.. 1 10-11 Nov., 2016 1 email:keiichi.r.ito@gmail.com, ito@kurims.kyoto-u.ac.jp ( ) 10-11 Nov., 2016 1 / 45
Clay Institute.1 Construction of 4D YM Field Theory (Jaffe, Witten) Jaffe, Balaban (1980).2 Solution of Navier-Stokes Equation (Feffermann) Sinai (2005) ( ) 10-11 Nov., 2016 2 / 45
.1 Boltzmann (Erdös, Yau ).2 Pauli-Fierz (Semi-Classical QED).3 ( σ model Ricci, Perelman s theory,) ( ) 10-11 Nov., 2016 3 / 45
.1..2, ( ).3.4 ( ) 10-11 Nov., 2016 4 / 45
=,,.1, (x, p),,.2,,..3, ( ) 10-11 Nov., 2016 5 / 45
.1 : KPZ, Wilson.2 :. ( ) 10-11 Nov., 2016 6 / 45
PKZ, h(x, t) KPZ eq.. Ξ white noise: t h = 2 x h + V ( x h) + Ξ V ( x h) = ( x h) 2 C ε E(Ξ(x, t)ξ(y, s)) = δ(x y)δ(s t) V ( x h). C ε. ( ): h = G (V + Ξ) G = ( t 2 x ) 1 = (4πt) 1/2 exp[ (x y) 2 /2t] ( ) 10-11 Nov., 2016 7 / 45
PKZ Spohn h 3 u = (u 1, u 2, u 3 ) t u = 2 x u + V (u) + Ξ V = ( x u, M x u) C R 3, Kupiainen Wilson M = (M (1), M (2), M (3) ), M (i) = 3 3 C R 3 = u = G (V (u) + Ξ) + e t u 0 G(x, t) = e t = 1 4πt exp[ x 2 /2t] G 0 < t < ε 2. χ(t) [0, 1] u = G ε (V (u) + Ξ) + e t u 0 G ε (x, t) = e t (x, 0)(1 χ(ε 2 t)) C = C ε ( ) 10-11 Nov., 2016 8 / 45
PKZ Kupiainen Wilson C ε > 0, Ξ, t(ξ) > 0 ε > 0 u ε (t, x) t [0, t(ξ)] u D ([0, t(ξ)] T ). u χ.. C ε = m 1 ε 1 + m 2 log ε 1 + m 3 ( ) 10-11 Nov., 2016 9 / 45
PKZ Block Spin s ε t ε 2 t,x εx. x = ε x = 1, x = 1 x = ε 1 = L N : v (ε) (φ) = ε 1/2 ( x φ, M x φ) + ε 3/2 C ε φ = G 1 (v ε (φ) + ξ) :.1 G 1 = G L 2 + (G 1 G L 2) x > L 1 < x < L.2 (G 1 G L 2) G L 2. s s L 1, G L 2 G 1 ( ) 10-11 Nov., 2016 10 / 45
PKZ ε 1, 1 ε 1 = L N, N eps 1 1 L L^2 1/eps=L^N 1 L L^(N-1) ( ) 10-11 Nov., 2016 11 / 45
PKZ φ = φ 1 + φ 2 φ 1 = G L 2 (v ε (φ 1 + φ 2 ) + ξ) φ 2 = (G 1 G L 2) (v ε (φ 1 + φ 2 ) + ξ) G sf (t, x) = L 1/2 f (t/l 2, x/l) ( s 1 f (t, x) = L 1/2 f (L 2 t, Lx) ) G L 2 G 1 : φ 1 = sφ, φ 2 = sζ ( ) 10-11 Nov., 2016 12 / 45
PKZ φ = s 1 G L 2 (v ε (s(φ + ζ)) + ξ) = G 1 (Sv ε (φ + ζ) + ξ) = G 1 (Sv ε (φ + ζ(φ )) +ξ) }{{} Rv ε (φ ) = G 1 (Rv ε (φ ) + ξ) ζ = Γ (Sv ε (φ + ζ) + ξ) Γ = G 1/L 2 G 1 = e t (x, 0)(χ(t) χ(l 2 t)) (Sv)(ϕ) = L 2 s 1 v(sϕ) v = ( x φ) k (k > 3) S( x φ) k = L (3 k)/2 ( x φ) k ( ) 10-11 Nov., 2016 13 / 45
