1 + i η )/ S 3 e ±iη P (+) e iη Q M Hermite Q M Q ζ = {H, R MN, Q M.Q M} (4.1.8) O λ1 λ l (η, ˆx) Euclid (4.1.11) Q ρ ρ µ Q ζ ζ µ Hermite O λ 1 λ l (η, ˆx) = O λ1 λ l (η, ˆx) {µ 1 µ l }; = lim η i e i η O µ1 µ l (η, ˆx) 0 (4..4) (4.1.14) 4.3 R S 3 MinkowskiR S 3

60 60 4 R S 3 R S 3 I = 1 dη dω 3 S 3 X ( η ) X S 3 1 X e iωη Y JM ω (J + 1) = 0 X = J 0 M 1 { φjm e i(j+1)η Y JM + φ } JMe i(j+1)η YJM (J + 1) P X = η X X [X(η, x), P X (η, x )] = iδ 3 (x x ) S 3 δ 3 (x x ) = YJM(x)Y JM (x ) J 0 M = 8δ(α α )δ(cos β cos β )δ(γ γ ) [φ J1 M 1, φ J M ] = δ J1 J δ M1 M Hamilton { 1 H = dω 3 : S 3 P X 1 } X ( 3 1) X : = (J + 1)φ JMφ JM (4.3.1) J 0 M T µν = 3 ˆ µ X ˆ ν X 1 3 X ˆ µ ˆ ν X 1 { ˆ λ X ˆ λ X + 1 6ĝµν 6 ˆRX } ˆR µν X

61 4.3. R S 3 61 T λ λ = 1 3 X( ˆ ˆR)X = 0 S 3 Hamilton (4.3.1) (4..3) Q M = P (+) 1 J 1,M 1 J,M 4 V 3 (J 1 + 1)(J + 1) {[ (J 1 + 1)(J + 1) + (J + 1) 1 ( φ J1 M 1 φ J M e i(j 1+J +)η dω 3 Y S 3 1 MY J 1 M 1 Y J M ] +ϵ M1 φ J 1 M 1 ϵ M φ J M e ) i(j 1+J +)η [ + (J 1 + 1)(J + 1) + (J + 1) 1 ] ( φ J1 M 1 ϵ M φ J M e i(j 1 J )η +ϵ M1 φ J 1 M 1 φ J M e )} i(j 1 J )η = C 1 M JM 1 (J + 1)(J + )ϵ,j+ 1 J 0 M 1,M M M1 φ J M 1 φ J+ 1 M C S 3 SU() SU()Clebsch-Gordan C JM J 1 M 1,J M = = V 3 dω 3 YJMY J1 M 1 Y J M S 3 (J1 + 1)(J + 1) CJ Jm J m 1,J m CJ Jm 1 m 1,J m (4.3.) C Jm J 1 m 1,J m Clebsch-Gordan J + J 1 + J J 1 J J J 1 + J M = M 1 + M Q M J = 1/ C C C JM J 1 M 1,J M C JM 00,JN = δ MN = C JM J M,J 1 M 1 = C J M J 1 M 1,J M = ϵ M C J 1M 1 JM,J M

62 6 4 R S 3 4 M {1,, 3, 4} 5 R 11 = (m + m )φ JMφ JM, R = (m m )φ JMφ JM, J>0 M J>0 M R 1 = (J + 1 m )(J + m )φ JMφ JM, J>0 M R 31 = (J + 1 m)(j + m)φ JMφ JM J>0 M M = (m, m 1) M = (m 1, m ) 1 i[q ζ, X] = ζ µ ˆ µ X ˆ µ ζ µ X Killing η µ Hamilton i[h, X] = η X ζ µ M (E.3.1) i[q M, X] (4.1.14) Q M 5 SU() SU()Clebsch-Gordan G JM J 1 (M 1 y 1 );J M (E..3) S 3 Killing ζ j MN = i( V 3 /) V,y G1/M 1/(V y);1/n Y j 1/(V y) J = 1/ [ E ] R MN = 1 ϵ V G 1 M 1 J 0 S 1,S V,y ( V y); 1 N GJS 1 1 (V y);js φ JS 1 φ JS G 1/M J(V y);jn = J(J + )C 1/m J+yv,Jn C1/m J yv,jn G JM 1/(V y);jn = J(J + )C1/+yv,Jn Jm CJm 1/ yv,jn G J = 1/ J 1 = J J 1 = 1/ J = J

63 4.3. R S 3 63 φ JM Q M [Q M, φ JM 1 ] = J(J + 1) M ϵ M C 1 M JM 1,J 1 M φ J 1 M Q M 1 φ 00 Z X X φ 00 L + J Φ [L] JN = L K=0 M 1,M f(l, K)C JN L KM1,KM φ L KM 1 φ KM Q M [Q M, Φ [L] JN ] = S L φ L K 1 K=0 M 1,M M φ 1 KM { (L K)(L K + 1)f(L, K)ϵ S C 1 M L K 1 M 1,L K S CJN L KS,KM ( + (K + 1)(K + )f L, K + 1 ) } ϵ S C 1 M KM,K+ 1 SCJN K+ 1 S,L K 1 M 1 (crossing relation) S 3 S 3 dω 3 Y J 1 M 1 Y J M Y J 3 M 3 Y J4 M 4 Y J1 M 1 Y J M = 1 V3 J 0 M C JM J 1 M 1,J M Y JM ϵ M C J 1M 1 J M,J MC J 3M 3 JM,J 4 M 4 = J 0 J 0 M M ϵ M C J 1M 1 J 4 M 4,J MC J 3M 3 JM,J M (4.3.3)

