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1 1 Constructive field theory and renormalization group Hiroshi WATANABE , Euclidean field theory φ 4 model φ 4 model Regularization Counterterms Block spin dynamics Hierarchical model Characteristic function Lee Yang property Triviality Department of Mathematics, Nippon Medical School 1

2 spin. ( 0 ) macroscopic. 0 spin. (renormalization).,. ( ) (functional integral).. formulation. complex phase., Euclidean field theory. Euclidean field theory, formal, t ix(x R), Minkowski Euclid. Euclid,. framework. 1 Euclidean quantum field theory 2. 2

3 .. parameter.. 4 model,. 4,,. Euclidean field theory, [Section 2], φ 4 [Section 3], [Section 4]. (hierarchical model), [Section 5]. [16]. 2 Euclidean field theory Euclidean field theory, R d x random variable φ(x), F (φ) = 1 F (φ)e V (φ) dµ(φ) (2.1) Z Z = e V (φ) dµ(φ) (2.2) formal., dµ(φ) Gaussian probability measure, V (φ) φ( ) functional potential. Gauss potential. Gauss dµ(φ) spin potential V (φ) spin V =0 3, (2.1) Gauss, ( ). 3 φ bilinear form 3

4 2.1 (2.1)(2.2). Regularization Gaussian measure µ sample field φ regularity, φ V. Gaussian measure regularization. µ = µ ɛ (2.3) ɛ>0, measure µ ɛ 1, sample field φ regularity, V (φ) well-defined, ɛ 0, µ ɛ µ. regularization sample field cutoff. R d bounded region regularization. cutoff potential V. (UV) : x R d y R d (IR) : x R d y R d, cutoff. Renormalization regularization, ɛ 0,., ɛ (counterterm) U,ɛ (φ), V,ɛ (φ) =V (φ)+u,ɛ (φ) (2.4), V (φ) V,ɛ (φ). (renormalization). counterterm U,ɛ V. counterterm. Ω={a + bi a, b Z} 2 pole P (z) = ω Ω 1 (z ω) 2 (2.5). (2.5) well-defined (cutoff) P ɛ (z) = ω Ω ω <1/ɛ 1 (z ω) 2 (2.6) 4

5 . ɛ 0 ( Pɛ ren (z) = ω Ω 0< ω <1/ɛ 1 (z ω) 2 1 ω 2 ) + 1 z 2 (2.7). P ren ɛ well-defined (z) = lim ɛ 0 P ren ɛ (z) (2.8). P ɛ (z) Pɛ ren (z) = 1 ω 2 (2.9) ω Ω 0< ω <1/ɛ ɛ 0 (2.5) well-defined. (2.9) counterterm. counterterm. counterterm., regularization renormalization 1 F (φ),ɛ = F (φ)e V,ɛ(φ) dµ ɛ (φ) (2.10) Z,ɛ Z,ɛ = e V,ɛ(φ) dµ ɛ (φ) (2.11),. lim R d lim F (φ),ɛ (2.12) ɛ 0 (1) Limit, limit ( 0 ) (2) Gaussian measure (triviality) (3) Non-Gaussian measure (nontriviality) Counterterm, (1). counterterm (2) Gauss., (3). Minkowski x R d 1 x 1 Re x 1 > 0 Euclid Euclidean field theory Minkowski x 1.. Euclidean field theory Osterwalder- Schrader axioms [16, P.10]. 5

6 2.2 Well-defined (2.10)(2.11), ɛ 0( R d ) non-trivial (non-gaussian) Osterwalder-Schrader axioms,., 2 3 model, 4 nontrivial field theory,. φ 4 model model 4 Gauss (triviality). Triviality counterterm model. ρ N R N ρ N L N L/N ρ N (Ω) N +1 ( ) R N (Ω). N 0 ρ N,R N N a>0 f(a) = e ax2 dx (2.13) a = ib, b R,. 3 φ 4 model φ 4 model regularization renormalization. 6

