$)\triangleleft\hat{g}$ $\mathcal{t}\mathcal{h}$ 106 ( ) - Einstein ( ) ( ) $R_{\mu\nu}- \frac{1}{2}g_{\mu\nu}r=\kappa T_{\mu\nu}$ bottom-up feedback

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$$+\mathrm{f}_{\mathrm{o}\mathrm{l}1\gamma}\mathrm{i}\mathrm{e}\mathrm{r}$ Galois 1507 RIMS 11 ( 2 2006 12 25 27 ) 1507 1507 6 5 1507. $\mathrm{v}\mathrm{s}$ / [.$

1 duality $-*$ (Izumi Ojima) Research Institllte for Mathematical Sciences Kyoto University 1? 3 ( 2-4 ) 1507 RIMS. ( ) ( ). $+\mathrm{f}_{\mathrm{o}\mathrm{l}1\gamma}\mathrm{i}\mathrm{e}\mathrm{r}$ Galois 1507 RIMS 11 ( ) $\mathrm{v}\mathrm{s}$ / [. ] IUMS ( B)

2 $)\triangleleft\hat{g}$ $\mathcal{t}\mathcal{h}$ 106 ( ) - Einstein ( ) ( ) $R_{\mu\nu}- \frac{1}{2}g_{\mu\nu}r=\kappa T_{\mu\nu}$ bottom-up feedback $T_{\mu\nu}$ $arrow$ $g_{\mu\nu}$ ( ) ( ) $\Gamma^{\lambda}=$ $\mu\nu$ $arrow$ top-down control [ (= ) ( $=$ ) ( $=$ ) Fourier-Pontryagin -Krein ( ) $(\mathcal{m}\aleph G)\aleph\hat{G}\simeq \mathcal{m}\otimes B(L^{2}(G))$ $\aleph G$ $arrow$ $\mathcal{m}xg$ $ [$ $ l$ $\mathcal{m}^{g}\aleph\hat{g}$ $arrow$ $\mathcal{m}^{g}$ 2 $G$ $\text{ }$ $\tau$ ( $G$ ) G- $\mathrm{v}\mathrm{s}$ 2.1 (top-down) G $=\mathfrak{u}$. (bottom-up) $G$ $G$ 75

3 $\mathcal{t}\mathcal{h}$ $\mathrm{o}_{d}^{g}$ 107 $\mathfrak{u}=s^{g}$? $\omega\in E_{\mathfrak{U}}$? $ad$ hoc $\mathrm{i}arrow$ $\mathcal{e}\mathcal{x}$ $\mathcal{t}\mathcal{h}$ $\backslash$ $\mathcal{t}\mathcal{h}_{1}$. $arrow$ $\mathcal{e}\mathcal{x}+$ $\nearrow$ 2.2? duality $\mathfrak{u}=s^{g}$ $S$ $G$ $\mathfrak{u}(=\text{ ^{}G})+\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}\text{ }[\text{ }\bigwedge_{\mathcal{t}}g]$. $\mathrm{d}\mathrm{o}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{r}-\mathrm{h}\mathrm{a}\mathrm{a}\mathrm{g}$-roberts something DHR selection critelion [1] ( ) Doplicher-Roberts $\mathfrak{u}$ $\mathcal{t}(\subset End(\mathfrak{U}))$ $\mathrm{c}^{*}$-tensor category Lie $G$ $\mathcal{t}$ $\mathcal{t}(\subset End(\mathfrak{U}))^{[2]}\simeq Rep(G)$ Rep $(G)$ DR category $[2]$ -Krein ( ) $\hat{g}$ $\mathfrak{u}$ $=$ $G$ -chargej parametrize ( $d$ ) Cuntz $O_{d}$ $\mathfrak{u}$ $\hat{g}$ $\mathfrak{u}x\hat{g}\simeq \mathfrak{u}\otimes O_{d}$ $O_{d}^{G}$

$. $\mathfrak{u}$ $\tau$ 108 $= \mathfrak{u}\bigotimes_{o_{d}^{g}}o_{d}\simeq \text{ ^{}G}\chi\hat{G}$ $G$ $\mathfrak{u}$ Galois 1 $G=Gal(S/\mathfrak{U})[2]$ [ + $\mathrm{o}g$] $\mathfrak{u}=\text{$