64 64 4 R S 3 [Q M, Φ [L] JN ] J = L L f(l, K) f ( L, K + 1 ) = (L K)(L K + 1) f(l, K) (K + 1)(K + ) L f f(l, K) = ( 1) K L (L K + 1)(K + 1) K (4.3.4) Q M Φ LN = Φ [L] LN Φ 00 = (φ 00) L = 0 Φ LN SU() SU()Clebsch-Gordan Q M Clebsch- Gordan Q M Φ LN(L Z 0 ) n φ n 00 0 (4..4) X n 4 Φ 1M 0 T µν 6 Φ LM 0 l = L L + (.3.) 7 6 Φ 1M 0 lim η i e i4η M 1,M C 1M 1/M 1,1/M Ê µ M 1 ÊM ν T µν 0 C 00 1/M 1,1/M = ϵ M δ M1 M = h M1M M 1,M h M1 M Ê µ M 1 Ê ν M = η µν 7 l = l +

65 4.3. R S 3 65 T µ1 µ l µ 1 T µ1 µ l = 0 Q κ = dω 3 κ µ1 µ l 1 T µ1 µ l 1 0 κ µ 1 µ l 1 Killing

66

67 Wess-Zumino Weyl δg µν = ωg µν Γ δ ω Γ = d 4 x gω { } acµνλσ + bg 4 + cr + d R + efµν C µνλσ Weyl G 4 Euler Cµνλσ = Rµνλσ Rµν R, G 4 = Rµνλσ 4Rµν + R (5.1.1) Fµν Weyl (counterterm) (bare action) Wess-Zumino (Wess- Zumino integrability condition)[δ ω1, δ ω ] = 0

68 68 5 (5.1.1) [δ ω1, δ ω ] Γ = 4c d 4 x gr ( ) ω ˆ 1 ω ω ˆ ω 1 c = 0 R 5. Riegert-Wess-Zumino g µν = e ϕ ḡ µν ϕ Euler Euler E 4 = G 4 3 R (5..1) Euler E 4 ge 4 = ḡ(4 4 ϕ+ Ē 4 ) S RWZ (ϕ, ḡ) = b ϕ 1 d 4 x dϕ ge (4π) 4 0 = b 1 d 4 x ḡ ( ϕ (4π) 4 ϕ + Ē4ϕ ) g 4 (self-adjoint) d 4 x ga 4 B = d 4 x g( 4 A)B 4 4 = 4 + R µν µ ν 3 R µ R µ (5..)

69 5.. Riegert-Wess-Zumino 69 Riegert-Wess-Zumino ϕ Riegert b 1 N X N W Weyl N A b 1 = ( N X + 11 N W + 6N A ) (5..3) f I Riegert ϕ I(f, g) = I(f, ḡ)( f ϕ ) Riegert ϕ g µν (= e ϕ 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 ω ḡ µν e is(ϕ ω,eωḡ) e iγ(eωḡ) = e is(ϕ ω,eωḡ) e is(ω,ḡ) e iγ(ḡ) e iγ(g) S S(ϕ ω, e ω ĝ) + S(ω, ĝ) = S(ϕ, ĝ) Wess-Zumino S RWZ (5..) [0, ϕ] [0, ω] [ω, ϕ]

70 70 5 (5..4) Riegert-Wess-Zumino Riegert-Wess-Zumino S RWZ Riegert ϕ (5..4) Γ(ḡ) Γ Riegert (g) = S RWZ (ϕ, ḡ) + Γ Riegert (ḡ) Γ Riegert (g) = b 1 (4π) d 4 x ge E 4 Riegert-Wess-Zumino S RWZ Wess-Zumino 4 Einstein 5.3 Weyl QED U(1) Weyl U(1) F µν Wess-Zumino S QED(ϕ, ḡ) = aϕ gfµν/4 ϕ

71 QED Riegert ϕ ϕ S QED ϕ Γ QED (ḡ) Γ QED (ḡ) = 1 { ( )} 1 e r k gf 4 1π log µ µν µ k = ḡ µν k µ k ν ḡ µν ḡ µν Minkowski QED 1 β e = e 3 r/1π e r Wess-Zumino (5..4) k e ω k Wess-Zumino a = e r/6π 1 { ( )} 1 + e r 4 6π ϕ e r k gf 1π log µ µν g µν (= e ϕ ḡ µν ) p = k e ϕ (5.3.1) Γ QED (g) = 1 { ( )} 1 e r p gf 4 1π log µ µν µ p k (comoving momentum) Weyl t Weyl (1/t ) d 4 x gc µνλσ ḡ µν Weyl { ( ) } 1 k gc Γ Weyl (g) = + β 0 log β 0 ϕ µνλσ (5.3.) t r µ

72 7 5 ḡ µν h µν ḡ µν Minkowski ḡ µν = η µν + th µν I int = d 4 xth µν T µν / T µν h µν β 0 β 0 = 1 40(4π) (N X + 3N W + 1N A ) t β t = β 0 t 3 r (5..4) Wess- Zumino p t r(p) = 1 β 0 log(p /Λ QG) (5.3.3) Γ Weyl (g) = 1 gc t µνλσ (5.3.4) r(p) Λ QG = µ exp( 1/β 0 t r) ( Minkowski ) T µν I int = t d 4 x ĝh µν T µν / β Adler-Bardeen 1

73 t (Riegert ) t t = 0

74

75 75 6 : / 4 Einstein I e ii 1 Einstein

76 76 6 I = d 4 x { g 1 t C µνλσ bg h ( )} 1 16πG R Λ + L M (6.1.1) (5.1.1) Weyl Euler t Euler b Euler G Λ Newton h Planck c 1 Wess-Zumino R 4 4 h Einstein 4 4 I Planck 4 Weyl t C µνλσ = 0 (conformal flat) g µν = e ϕ ḡ µν, ḡ µν = (ĝe th ) µν = ĝ µν + th µν + t h µλh λ ν + (6.1.) h µν h µ µ = ĝ µν h µν = 0 ĝ µν Riegert ϕ Wess-Zumino

