ver.1, 1994 ver.2, 1996 ver.3.0.7, ver.3.1.0, ver.3.2.0, I HP ϕ

Transcription

1 ver.1, 1994 ver., 1996 ver.3.0.7, ver.3.1.0, ver.3..0, I HP ϕ Mean Field Theory MF Block Spin Transformation (BST) BST i.i.d BST ϕ θ

2 scaling limit effective theory ϕ BST (Weak) ϕ 4 d- d Fractal KT-type? Coulomb Gas? A 43 A A ϕ 4 - d- Z d {x =(x 1,x,...,x d ):x j R} Z d site x ϕ x ρ({ϕ}) marginal distribution Z d Λ Λ L Periodic Boundary Condition, P.B.C. Λ {(x 1,x,..., x d ) x j Z, L <x j L} (1.1.1) x j = L x j = L x Λ ϕ x ρ Λ ρ Λ 1 1 universality

3 Λ ( F ({ϕ x }) ρλ dϕ x )ρ Λ ({ϕ x }) F ({ϕ x }) (1.1.) x Λ Φ {ϕ x } x Λ dφ x Λ dϕ x [ ρ Λ (Φ) 1 exp Z Λ J 4 x,y Λ x y =1 (ϕ x ϕ y ) + H ] ϕ x η(ϕ x ) (J, H 0) (1.1.3) x Λ x Λ δ(ϕ K ) K>0 (Ising model) η(ϕ) = exp [ µ ϕ λ 4! ϕ4] µ R, λ 0 (ϕ 4 model) Z Λ 1 ρλ =1 Remarks: 1. Ising Model ϕ 4 -model µ = λk 6 λ. ϕ 4 -model (J, µ, λ, H) ϕ trivial (1.1.4) ϕ ϕ aϕ (1.1.5) (J, µ, λ, H) (J,µ,λ,H ) (a J, a µ, a 4 λ, ah) (1.1.6) 3. ρ = exp( βh) H 4. J /(kt) H /(kt) 1.1. Λ Z d ρ lim Λ Z d ρλ (1.1.7) P.B.C. J >0 ρλ Λ- 3

4 1. Definition X 1, X,..., X n truncated expectation X 1 ; X ;...; X n [ n log exp h i X i] (1..1) h 1 h h n i=1 h1=h =...=h n=0 [ n ] exp h i X i X i i=1 X 1 X...X n = Xi1 ; X i ;...; X ip P p=(i 1,i,...,i p) P (1..) truncated expectation P 1,,...,n Definition 1... Definition 1..1 X i = ϕ xi function n- Ursell function connected correlation u n (x 1,x,...,x n )= ϕ x1 ; ϕ x ;...; ϕ xn (1..3) (J, µ, λ, H) G(x, y) ϕ 0 ; ϕ x ϕ 0 ϕ x ϕ 0 ϕ x two point function (1..4) M s lim ϕ 0 H 0 spontaneous magnetization (1..5) χ ϕ 0 ; ϕ x x susceptibility (1..6) 1 ξ lim log ϕ 0 ϕ ne1 n n ξ correlation length (1..7) (1..7) e 1 (1..7) reflection positivity [1] u 4 ϕ 0 ; ϕ x ; ϕ y ; ϕ z { ϕ 0 ϕ x ϕ y ϕ z ϕ 0 ϕ x ϕ y ϕ z ϕ 0 ϕ y ϕ x ϕ z ϕ 0 ϕ z ϕ x ϕ y } (1..8) x,y,z x,y,z H = λ =0I A ρ(φ) = 1 Z exp 1 ϕ x A x,y ϕ y + H ϕ x (1.3.1) x,y I x I 4

5 A ϕ x = H y I ( A 1 ) x,y (1.3.) ϕ x ; ϕ y ϕ x ϕ y ϕ x ϕ y = ( A 1) x,y (1.3.3) Wick Theorem (Wick ). ϕ u n 0 (n 3) (1.3.4) ϕ 4 - (1.1.3) (1.1.4) λ =0 A x,y =(µ +dj)δ x,y Jδ x y,1, ˆD(k) 1 d ϕ x ; ϕ y = ( A 1) x,y = µ>0,h =0 [ π,π) d d d k (π) d d cos k j (1.3.5) j=1 e ik (x y) µ +dj{1 ˆD(k)} [ π,π) d d d k (π) d eik (x y) Ĉ(k) (1.3.6) χ = 1 µ C(0,x) ϕ 0 ϕ x x (1.3.7) d d k e ikx (π) d µ + J k µ x e const. = ξ µ 1/ (1.3.8) µ = H =0 const C(0,x) x d (d>) (d ) (1.3.9) µ c =0, γ =1, ν = 1, η =0 (1.3.10) 1.3. ϕ 4 - ϕ 4 -model (1.1.3) (1.1.4) J>0,λ>0 µ Simon[] M s χ µ c ξ µ 5

6 Theorem d>1 J, λ µ c (J, λ) µ>µ c (J, λ) ϕ 3 G(0,x) Ce m x, ( C(µ, J, λ), m(µ, J, λ) > 0) (1.3.11) M s =0, χ <, ξ < (1.3.1) µ<µ c (J, λ) 4 G(x, y) ɛ(µ, J, λ) > 0 ( x, y Z d ) (1.3.13) χ =, ξ =, M s > 0, (1.3.14) χ, ξ as µ µ c (1.3.15) µ µ = µ c µ c µ c (critical phenomena, critical behaviour) critical exponents γ,ν,β,η,δ, 4 χ(µ) (µ µ c ) γ, ξ(µ) (µ µ c ) ν, u 4 (µ µ c ) ( 4+γ) (µ µ c ; H 0) (1.3.16) M s (µ) (µ c µ) β, (µ µ c ; H 0) (1.3.17) G(0,x) x d+ η ( x ; µ µ c,h 0) (1.3.18) ϕ 0 H H 1/δ (H 0; µ µ c ) (1.3.19) log f(x) x a f(x) g(x) lim x a log g(x) =1 ϕ 4 d ϕ 4 universality d >4 Mean Field Values γ =1, ν = β = 1, η =0, δ =3, 4 = 3 (1.3.0) ( η)ν = γ, γ +β = β(δ +1) (scaling law d ) (1.3.1) dν = 4 γ = γ +β (hyperscaling law d <4 ) (1.3.) 3 µ 1 a = 1/µ (J/µ, 1, λ/µ ) (0, 1, 0) ρ(φ) x exp( ϕ x/) product measure 4 a = µ/(λ) ( J µ λ, µ λ, µ 4λ ) Peierls argument 6

