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

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1 Hierarchical model and triviality of $\phi_{4}^{4}$ abstract (Takashi Hara) (Tetsuya Hattori) $\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{b}\mathrm{e}$ (Hiroshi Hierarchical Ising model 4 1 Hierarchical model $\backslash$ ) characteristic function Gauss approach $\phi^{4}$ 4 triviality trajectory $\phi^{4}$ model Ising model hierarchical [ ] 11 Hierarchical model Ising model 1 spin long range interaction Dyson [1] hierarchical model spin Hierarchical model Gaussian measure model \langle $2^{N}$ spin Hamiltonian $\phi_{\theta}$ $=$ $\phi_{\theta_{n}\ldots\theta_{1}}$ $\theta=(\theta_{n} \ldots \theta_{1})\in\{01\}^{n}$ (11) $H_{N}(\phi)$ $=$ $- \frac{1}{2}\sum_{n=1}^{n}(\frac{c}{4})^{n}\sum_{n+1}\theta_{n}\ldots\theta(\sum_{\theta_{n}\ldots\theta_{1}}\phi_{\theta}n\ldots\theta 1l^{2}$ (12) $h(\phi_{\theta})$ single spin distribution $<F>_{Nh}$ $=$ $\frac{1}{z_{nh}}\int d\phi F(\phi)\exp(-\beta H_{N}(\phi))\square h(\phi_{\theta})\theta$ (13) $Z_{Nh}$ $=$ $\int d\phi\exp(-\beta H_{N}(\phi))\prod_{\theta}h(\emptyset\theta)$ (14)

2 $\lambda$ 71 $\int_{\mathrm{r}}h(x)dx=1$ (15) (hierarchical model) $s$ Ising spin measure $h_{\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(x)$ $=$ $\frac{1}{2}(\delta(x-s)+\delta(_{x+}s))$ (16) $\phi^{4}$ single spin measure $h_{\mu\lambda}(x)$ $=$ const $\exp(-\mu x-2\lambda x)4$ (17) $\mu=-2\lambda S^{2}$ $arrow$ $\infty$ (18) \langle hierarchical Ising model Hiercarchial Ising model infinite volume limit $0<c<2$ (19) $1<c<2$ (110) $\phi$ [1] spin $\beta>0$ $\beta=\frac{1}{c}-\frac{1}{2}$ (111) 12 Block spin Block spin $\phi $ $\phi_{\tau} $ $=$ $\frac{\sqrt{c}}{2}\sum_{1\theta_{1}=0}\phi_{\mathcal{t}\theta}1 \tau=(\tau_{n}-1 \ldots\tau_{1})$ (1J2) $\sum_{\theta_{n}\ldots\theta_{1}}\emptyset\theta_{n}\ldots\theta_{1}$ $= \sum_{\theta_{n\cdot\theta_{2}}}\frac{\sqrt{c}}{2}\phi \theta\theta N\ldots2$ (113)

3 72 $H_{N}(\phi)$ $=$ $H_{N-1}( \emptyset )-\frac{1}{2}\sum_{\mathcal{t}}\emptyset_{\mathcal{t}} 2$ (114) block spin $F(\phi)$ $F(\phi)$ $=$ $F (\emptyset^{j})$ (115) $<F>_{Nh}$ $=$ $<F >_{N-1iRh}$ (116) $\mathfrak{r}h(x)$ $=$ connst $\exp(\frac{\beta}{2}x)2i\mathrm{r}dyh(\frac{x}{\sqrt{c}}+y)h(\frac{x}{\sqrt{c}}-y)x\in \mathrm{r}$ (117) $h$ \langle $\mathfrak{r}$ h trajectory $h_{n}$ $=$ $\mathfrak{r}^{n}h_{0}$ $\mathfrak{r}$ 13 Gaussian trajectory Single spin measure Gauss trajectory $n=012$ $\ldots$ (118) $h_{0}(x)$ $=$ $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\exp(-\frac{\alpha_{0}}{2}x)2$ (119) $\mathrm{j}\mathfrak{i}^{n}h_{0}(x)$ $=$ $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\exp(-\frac{\alpha_{n}}{2}x)2$ (120) $\alpha_{n+1}$ $=$ $\frac{2}{c}\alpha_{n}-\beta$ (121) (111) $\alpha_{n}$ $=$ $( \frac{2}{c})^{n}(\alpha_{0^{-\frac{1}{2}}})+\frac{1}{2}$ (122) $\alpha_{n}>0$ (123) we -defined $\alpha_{0}>\frac{1}{2}$ (124) $\alpha_{0}=\frac{1}{2}$ infinite volume limit $h_{g}(x)$ $=$ $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\exp(-_{4}-x^{2})\perp$ (125) fixed point massless Gaussian measure