77 g µν ĝ µν e iγ = = [dgdf] g exp{ii(f, g)} [dϕdhdf]ĝ exp {is(ϕ, ḡ) + ii(f, g)} (6.1.3) S Wess-Zumino f Wess-Zumino t ϕ Riegert-Wess- Zumino (5..) (6.1.3) t β t = β 0 t 3 r h µν β 0 = 1 { 197 (4π) } 40 (N X + 3N W + 1N A ) ϕ h µν 1/15 199/30 Riegert-Wess-Zumino Weyl C µνλσ = 0 Riegert-Wess-Zumino b 1 t ( ) I Weyl R R R

78 78 6 b 1 = ( N X + 11 ) 360 N W + 6N A (6.1.4) 7/90 Riegert ϕ 87/0 Riegert-Wess-Zumino 3 ĝ µν 4 Riemann Weyl Schwarzschild Riemann t ĝ µν 4 I 4DQG = S RWZ (ϕ, ĝ) + I(f, g) t 0 (6.1.5) Weyl t h µν ḡ µν ĝ µν 3 Euclid e I I > 0 Weyl Euclid I = (1/t ) gc µνλσ 4

79 Planck Riegert ϕ ϕ ω ĝ µν e ω ĝ µν 6. (contravariant vector)ξ µ δ ξ g µν = g µλ ν ξ λ + g νλ µ ξ λ g µν (6.1.) e ϕ ḡ µν δ ξ ϕ = ξ λ λ ϕ ˆ λ ξ λ, δ ξ ḡ µν = ḡ µλ ν ξ λ + ḡ νλ µ ξ λ 1 ḡµν ˆ λ ξ λ λ ξ λ = ˆ λ ξ λ δ ξ h µν = 1 ( ˆ µ ξ ν + t ν ξ µ 1 ˆ ) λ ξ λ + ξ λ ˆ λ h µν ĝµν + 1 h ( µλ ˆ ν ξ λ ˆ ) λ 1 ξ ν + h ( νλ ˆ µ ξ λ ˆ ) λ ξ µ + (6..1) (covariant vector) ξ µ = ĝ µν ξ ν t Weyl κ = ξ/t t 0 δ κ h µν = ˆ µ κ ν + ˆ ν κ µ 1 η µν ˆ λ κ λ (6..)

80 80 6 δ κ ϕ = 0 (6..) Killing ˆ µ ζ ν + ˆ ν ζ µ 1 η µν ˆ λ ζ λ = 0 (6..3) ξ µ = ζ µ (6..1) δ ζ ϕ = ζ λ λ ϕ ˆ λ ζ λ, δ ζ h µν = ζ λ ˆ λ h µν + 1 h µλ ( ˆ ν ζ λ ˆ ) λ 1 ζ ν + h ( νλ ˆ µ ζ λ ˆ ) λ ζ µ (6..4) ĝ µν Riegert (.3.) 6.3 ĝ µν t Weyl ĝ µν Minkowski η µν = ( 1, 1, 1, 1)

81 Riegert (6..) κ µ (6..4) ζ µ Riegert Riegert-Wess-Zumino (5..) Minkowski (b 1 /8π ) d 4 xϕ 4 ϕ Riegert ϕ Dirac Riegert { S RWZ = d 4 x b 1 8π χ = η ϕ (6.3.1) [ ( η χ) + χ χ + ( ϕ ) ] } + v ( η ϕ χ) Lagrange (Lagrange multiplier) χ ϕ v P χ P ϕ P v Poisson {χ(η, x), P χ (η, x )} P = {ϕ(η, x), P ϕ (η, x )} P = {v(η, x), P v (η, x )} P = δ 3 (x x ) χ P χ = (b 1 /4π ) η χ ϕ v φ 1 = P ϕ v 0, φ = P v 0 5 Lagrange (v η ϕ ϕ η v)/ φ 1 = P ϕ v/ φ = P v + ϕ/

82 8 6 ϕ χ v P ϕ P χ P v Poisson C ab = {φ a, φ b } P = det C ab 0 Dirac Dirac {F, G} D = {F, G} P {F, φ a } P C 1 ab {φ b, G} P Dirac Poisson F {F, φ a } D = 0 Dirac Poisson F Hamilton φ a = 0 0 Dirac Dirac {χ(η, x), P χ (η, x )} D = {ϕ(η, x), P ϕ (η, x )} D = δ 3 (x x ) Hamilton { H = d 3 x π b 1 P χ + P ϕ χ + b 1 8π [ χ χ + ( ϕ ) ]} (6.3.) η ϕ = {ϕ, H} D = χ, η χ = {χ, H} D = 4π b 1 P χ, η P χ = {P χ, H} D = P ϕ b 1 π χ, η P ϕ = {P ϕ, H} D = b 1 4π 4 ϕ (6.3.3)

83 Dirac [ϕ(η, x), P ϕ (η, x )] = [χ(η, x), P χ (η, x )] = iδ 3 (x x ) (6.3.4) (6.3.3) P χ = b 1 4π ηχ, P ϕ = η P χ b 1 π χ (6.3.5) Riegert 4 ϕ = 0 η P ϕ = (b 1 /4π ) 4 ϕ k µ x µ = ωη + k x e ik µx µ ηe ik µx µ ω = k Riegert ϕ = ϕ < + ϕ > ϕ < (x) = π b1 d 3 k 1 µxµ {a(k) + iωηb(k)} (π) 3/ eik ω3/ ϕ > = ϕ < (6.3.1) (6.3.5) χ < (x) = i π d 3 k 1 µxµ {a(k) + ( 1 + iωη)b(k)} b1 (π) 3/ eik, ω1/ b1 d 3 k P χ< (x) = 4π (π) 3/ ω1/ ik µxµ {a(k) + ( + iωη)b(k)} e, b1 d 3 k P ϕ< (x) = i 4π (π) 3/ ω3/ ik µxµ {a(k) + (1 + iωη)b(k)} e (6.3.4) [ a(k), a (k ) ] = δ 3 (k k ), [ a(k), b (k ) ] = [ b(k), a (k ) ] = δ 3 (k k ), [ b(k), b (k ) ] = 0