7 .1 Mean Field Theory MF nearest neighbour model (1.1.) 1 ( J dη(ϕ x ) exp ϕ x ϕ y + H ) ϕ x ( ) (.1.1) Z 4 x x y =1 x η (.1.1) 0 Z d M ϕ 0 MF (.1.1) ϕ 0 ϕ 0 H eff = J ϕ y + H y: x y =1 ϕ y = ϕ 0 = M ϕ y M R ϕ y (.1.1) ( J ϕ 0 exp ϕ x ϕ y + H ) ϕ x dρ 0 (ϕ x ) x y =1 x x ( J = dρ 0 (ϕ y ) exp ϕ x ϕ y + H ) [ ( ] ϕ x dρ 0 (ϕ 0 ) exp ϕ 0 ϕ 0 Jϕ y + H )ϕ 0 y:y 0 x 0 x,y: x y =1 x,y 0 ( J dρ 0 (ϕ y ) exp ϕ x ϕ y + H ) [ ϕ x y:y 0 x 0 ϕ 0 = y:y 0 ( J dρ 0 (ϕ y ) exp x,y: x y =1 x,y 0 x,y: x y =1 x,y 0 ϕ x ϕ y + H x 0 ϕ x ) [ ( dρ 0 (ϕ 0 ) exp ϕ 0 ( dρ 0 (ϕ 0 ) exp ϕ 0 y: y =1 y: y =1 y: y =1 ] ϕ 0 Jϕ y + H )ϕ 0 (.1.) ) ] (.1.3) ϕ 0 Jϕ y + H ϕ 0 H eff J ϕ y + H y: x y =1 (.1.4) ϕ 0 ϕ y M Remark. 7

8 ϕ y M (.1.4) y: x y =1 ϕ y dm ϕ y M y: x y =1 ϕ y dm {ϕ y } y: y =1 {ϕ y } y: y =1 d...1 H = H 0 + V dφ e H F F H dφ e H (..1) V ρ 0 (Φ) = e H0(Φ) dρ0 (Φ) ( ), ρ0 (..) dρ0 (Φ) 1..1 truncated expectation X 1,X X 1 ; X 1 ; ; X }{{ 1 ; X } ; X ; ; X }{{ = m n } h m 1 h n log e h1x1+hx h1=h =0 m n log e h1x1+hx = h m 1 h n X 1 ; X 1 ; ; X 1 m! n! m,n 0 m+n 1 } {{ } m ; X ; X ; ; X }{{} n (..3) (..4) (..4) Fe V ρ F H = 0 e V = ρ0 h log e hf λv ρ 0 h=0,λ=1 = h m ( λ) n F ; F ; ; F ; V ; V ; ; V ( 1) n h m! n! }{{}}{{} ρ 0 = F ; V ; V ; ; V m,n=0 h=0,λ=1 n! }{{} ρ 0 m n n=0 n (..5) 8

9 / h m =1 log e V ρ 0 = ( 1) n V ; V ; ; V (..6) n! }{{} ρ 0 n=1 n.. ϕ 4 H 0 J x ϕ y ) 4 x y =1(ϕ, V x ( µ ϕ x + λ ) 4! ϕ4 (..7) ρ 0 = e H0 (..5) Wick ϕ ρ 0 C xy ϕ x ϕ y ρ0 ϕ 0 ϕ x = C 0x ( ) λ C 0y C yx C 00 + µ + O(λ ) (..8) y d 4 d > Block Spin Transformation (BST) Renormalization Group Transformation, RGT BST marginal distribution 5 RGT BST BST marginal distribution 6 marginal distribution 5 6 x, y ρ(x, y) y x ρ x marginal distribution ρ(x, y) y η(x) ρ(x, dy) marginal distribution 9

10 BST ρ ρ R L,θ Λ L N, N 1 sites L L>1 Lx { B x x Z d : x Lx < L } (3.1.1) Lx Λ L x L N 1 x } Λ {x Z d : x < LN 1 (3.1.) x' y' z' Lx' Ly' Lz' Λ L = 3 Λ' Λ {ϕ x } x Λ {ϕ x } x Λ ϕ x L θ (3.1.3) y B x ϕ y θ R {ϕ x } x Λ [ ρ ({ϕ }) (R L,θ ρ)({ϕ }) ρ({ϕ}) (ϕ x ) ][ ] L θ ϕ x dϕ x x Λ δ x B x x Λ ρ (3.1.4) δ δ-ρ ρ {ϕ} {ϕ } ρ (3.1.3) Remark. 1.. B x BST 10

11 (a) L θ+d (b) x L 1/L 3. θ θ = d θ = d/ θ BST 3.1. BST R L,θ R L1,θ = R LL 1,θ (3.1.5) ( { F ({ϕ } x Λ ) ρ ({ϕ }) F = L θ } ϕ y y B x x Λ ) ρ({ϕ}) (3.1.6) BST ρ ϕ 0 ϕ 0 ϕ 0 = 1 ϕ x (3.1.7) Λ x Λ BST [3, 4] BST BST BST n ρ (n) R n ρ L nd ρ (n) n BST BST fixed point flow flow

12 3.1.4 R L,θ ρ R L,θ (ρ )=ρ (3.1.8) RGT fixed point θ θ relevant, irrelevant, marginal operators ρ ρ = ρ + δρ ρ = ρ (1 + η) RGT 8 Definition ( ). ρ BST R H e H ρ, R(e H )=e H (3.1.9) R(e (H +ɛf) )=e (H +αɛf+o(ɛ )), α 0 (3.1.10) f BST R ρ f Φ α α L κ κ R Definition 3.1. ( ). ρ BST R R(ρ f)=αρ f (3.1.11) f BST R ρ f Φ ρ f Φ ρ (Φ)f(Φ) ρ f Remark. ɛ 9 R ρ (3.1.10) ɛ ( ) LHS of (3.1.10) = R e H ɛf e H + O(ɛ ) = R(e H ) ɛr(f e H )+O(ɛ ) (3.1.1) RHS of (3.1.10) = e [ H 1 αɛf + O(ɛ ) ] = e H αɛfe H + O(ɛ ) (3.1.13) ρ = e H (3.1.11) α α>1 κ>0 relevant α =1 κ =0 marginal 0 α<1 κ<0 irrelevant (3.1.14) ρ BST irrelevant 10 8 BST 9 η O(ɛ ) ρ f = F ρ Gaussian fixed point 10 irrelevant dangerously irrelevant operators