4 . $\mathfrak{u}$ $\tau$ 108 $= \mathfrak{u}\bigotimes_{o_{d}^{g}}o_{d}\simeq \text{ ^{}G}\chi\hat{G}$ $G$ $\mathfrak{u}$ Galois 1 $G=Gal(S/\mathfrak{U})[2]$ [ + $\mathrm{o}g$] $\mathfrak{u}=\text{ ^{}G}+$ [ ] $\mathfrak{u}$ DHR criterion Galois $\cdot$. 2.3 $=$ [ ] $=$ [ ] $=$? $\pi_{1}\approx\pi_{2}[=\mathrm{l}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}$ (multiplicity) equivalence llp to multiplicity] $\pi$ $\Pi\overline{\mathrm{p}}$ von Neumann $\pi(\mathfrak{u}) $ $\pi_{1}\approx\pi_{2}\leftrightarrow\pi_{1}(\mathfrak{u}) \simeq\pi_{2}(\mathfrak{u}) $ [3] factor = [centre ] $C$ $C$ centre $3(C)=C\cap C $ centre $3(\mathcal{M})=\mathbb{C}1$ factor $=$ Factor centre factor ( ) $\pi(\mathfrak{u}) $ $=\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}$ I disjointness (!?) 1 Galois [ ] Galois

$109 $\pi_{1}0\pi_{2} \Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}(\pi_{1} \pi_{2})=\{tfl_{\pi_{1}}arrow\ovalbox{\tt\small REJECT}_{\pi_{2}} ; T\pi\iota(A)=\pi_{2}(A)T\}=0$ [ ] disjoint ] $-$ $\pi$$

5 109 $\pi_{1}0\pi_{2} \Leftrightarrow \mathrm{d}\mathrm{e}\mathrm{f}(\pi_{1} \pi_{2})=\{tfl_{\pi_{1}}arrow\ovalbox{\tt\small REJECT}_{\pi_{2}} ; T\pi\iota(A)=\pi_{2}(A)T\}=0$ [ ] disjoint ] $-$ $\pi$ $n \pi=\frac{n}{\pi\oplus\cdots\oplus\pi}$ $n$ $n\pi\not\cong m\pi(n\neq m)$ Factor 2 [4] $=$ $=$ [centre factor ] [factor $=$ ] [ centre $=$ ] centre $=$ (= centre ) $=$ ( ) centre $=$ disjointness = $=$ [ ] [ superselection sectors ] $=$ [ ] $=$ centre ] $=$ [ ]... [5] [4] Stonevon Neumann- centre emerge = d oint centre ( ) ( $=$ ) 2.

6 $\downarrow$ $\varphi_{\lambda}$ $\Uparrow$ $\uparrow$ $\Downarrow$ $-\mathrm{b}_{\grave{\mathrm{j}}}\mathrm{f}^{\mathrm{a}}\cdot\ovalbox{\tt\small REJECT}\star \text{ }$ 110 A) $[7 6]$ B) DHR-DR $[1 2]$ [4] $(\mathrm{s}\mathrm{s}\mathrm{b})^{\text{ }}\mathrm{b})$ C) [4] D). $[6 4]$ E) [?] F) $[5 16]$ $\backslash$ $[\mathrm{b})$ \not\in Ppx m& $]arrow$ $arrow$ $/\pi\backslash ) 1*q $) $[\mathrm{e}$ $\text{ ^{}\frac{*_{\backslash }}{\mathrm{t}}}j\backslash \#\backslash 0\mathrm{i}R\text{ }]arrow\cdots$ D) ( ) ( ) i) $\Rightarrow\uparrow\uparrow \mathrm{i}\mathrm{i}$) i ) i) ii) iv) $adjunction\leftrightarrow[$ $\mathrm{i}\mathrm{i}$) $\text{ }\tau^{\backslash }<\mathrm{i}$ ) $\sigma)_{\overline{\overline{\mathrm{p}}}}^{-}q- c$ $\Rightarrow \mathrm{i}\mathrm{i}$ $\mathrm{r}hannel\mathrm{i}$) ). $]$ selection criteria $[6 4]$ $[7 6]$ Example 1 $\{(U_{\lambda\varphi_{\lambda}} ; U_{\lambda}arrow \mathrm{r}^{n})\}$ $M$ $\mathrm{r}^{n}$ $U_{\lambda}$ $i)=$ $ii$) $=$ $U_{\lambda}arrow \mathrm{r}^{n}$ $iii)=$ $iv)=$ - $K$. $M$

$$\ovalbox{\tt\small REJECT}$ 111 Example 2 $[7 \mathit{6}]$ $i)=$ $\omega((1+h_{\mathcal{o}})^{m})<\infty$ $\omega$ $E_{x}$ $ii)=$ $(\beta \mu)$ $B_{K}$ $(\beta \mu)$ $B_{K}$ $\rho\in NI+(B_{K})=$Th$