84 84 6 Hamilton H = d 3 kω { a (k)b(k) + b (k)a(k) b (k)b(k) } Hamilton (6.3.) 6.3. δ κ h µν (6..) Coulomb i h i µ = 0 h 00 (= h i i) h 00 = 0, h 0j = h j, h ij = h ij h j h ij i h i = 0 i h ij = h i i = 0 κ µ ζ µ (6..4) Riegert Dirac u ij = η h ij u ij Weyl I = d 4 x { 1 ( hij 4η η + 4) h ij + h j ( η + ) } h j { = d 4 x 1 ηu ij η u ij u ij u ij 1 h ij h ij + λ ij ( η h ij u ij ) } + η h j η h j + h j h j λ ij Lagrange

85 λ ij u ij h ij h j P ij u = η u ij, P ij h = ηp ij u u ij, P j = η h j [ h ij (η, x), P kl h (η, y) ] = [ u ij (η, x), P kl u (η, y) ] = iδ ij,kl 3 (x y), [ h i (η, x), P j (η, y) ] = iδ ij 3 (x y), (6.3.6) δ ij 3 (x) = ij δ 3 (x) δ ij,kl 3 (x) = ij,kl δ 3 (x) ij = δ ij i j, ij,kl = 1 ( ik jl + il jk ij kl ) i ij = 0 j j = i ij,kl = 0 i i,kl = 0 ik k j = ij ij,kl kl, mn = ij,mn Hamilton H = { d 3 x : 1 Pij u P u ij + P ij h u ij + u ij u ij + 1 h ij h ij Pj P j h j h j }: = 1/ : : 4 h ij = 0 η P ij h = 4 h ij Riegert h ij = h ij < + h ij > h ij <(x) = d 3 k 1 { c ij (k) + iωηd ij (k) } ik µxµ e (π) 3/ ω 3/

86 86 6 h ij > = h ij < h j = 0 η P j = 4 h j h j = h j < + h j > h j > = h j < h j <(x) = d 3 k 1 (π) 3/ ω e ik µxµ j(k)e 3/ u ij <(x) = i P ij u<(x) = P ij h< (x) = i P j <(x) = i d 3 k 1 { c ij (k) + ( 1 + iωη)d ij (k) } e ikµxµ, (π) 3/ ω 1/ d 3 k ω 1/ (π) 3/ { c ij (k) + ( + iωη)d ij (k) } e ikµxµ, d 3 k ω 3/ { c ij (k) + (1 + iωη)d ij (k) } e ik µx µ, (π) 3/ d 3 k (π) 3/ ω3/ e j ik µxµ (k)e (6.3.6) [ c ij (k), c kl (k ) ] = δ ij,kl 3 (k k ), [ c ij (k), d kl (k ) ] = [ d ij (k), c kl (k ) ] = δ ij,kl 3 (k k ), [ d ij (k), d kl (k ) ] = 0, [ e i (k), e j (k ) ] = δ ij 3 (k k ) δ ij 3 (k) δ ij,kl 3 (k) δ 3 (k) ij (k) = δ ij k ik j k, ij,kl (k) = 1 { ik (k) jl (k) + il (k) jk (k) ij (k) kl (k) } ε i (a) (a = 1, ) ε ij (a) (a = 1, ) k iε i (a) = 0

87 k i ε ij (a) (k) = ε (a)i i (k) = 0 a=1 ε i (a) (k)εj (a) (k) = ij (k) ε j (a) (k)ε (b)j(k) = δ ab a=1 ε ij (a) (k)εkl (a) (k) = ij,kl (k) ε ij (a) (k)ε (b)ij(k) = δ ab c ij (k) = e j (k) = a=1 a=1 ε ij (a) (k)c (a)(k), d ij (k) = ε j (a) (k)e (a)(k) a=1 ε ij (a) (k)d (a)(k), [ c(a) (k), c (b) (k ) ] = δ ab δ 3 (k k ), [ c(a) (k), d (b) (k ) ] = [ d (a) (k), c (b) (k ) ] = δ ab δ 3 (k k ), [ d(a) (k), d (b) (k ) ] = 0, [ e(a) (k), e (b) (k ) ] = δ ab δ 3 (k k ) Hamilton H = d 3 kω { c (a) (k)d (a)(k) + d (a) (k)c (a)(k) d (a) (k)d (a)(k) e (a) (k)e (a)(k) } a=1 6.4 δ ζ (6..4) Riegert D. I 4DQG T µν = ĝ δi 4DQG δĝ µν ˆT µν = ĝ µλ ĝ νσ ˆT λσ Minkowski

88 88 6 Riegert T µν = b { 1 4 ϕ 8π µ ν ϕ + µ ϕ ν ϕ + ν ϕ µ ϕ µ λ ϕ ν λ ϕ 4 3 µ ν λ ϕ λ ϕ + η µν ( ϕ ϕ 3 λ ϕ λ ϕ ) 3 λ σ ϕ λ σ ϕ 3 µ ν ϕ + } 3 η µν 4 ϕ Riegert-Wess-Zumino (5..) T λ λ = (b 1 /4π ) 4 ϕ = 0 µ T µν = (b 1 /4π ) 4 ϕ ν ϕ = 0 (00) T 00 = π P χ + P ϕ χ P χ ϕ k P χ k ϕ b 1 + b ( 1 8π 3 χ χ 4 3 kχ k χ + ϕ ϕ 3 k ϕ k ϕ 3 k l ϕ k ϕ) l P χ + b 1 1π 4 ϕ (0j) T 0j = 3 P χ j χ 1 3 jp χ χ + P ϕ j ϕ + b 1 8π ( 4 j χ ϕ 8 3 kχ j k ϕ χ j ϕ + χ j ϕ j k χ k ϕ 1 3 jp ϕ b 1 1π j χ (..5) Q ζ P 0 = H = d 3 xa, P j = d 3 xb j A B j A = π :P χ : + :P ϕ χ: + b ( 1 :χ χ: + : ϕ ϕ: ), b 1 8π B j = :P χ j χ: + :P ϕ j ϕ: )