13 3. BST i.i.d Trivial i.i.d.- CLT i.i.d. identical independent distribution BST (1.1.3) J 0 ρ(φ) = x η(ϕ x ) (3..1) ϕ x BST ϕ x θ (ϕ x + ϕ x+1 ) (3.1.4) L = ρ (Φ )= η (ϕ x), η (ϕ )= θ dϕ η(ϕ) η( θ ϕ ϕ) (3..) x BST n- η η n+1 (ϕ )= θ dϕ η n (ϕ) η n ( θ ϕ ϕ) (3..3) Fourier CLT ˆη n+1 (k) = [ˆη n ( θ k) ] ], [ˆη(k) dϕ e ikϕ η(ϕ) (3..4) g n+1 (k) =g n ( θ k), [g n (k) log ˆη n (k)] (3..5) g n (k) = n g 0 ( nθ k) (3..6) CLT η θ =1/ g n (k) g 0 (0) σ k CLT σ ϕ; ϕ (3..7) BST 3.. CLT [5] Section II BST CLT fixed point fixed point 3..3 BST BST 13

14 (3.1.6) BST Φ BST BST Wick mean covariance ϕ 0; ϕ x = L θ ϕ y ; ϕ z y, z Lx < L = L (θ+d) [ πl,πl] d n- ϕ (n) 0 ; ϕ(n) x = L (θ+d)n d d k (π) d [ πl n,πl n ] d e ik x µ +dj{1 ˆD(kL 1 )} d d k (π) d e ik x d j=1 µ +dj{1 ˆD(kL n )} [ ] k sin j (3..8) sin kj L d j=1 [ ] k sin j (3..9) sin kj L n n- BST mean, covariance 11 [ ρ (n) (Φ (n) )= 1 Z exp 1 ( ϕ (n) x A (n)) x,y ϕ(n) y + HL ] θ ϕ (n) x (3..10) x,y x [(A (n))] 1 = ϕ (n) 0 ; x,y ϕ(n) x of (3..9) (3..11) θ ρ (n) ρ θ H =0 A (n) ϕ (n) covariance (3..9) n θ (3..9) x = O(1) k O(1) n ϕ (n) 0 ; ϕ(n) x L (θ d)n R d d d k (π) d Case (1) µ>0 θ = d ϕ (n) 0 ; ϕ(n) x 1 d d k eik x µ R (π) d d d j=1 e ik x µ + J k L n d j=1 [ ] k sin j (3..1) k j [ ] k sin j (3..13) k j BST i.i.d Case ( ) µ =0 θ = d+ ϕ (n) 0 ; d d k e ik x ϕ(n) x R (π) d J k d d j=1 [ ] k sin j (3..14) k j covariance d d k e ik x C cont (0,x)= (π) d J k R d 11 H (3.4.8) (3..15) 14

15 0, x 1 µ >0 θ = d+ BST ϕ (n) 0 ; ϕ(n) x R d d d k (π) d e ik x µl n + J k d j=1 [ ] k sin j (3..16) k j µ µl n µ ρ* θ = d +, κ = (3..17) BST ϕ 4 - ϕ 4 - BST ϕ 4 - ϕ 4 - BST 3.3. ϕ 4-1 d ϕ ϕ 4 - flow 15

16 λ (n) λ (n) λ 0 µ (n) 0 µ µ (n) 1: ϕ 4 flow d<4 d>4 1.. ϕ θ θ θ Rρ = ρ (3.4.1). ρ relevant R(ρ f 1 )=L κ ρ f 1 (3.4.) ϕ (3.1.11) α α L κ κ marginal irrelevant ρ µ µ µ µ = µ ρ crit BST n- ρ (n) crit ρ(n) n crit ρ 16

17 . µ µ ρ (n) ρ (n) relevant operator (µ µ )L κn ϕ 4 - relevant ϕ θ η µ = µ µ = µ c crit ϕ ϕ (n) ϕ (n) x (n) L nθ x: x L n x (n) < L ϕ x (3.4.3) (3.1.6) ϕ (n) 0 ϕ(n) y = L nθ ρ (n) crit x: x < Ln ϕ x ϕ z ρcrit (3.4.4) z: z L n y < Ln ϕ x ϕ z ρcrit y, z ϕ x ϕ z ρcrit ϕ 0 ϕ Ln y ϕ 0 ϕ L n y ρcrit L n(d θ) ϕ (n) 0 ϕ(n) y (3.4.5) ρ (n) crit ρ (n) crit ρ y n (3.4.5) ϕ (n) 0 ϕ(n) y ϕ 0ϕ y ρ ρ (n) crit (3.4.6) ρ (n) crit ρ reasonable 0 < ϕ 0 ϕ y < ρ (3.4.7) ϕ 0 ϕ Ln y ρcrit L n(d θ) O(1) (3.4.8) x = L n y x ϕ 0 ϕ x ρcrit x (d θ) as x (3.4.9) η η =θ d (3.4.10) l = L N 1 17

18 3.4. γ ν I off-critical t µ µ µ = µ + t ρ ρ t [ ρ t ρ crit exp t ] ϕ x (3.4.11) x ρ t t 1 BST n- n 1 tl nκ O(1) ρ (n) t ρ exp [ tlnκ ( ) ] ϕ (n) x x (3.4.1) n = n(t) tl nκ = O(1) nκ log L t (3.4.13) n- BST [ ρ (n) t ρ exp O(1) ( ) ] ϕ (n) x (3.4.14) x ρ (n) t critical ϕ O(1) ξ (n) ξ (n) = O(1) (3.4.15) ξ t = L n ξ (n) = O(L n ) O(t 1/κ ) (3.4.16) ν = 1 κ (3.4.17) γ θ ϕ (n) ρ (n) t ρ ρ (n) t O(1) ϕ (n) χ (n) ϕ (n) 0 ϕ(n) = O(1) (3.4.18) x ρ (n) (n) x (n) (3.1.6) χ = L n(θ d) χ (n) = O(L n(θ d) )=O(t (θ d)/κ ) (3.4.19) γ = θ d κ (3.4.0) 18