7 $\ovalbox{\tt\small REJECT}$ 111 Example 2 $[7 \mathit{6}]$ $i)=$ $\omega((1+h_{\mathcal{o}})^{m})<\infty$ $\omega$ $E_{x}$ $ii)=$ $(\beta \mu)$ $B_{K}$ $(\beta \mu)$ $B_{K}$ $\rho\in NI+(B_{K})=$Th $iii)=(i)$ ) 1 $x$ $\omega$ $\omega_{\rho}=c^{*}(\rho)=\int_{b}..d\rho(\beta \mu)\omega_{\beta\mu}$ $\omega\equiv$ $C^{*}(\rho)\mathcal{T}_{x}$ $\mathit{0}$ $iv)=$ (. ) adjunction $=$ $[E_{x}/\mathcal{T}_{x}](\omega C^{*}(\rho))^{q\Leftrightarrow c}\simeq[th/c(\mathcal{t}_{x})]((c^{*})^{-1}(\omega) \rho)$. $C^{*}$ $carrow q$ channel ( ) $ {}^{t}(c^{*})^{-1}$ $qarrow \mathrm{c}$ channel $a$) $\omega$ $\omega$ $C^{*}(\rho)$ $b$) $\omega_{\mathcal{t}_{x}}\equiv C^{*}(\rho)$ $\rho\in Th$ $\rho$ $\equiv$ $(C^{*})^{-1}(\omega)$ $C^{*}(\mathcal{T}_{x})$ $[\omega_{t_{l}}\equiv. C^{*}(\rho)]\Leftrightarrow[(C^{*})^{-1}(\omega)_{c_{(\mathcal{T}_{x})}^{\equiv}}.\rho]$ $\mathrm{v}\mathrm{s}$ 3. centre= clear-cut? (1 factor M ) (MASA) $A=A $ $\mathcal{m} \subset A =A\subset \mathcal{m}$ $\mathcal{m} =\mathcal{m} \cap \mathcal{m}=3(\mathcal{m})$ type I $A=A \cap \mathcal{m}$ 3.1 MASA $A$ $A$ $A$ $\mathcal{m}\otimes A$ couple centre $=$ $A$ $3(\mathcal{M}\otimes A)=3(\mathcal{M})\otimes A=1\otimes L^{\infty}(Spe\mathrm{c}(A))$. $A$ couple $\mathcal{m}\otimes A$!

8 112 $A$ coupling dynamics coupling $\text{ }$ $\mathcal{m}\otimes A$ Hilbert von Neumann $A$ 1 $A_{0}=A_{0}^{*}\in A$ $A=\{A_{0}\} [8]$ $A$ $\mathcal{u}(a)$ $A$ Lie ( $du$ ) $\mathcal{u}\subset \mathcal{u}(a)$ $A=\mathcal{U} 0$ MASA $A=A \cap \mathcal{m}$ $A=\mathcal{M}\cap A =\mathcal{m}\cap \mathcal{u} =\mathcal{m}^{\alpha(\mathcal{u})}$ MASA $A$ $\mathcal{m}\ni$ $\alpha_{u}=ad(u)$ $x-uxu^{*}$ [5] Galois Kac\leftrightarrow ( K-T ) $[9 10]$ 3.2 instrument $G$ K-T von $\Gamma$ Neumann $M=L^{\infty}(Gdg)$ ( $dg$ ) $Marrow M\otimes M$ $(f\in \mathrm{a}^{\mathit{1}}i st\in G)$ $\Gamma(f)(st)=f(st)$ $\Gamma(X)=V^{*}(1\otimes X)V(X\in M)$ $\otimes$ $(V\xi)(s f)=\xi(s s^{-1}t)$ $(\xi\in \mathfrak{h}\otimes ff$ $s$ $t\in G)$ $=L^{2}(G dg)[91011]$ $\Gamma$ $\otimes$ - $\otimes$ 5 $V_{12}V_{13}V_{23}=V_{23}V_{12}$ convolution $\omega_{1}*\omega_{2}=\omega_{1}\otimes\omega_{2}\circ\gamma$ predual $\lambda$ $\mathrm{a} I_{*}=L^{1}(G)$ Fourier $l\downarrow/i_{*}\ni\omega\mapsto\lambda(\omega)=(i\otimes\omega)(v)\in$ $\hat{m}=\lambda(g) $ t $G$ $(\lambda \mathfrak{h})$ $\lambda(\omega_{1}*\omega_{2})=\lambda(\omega_{1})\lambda(\omega_{2})$ $\lambda^{\otimes m}\approx\lambda^{\otimes n}$ $\lambda^{\otimes n}=\lambda\otimes\cdots\otimes\lambda$ $(\forall m n\in \mathrm{n})$ intertwiner $V( \lambda\bigotimes_{\vee}\iota)=(\lambda\otimes\lambda)v$ $\lambda$ $G$ [12] ( Kac ) $Vrightarrow\hat{V}=\sigma V^{*}\sigma$ $\Lambda\hat{i}$ $(\sigma(\xi\otimes\eta)=\eta\otimes\xi \xi \eta\in \mathfrak{h})$ $M$ [11] MASA $A=L^{\infty}(Spec(A))$ $\mathcal{u}(\subset A)$ $\chi$ Dirac $\hat{\mathcal{u}}$ $V$ $M=L^{\infty}(\hat{\mathcal{U}})=\lambda(\mathcal{U}) $ $\mathcal{u}\ni u-\chi(u)\in \mathrm{t}$ $V \gamma$ $\chi\rangle= \gamma\gamma\chi\rangle$ $(\gamma \chi\in\hat{\mathcal{u}})$. (1) $Aarrow \mathbb{c}$ $\chi$ $A$ $\chi\in Spec(A)$ $\chi \mathrm{r}_{\mathcal{u}}\in\hat{\mathcal{u}}$ Spec $(A)$ $\hat{\mathcal{u}}$ $\iota\in\hat{\mathcal{u}}$ Spec $(A)arrow$ $\iota(u)\equiv 1(\forall u\in \mathcal{u})$ $\iota\in\hat{\mathcal{u}}$ $L^{2}(\mathcal{U})$ ( $m_{\mathcal{u}}$ )