89 Lorentz M 0j = d 3 x { ηb j x j A :P χ j ϕ:}, M ij = d 3 x {x i B j x j B i } Dilatation D = d 3 x { } ηa + x k B k + :P χ χ: +P ϕ K 0 = K j = (η d x{ 3 + x ) A + ηx k B k + η :P χ χ: +x k :P χ k ϕ: b ( 1 :χ : + : 4π k ϕ k ϕ: ) } + ηp ϕ + P χ, ( η d x{ 3 + x ) B j x j x k B k ηx j A x j :P χ χ: η :P χ j ϕ: b } 1 π :χ jϕ: x j P ϕ M 0j D K µ η η M 0j = η D = η K µ = 0 D K µ δ ζ ϕ(6..4) 6.5 D.1 Hermite Hermite A B (operator product) A(x)B(y) = 0 A(x)B(y) 0 + : A(x)B(y): 0 A(x)B(y) 0 = [A < (x), B > (y)]

90 90 6 A < A B > B :A(x)B(y):=:B(y)A(x): [A(x), B(y)] = 0 A(x)B(y) 0 0 B(y)A(x) 0 0 B(y)A(x) 0 = 0 A(x)B(y) 0 Riegert 0 ϕ(x)ϕ(x ) 0 = π b 1 ω>z d 3 k 1 (π) 3 ω {1 + iω (η 3 η )} e iω(η η iϵ)+ik (x x ) = 1 log {[ (η η iϵ) + (x x ) ] z e γ } 4b 1 1 iϵ 4b 1 x x log η η iϵ x x (6.5.1) η η iϵ + x x ϵ 6 Riegert z z 7 z 0 ϵ (6.5.1) [ϕ(η, x), P ϕ (η, x )] = 0 ϕ(η, x)p ϕ (η, x ) 0 0 ϕ(η, x)p ϕ (η, x ) 0 = i 1 π ϵ [(x x ) + ϵ ] 6 ϵ 7 Einstein (6.1.) Riegert

91 δ 3 (x) = d 3 k (π) 3 eik x ϵω = 1 π ϵ (x + ϵ ) (6.5.) χ P χ ϵ Wick [:AB(x):, : C k (y):] k = [A(x), C i (y)] :B(x) C k (y): + [B(x), C i (y)] :A(x) C k (y): i k( i) i k( i) + { 0 A(x)C i (y) 0 0 B(x)C j (y) 0 H.c.} : C k (y): i,j(i j) k( i,j) H.c. { } Hermite [ D.1] Riegert (..1) :ϕ n : A [A(x), :ϕ n (y):] = inδ 3 (x y) :χϕ n 1 (y): = iδ 3 (x y) η :ϕ n (y): (6.5.3) B j [B j (x), :ϕ n (y):] = iδ 3 (x y) j :ϕ n (y): +i 1 b 1 n(n 1)e j (x y) :ϕ n (x): (6.5.4)

92 9 6 e j (x) e j (x) = 1 ϵx j [1 h(x)] iϵ iϵ + x, h(x) = log π x (x + ϵ ) x iϵ x h h (x) = h(x) lim x 0 h(x) = 1 ( ) (6.5.3) (6.5.4) [:P χ j ϕ(x):, :ϕ n (y):] = 0 Lorentz i[p µ, :ϕ n (x):] = µ :ϕ n (x):, i[m µν, :ϕ n (x):] = (x µ ν x ν µ ) :ϕ n (x): Lorentz e j d 3 x e j (x) = 0 d 3 x x i e j (x) = 1 3 δ ij dilatation 0 4πx dx 1 π ϵ[1 h(x)] (x + ϵ ) = 1 6 δ ij (6.5.5) i[d, :ϕ n (x):] = x µ µ :ϕ n (x): +n :ϕ n 1 (x): 1 4b 1 n(n 1) :ϕ n (x):, i[k µ, :ϕ n (x):] = ( ) x µ x µ x ν ν :ϕ n (x): ( x µ n :ϕ n 1 (x): 1 ) n(n 1) :ϕ n (x): 4b 1 :ϕ n 1 : P ϕ 1/b 1 D K 0 (6.5.5) K j d 3 x { x e j (x y) x j x k e k (x y) } = y j n = 1 Riegert (6..4) i[q ζ, ϕ] = δ ζ ϕ

93 V α (x) =:e αϕ(x) := n=0 α n n! :ϕn (x): (6.5.6) α Riegert :ϕ n : V α i[p µ, V α (x)] = µ V α (x), i[d, V α (x)] = (x µ µ + h α ) V α (x), i[m µν, V α (x)] = (x µ ν x ν µ ) V α (x), i[k µ, V α (x)] = ( x µ x µ x ν ν x µ h α ) Vα (x) h α = α α 4b 1 (6.5.7) 1/b 1 Lorentz R 1 β = R β = n=0 n=0 β n ( ) 4π n! :ϕn ϕ: = :e βϕ P χ + ϕ :, b 1 β n n! :ϕn λ ϕ λ ϕ: = :e βϕ ( χ + k ϕ k ϕ ) : Lorentz Dilatation i[d, R 1, β (x)] = (xµ µ + h β + ) R 1, β (x) i[k µ, R 1 β(x)] = { x µ x µ x λ λ x µ (h β + ) } R 1 β(x) + 4 : µ ϕe βϕ (x):, i[k µ, R β(x)] = { x µ x µ x λ λ x µ (h β + ) } R β(x) 4 h β β : µϕe βϕ (x):