19 3.4.3 II renormalized coupling g ren u 4 χ ξ d (3.4.1) BST g (n) ren = g ren. n- BST u (n) 4 ( ϕ (n)) 4 λ (n) Case (1): λ (n) λ > 0 γ, ν n- BST (n) u 4 λ = O(1). (3.1.6) u 4 = L (4θ d)n u (n) = O(L (4θ d)n )=O(t 4θ d κ ) (3.4.) 4 + γ = 4θ d κ 4 4 = θ κ (3.4.3) Case (): λ (n) 0 ϕ 4 irrelevant ρ (n) ρ gauss θ, κ θ = d +, κ = λ (n) gaussian fixed point perturbation (3.4.4) λ (n) λ (n 1) L 4(θ d)+d λ (n 1) L 4 d O(L (4 d)n ) (3.4.5) u (n) 4 λ(n) = O(L (4 d)n ) (3.1.6) u 4 = L (4θ d)n (n) u O(L 8n ) O(t 4 ) (3.4.6) γ =4, 4 = 3 (3.4.7) δ µ = µ c ϕ x = L θ ϕ x H ϕ x = HL θ ϕ x (3.4.8) x: x Lx <L/ x x n- BST HL nθ [ ρ (n) ρ (n) crit exp HL ] nθ ϕ (n) x (3.4.9) x Case (1): λ (n) λ > 0 n H (n) = HL nθ = O(1) (3.4.30) 19

20 ϕ (n) effective potential λ (n) ( ϕ (n)) 4 H (n) ϕ (n) (3.4.31) ϕ (n) 0 ρ (n) ( H (n) λ(n) )1/3 = O(1) (3.4.3) (3.1.6) ( ϕ 0 ρ = L (d θ)n ϕ (n) 0 = O L (d θ)n) ( = O H (d θ)/θ) (3.4.33) ρ (n) δ = θ d θ (3.4.34) Case (): λ (n) 0 (3.4.4) H (n) = O(1) BST ϕ (n) effective potential ( ) H ϕ (n) (n) 1/3 λ(n) ( ) 1/3 1 (3.4.35) λ(n) Large Field Problem ϕ ϕ 6, ϕ 8 ϕ 4 - ϕ (n) = O(1) n H (n) λ (n) = HL nθ λl (4 d)n = L (d 4+θ)n λ/h (3.4.36) n (3.4.4) ϕ 0 ρ L (d θ)n ϕ (n) ρ(n) 0 (L = O d n) ( = O H 1/3) (3.4.37) δ = β ν, γ t L nκ = O(1) (3.4.38) n- BST χ (n) = O(1), ξ (n) = O(1) (3.4.39) ν = 1 κ, γ = θ d κ (3.4.40) β 0

21 Case (1): λ (n) λ > 0 n ( t)l nκ = O(1) (3.4.41) ϕ (n) effective potential λ (n) ( ϕ (n)) 4 + tl nκ ( ϕ (n)) (3.4.4) ϕ (n) 0 ρ (n) ( ) ( t)l nκ 1/ = O(1) (3.4.43) λ(n) ( ϕ 0 ρ = L (d θ)n ϕ (n) ρ(n) 0 = O L (d θ)n) ( = O t (d θ)/κ) (3.4.44) β = d θ κ (3.4.45) Case (): λ (n) 0 (3.4.4) ( µ n /λ n ) 1/3 = O(1) BST ϕ (n) effective potential ( ) ϕ (n) ( t)l nκ 1/ ( ) 1/ 1 0 (3.4.46) ρ (n) λ(n) λ(n) ϕ (n) = O(1) n ( t)l nκ λ (n) = ( t)l nκ λl (4 d)n = L (d )n λ t (3.4.47) n θ = d+ ϕ 0 ρ L (d θ)n ϕ (n) 0 = O (L d n) ( = O t 1/) (3.4.48) β =1/. ρ (n) Universality, Scaling, Hyperscaling θ, κ ρ ρ ρ ρ universality θ, κ ( η)ν = γ (3.4.49) βδ = 4 = β + γ (3.4.50) γ +β = β(δ +1)= 4 γ = 4 + β (3.4.51) scaling laws d <4 dν =δ 4 γ = γ +β (3.4.5) hyperscaling law

22 3.4.7 Tricritical Behaviour Relevant 3.5 θ {ϕ x } x Φ Φ Ω Φ ρ A Ω Prob(Φ A) T :Ω Ω A ρ(φ)dφ (3.5.1) T T L,γ : {ϕ x } x {L γ ϕ x } x {ϕ T x } x (3.5.) L γ Φ Φ T Prob(Φ T TA)=Prob(T Φ TA)=Prob(Φ A) (3.5.3) Ω A Φ T ρ T (Φ T ) Prob(Φ T A) ρ T (Φ T )dφ T (3.5.4) A ρ T ρ T (Φ T )dφ T = Prob(Φ T TA)=Prob(T Φ TA)=Prob(Φ A) = TA A ρ(φ)dφ (3.5.5) LHS Φ T = T Φ Φ LHS = ρ T (T Φ) ΦT dφ = ρ T (T Φ) (det T ) dφ (3.5.6) Φ A ΦT =T Φ A (det T ) ρ T (T Φ) = ρ(φ) (3.5.7) ρ T (Φ T )= 1 det T ρ(t 1 Φ T )=L γ Λ ρ(l γ Φ T ) (3.5.8) ρ T (Φ T )= ρ(φ)δ(φ T L γ Φ)dΦ (3.5.9) ad hoc

23 L>0, γ R T L,γ T L,γ : ϕ x L γ ϕ x ( x Λ) (3.5.10) ρ T L,γ Φ T (Φ )=(T L,γ ρ)(φ )= ρ(φ)δ(φ L γ Φ)dΦ =L γ Λ ρ(l γ Φ ) (3.5.11) ρ 1 ρ L, γ T L,γ ρ 1 = ρ (3.5.1) ρ 1 ρ ρ 1 ρ (3.5.13) H = J x ϕ y ) 4 x y =1(ϕ + [ µ ϕ x + λ ] 4! ϕ4 x x (3.5.14) J, µ, λ µ/j, λ/j J ρ 1, ρ T L,γ T L,γ (ρ 1 ρ )=L γ Λ T L,γ (ρ 1 ) T L,γ (ρ ) (3.5.15) 3.5. BST θ BST T L,γ ρ R L,θ T L,γ = R L,θ+γ = T L,γ R L,θ (3.5.16) Proof. (ŜΦ) x y B x ϕ y (R L,θ T L,γ ρ)(φ )=(R L,θ (T L,γ ρ)) (Φ )= (dφ )(T L,γ ρ)(φ ) δ(φ L θ ŜΦ ) [ ] ( ) = (dφ ) (dφ)δ(φ L γ Φ)ρ(Φ) δ(φ L θ ŜΦ )= (dφ)δ Φ L θ Ŝ(L γ Φ) ρ(φ) ( ) = (dφ)δ Φ L θ γ Ŝ(Φ) ρ(φ) = (R L,θ+γ ρ)(φ ) (3.5.17) 3