7 3.1 φ 4 model d φ 4 model formal 4. F (φ) = 1 Z F (φ)e V (φ) dµ C (φ) (3.1) Z = e V (φ) dµ C (φ) (3.2) V (φ) = λ φ(x) 4 dx R d (3.3) λ>0, φ( ) Gaussian random field, µ C mean : φ(x)dµ C (φ) =0, x R d (3.4) covariance : φ(x)φ(y)dµ C (φ) =( + m 2 ) 1 (x, y) =C(x, y), x, y R d,m >0 (3.5). Gaussian covariance C, { C(x, y) const. log x y, d =2, const. x y d+2, d > 2, x y 0 (3.6) C(x, y) const. x y (d 1)/2 e m x y, x y (3.7), formal [2]. m mass. C(x, y) = Rd e ip(x y) p 2 dp (3.8) + m2, F (φ) = 1 Z dφf (φ)e A(φ) (3.9) A(φ) = 1 2 (φ, ( + m2 )φ)+ λφ(x) 4 dx (3.10) R d = ( 1 R d 2 φ(x) m2 φ(x) 2 + λφ(x) 4 )dx (3.11). A (action) A 0 φ 5. ( + m 2 )φ(x)+4λφ(x) 3 = 0 (3.12) 4 potential V φ 4, 2, (P (φ) 2 [8]). potential, Euclidean classical field theory 7

8 Feynman-Kac formula d =1, x s (3.9), ( 1 dφf (φ) exp ( 1 ) Z R 2 φ(s) 2 + m2 2 φ(s)2 + λφ(s) 4 )ds Feynman-Kac formula. Feyman-Kac formula (3.13) H = + U (3.14) Wiener process b(t) E ( t ) e th f(x) = E f(x + b(t)) exp( U(x + b(s))ds) formal =. 6 Schwinger function 1 f(b(t)) exp( Z b(0)=x 0 t (3.1) (3.3). F (φ) field φ(x 1 )φ(x 2 ) φ(x n ), 0 (3.15) ( 1 2ḃ(s)2 + U(b(s)))ds)db (3.16) S n (x 1,x 2,,x n )= φ(x 1 )φ(x 2 ) φ(x n ) (3.17) Schwinger function. Model ( ), Schwinger function [2, 1]., (3.17)., regularize well-defined. 3.2 Regularization Gaussian measure µ C sample field φ 1., field φ(x) 4, φ potential V. Sample field (regularity), regularization. UV cutoff Gaussian covariance C momentum space cutoff Rd e ip(x y) C ɛ (x, y) = p 2 + m 2 χ ɛ(p)dp (3.18) χ ɛ (p) = e ɛ2 p 2, ɛ > 0 (3.19),. { const. log(1/ɛ), d =2, C ɛ (x, y) < const.ɛ d+2, d > 2, (3.20) χ ɛ. p (UV region) smooth cutoff. 6 (3.15) counterterm. d =1,. H Schrödinger. 8

9 Theorem 3.1. [6, P.28] Gaussian measure µ Cɛ sample field, 1 C S. IR cutoff UV cutoff, sample field φ(x) 4, V (φ). R d, potential V (φ) V (φ) =λ φ(x) 4 dx (3.21) IR cutoff, (3.1) well-defined. 1 F (φ),ɛ = F (φ)e V(φ) dµ Cɛ (φ) (3.22) Z,ɛ Z,ɛ = e V(φ) dµ Cɛ (φ) (3.23) ɛ 0 UV cutoff. ɛ 0. counterterm. 3.3 d φ 4 model. counterterm. Field φ Gauss dµ C = dy dzc ɛ (x 1,y)C ɛ (y, z) 3 C ɛ (z,x 2 ) (3.24) R d R d = dy C ɛ (x 1,y)C ɛ (x 2,y)C ɛ (y, z) 2 C ɛ (z,x 3 )C ɛ (z,x 4 ) (3.25) R d R d covariance C convolution. (3.24),(3.25) covariance line Feynman graph amplitude. Gauss Feynman graph. n j>0 connected Feynman graph G ω(g) =(4 d)j + d 2 2 n d (3.26). (3.24) n =2,j =2,ω =6 2d (3.25) n =4,j =2,ω = 4 d. Theorem 3.2. [16] ω(g) 0, G amplitude ɛ 0.., G subgraph, G amplitude. G subgraph (G ) connected F, ω(f ) ω(g)., F F, F G F F. 9