94 94 6 h β (6.5.7) R β = R 1 β + β R β =:e ( βϕ ϕ + β ) λ ϕ λ ϕ : (6.5.8) h β h β R β i[p µ, R β (x)] = µ R β (x), i[m µν, R β (x)] = (x µ ν x ν µ ) R β (x), i[d, R β (x)] = ( x λ λ + h β + ) R β (x), i[k µ, R β (x)] = { x µ x µ x λ λ x µ (h β + ) } R β (x) h β + m R [m] γ =:e γϕ ( ϕ + γ h γ λ ϕ λ ϕ) m : h γ +m m = 0, 1 V α R β 6.6 [Q ζ, d 4 xo(x)] = 0 (6.6.1) 4 Killing ζ µ i[q ζ, V α (x)] = µ {ζ µ V α (x)}

95 6.7. BRST 95 O h α = 4 V α (6.5.6) h α = 4 Riegert ) α = b 1 (1 1 4b1 (6.6.) ( N ) b 1 α 4 V α g V α (6.1.4) b 1 > 4 α h β = R β b 1 Riegert ) β = b 1 (1 1 b1 (6.6.3) R β Ricci β β/h β 1 (6.5.8) Ricci gr h γ = 4 m Riegert γ = b 1 (1 1 (4 m)/b 1 ) R [m] γ Ricci m gr m 6.7 BRST (6..4) BRST BRST 15 ζ µ c µ

96 Grassmann c λ cµν c c λ + c λ = c µ ( ) ) ( ) ζ λ T ζ λ S + ( µ cµν ζl λ + µν cζλ D + c µ + = c λ + x µ c µλ + x λ c + x c λ + x λ x µ c µ + c µν Hermite c c µν c µ c µ b λ b µν b b λ + {c, b} = 1, {c µν, b λσ } = η µλ η νσ η µσ η νλ, {c µ, b ν +} = {c µ +, b ν } = η µν µ (..1) P µ gh = i ( bc µ + + b µ +c + b µ λ cλ + + b λ +c µ λ), M µν gh = i ( b µ +c ν b ν +c µ + b µ c ν + b ν c µ + + b µλ c ν λ b νλ c µ λ), D gh = i ( b λ c +λ b λ +c λ ), K µ gh = i ( bc µ b µ c + b µ λ cλ + b λ c µ λ gh BRST ( Q BRST = c µ P µ + 1 ) ( P µ gh + c µν M µν + 1 ) M µν gh ( +c µ + K µ + 1 ) Kgh µ ) + c (D + 1 ) Dgh = c ( D + D gh) + c µν ( M µν + M gh µν ) bn b µν N µν + ˆQ P µ M µν D K µ N = ic µ +c µ, N µν = i ˆQ = c µ P µ + c µ +K µ ( c µ + c ν + c µ c ν + ) + ic µλ c ν λ,

97 6.7. BRST 97 BRST P µ M µν D K µ Q BRST = ˆQ ND ic µ +c ν M µν = 0 BRST {Q BRST, b} = D + D gh, {Q BRST, b µν } = ( M µν + M µν gh ), {Q BRST, b µ } = K µ + K µ gh, {Q BRST, b µ +} = P µ + P µ gh [Q BRST, D+D gh ] = 0 BRST ζ µ Riegert BRST i[q BRST, ϕ(x)] = c µ µ ϕ(x) µc µ (x) V α R β O BRST i[q BRST, O(x)] = c µ µ O(x) + 4 µc µ O(x) = 4 i[q BRST, d 4 xo(x)] = d 4 x µ {c µ O(x)} = 0 BRST (6.6.1) BRST ω = 1 4! ϵ µνλσc µ c ν c λ c σ BRST i{q BRST, c µ (x)} = c ν ν c µ (x)

98 98 6 ω i[q BRST, ω(x)] = c µ µ ω(x) = ω µ c µ (x) c µ ω = 0 ω = 4 i[q BRST, ωo(x)] = 1 4 ( 4) ω µc µ O(x) = 0 BRST BRST 4 BRST 6.8 p Riegert Euclid Wick S RWZ + µv α µ V α = d 4 xv α O γ = d 4 xo γ Riegert σ ϕ ϕ + σ σ Euler d 4 x ĝĝ4/3π = σ S RWZ S RWZ + 4b 1 σ A = e ασ O γ A γ/α O γ σ O γ1 O γn = 1 α 0 da A A s O γ1 O γn e µav α = µ s Γ( s) α O γ 1 O γn (V α ) s (6.8.1)

99 s = 4b 1 α n 4 i=1 O n = A n e γnϕ A n n n Weyl d 4 xo n ω 4 n d 4 xo n n = 0 Weyl Riegert ϕ 0 ϕ 0 + (4/γ 0 ) ln ω e γ nϕ 0 O n (n > 0) d 4 xo n ω 4γ n/γ 0 d 4 xo n γ i α n = 4 4 γ n γ 0 N b 1 n n 1970 (6..)

100 100 6 n = 1970 Lee Wick 4 (6.1.1) h

101 101 7 R S 3 Minkowski M 4 R S 3 (6..4) ( ) 7.1 R S 3 R S 3 S 3 1 Euler x i = (α, β, γ) 1 ds R S 3 = ĝ µνdx µ dx ν = dη + ˆγ ij dx i dx j = dη (dα + dβ + dγ + cos βdαdγ) ˆR ijkl = (ˆγ ikˆγ jl ˆγ ilˆγ jk ), ˆRij = ˆγ ij, ˆR = 6 Ĉ µνλσ = Ĝ4 = 0 V 3 = dω 3 = dω 3 = d 3ˆx ˆγ = 1 sin βdαdβdγ 8 π 0 π 4π dα dβ dγ 1 sin β = π Euler ˆx j