PKZ ζ = Γ (Sv ε (φ + ζ) + ξ) Γ Sv ε (φ ) + Γ ξ φ = G 1 (Sv ε {φ + θ + Γ (Sv ε (φ )} + ξ) G 1 (Sv ε (φ + θ)) = L 1/2 ε 1/2 ( x (φ + θ), M x (φ + θ)) + L 3/2 ε 3/2 C ε θ = Γ ξ ( ) 10-11 Nov., 2016 14 / 45
PKZ, ( x φ) k. Γ = e t (χ(t) χ(l 2 t)) 1 L 1, ζ φ + ζ = φ + ζ(φ ),. Γ. ζ θ (Γ ξ), ε = L N N ( x φ + θ, M( x φ + θ)) C C E(θ(t, x)θ(s, y)) = m 1 L N + m 2 N + O(1) ( ). ( ) 10-11 Nov., 2016 15 / 45
PKZ 4D, 2D, NV.1,..2, (relevant ) ( ) 10-11 Nov., 2016 16 / 45
.1 :, R.Peierls (1936), L.Onsager (1944).2 XY (O(2) ) Kosterlitz-Thouless (Nobel prize 2016) J.Fröhlich and T.Spencer (1982).3 2 (SO(N) ) ( ) ( ) ( ) 10-11 Nov., 2016 17 / 45
[ ] Peierls : +1, 1. {s x ±1} exp[β nn (s i s j 1)] = exp[ 2β i γ i ], γ 0 {0} s 0 1 γ : Prob(s 0 = 1) < γ 0 exp[ 2β γ ] < 1/2, β 1 s 0 > 0 [ ], Ising.! Peierls 85! ( ) 10-11 Nov., 2016 18 / 45
[ ] γ: : s i = 1, : s i = +1 [ O(N) ] flip-flop. ( ) 10-11 Nov., 2016 19 / 45
2 2 =Landau-Ginzbrg(2 ) f (ϕ) = f (ϕ) exp[ W 0 (ϕ)] x d N ϕ(x) W 0 = 1 2 ϕ, ( + m2 0 )ϕ + g 0 2N : ϕ2 : G, : ϕ 2 : G N : ϕ 2 : G (x) = (x) NG(0), β = G(0) i=1 ( ) xy = 4δ xy δ 1, x y, Lattice Laplacian G(0) = β m0 2 32e 4πβ : ϕ 2 i G(x) = 1 + m0 2 (x) = π π π π e ipx m0 2 + 2 dpi (1 cos p i ) 2π ( ) 10-11 Nov., 2016 20 / 45
2 G(0) = β m 2 0 32e 4πβ : G(x) = = 1 + m0 2 (x) π π π π e ipx m 2 0 + 2 (1 cos p i ) dpi 2π exp[ m 0 x ], m 0 0.. β,. ( ) 10-11 Nov., 2016 21 / 45
2 ) 1 G 0 (x, y) = + m0 2 (x, y) G n (x, y) = 1 L 4 1 x y d 2 ζ,ξ 0 G n 1 (Lx + ζ, Ly + ξ) ϕ n (x) = (Cϕ n 1 )(x) = 1 L 2 C = = (L 2 ) + (Lx x): ζ 0 ϕ n 1 (Lx + ζ) ( ) 10-11 Nov., 2016 22 / 45
2 : ϕ n (x)ϕ n (y) = G n (x, y) ϕ n (x) = A n+1 ϕ }{{ n+1 (x) + z } n (x) }{{} ϕ n, Gn 1 ϕ n Λn = ϕ n+1, G 1 n+1 ϕ n+1 Λn+1 + z n, Gn 1 z n Λn A n+1 z n Λ n = L n Λ CA n+1 = 1, Cz n = 0 A n+1 = G n C + G 1 n+1 : RΛ n+1 R Λn ( ) 10-11 Nov., 2016 23 / 45