10 Theorem 3.3. [6, P.65] Graph G subgraph F, connected,, G amplitude ɛ 0. ω(f ) > 0 (3.27) (3.27) subgraph,. 3.4 Counterterms subgraph j 1 (3.28). subgraph (3.29), Wick product renormalize [16, P.27]. renormalized potential. V,ɛ (φ) =λ dx : φ(x) 4 : Cɛ (3.30) counterterm µ (0) ɛ. well-defined dxφ(x) 2, µ (0) ɛ = 6λC ɛ (0) (3.31) F (φ),ɛ = 1 e V F (φ) Cɛ Z,ɛ (3.32) Z,ɛ = e V Cɛ (3.33) UV limit ɛ 0, IR limit R 2. subgraph. subgraph j n 3 (3.34) (j, n) =(1, 2) : (3.35) (j, n) =(2, 2) : (3.36) 10

11 . counterterm, Wick product (3.31), counterterm µ (1) ɛ dxφ(x) 2, µ (1) ɛ =48λ 2 dyc ɛ (x, y) 3 (3.37). 3 φ 4 model counterterm µ ɛ dxφ(x) 2, µ ɛ = µ (0) ɛ + µ (1) ɛ (3.38) potential V,ɛ (φ) = λ = λ dxφ(x) 4 + µ ɛ dxφ(x) 2 (3.39) dx : φ(x) 4 :+µ (1) ɛ dx : φ(x) 2 : +const. (3.40). n 4 subgraph, n =2, 4., subgraph counterterm. λ ɛ dxφ(x) 4 + µ ɛ dxφ(x) 2 + ζ ɛ dx φ(x) 2 (3.41) 2 4, explicit., Section 4. 4 φ 4 model potential, V,ɛ (φ) = λ ɛ dxφ(x) 4 + µ ɛ dxφ(x) 2 + ζ ɛ dx φ(x) 2 (3.42). d>4 d>4, n j, ω(g) 0., counterterm. [7] 4, counterterm,. λ ( )... Schwinger function (3.17),.. 3 Section

12 3.1. (3.24) Feynman graph R d dx 1 Fourier kernel R d dx 2 exp (ip 1 x 1 + ip 2 x 2 ) A(x 1,x 2 )=δ(p 1 + p 2 )Â(p 1) (3.43). Â(p 1 ) UV limit (ɛ 0) d =2 d (3.37) µ (1) ɛ 48λ 2 µ (1) ɛ dyc ɛ (x 1,y)C ɛ (y, x 2 ) (3.44) Fourier kernel UV limit d =3 d (3.40) potential S,ɛ (x 1,x 2 ) = φ(x 1 )φ(x 2 ) exp( V,ɛ ) Cɛ exp( V,ɛ ) Cɛ, x 1 x 2 (3.45). λ λ 2 d =3 S,ɛ UV limit. 4 model scale. φ 4 model. 4.1 Block spin. S N = N (4.1) N. 1 2k k = 1 k + 1 (2k 1)(2k), k N (4.2) 0 <α<β. S 2N = S N + r N (4.3) α<r N <β (4.4) 1+αn S 2 n 1+βn, n N (4.5) scale. N R : N N R(2k 1) = R(2k) =k, k N (4.6) 12

13 N a : N R R (Ra)(k) = a(j) (4.7).. 2 n k=1 j:r(j)=k a(k) =(R n a)(1) (4.8) 2 n. a(k) =1/k R b scale Ra = a + b (4.9) α<(r n b)(1) <β (4.10). (R n a)(1) O(n). Z d R d random field. Block spin spin. Ising spin ( 4.1) (1) up spin down spin (a) (2) 3 3 block up spins gray level (b). (3) 1/3 (c) (4) up/down (a) (d). 7 spin microscopic macroscopic. gray level block spin gray level block spin (Section 5 ). momentum space scale. 4.2 Scale D covariance Gauss dµ D (φ) H(φ). dµ D (φ)h(φ) (4.11) 7 (c) (d) (b). 13

14 (a) Ising spin (b) Block spin (c) Block spin (d) Ising spin 4.1: Ising spin block spin D D = D 0 + D 1 (4.12) D 0,D 1 covariance Gaussian variable φ 0,φ 1 H (φ 0 ) = dµ D1 (φ 1 )H(φ 0 + φ 1 ) (4.13) dµ D (φ)h(φ) = dµ D0 (φ 0 )H (φ 0 ) (4.14). φ 1 UV ( ), φ 0 IR ( ),, H H (4.15) UV, IR marginal distribution. Iteration Gaussian covarinace D D = N D k (4.16) k=0 14