102 10 7 S 3 n (symmetric transverse traceless, ST ) SU() SU() (J + ε n, J ε n ) Y i 1 i n J(Mε n ) ε n = ±n/ 3 Y i 1 i n J(Mε n) = { J(J + ) + n}y i 1 i n J(Mε n) 3 = ˆγ ij ˆ i ˆ j S 3 J( n/) M = (m, m ) m = J ε n, J ε n + 1,, J + ε n 1, J + ε n, m = J + ε n, J + ε n + 1,, J ε n 1, J ε n n > 0 (J + n + 1)(J n + 1) n = 0 (J + 1) ST Y i 1 i n J(Mε n) = ( 1)n ϵ M Y i 1 i n J( Mε n), S 3 dω 3 Y i 1 i n J 1 (M 1 ε 1 n) Y i 1 i n J (M ε n ) = δ J1 J δ M1 M δ ε 1 n ε n δ M1 M = δ m1 m δ m 1 m ϵ M = ( 1) m m ϵ M = 1 n 4 y = ε 1 = ± 1, x = ε = ±1, z = ε 3 = ± 3, w = ε 4 = ±

103 7.1. R S R S Coulomb ˆ i A i = 0 R S 3 I = { 1 dη dω ( 3 S 3 Ai η + 3 ) A i 1 } A 0 3 A 0 A i = ˆγ ij A j A 0 A 0 = 0 A i e iωη Y i J(my) ω (J + 1) = 0 A i = J 1 M,y 1 { qjm e i(j+1)η YJ(My) i + q } JMe i(j+1)η YJ(My) i (J + 1) P i A = η A i [A i (η, x), P j A(η, y)] = iδ ij 3 (x y) S 3 M,y Y i J(My) (x)y j J(My) (y) δ ij 3 (x y) = J 1 [q J1 (M 1 y 1 ), q J (M y ) ] = δ J 1 J δ M1 M δ y1 y Hamilton { 1 H = dω 3 : S 3 PiAP Ai 1 } Ai ( 3 ) A i : = (J + 1)q J(My) q J(My) (7.1.1) J 1 M,y

104 104 7 Weyl h 00 = h, h 0i = h i, h ij = h tr ij ˆγ ijh h tr ij (htri i = 0) δ κ h = 3 ηκ ˆ k κ k, δ κ h i = η κ i + ˆ i κ 0, δ κ h tr ij = ˆ i κ j + ˆ j κ i 3 ˆγ ij ˆ k κ k ˆ i h i = ˆ i h tr ij = 0 h T i h TT ij h i = h T i, h tr ij = h TT ij Riegert-Wess-Zumino Weyl R S 3 I 4DQG = { dη dω 3 b 1 S 3 (4π) ϕ ( ) 4 η 3 η η ϕ 1 ( htt ij 4 η 3 η η ) h ij TT +h T i ( 3 + ) ( η + 3 ) h i T 1 } 7 h ( ) 3 h (7.1.) h δ κ ( ˆ i h i ) = δ κ ( ˆ i h tr ij ) = 0 h = 0 3 κ 0 = 0 κ 0 (η) ( 3 + )h T i = 0 J = 1/

105 7.1. R S h T i J= 1 = 0 (7.1.3) + Killing Dirac χ = η ϕ(6.3.1) Riegert { I = dη dω 3 b } [ 1 ( η χ) + χ S 3 8π 3 χ 4χ + ( 3 ϕ) ] + v( η ϕ χ) S 3 1 Dirac [χ(η, x), P χ (η, y)] = [ϕ(η, x), P ϕ (η, y)] = iδ 3 (x y) P χ = b 1 4π ηχ, P ϕ = η P χ b 1 π 3χ + b 1 π χ Hamilton { H = dω 3 : π P χ + P ϕ χ + b } [ 1 χ 3 χ 4χ + ( b 1 8π 3 ϕ) ] : Riegert-Wess-Zumino (7.1.) Riegert ϕ e iωη Y JM {ω (J) }{ω (J + ) } = 0 Riegert ϕ = { π (ˆq + ˆpη)Y 00 b J 1 M J(J + 1) + 1 J 0 M (J + 1)(J + 1) ( ajm e ijη Y JM + a ) JMe ijη YJM ( bjm e i(j+)η Y JM + b ) } JMe i(j+)η YJM

106 106 7 Y 00 = 1/ V 3 = 1/ π [ˆq, ˆp] = i, [a J1 M 1, a J M ] = δ J1 J δ M1 M, [b J1 M 1, b J M ] = δ J1 J δ M1 M a JM b JM Hamilton H = 1 ˆp + b 1 + {Ja JMa JM (J + )b JMb JM } (7.1.4) J 0 M b 1 R S 3 h TT ij Dirac h T i h ij TT e iωη Y ij J(Mx) hi T e iωη Y i J(My) (7.1.) {ω (J) }{ω (J + ) } = 0 (J 1)(J + 3){ω (J + 1) } = 0 h ij TT = 1 4 h i T = J 1 M,x J 1 M,x J 1 M,y 1 J(J + 1) 1 (J + 1)(J + 1) { cj(mx) e ijη Y ij J(Mx) + c J(Mx) eijη Y ij } J(Mx) i (J 1)(J + 1)(J + 3) { dj(mx) e i(j+)η Y ij J(Mx) +d J(Mx) ei(j+)η Y ij J(Mx)}, { ej(my) e i(j+1)η Y i J(My) e J(My) ei(j+1)η Y i J(My) } (7.1.5) h i T Q M

107 J = 1/ ( 3 + )h i T J=1/ = 0 [c J1 (M 1 x 1 ), c J (M x ) ] = [d J 1 (M 1 x 1 ), d J (M x ) ] = δ J 1 J δ M1 M δ x1 x, [e J1 (M 1 y 1 ), e J (M y ) ] = δ J 1 J δ M1 M δ y1 y c J(Mx) d J(Mx) e J(My) Hamilton H = {Jc J(Mx) c J(Mx) (J + )d J(Mx) d J(Mx)} J 1 M,x (J + 1)e J(My) e J(My) (7.1.6) J 1 M,y 7. / Q ζ = {H, R MN, Q M, Q M} 3.4 T µν = F µλ F λ ν 1 4ĝµνF λσ F λσ F µ ν = ĝ µλ F λν A 0 = ˆ i A i = 0 Hamilton (7.1.1) Q M = J 1 D 1 M J(M 1 y 1 ),J+ 1 M 1,y 1,M,y (M y ) (J + 1)(J + ) ( ϵ M1 )q J( M 1 y 1 ) q J+ 1 (M y )