4 73 14 Dimensionality $m=12$ $\ldots$ $N$ $M_{m}(\phi)$ $=$ $\sum_{\theta_{m}\ldots\theta_{1}}\phi\theta_{n}\ldots\theta_{1}$ (126) $\chi m_{)}nh$ $=$ $\frac{1}{2^{m}}<m_{m}(\phi)>_{nh}$ (127) $xnnh$ spin susceptibility $\chi_{\lambda}$ $=$ $\sum_{x\in\lambda}<\emptyset(\mathrm{o})\emptyset(x)>$ (128) Block spin $\chi_{mnh}$ $=$ $\frac{2}{c}x_{m-1n-1\re}h$ (129) $h=h_{g}$ infinite volume limit $Narrow\infty$ $\chi_{m\infty}h_{g}$ $=$ $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}(\frac{2}{c})m$ (130) $\mathbb{z}^{d}(d>2)$ massless Gaussian model correlation decay $<\emptyset(x)\phi(y)>$ $\sim$ $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t} x-y ^{-d}+2$ $ x-y arrow \mathrm{o}\mathrm{o}$ (131) <r $<\phi(0)\emptyset(x)>$ const $r^{2}$ (132) $( \frac{2}{c})^{m}$ $=$ $r^{2}$ (133) $2^{m}\sim r^{d}$ spin $c=$ $2^{1-2/d}$ (134) $\beta=$ $\frac{1}{2}(2^{2/d}-1)$ (135) $c$ (134) hierarchical model $d$

5 $d\geq $d<4$ $d\geq $\mathcal{r}$ Fixed points (125) (Gaussian) fixed point (non-gaussian) fixed point [2 3 4] 4$ $\varphi_{g}$ non-gaussian fixed point 4 $d$ $\varphi_{g}$ non-gaussian fixed point 4$ $\varphi_{g}$ trajectory trivial $\phi^{4}$ $\phi^{4}$ Ising model triviality Ising model ( Gauss ) trajectory $[2 4]$ Gauss trajectory characteristic function 2 Characteristic function Single spin distribution $h_{n}\text{ }$ characteristic function $\varphi_{n}(\xi)$ $=$ $\int_{1\mathrm{r}}dxe^{-}hi\epsilon x(nx)$ (21) $\mathcal{r}\varphi_{n}$ $\varphi_{n+1}$ $=$ (22) $=$ $\mathcal{t}s$ (23) $Sg(\xi)$ $=$ $g( \frac{\sqrt{c}}{2}\xi)^{2}$ (24) $\mathcal{t}g(\xi)$ $=$ $\exp(-\frac{\beta}{2}\triangle)g(\xi)$ (25) Ising spin $h_{0}(x)$ $=$ $\frac{1}{2}(\delta(x-s)+\delta(x+s))$ (26) $\varphi_{0}(\xi)$ $=$ $\cos(s\xi)$ (27)

6 Dual $d=\infty$ $c=2\beta=0$ recursion $\varphi_{n+1}(\xi)$ $=$ $\varphi_{n}(\frac{\xi}{\sqrt{2}})^{2}$ (28) Ising spin $\varphi_{n}(\xi)$ $=$ $\cos^{2^{n}}(\frac{s\xi}{2^{n/2}})$ (29) $arrow$ $\exp(-\frac{s^{2}}{2}\xi 2)$ $narrow\infty$ (210) trajectory Gauss (trivial) ( ) (210) $ \xi <2^{n/2_{\frac{\pi}{2s}}}$ $\xi$ $\varphi_{n}(\xi)$ $=$ $\exp(-v_{n}(\xi))$ (211) $V_{n}(\xi)$ $=$ $\sum_{j=1}^{\infty}\mu^{()}2\mathrm{j}\xi^{2j}n$ $(\angle1\wedge 2)$ $\text{ }$ $V_{n}$ dual potential recursion $\mu_{2j}^{(n)}$ $=$ $2\mathrm{J}-j\mu_{2j}(n-1)$ $j=123$ $\ldots$ (213) $\xi^{2}$ marginal 4 irrelevant potential $V_{n}(\xi)$ $ \xi $ $<2^{n/2_{\frac{\pi}{2s}}}$ (214) continuum limit $\lim_{narrow\infty}v_{n}(\xi)$ $=$ $\frac{s^{2}}{2}\xi^{2}$ (215) $\mathit{1}\mathrm{r}$ continuum limit characteristic function $\xi=0$ characteristic function large field problem Ising spin (26) dual potential $V_{0}(\xi)$ $=$ $\frac{s^{2}}{2}\xi^{2}+\frac{s^{4}}{12}\xi^{4}+\frac{s^{6}}{45}\xi^{6}+\frac{17s^{8}}{2560}\xi^{8}+\cdots$ (216) (213) ( $d=\infty$ ) \langle