15 . D k covariance Gaussian variable φ k (x), φ k,0 = D k,0 = k φ j, k =0, 1, 2,,N (4.17) j=0 k D j, k =0, 1, 2,,N (4.18) j=0, φ k,0 D k,0 covariance Gaussian variable. Gauss (4.11) H N = H (4.19) H k 1 (φ k 1,0 ) = dµ Dk (φ k )H k (φ k + φ k 1,0 ), k = N,N 1,, 1 (4.20) dµ D (φ)h(φ) = dµ D0 (φ 0 )H 0 (φ 0 ) (4.21). (4.15), φ 0 H, φ 0 φ 1,. k, φ k (x)., φ k (x) covariance.,,. H k φ k (x) =L k(d 2)/2 φ k (L k x) (4.22) D k (x, y) =L k(d 2) D k (L k x, L k y) (4.23) φ k,0 (x) = L k(d 2)/2 φ k,0 (L k x) (4.24) D k,0 (x, y) = L k(d 2) D k,0 (L k x, L k y) (4.25) φ k,0 (x) = φ k (x)+l (d 2)/2 φk 1,0 (L 1 x) (4.26) D k,0 (x, y) = D k (x, y)+l (d 2) Dk 1,0 (L 1 x, L 1 y) (4.27) H k (φ k,0 )= H k ( φ k,0 ) (4.28) 15

16 H k 1 ( φ k 1,0 ) = dµ Dk ( φ k ) H k ( φ k ( )+L (d 2)/2 φk 1,0 (L 1 )) (4.29) H N ( φ N,0 ) = H(φ N,0 ) (4.30) dµ D (φ)h(φ) = dµ D0 ( φ 0 ) H k ( φ 0 ) (4.31). H k 1 H k,.. UV limit UV cutoff covariance. N ( ). covariance D (4.16) H k H k 1 (4.32) Rd e ip(x y) D(x, y) = p 2 + m 2 χ(l N p)dp (4.33) χ(p) = e p2 (4.34) D k (x, y) = D 0 (x, y) = Rd e ip(x y) p 2 + m 2 (χ(l k p) χ(l (k 1) p))dp, k =1, 2,,N (4.35) Rd e ip(x y) p 2 χ(p)dp (4.36) + m2 φ k, momentum scale L k ( L k ) Gaussian variable. (4.23) covarinace D k (x, y) = Rd e ip(x y) p 2 +(m/l k (χ(p) χ(lp))dp (4.37) ) 2, D k k, φ k k O(1). UV limit UV cutoff (4.33) N. UV limit. H n H n 1 H 0 (4.38) 16

17 4.3 dynamics (3.40) (3.42) potential V (φ) V N (φ) = V (φ) (4.39) H N (φ) = exp( V N (φ)) (4.40). H N (φ) H N ( φ N,0 ) = exp( ṼN( φ N,0 )) (4.41). H N 1 ( φ N 1,0 ) = dµ DN ( φ N ) H N ( φ N ( )+L (d 2)/2 φn 1,0 (L 1 )) (4.42), scale dµ DN ( φ N ) H N (L (d 2)/2 φn 1,0 (L 1 )) = H N (L (d 2)/2 φn 1,0 (L 1 )) (4.43),, Gauss contract., step,, scale contract. Relevant part scale. φ m R d dx φ k,0 (x) m (4.44) scaled term ( ) m dx L (d 2)/2 φk 1,0 (L 1 x) = L d m(d 2)/2 dx φ k 1,0 (x) m (4.45) R d R d d m(d 2)/2 > 0 (relevant) d m(d 2)/2 =0 (marginal) d m(d 2)/2 < 0 (irrelevant) d>2, irrelevant d =2, relevant. d>2, relevant/marginal terms explicit 17

18 . Gaussian contraction (4.42) Gauss contract. (4.41) Taylor H N ( φ N,0 ) = ( ṼN( φ N,0 )) k k=0 k! (4.46) contraction. k =2 = dx dy φ N 1,0 (x) φ N 1,0 (y) D N (x, y) 3 (4.47) nonlocal interaction. H n Taylor (4.44). H n H n 1. counterterm.. (4.44) H n a n n. 1 a n 1. scale L α a n b n. a n 1 = L α a n + b n, n = N,N 1,, 1 (4.48) a n relevant α>0 marginal α =0 irrelevant α<0. a 0 a 0 = L Nα a N + N L (n 1)α b n (4.49) n=1. UV limit N lim a 0 = lim N N LNα a N + L (n 1)α b n (4.50). a N N. 8 b n n ( ) 0 a n irrelevant (α <0) (4.50) 2. (4.50) a N 2. relevant marginal (α 0) (4.50) 2. a N. 8 a n,b n N a n,n,b n,n,n = N,N 1,, 0 lim N a n,n. n=1 18