2 Q : R Λ n R Λ n, CQ = 0 Q + Gn 1 Q = Γ 1 n z n = QΓ 1/2 n ξ ϕ, ( + m0 2)ϕ {z n = QΓ 1/2 n ξ n }. ϕ, G 1 0 ϕ = ϕ 1, G 1 1 ϕ 1 + ξ 0, Q + G 1 0 }{{ Q ξ 0 } Γ 1 0 = ϕ 2, G 1 2 ϕ 2 Λ2 + ξ 1, Q + G 1 1 }{{ Q ξ 1 Λ1 + ξ 0, Q + G 1 0 } Q ξ 0 Λ0 }{{} Γ 1 1 Γ 1 0 Γ n = Q + Gn 1 Q : ( ) 10-11 Nov., 2016 24 / 45
2 :. L n m 1 m. exp[ W n+1 (ϕ n+1 )] = exp[ W n (A n+1 ϕ n+1 + z n )]dµ 0 (ξ) W 0 = 1 2 ϕ 0, ( )ϕ 0 + g 0 2N : ϕ2 0 :, : ϕ2 0 : z 0 = QΓ 1/2 0 ξ, ξ N(0, 1) Γ 0 = (Q + ( )Q) 1 ( ) 10-11 Nov., 2016 25 / 45
: 2 ϕ n+1 : ϕ 2 n+1 : D : ϕ n+1(x)ϕ n+1 (y) : R, or K ( ) (n + 1)th Gibbs = exp X= X i i g D R X i [ ] Wn+1 K (ϕ n+1) F K \X (ϕ n+1 ) X i ξ D R X i X j =, X i D R F K \X (ϕ n+1 ) = X ϕ n+1 ( ) 10-11 Nov., 2016 26 / 45
2 W K n+1 W K n+1 = 1 2 ϕ n+1, G 1 n+1 ϕ n+1 + 1 2N : ϕ2 n+1 : G n+1, D n+1 : ϕ 2 n+1 : G n+1 K + 1 2 γ n ϕ 2 n+1, E G 1 n+1 E ϕ 2 n+1 K (i) D n+1 D > 0. (ii) g D R X i X gd R X i D n D > 0. (D, R ) ( ) 10-11 Nov., 2016 27 / 45
2 次元シグマ模型と繰り込み群 1/2 球面に拘束される揺動場 zn = QΓn ξn ] [ g exp ((ϕn+1 + zn )2 Nβn )2 かつ ϕ2n+1 Nβn+1 = O(1) N 0 1 2 0-1 -2 1 2-1 -2-2 -1 0 1 2 15 15 10 10 5 5 0 0-2 0-1 1 2 揺動場 ξn (x) は背景場のブロックスピン ϕn+1 に影響される. ξn は井戸の 底に生きて かつ ϕn+1 に鉛直にすすむ. ξn T (S N 1 ). 伊東恵一 (立教大学 数理物理学研究センター) 繰りこみ群とミレニアム問題.... 10-11 Nov., 2016. 28 / 45
Mathematical Meanings of RG D(ϕ n ) = Large and/or non-smooth configuration of ϕ n = D 0 (ϕ n+1 ) D w (ϕ n+1 ) R(z n = Qξ n ) = + D w = ϕ n (x)ϕ n (y) Nβ n > τ 0 N 1/2 exp[(c/10) x y ] x D w, y D w 1/2 : ξ 2 i : D w ϕ n (x)ϕ n (y) Nβ n < τ 0 N 1/2 exp[(c/10) x y ] x D c w, y D c w (D w ) c ϕ n (x)ϕ n (y) = NG n (x, y) ( ) 10-11 Nov., 2016 29 / 45
Mathematical Meanings of RG ϕ n = A n+1 ϕ n+1 + Qz n )A n+1 ϕ n+1 Qz n 10 5 0 5 10 2 0 2 10 10 5 10 0 5 0 5 5 10 1.0 0.5 0.0 0.5 1.0 10 5 0 5 10 ( ) 10-11 Nov., 2016 30 / 45
Mathematical Meanings of RG = 10 5 0 5 10 2 1 0 1 2 10 5 0 5 10 ϕ n (x) ξ n (x) N 1 (tangent vector ) ( ) 10-11 Nov., 2016 31 / 45
Mathematical Meanings of RG : Z (x, y) = : ϕ n (x)ϕ n (x) : : φ n+1 (x)φ n+1 : = φ n+1 (x)z n (y) + φ n+1 (y)z n (x)+ : z n (x)z n (y) : K = {{ξ n (u) R N ; u K }; Z (x, y) τ 0 N 1/2 exp[(c/10) x y ]} = ( ) ( ) 10-11 Nov., 2016 32 / 45