108 108 7 SU() SU()Clebsch-Gordan D D 1 M J(M 1 y 1 ),J+ 1 (M y ) = = V 3 dω 3 Y S 3 1 MY J(M i 1 y 1 )Y ij+ 1 (M y ) J(J + 3)C 1 m J+y 1 m 1, J+ 1 +y m C 1 m J y 1 m 1, J+ 1 y m D E. S 3 Riegert Riegert-Wess-Zumino Riegert T µν = b { 1 4 8π ˆ ϕ ˆ µ ˆ ν ϕ + ˆ ˆ µ ϕ ˆ ν ϕ + ˆ ˆ ν ϕ ˆ µ ϕ ˆ µ ˆ λ ϕ ˆ ν ˆ λ ϕ 4 3 ˆ µ ˆ ν ˆ λ ϕ ˆ λ ϕ + 4 ˆR µλνσ ˆ λ ϕ ˆ σ ϕ +4 ˆR µλ ˆ λ ϕ ˆ ν ϕ + 4 ˆR νλ ˆ λ ϕ ˆ µ ϕ 4 3 ˆR µν ˆ λ ϕ ˆ λ ϕ 4 3 ˆR ˆ µ ϕ ˆ ν ϕ 3 ˆ µ ˆ ν ˆ ϕ 4 ˆR µλνσ ˆ λ ˆ σ ϕ ˆR µν ˆ ϕ + ˆR ˆ µ ˆ ν ϕ 4 ˆR µλ ˆ λ ˆ ν ϕ 4 ˆR νλ ˆ λ ˆ µ ϕ 1 3 ˆ µ ˆR ˆ ν ϕ 1 3 ˆ ν ˆR ˆ µ ϕ +ĝ µν [ ˆ ϕ ˆ ϕ 3 ˆ λ ˆ ϕ ˆ λ ϕ 3 ˆ λ ˆ σ ϕ ˆ λ ˆ σ ϕ 8 3 ˆR λσ ˆ λ ϕ ˆ σ ϕ + 3 ˆR ˆ λ ϕ ˆ λ ϕ + 3 ˆ 4 ϕ + 4 ˆR λσ ˆ λ ˆ σ ϕ ˆR ˆ ϕ ˆ λ ˆR ˆ λ ϕ T λ λ = (b 1 /4π ) ˆ 4 ϕ = 0 R S 3 Riegert Hamilton (7.1.4) Q M = ( b1 iˆp ) a 1 + J 0 M C 1 M JM 1,J+ 1 M 1,M M { α(j)ϵm1 a J M 1 a J+ 1 M +β(j)ϵ M1 b J M 1 b J+ 1 M + γ(j)ϵ M a J+ 1 M b JM 1 }

109 C (4.3.) α(j) = J(J + ), β(j) = (J + 1)(J + 3), γ(j) = 1(7..1) SU() SU()Clebsch-Gordan (4.3.3) Q M Q N Riegert i[q ζ, ϕ] = ζ µ ˆ ϕ ˆ µ ζ µ (7..) (E.3.1) + + ˆ i h i = ˆ i h tr ij = 0 h = 0 δ κ h = (3 η κ 0 + ψ)/ = 0 δ κ ( ˆ i h i ) = η ψ + 3 κ 0 = 0 δ κ ( ˆ i h tr ij ) = ( 3 + )κ j + ˆ j ψ/3 = 0 ψ = ˆ λ κ λ Killing 15 Killing (4..1) S 3 Killing η κ i 0 f(η) κ µ = (0, f(η)y i 1/(My) ) h T i J = 1/ (7.1.3) + Killing

110 110 7 Hamilton H (7.1.6) Q M = E 1 M { J(M 1 x 1 ),J+ 1 J 1 M 1,x 1,M,x (M α(j)ϵm1 x c ) J( M1x1) c J+ 1 (M x ) +β(j)ϵ M1 d J( M 1 x 1 ) d J+ 1 (M x ) + γ(j)ϵ M c J+ 1 ( M x ) d J(M 1 x 1 ) + H 1 M J(M 1 x 1 );J(M y ){ A(J)ϵM1 c J( M 1 x 1 ) e J(M y ) J 1 M 1,x 1,M,y +B(J)ϵ M e J( M y ) d J(M 1 x 1 ) + D 1 M C(J)ϵ J(M 1 y 1 ),J+ 1 J 1 M 1,y 1,M,y (M y ) M 1 e J( M 1 y 1 ) e J+ 1 (M y ) α(j) β(j) γ(j) Riegert (7..1) 4J A(J) = (J 1)(J + 3), B(J) = (J + ) (J 1)(J + 3), C(J) = (J 1)(J + 1)(J + )(J + 4) J(J + 3) SU() SU()Clebsch-Gordan E 1 M J(M 1 x 1 ),J+ 1 (M x ) = = V 3 dω 3 Y S 3 1 (J 1)(J + )C 1 m J+x 1 m 1,J+ 1 +x m C 1 m J x 1 m 1,J+ 1 x m, MY ij J(M 1 x 1 ) Y ijj+ 1 (M x ) H 1 M J(M 1 x 1 );J(M y ) = V 3 dω 3 Y S ij 3 1 MY ˆ J(M 1 x 1 ) i Y jj(m y ) = (J 1)(J + 3)C 1 m J+x 1 m 1,J+y m C 1 m J x 1 m 1,J y m (E..) (E..4) Weyl α β γ A B C } }