19 a N counterterm. counterterm (4.50) (4.49). (4.48) 1 2 inductive. φ 4 model a n b n. Large field bound Taylor., φ 4 potential 1 I(λ) = exp( λx 4 1 R 2 x2 )dx (4.51).. ( λ) n I(λ) x 4n exp( 1 n! R 2 x2 )dx = λ n a n (4.52) n=0 n=0 a n =( 1) n 2π(4n 1)!!/n! (4.53), (4.52) 0. λ. potential λ>0. φ 4 model 1 λ., φ 4 potential φ. large field bound., (relevant/marginal part) (irrelevant part) Large field bound 3., ( ) large field bound.,,,. coupling constant λ, formulation. [4, 16] (4.5) (4.10). 5 4 φ 4 triviality, (λ ) trajectory., φ 4 model (λ ) Ising model hierarchical [9, 10, 11, 12, 13],. [ [15] ] 19

20 5.1 Hierarchical model, 1 Ising model. spin long range interaction. Dyson [9], hierarchical model spin,. Hierarchical model, Gaussian measure model,,. Hierarchical model 2 N spin φ θ = φ θn,...,θ 1, θ =(θ N,..., θ 1 ) {0, 1} N (5.1), Hamiltonian H N (φ) = 1 2 N ( c n 4) n=1 θ N,...,θ n+1 θ n,...,θ 1 φ θn,...,θ 1 2 (5.2) single spin distribution h(φ θ ) 1 F N,h = dφf (φ) exp( βh N (φ)) h(φ θ ) (5.3) Z N,h θ Z N,h = dφ exp( βh N (φ)) h(φ θ ) (5.4) θ., h(x)dx = 1 (5.5) R. (hierarchical model)., s Ising spin measure, φ 4 single spin measure h Ising (x) = 1 (δ(x s)+δ(x + s)) (5.6) 2 h µλ (x) = const. exp( µx 2 λx 4 ) (5.7) µ = 2λs 2, λ (5.8), hierarchical Ising model. Hiercarchial Ising model, infinite volume limit, 0 <c<2 (5.9) 1 <c<2 (5.10) 20

21 , [9]. spin φ, β>0, β = 1 c 1 2 (5.11). Block spin Block spin φ, φ τ = c 2 φ τθ1, τ =(τ N 1,..., τ 1 ) (5.12) θ 1 =0,1 θ n,...,θ 1 φ θn,...,θ 1 = θ n,...,θ 2 c 2 φ θ N,...,θ 2 (5.13) H N (φ) = H N 1 (φ ) 1 2 τ φ 2 τ (5.14), F (φ) block spin F (φ) = F (φ ) (5.15), F N,h = F N 1,Rh (5.16) Rh(x) = const. exp( β 2 x2 ) dyh( x + y)h( x y),x R (5.17) R c c., h R., h 0, R trajectory. Gaussian trajectory Single spin measure Gauss, trajectory h n = R n h 0, n =0, 1, 2,... (5.18) h 0 (x) = const. exp( α 0 2 x2 ) (5.19) R n h 0 (x) = const. exp( α n 2 x2 ) (5.20) α n+1 = 2 c α n β (5.21) 21

22 ., (5.11) α n = ( 2 c )n (α )+1 2 (5.22)., α N > 0 (5.23) well-defined, α 0 > 1 2 (5.24) infinite volume limit. α 0 = 1 2, h G (x) = const. exp( 1 4 x2 ) (5.25) R fixed point, massless Gaussian measure. Dimensionality m =1, 2,..., N,. χ N,N,h spin susceptibility M m (φ) = φ θn,...,θ 1 (5.26) θ m,...,θ 1 χ m,n,h = 1 2 m M m (φ) N,h (5.27) χ = x φ(0)φ(x) (5.28). Block spin, χ m,n,h = 2 c χ m 1,N 1,Rh (5.29). h = h G, infinite volume limit N,. χ m,,hg = const.( 2 c )m (5.30) Z d (d>2) massless Gaussian model correlation decay φ(x)φ(y) const. x y d+2, x y (5.31) φ(0)φ(x) const.r 2 (5.32) x <r ( 2 c )m = r 2 (5.33) 22