Mathematical Meanings of RG RG Banach Space H K n K 1 K 2 K n K n = (.1 ϕ n (x)ϕ n (y) Nβ n < τ 0 N 1/2 exp[(c/10) x y ] x, y K.2 ϕ n (x) 2 Nβ n < τ 0 N 1/2.3 ϕ n (x) < τ 0 N 1/2 ( ) 10-11 Nov., 2016 33 / 45
Mathematical Meanings of RG.1 ϕ 4 ϕ n (x) = A n ϕ n+1 + Qξ(x) ϕ n+1 ([x/l]) + Qξ(x) ϕ n (x), x < L/2 L 2 ϕ n+1 ([x/l]) ϕ 2 n(x) L 2 ϕ 2 n+1 (x) x x (: ϕ 2 n : Gn (x)) 2 L 2 (: ϕ 2 n+1 (x) : G n+1 ) 2 x x ϕ 4 n,..2 ( ), ( ) 10-11 Nov., 2016 34 / 45
Main Theorem on RG flow Gibbs ϕ n+1 K / D, : ξ. K W n Wn K (ϕ n, ψ n ) = 1 2 ϕ n, Gn 1 ϕ n + g n 2N : ϕ2 n : Gn, : ϕ 2 n : Gn + 1 2 γ n < ϕ 2 n, E G 1 n E ϕ 2 n >.1 G 1 n = + m 2 n, m 2 n = L 2n m 2 0.2 γ n = (Nβ n ) 1..3 g n g = O(1) > 0 ( ).4 E = N (C) = {f ; Cf = 0} ( ) 10-11 Nov., 2016 35 / 45
Main Theorem on RG flow (marginal term) (irrelevant term) (: ϕ 2 n :) 2 (relevant term) g n 3 mn 2 = L 2n m0 2 exp[ 4πβ + 2n log L] O(1), β n = β const.n O(1) γ n = O((β n N) 1 ) g n = O(1) :,. ( ) 10-11 Nov., 2016 36 / 45
3 :.1,.2,.3 ϕ n = φ n+1 + z n, φ n+1 = A n+1 ϕ n+1, z n = Qξ n < ϕ n, G 1 n ϕ n > = < ϕ n+1, G 1 n+1 ϕ n+1 > + < ξ n, Γ 1 n ξ n >, Γ 1 n = Q + Gn 1 Q + ( Λ)Q > O(1) : ϕ 2 n(x) : Gn = : ϕ 2 n+1 (x) : G n+1 +q(x) q(x) = 2ϕ n+1 (x)z n (x)+ : z(x) 2 n : Γn ( ) 10-11 Nov., 2016 37 / 45
: ϕ 2 n(x) : Gn : φ 2 n+1 (x) : G n+1 = q(ξ) = 2φ n+1 (x)z n (x)+ : z(x) 2 n : Γn dµ(ξ) ( q(ξ)dµ = 0, ): P(φ n+1, p) = [ ] i exp (p q(ξ)) x λ x dµ(ξ) dλ x N x [ ] i = exp (p q(ξ)), λ dµ(ξ) dλ x N dµ(ξ) = exp[ ξ, Γ 1 n ξ ] dξ x ( ) 10-11 Nov., 2016 38 / 45
1: p ( ) P(p, φ) = exp[ 1 4N p, 1 M p ] M = Γ 2 n + 2 N (ϕ nϕ n ) Γ n = Γ 2 n + 2β n Γ n + : ϕ n ϕ n : }{{} Γ n /N (Γ n )(x, y) = (QGn 1 Q + )(x, y) exp[ x y ] ((ϕϕ) Γ)(x, y) = (ϕ(x)ϕ(y))γ(x, y) NG(x, y)γ(x, y) spec M = { κ }{{} 0, κ 1,, κ L 2 1} }{{} O(1)>0 O(β n ) ( ) 10-11 Nov., 2016 39 / 45
2:, : ϕ n (x)ϕ n (y) : Γ n (x, y) < N 1/2+ε 1 ( p, 1 M p = 1 p U, E 0 + κ 0 blocks:u Λ n i ( ) 1 p U, E 0 P U κ 0 blocks:u Λ n 1 = (E 0 p U ) 2 κ 0 blocks:u Λ n ) 1 E i p U κ i E 0 = E 1 = ( ) 10-11 Nov., 2016 40 / 45