24 ρ = T L,γ ρ 1 (3.5.18) R L,θ (ρ )=T L,γ (R L,θ ρ 1 ) (3.5.19) ρ 1 ρ Rρ 1 Rρ. BST R L,θ1 (ρ) R L,θ (ρ) (3.5.0) R L,θ ρ = T L,(θ θ 1) (R L,θ1 ρ) (3.5.1) θ BST ρ Proof. Prop R L,θ ρ = R L,θ (T L,γ ρ 1 )=T L,γ R L,θ ρ 1 Prop ρ ρ BST R L R L ( ρ) R L,θ (ρ) R L,θ (ρ) (3.5.) θ BST ρ ρ R L ( ρ) = ρ (3.5.3) R L θ L J 1 1. ρ J = 1 θ BST ρ. ϕ J 1 ρ 3. ρ ρ θ ρ ρ θ 4

25 3.5.3 flow ρ R L,θ R L,θ (ρ )=ρ (3.5.4) 1. ρ ρ ρ θ R L,θ (T L,γ ρ )=T L,γ (R L,θ ρ )=(T L,γ ρ ) (3.5.5). θ θ BST ρ R L,θ ρ = T L,θ θ R L,θ ρ = T L,θ θ ρ (3.5.6) ρ θ Proof. Prop R R( ρ )= ρ θ,ρ ρ ρ R L,θ ρ = ρ (3.5.7) Proof. ρ ρ θ ρ ρ θ R L,θ ρ = ρ θ (3.5.8) ρ ρ θ γ(θ), s.t. ρ θ = T L,γ ρ (3.5.9) R L,θ ρ = T L,γ ρ (3.5.30) Prop T L,γ R L,θ γ ρ = T L,γ ρ = R L,θ γ ρ = ρ (3.5.31) θ = θ γ θ Prop ρ R L ( ρ )= ρ (3.5.3) Relevant, Irrelevant 5

26 ρ η R L,θ (ρ )=ρ, R L,θ (ρ η)=α(ρ η) (3.5.33) ρ 1 T L,γρ, η 1 T L,γ η (3.5.34) R L,θ (ρ 1 )=ρ 1, R L,θ (ρ 1 η 1)=α(ρ 1 η 1) (3.5.35) T L,γ ρ T L,γ Proof. (3.5.15) ρ 1 η 1 = T (ρ 1 )T (η 1)=L γ Λ T (ρ 1 η 1) (3.5.36) R(ρ 1 η 1)=R(L γ Λ T (ρ 1 η 1)) = L γ Λ RT (ρ 1 η 1)=L γ Λ T R(ρ 1 η 1) = L γ Λ Tα(ρ 1η 1 )=αl γ Λ T (ρ 1η 1 )=αρ 1η 1 (3.5.37) R L,θ φ = αφ R L φ = α φ (3.5.38) θ θ φ = ρ η Proof. φ φ 1, φ θ R L,θ (φ 1 ) αφ = γ, R L,θ (φ 1 )=T L,γ (αφ ) (3.5.39) φ φ 1, φ γ 1 φ = T L,γ1 (φ 1 ) (3.5.40) R L,θ (φ 1 )=T L,γ (αt L,γ1 (φ 1 )) = αt L,γ1+γ (φ 1 ) (3.5.41) Prop R L,θ γ1 γ (T L,γ1+γ φ 1 )=αt L,γ1+γ (φ 1 ) (3.5.4) φ T L,γ1+γ (φ 1 ), θ = θ γ 1 γ θ 6

27 4 4.1 Here we concentrate on three axioms of QFT, which will be relevant for our later studies. A general reference is [6] Mathematical Definitions Minkowski space: A space R d = {(x 0,x 1,..., x d 1 )}, with a special inner product (x, y) =x 0 y 0 d 1 i=1 xi y i. Note that this is not a proper mathematical inner product (indefinite metric). Lorentz transformation: A linear transformation Λ:x =(x 0,x 1,..., x d 1 ) R d x, x i =Λ i j x j on a Minkowski space, which leaves the Minkowski inner product invariant. The set of all Lorentz transformations (on a given Minkowski space) is called the full Lorentz group. Proper Lorentz group A subset of the full Lorentz group. Its elements are those Lorentz transformations Λ which satisfy Λ and det Λ = 1. This is written as L + Proper Poincare group: The set of all transformations (a, Λ) where a R d and Λ L +, defined by (a, Λ)x = λx + a. The set is written as P Wightman-Garding Axioms This is a very natural starting point for Axiomatic QFT. In essence, it proposes a framework of QFT in a usual Hilbert space, operator language. Axiom GW A quantum field theory (of neutral scalar field) consists of the following four: (1) H, aseparable Hilbert space, () Ω H,aunit vector (vacuum), (3) φ, amapping from S(R d )tothe set of linear operators on H (field operators), (4) U, astrongly continuous unitary representation of P + on H. Moreover, these four should satisfy the following axioms (GW1 GW5). (GW1) [This part lays foundation of QFT as a usual quantum mechanics.] There exists D 0 H, dense in H. And (we denote the inner product in H by <, >) (a) For each f S(R d ), the domain of φ(f) [as an operator on H] includes D 0. Ψ 1, Ψ D 0, Ψ 1,φ( )Ψ is a tempered distribution. (b) For real f S(R d ), φ(f) isasymmetric operator on D 0. I.e. Moreover, for any Ψ 1,φ(f)Ψ = φ(f)ψ 1, Ψ. (c) D 0 is an invariant subspace of H for each f S(R d ). I.e. Ψ D 0 implies φ(f)ψ D 0. (d) Ω D 0. Moreover, D 0 is a linear span of {Ω,φ(f 1 )Ω,φ(f 1 )φ(f )Ω,... f i S(R d )} 7