23 , spin 2 m r d c = 2 1 2/d (5.34) β = 1 2 (22/d 1) (5.35). c (5.34), d hierarchical model. Fixed points (5.25) (Gaussian) fixed point, (non-gaussian) fixed point,.[10, 11, 12] d 4, ϕ G non-gaussian fixed point. d<4 d 4, ϕ G non-gaussian fixed point., d 4, ϕ G trajectory, trivial., Ising φ 4 model triviality, Ising φ 4 model ( Gauss ) trajectory., [10, 12] Gauss, trajectory., characteristic function. 5.2 Characteristic function Single spin distribution h n characteristic function ϕ n (ξ) = dxe iξx h n (x) (5.36) R. Ising spin, ϕ n+1 = Rϕ n (5.37) R = TS (5.38) c Sg(ξ) = g( 2 ξ)2 (5.39) T g(ξ) = exp( β )g(ξ) (5.40) 2 h 0 (x) = 1 (δ(x s)+δ(x + s)) (5.41) 2 ϕ 0 (ξ) = cos(sξ) (5.42) 23

24 . d = c =2,β =0, recursion ϕ n+1 (ξ) = ϕ n ( ξ 2 ) 2 (5.43). Ising spin ϕ n (ξ) = cos 2n ( sξ ) (5.44) 2n/2 exp( s2 2 ξ2 ), n (5.45), trajectory Gauss (trivial). (.) (5.45), ξ < 2 n/2 π 2s ξ ϕ n (ξ) = exp( V n (ξ)) (5.46) V n (ξ) = µ (n) 2j ξ2j (5.47), V n dual potential. V n, recursion j=1 µ (n) 2j = 2 1 j µ (n 1) 2j, j =1, 2, 3,... (5.48)., ξ 2 marginal, 4 irrelevant. Dual potential V n (ξ) ξ < 2 n/2 π 2s (5.49), continuum limit lim V n(ξ) = s2 n 2 ξ2 (5.50), R., continuum limit, characteristic function ξ =0., characteristic function, large field problem. Ising spin (5.41), dual potential V 0 (ξ) = s2 2 ξ2 + s4 12 ξ4 + s6 45 ξ6 + 17s ξ8 + (5.51)., (5.48), (d = ). d< d< (β >0), (5.40) operator T, dual potential., relevant/irrelevant terms. 24

25 d>4, ξ 2 relevant, ξ 4 irrelevant. d =4, ξ 2 relevant, ξ 4 marginal, ξ 6 irrelevant. d<4, ξ 2,ξ 4 relevant, d, relevant part., non-gaussian fixed point., trajectory,. [.] (a) d=2 (b) d= (c) d=4 (d) d=5 5.1: Ising trajectory, potential (µ 2,µ 4 /µ 2 2 ) plot. Gaussian fixed point (5.25) (1, 0). trajectory s, s =1.0, 1.1, 1.2,...,, d>2, µ (0) 2 = s 2 /2, µ (0) 4 /µ(0)2 2 =1/3 (5.52) µ (0) 2, trajectory (0, 0) ( ). µ (0) 2, trajectory ( ) µ (0) 2 (critical point), trajectory (0, 0) (critical trajectory). 25

26 5.3 Lee Yang property Newman s bound Ising dual potential., µ (n) 2j 0, j 1 (5.53) µ (n) 2j 1 j (2µ(n) 4 )j/2, j 3 (5.54) Characteristic function [ spin Lee-Yang property] [14]. Newman s bound (5.54) V n Taylor, n µ (n) 4 0,. µ (n) 4 0, 0, triviality. Gauss µ 2j., critical trajectory Gauss, computer. (5.53), operator T. (5.40). g t., V t g t (ξ) = exp( t )g(ξ) (5.55) T g = g β/2 (5.56) d dt g t(ξ) = g t (ξ) (5.57) g 0 (ξ) = g(ξ) (5.58) g t (ξ) = exp( V t (ξ)) (5.59). d dt V t = ( V t ) 2 V t (5.60) V t Taylor, (5.60) Taylor upper bound. [ upper bound, (5.60) lower bound.] 26