7 terms $\tilde{\mu}_{2j}^{(n)}$ $d<\infty$ $\mathcal{t}$ $d<\infty(\beta>0)$ (25) operator dual potential $\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{n}\mathrm{t}/\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{n}\mathrm{t}$ $\xi^{2}$ $\xi^{4}$ $d>4$ relevant irrelevant $\xi^{2}$ $\xi^{4}$ $d=4$ relevant marginal $\xi^{6}$ irrelevant $\xi^{2}$ $\xi^{4}$ $d<4$ relevant $d$ relevant part non-gaussian fixed point \langle trajectory [ ] 23 Dual potential $(n)$ $=$ $\frac{\mu_{2j}}{(\mu_{2}^{(n)})^{j}}$ $j=23$ $\ldots$ (217) $\tilde{\mu}_{2j}^{(n)}$ Ising trajectory dual potential plot : $(\mu_{2}\tilde{\mu}_{4})$ : $(\mu_{2}\tilde{\mu}_{4}\tilde{\mu}_{6})$ $d=2$

8 trajectory $\tilde{\mu}_{4}^{(0)}$ $\tilde{\mu}_{6}^{(0)}$ 77 $d_{=}$ -q $d_{-=}4$ $d_{=_{\sim}^{t}}\mathrm{i}$ $s$ $s=101112$ $\ldots$ $\mu_{2}^{(0)}$ $=$ $s^{2}/2$ (218) $=$ 1/3 (219) $=$ 8/45 (220) Gaussian fixed point (125) $\varphi_{g}(\xi)$ $ \exp(-\xi^{2})$ (221) (22) fixed point $(1 0)$ (or (1 ) $0$ $\mathrm{o})$

9 $d\geq 78 $d>2$ $\mu_{2}^{(0)}$ trajectory $(00)$ ( ) $\mu_{2}^{(0)}$ trajectory ( ) $\mu_{2}^{(0)}$ (critical point) trajectory $(00)$ (critical trajectory) (non-)triviality $d<4$ critical trajectory non-gaussian fixed point [ trajectory nontrivial continuum limit ] 4$ critical trajectory Gaussian fixed point (triviality) 3 Triviality $d\geq 4$ Characteristic function [7] hierarchical Ising model triviality idea (1) Characteristic function Gauss \langle (2) Ising critical trajectory Gauss computer 31 Characteristic function Ising dual potential Dual potential Taylor : $\mu_{2j}^{(n)}$ $\geq$ $0$ $j\geq 1$ (31) bound Dual potential Taylor 4 bound] : [Newman s $\mu_{2j}^{(n)}$ $\leq$ $\frac{\perp\tau}{j}(2\mu_{4}^{(n)})j/2$ $j\geq 3$ (32) $\mathrm{p}\mathrm{r}\mathrm{o}_{\mathrm{p}^{-}}$ Characteristic function [ spin Lee-Yang erty] [6]

10 79 32 Newman s bound (32) $V_{n}$ Taylor $narrow\infty$ $\mu_{4}^{(n)}arrow 0$ $\mu_{4}^{(n)}$ $0$ $0$ triviality Gauss $\mu_{2j}$ critical trajectory Gauss computer 33 Taylor (31) operator $T$ (25) $g_{t}$ $g_{t}(\xi)$ $=$ $\exp(-t\triangle)g(\xi)$ (33) $Tg$ $=$ $g_{\beta/2}$ (34) $\frac{d}{dt}g_{t}(\xi)$ $=$ $-\triangle g_{t}(\xi)$ (35) $g_{0}(\xi)$ $=$ $g(\xi)$ (36) $g_{t}(\xi)$ $=\exp(-v_{t}(\xi))$ (37) $V_{t}$ $\frac{d}{dt}v_{t}$ $=$ $(\nabla V_{t})^{2}-\triangle Vt$ (38) $V_{t}$ Taylor (38) 2 Taylor [ upper bound (38) lower bound ] $d\geq 4$ hierarchical Ising model triviality computer as- [7] sisted proof upper bound

11 80 References [1] F J Dyson Exisitence of a Phase-Transision in a One-Dimensional Ising Ferromagnet Commun Math Phys [2] Ya G Sinai Theory of Phase Transition: Rigorous Results Pergamon Press 1982 [3] P Collet J-P Eckmann A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics Springer Lecture Note in Physics 74 [4] K Gawedzki A Kupiainen Non-Gaussian Fixed Point of the Block Spin Transformation Hierarchical Model Approximation Commun Math Phys [5] HKoch PWittwer A Non-Gaussian Renormalization Group Fixed Point for Hierarchical Scalar Lattice Field Theories Commun Math Phys [6] $\mathrm{c}\mathrm{m}$newman Inequalities for Ising medels and field theories which obey the Lee- Yang theorem Commun Math Phys [7] THara THattori HWatanabe in preparation

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untitled 1 Constructive field theory and renormalization group Hiroshi WATANABE 1 2001 7,. 1 2 2 Euclidean field theory 3 2.1...................................... 4 2.2..................................... 6 3