28 (GW) [This takes care of Poincare invariance.] (a) D 0 is an invariant subspace of U(a, Λ) for each (a, Λ) P +. I.e. U(a, Λ)Ψ D 0 if Ψ D 0. (b) U(a, Λ)Ω = Ω for all (a, Λ) P +. (c) The following identity holds as operators on D 0 :Forall (a, Λ) P + and for all f S(Rd ), U(a, Λ)φ(f)U(a, Λ) 1 = φ(f a,λ ) In the above, f a,λ f(λ 1 (x a)). (GW3) [Spectral Condition] Consider the set of all translations {U(a, I) a R d } P +. The joint spectrum of the generators of these translations is contained in the closure of the forward light cone: V + {p R d p 0 > 0, (p, p) > 0}. (GW4) [Locality or causality] If f,g S(R d ) are space-like separated, i.e. (x y, x y) < 0 for all x suppf and y suppg, then (as operators on D 0 ) [φ(f),φ(g)] φ(f)φ(g) φ(g)φ(f) =0. (GW5) [Uniqueness of the vacuum] Ωisthe unique translation invariant vector. I.e. it is the only vector which satisfies Ω = U(a, I)Ω Wightman s Axioms for Wightman Functionals (Green s functions) and Wightman Reconstruction Theorem In 1956, Wightman gave an another set of axioms, which he proved to be equivalent to the above one. It is written in terms of Green s functions of the field theory [called Wightman function(al)s]. It is stated as follows: Axioms for Wightman functions: A QFT (of neutral scalar fields) is a set of tempered distributions {W n S (R nd )} n=0, which satisfy the following (W1 W6). By convention, we always understand W 0 1. In the following, W n (x 1,x,..., x n ) represents the value of W n at (x 1,..., x n )asatempered distribution. I.e. W n (f) = W n (x 1,x,..., x n )f(x 1,x,..., x n )dx 1 dx dx n. We denote the complex conjugate of z by z. (W1) [neutrality] For any f S(R dn ), W n (f) =W n (f ) where f (x 1,x,..., x n ) f(x n,x n 1,..., x 1 ). (W) [Poincare covariance] For any (a, Λ) P + and for any f S(Rdn ), W n (f) =W n (f a,λ ) 8

29 (W3) [Positivity] We define a tensor product f g for f S(R dm ) and g S(R dn )as (f g)(x 1,..., x m+n )=f(x 1,..., x m )g(x m+1,..., x m+n ). Then, for any f 0 C,f 1 S(R d ),..., wehave N m,n=0 for N =0, 1,,... W m+n (f m f n ) 0 (W4) [Spectral Condition] The support of the Fourier transform F of W, defined by n 1 W n (x 1,..., x n )= F n 1 (p 1,..., p n 1 ) exp{i (p j,x j+1 x j )}dp 1 dp n 1 j=1 is contained in (a tensor product of) forward light cone. (W5) [Locality] For any n and for any x j,x j+1 such that (x j x j+1,x j x j+1 ) < 0, we have W n (x 1,..., x j,x j+1,...x n )=W n (x 1,..., x j+1,x j,...x n ) (W6) [Clustering Property] For any spacelike vector a R, we have lim λ W n(x 1,..., x j,x j+1 + λa,..., x n + λa) =W j (x 1,..., x j )W n j (x j+1,...x n ) Osterwalder-Schrader Axioms This is the set of axioms which will be directly relevant for our later studies. Euclidean Green s functions (called Schwinger functions). We begin with some definitions. It is a set of axioms about S (R dn ) {f S(R dn ) f and all its partial derivatives are zero on a hyperplane y i = y j } (4.1.1) S (Rdn ) {linear functionals on S (R dn )} (4.1.) S + (R dn ) { f S (R dn ) suppf {((t 1, x 1 ),..., (t n, x n )) R nd, 0 <t 1 <... < t n } } (4.1.3) Osterwalder-Schrader Axioms: Schwinger functions {S n } n=1 should satisfy: (OS1) [Temperedness] Each S n S (Rdn ). And S n (f) =S n (Θf ) for any f S (R dn ). Here (Θf)((t 1, x 1 ),..., (t n, x n )) = f(( t 1, x 1 ),..., ( t n, x n )) means time reversal. (OS) [Euclidean Invariance] Forany Λ, a rotation of R d, and for any a R d, S n (f) =S n (f (a,λ) ), f S (R dn ). (OS3) [Positivity] For any f 0 C, f j S + (R dj ), N S m+n (Θfn f m ) 0. m,n=0 9

30 (OS4) [Symmetry] S n is symmetric. I.e. S n (f) =S n (f π ) for any f π, which is obtained from f by changing the order of its arguments. (OS5) [Clustering Property] lim S m+n(f T t g)=s n (f)s m (g) t for f S (R dn ) C 0 (R dn ), g S (R dm ) C 0 (R dm ). Here (T t g)((t 1, x 1 ),..., (t n, x n )) g((t 1 t, x 1 ),..., (t n t, x n )) is a time translation. Important!! The above three axioms are equivalent!! 4. scaling limit OS Schwinger- Schwinger- 14 [6] 4..1 R d ɛz d Z d d- Z d {x =(x 1,x,..., x d ) R d x µ Z, 1 µ d} d- ɛ d- ɛ >0 ɛz d { } ɛz d x =( x 1, x,..., x d ) R d x µ ɛ Z, 1 µ d Z d x, y,... ɛz d R d x, ỹ,... x Z d ɛx µ- ɛx µ ɛz d ɛ ɛ 0 x R d x µ x µ /ɛ x Z d ɛx ɛz d ɛ 0 R d ɛz d 15 (a) 4.. ɛz d Schwinger- R d Schwinger- ɛz d ɛ >0 ɛ 0 Schwinger- Schwinger- OS ɛz d Schwinger- OS R d

31 ε = 1 cm ε = 1/ cm ε = 1/4 cm ε =0 cm (a) 1 cm 1 cm 1 cm 1 cm (b) 1 cm 1 cm 1 cm 1 cm (c) : (a) (b) I: (c) II: ɛ 31