27 5.4 Triviality Characteristic function, [15]. d 4 hierarchical Ising model triviality idea. (1) Characteristic function Gauss,. (2) Ising critical trajectory, Gauss, computer. Model critical N =0, 1, 2,. 1 µ 2,N µ 4,N (5.61) s N = inf{s >0 µ 2,N 1}, (5.62) s N = inf{s >0 µ 2,N min{1+ 3 µ 4,N, 2+ 2}}. 2 (5.63) Bleher Sinai argument [10]. Proposition 5.1. N 0 N 1 N 0 N 1 s [s N1, s N1 ] 0 µ 4,N , (5.64) 1.6µ 2 4,N 0 µ 6,N0 6.07µ 2 4,N 0, (5.65) 0 µ 8,N µ 3 4,N 0, (5.66) µ 2,N < 2+ 2, N 0 N<N 1, (5.67). s c [s N1, s N1 ] s = s c. lim 4,N =0, N (5.68) lim 2,N = 1 N (5.69). Proposition 5.2. Proposition 5.1 N 0 =70and N 1 = 100. s N1 and s N s N1, s N (5.70) Proposition 5.1,Proposition 5.2. Theorem 5.3. d =4 s critical value s c [ , ] (5.71) s = s c trajectory Gauss. h (x) = exp( x 2 /4) (5.72) 27

28 s=s c µ 4 s=s -- N 1 s=s -- N N 0 N 0 N 1 N µ 2 5.2: (µ 2,µ 4 ) (Proposition 5.1). Gauss (1.0, 0) s = s N1 s = s N1 s = s c. (5.61),(5.64),(5.67) (5.2) H N (φ) = θ,θ J θθ (φ θ φ θ ) 2 θ µφ 2 θ (5.73) µ Ising measure ϕ(ξ) = cos(sξ) potential Taylor (5.51) ( 0). [1] J.Glimm, A.Jaffe, T.Spencer, The Particle Structure of the Weakly Coupled P (φ) 2 Model and Other Applications of High Temparature Expansions, in G. Velo, A. Wightman (eds.), Constructive Quantum Field Theory, Lecture Notes in Physics 25, Springer, 1973, [2] J.Glimm, A.Jaffe, Quantum Physics, Second Edition, Springer, [3] R. Fernández, J. Fröhlich, A. D. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory, Springer, [4] D.Brydges, J.Dimock, T.R.Hurd, The short distance behavior of (φ 4 ) 3, Commun. Math. Phys., 172, 1995, [5] K.Gawedzki, A.Kupiainen, Massless Lattice φ 4 4 theory: Rigorous Control of a Renormalizable Asymptotically Free Model, Commun. Math. Phys., 99, 1985, [6] V. Rivasseau, From Perturbative to Constructive Renormalization, Princeton University Press,

29 [7] C. Itzykson, J-B. Zuber, Quantum Field Theory, McGraw-Hill, [8],,,, [9] F. J. Dyson, Exisitence of a Phase Transision in a One Dimensional Ising Ferromagnet, Commun. Math. Phys., 12, 1969, [10] Ya. G. Sinai, Theory of Phase Transition: Rigorous Results, Pergamon Press, [11] P. Collet, J.-P. Eckmann, A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics, Springer Lecture Note in Physics 74 [12] K. Gawedzki, A. Kupiainen, Non-Gaussian Fixed Point of the Block Spin Transformation. Hierarchical Model Approximation, Commun. Math. Phys., 89, 1983, [13] H.Koch, P.Wittwer, A Non-Gaussian Renormalization Group Fixed Point for Hierarchical Scalar Lattice Field Theories, Commun. Math. Phys., 106, 1986, [14] C.M.Newman Inequalities for Ising medels and field theories which obey the Lee Yang theorem, Commun. Math. Phys., 41, 1975, 1-9. [15] T.Hara, T.Hattori, H.Watanabe, Triviality of hierarchical Ising model in four dimensions, Commun. Math. Phys., 220, 2001, [16] 22,1997,

Hierarchical model and triviality of $\phi_{4}^{4}$ abstract (Takashi Hara) (Tetsuya Hattori) $\mathrm{w}\mathrm{a}\mathrm{t}\mat

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