2 ϕ n (x)ϕ n (y) Nβ n > τ 0 N 1/2 exp[(c/10) x y ] x D w, y D w D w exp[ 1 2 φ n, G 1 0 φ n Dw ]dµ(ξ) < exp[ τ 0 D w ] : g D R, polymer. : D w φφ NG n ( ) 10-11 Nov., 2016 41 / 45
3 exp[ g n 2N : φ2 n :, : φ 2 n : + (...)]dµ(ξ) = exp[ g n 2N : φ2 n+1 : +p, : φ2 n+1 : +p ]P(p, φ) dp P(p) = exp[ 1 4N p, M 1 p ] = exp[ 1 ( ) 1 4N p, E 0 p ] = exp[ 1 (E0 p U ) 2 ] κ 0 4Nκ 0 E 0 p {p} (block spin type.) ( ) 10-11 Nov., 2016 42 / 45
p : p : φ 2 n+1 : 2. g n : g n 2N = g n 2N (: φ 2 n+1 : +p)2 + 1 (E 0 p) 2 2N x x [(E 0 (: φ 2 n+1 : +p))2 + ((1 E 0 )(: φ 2 n+1 : +p))2] + 1 2N x (E 0 p) 2 x E 0 p (1 E 0 )p, steepest descent + ). P(p) = P n (p) n, 4 g n : g n g ( ) 10-11 Nov., 2016 43 / 45
( ). A.Jaffe and E.Witten, Quantum Yang-Mills Theory, in Millennium Problems, Clay Mathematical Institute. T.Balaban, A low temperature expansion for classical N-vector model I, Commun.Math.Phys., 167 103 (1995); Variational problems for classical N-vector model, Commun.Math.Phys., 175, 607 (1996) J.Dimock, The Renormalization Group According Balaban, I, arxiv 1108/1335; II, arxiv:1212.5562 J.Fröhlich and T.Spencer, The Kosterlitz-Thouless transitions in Two-Dimensional Abelian Spin Systems and the Coulomb Gas, Commun.Math.Phys., 81, 527 (1981) ( ) 10-11 Nov., 2016 44 / 45
M.Hairer et al., A class of growth models rescaling to KPZ, arxiv:1512.0784 A.Kupiainen, Renormalization of Generalized KPZ equation, arxiv:1604.0872 K.Gawedzki and A.Kupiainen, Massless lattice ϕ 4 4 theory: Rigorous control of a renormalizable asymptotically free model, Commun.Math.Phys. 99, 197 (1985), and references cited therein.,, SGC 81 (2011), K.R.Ito, Renormalization Group Flow of 2D O(N) Spin Model with large N, Absence of Phase Transitions, Paper in Preparation K.R.Ito, Origin of Asymptotic Freedom in Non-Abelian Field Theories, Phys.Rev.Letters, 58 (1987) 439 ; Renormalization Group Flow of 2D Hierarchical Heisenberg Model of Dyson-Wilson Type, Commun. Math.Phys. 137(1991) 45 ( ) 10-11 Nov., 2016 45 / 45