32 Minkowski- Ṽ n- { W n ( x 1, x,..., x n ) [Dϕ]ϕ( x 1 )ϕ( x ) ϕ( x n ) exp i d d x ( ) } µ ϕ( x) µ ϕ( x) Ṽ (ϕ( x)) (4..1) [ ( d )] S n ( x 1, x,..., x n ) [Dϕ]ϕ( x 1 )ϕ( x ) ϕ( x n ) exp d d x { µ ϕ( x)} + Ṽ (ϕ( x)) µ=1 (4..) R d ɛz d S n ( x 1, x,..., x n ) [Dϕ]ϕ x1 ϕ x ϕ xn exp ɛ ( d ) d {ϕ x+ɛeµ ϕ x } + Ṽ (ϕ( x)) (4..3) ɛ 0 Schwinger N ɛ x ɛz d S n ( x 1, x,..., x n )=lim ɛ 0 N N ɛ ϕ x1,...,ϕ xn ρɛ (4..4) (4..3) ɛz d ϕ x (4..3) 17 (4..) (4..1) OS 18 ɛ 0 (4..3) (4..) n (b) (c) ɛ ɛ 0 µ=1 ɛ 4..4 (4..3) Schwinger m phys S (0, x) C exp ( m phys x ) (4..5) ɛz d x ɛz d x x ɛ Zd (4..6) 16 OS Schwinger- OS Schwinger- 17 Z d OS 3

33 x Z d x (4..5) ϕ 0 ϕ x ɛ C exp ( m phys x ) C exp ( m phys ɛ x ) (4..7) Z d ξ ξ = 1 m phys ɛ (4..8) ɛ 0 m phys (4..8) ξ Ṽ ɛ Zd - ϕ 4 - ɛ (4..) ϕ (4..8) Schwinger ϕ ϕ 0 ϕ x ɛ ɛ α exp ( m phys x ) (4..9) α >0 ϕ x ϕ x ɛ α ϕ x (4..10) S (ɛ) (0,x) ɛ α ϕ 0 ϕ x ɛ (4..11) ɛ α (4..4) N ɛ = ɛ α n- ɛ 4..6 Trivial 33

34 nontrivial Step 1 ɛ ɛz d ɛ dρ ɛ ({ϕ})( ) (4..1) Step ρ ɛ ɛ ɛ 0 Step 3 OS, Wightman Minkowski ρ ɛ 4.3 effective theory ρ ɛ effective theory ρ ɛ ρ ɛ ɛ 0 ɛ 1 ρ ɛ (4..4) Z d l ρ ɛ l/ɛ = ɛ l l cont ɛ lattice (4.3.1) ρ ɛ L n n >0 ρ (n) ɛ R n ρ ɛ (4.3.) ρ (n) ɛ n- = L n = ɛl n (4.3.3) (4.3.1) n- BST l l n-lattice cont ɛl n (4.3.4) 34

35 ε = 1 cm ε = 1/ cm ε = 1/4 cm ε =0 cm (a) 1 cm 1 cm 1 cm 1 cm 1 BST BST's BST's (b) 3: (a) (b) ɛl N 1 N L = BSR 3 ɛ 0 (4.3.3) BST ɛl n ρ (n) ɛ ɛl n effective theory 4.3. effective theory O(1) m ɛ 0 l ρ ɛ ɛl N 1 N log L ɛ (4.3.5) N N- BST (4.3.3) N- ɛl N 1 ρ (N) ɛ O(1) effective theory Schwinger (N) S (ɛ) ( x 1,..., x n ) ϕ (N) x 1,...,ϕ (N) x n (4.3.6) ɛ 19 x j O(1) (4.3.6) ρ (N) ɛ ρ ɛ ρ (N) ɛ 19 n- x j /(ɛl n ) ɛl N 1 x j (4..4) N ɛ N ɛ (4.3.6) N ɛ = L N(d θ) = ɛ θ d 35

36 λ ε=l ε=l 1 ρ eff 14 µ 4: ρ ɛ n =3, 4,... ρ L n ρ ɛ ρ ɛ 1. effective theory ρ eff. ɛ>0 ρ ɛ ρ (N) ɛ ρ eff N (4.3.5) 3. ɛ 0 ρ eff flow ρ ɛ ϕ nontrivial lim ɛ = ρ eff ɛl N 1 ɛ 0 ρ (N) (4.3.7) {ρ ɛ } ɛ>0 ɛ 0 N ρ eff BST ρ eff (4.3.7) flow ρ ρ flow Remark. ρ flow ρ 1 flow ρ ρ 1 flow 0 36

37 WF λ λ G µ G µ (a) (b) 5: ρ eff (a) ϕ 4 3 flow Gaussian fixed point ρ eff n =0, 1,,... 0 ρ eff ɛ = L n ρ ɛ (b) ϕ 4 5 flow Gaussian fixed point λ- ρ eff Gaussian fixed point µ- λ ϕ ϕ 4 d 5 ϕ 4 3 flow Gaussian fixed point ρ eff 5 (a)uv IR ρ ɛ ρ WF ρ WF ρ G ρ WF ρ G ρ ɛ IR ρ WF ɛ 0 ρ WF ρ ɛ 0, 1,,... IR ρ eff UV gaussian fixed point G UV asymptotic free ρ G ρ WF ρ ɛ ɛ 0 ρ G IR ρ WF UV ρ G UV asymptotic free ϕ 4 5 flow Gaussian fixed point λ- ρ eff Gaussian fixed point µ- 5 (b) remark λ ϕ 4 - d- ϕ 4 37

38 4.4.1 ϕ µ ɛ,λ ɛ 5 (a) ρ ɛ recursion ϕ µ + O(λ) Wick ordering recursion λ = Lλ{1+O(λ)}, µ = L µ c 1 λ c µλ + O(λ 3,µ 3 ) (4.4.1) c 1,c n n ɛ λ ɛ,µ ɛ N effective couplings N log L ɛ µ (N) ɛ µ eff [ 0.1], λ (N) ɛ λ eff [ ] (4.4.) (4.4.1) 1 (4.4.) λ (N) (4.4.1) k λ (k) 1 λ (k) = L 1 λ (k+1) {1+O(λ (k) )} = L 1 λ (k+1) {1+O(λ (k+1) )} (4.4.3) λ (N) = λ eff λ (k) λ (N) L (N k) N l=k+1 {1+O(λ (l) )} (4.4.4) λ (k) λ (N) (L/) (N k) (4.4.4) λ (k) λ eff L (N k) {1+O(λ eff )} (4.4.5) λ ɛ = λ (0) λ eff L N {1+O(λ eff )} ɛλ eff {1+O(λ eff )} (4.4.6) µ µ (k) = µ(k+1) + c 1 ( λ (k) ) + O(λ 3,µ 3 ) L c λ (k) 1 L µ(k+1) + c 1 L λ eff L (N k) + c λ eff {1+O(λ eff )}L (N k) µ (k+1) + O(λ 3,µ 3 ) (4.4.7) (4.4.5) ζ (k) µ (k) L (N k) ζ (k) = ζ (k+1)[ 1+c λ eff {1+O(λ eff )}L (N k)] + c 1 L (λ eff) + O(L (N k) ) (4.4.8) N 1 [ ζ (k) = ζ (N) + ζ (l+1) c λ eff {1+O(λ eff )}L (N k) + c ] 1 L (λ eff) l=k (4.4.9) ζ (N) = µ (N) = µ eff ζ (k) µ eff + O(λ eff (N k)) N,k ζ (k) = µ eff + O(λ eff )+ c 1 (N k) (4.4.10) L λ eff 1 ϕ 6 38

39 O(λ eff ) O(µ eff λ eff + λ eff ) µ ɛ = µ (0) = L N ζ (0) = L N µ eff + L N O(µ eff λ eff + λ eff)+l N c 1 λ effn = ɛ [ µ eff + O(µ eff λ eff + λ eff )+c 1L λ eff log L ɛ ] (4.4.11) µ ɛ,λ ɛ λ ɛ = ɛλ eff {1+O(λ eff )} (4.4.1) µ ɛ = ɛ [ µ eff + O(µ eff λ eff + λ eff )+c 1L λ eff log L ɛ ] (4.4.13) 4.4. ϕ φ 4 3 ϕ 4 Wick ordering ρ ɛ H ɛ H ɛ = 1 x ϕ y ) 4 x y =1(ϕ + x [ µ ɛ ϕ x + λ ɛ 4! : ϕ4 x :] (4.4.14) (4.4.1) (4..4) N ɛ S ɛ (0, x) N ɛ ϕ 0ϕ x ρɛ (4.4.15) ϕ( x) N ɛ ϕ x ϕ( x) N ɛ ρ eff S ɛ (0, x) ϕ (N) 0 ϕ (N) x (4.4.16) ρ (N) ɛ (4.3.6) ϕ (N) 0 ϕ (N) x L (d θ)n ϕ 0 ϕ x ρɛ (4.4.17) ρ (M) ɛ (4.4.15) (4.4.17) N ɛ = L (d θ)n = L N/ = ɛ 1/ (4.4.18) d =3,θ =(d +)/ ρ ɛ ϕ ϕ ϕ( x) ɛ 1/ ϕ x (4.4.19) x = ɛx 39

40 (4.4.1) (4.4.19) = ɛ d x (4.4.14) Z B Nɛ ɛ d =1 µ B Nɛ ɛ d µ ɛ = ɛ µ ɛ = µ eff + O(µ eff λ eff + λ eff)+c 1 L λ eff log L ɛ λ B Nɛ 4 ɛ d λ ɛ = ɛ 1 λ ɛ = λ eff (4.4.0) H ɛ d 3 x [ ZB { ϕ( x) } µ B + ϕ( x) + λ B 4! ] : ϕ( x) 4 : (4.4.1) wave function ϕ 4 4- triviality ϕ 4 4- trivial 5 BST 5.1 (Weak) ϕ 4 d- d Λ Λ 1 L 1 Λ Z d BST Λ Λ\LΛ 1. Λ x, y Λ 1 x 1,y 1 A, G site Φ (Aφ) x y Λ A x,yϕ y (Φ,AΦ) x,y Λ ϕ xa x,y ϕ y. Covariance G x,y N [ dµ G (Φ) N(dΦ) exp 1 ( ϕ x G 1 ) ] ϕ x,y y x,y (5.1.1) mean zero Λ 1 Λ Ĉ { Ĉ x1,y L (d+)/ 0 otherwise ( y Lx 1 <L/) (5.1.) covariance G 0 L, θ =(d +)/ BST 40

41 BST Φ (1) = ĈΦ covariance G 1 = ĈT G 0 Ĉ (5.1.3) Φ ϕ x = ψ x + Z x (5.1.4) ψ x Φ (1) Z x fluctuation 5.1. ψ ψ Φ (1) dµ G (Φ)/dΦ Φ Λ Λ 1 A A G 0 Ĉ T G 1 1 ψ x (Aϕ) x = A x,y1 ϕ (1) y 1 y 1 Λ 1 (5.1.5) (5.1.6) Ψ Ψ covariance AG 1 A T = G 0 Ĉ T G 1 1 G 1G 1 1 ĈG 0 = G 0 Ĉ T G 1 1 ĈG 0 (5.1.7) {Z} {Z} covariance Γ G 0 G 0 Ĉ T G 1 1 ĈG 0 Λ normal variables {z u } u Λ Λ Λ δ x,u (x LZ d ) Q x,u 1 (x LZ d, u Lx <L/) 0 (otherwise) Γ Λ Λ ( Q x,v Γ 1/) v,u M x,u v Λ (5.1.8) (5.1.9) (5.1.10) Γ Γ Λ Λ Z Z x = u Λ M x,u z u (5.1.11) 41

42 A L (d )/ A = L (d )/ GĈT G 1 1 (5.1.1) ψ Aϕ (1) = L (d )/ ψ (5.1.13) ϕ = L (d )/ ψ + Mz (5.1.14) Kernel A, M L d A x,y1 = δ x1,y 1, x B x1 L d x B x1 M x,y =0 (5.1.15) z 1 Λ 1 A x,z1 =1 (5.1.16) A, M β A = O(1), β Γ = O(L 1 ), C A = O(1), C Γ = O(L d+ ) A x,y1 C A exp [ β A L 1 x y 1 ] (5.1.17) C Γ exp [ β Γ u v ] (5.1.18) Γ 1/ u,v (5.1.15) ψ ϕ (1) ϕ x L (d )/ ϕ (1) x 1 + Z x (5.1.19) x B x Fractal 5.4 KT-type? Coulomb Gas? Non-gaussian fixed points 5.5. Non-perturbative Analysis (Global picture) Hierarchical Model Koch-Wittwer ϕ 4 3 ϕ 4 4 4

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