$$l^{\backslash } \backslash \hat{\overline{\pi}}\lambda^{\backslash } \text{ _{}\#\mathcal{m}\text{ ^{}\backslash }\text{ }^{}\tau_{\backslash }}\text{ }\mathrm{t}\mathrm{b}\text{ }[13] \text{$

9 $l^{\backslash } \backslash \hat{\overline{\pi}}\lambda^{\backslash } \text{ _{}\#\mathcal{m}\text{ ^{}\backslash }\text{ }^{}\tau_{\backslash }}\text{ }\mathrm{t}\mathrm{b}\text{ }[13] \text{ }\xi_{\gamma}k^{\backslash }ffi\mathfrak{l}\mathrm{j}\text{ }\mathrm{g}\mathrm{g}\sigma)_{\text{ }\backslash \theta\mathrm{b}^{\backslash }\text{ }\dot{\mathrm{x}}\text{ }^{}\frac{)}{\mathrm{t}}}\backslash \not\in_{7^{--\text{ }\gamma \text{ }}^{}\in\hat{g}}arrow \text{ ^{}\prime}\supset \mathrm{k}^{\backslash }\text{ }E_{*}(V(\xi\bigotimes_{\backslash } \iota\rangle)=\mathrm{f}\sum_{\mathrm{f}\mathrm{i}^{1\mathrm{j}}\backslash }c_{\gamma}\otimes \gamma\rangle$ 113 $\mathcal{u}arrow Aarrow \mathcal{m}$ $E\mathcal{U}arrow \mathcal{m}$ $E(u)= \int_{\chi\in Spec(A)\subset\hat{\mathcal{U}}}\overline{\chi(u)}dE(\chi)(u\in \mathcal{u})$ $de$ ( $fl_{\mathcal{m}}$ $\ovalbox{\tt\small REJECT}_{\mathcal{M}}\otimes L^{2}(\hat{\mathcal{U}})$ Hilbert ) $\cong L^{2}(\mathcal{M})$ $V$ $E_{*}(V)=$ (1) $E_{*}(V)$ $\hat{\mathcal{u}}$ $\int_{\chi\in Spec(A)}dE(\chi)\otimes\lambda_{\chi}$ $E_{*}(V)( \xi\otimes \gamma\rangle)=.\int_{\chi\in Spec(A)}dE(\chi)\xi\otimes \chi\gamma\rangle$ for $\gamma\in\hat{\mathcal{u}}$ $\xi\in L^{2}(\mathcal{M})$ $(2)$ E*(V)I2E*(V)I3V23 $=V_{23}E_{*}(V)_{12}$ 5 ( ) $ L\rangle$ $\xi=\sum_{\gamma\in\hat{g}}c_{\gamma}\xi_{\gamma}\in \mathfrak{h}_{\mathcal{m}}$ coupling $E_{*}(V)$ $\text{ ^{}\backslash }.\text{ }\xi\otimes \iota\rangle 1 \text{ }f_{l}$ 1 1 [5] [14] coupling MASA $A$ $\hat{\mathcal{u}}$ K-T $V$ $E_{*}(V)$ [5] instrument $0$ $2(\Delta \omega_{\xi})(b)=(\omega_{\xi}\otimes \iota\rangle\langle\iota )(E_{*}(V)^{*}(B\otimes x\delta)e_{*}(v))$ $=((\xi \otimes\langle\iota )E_{*}(V)^{*}(B\otimes\chi_{\Delta})E_{*}(V)( \xi\rangle\otimes \iota\rangle)$ [14] [5] $\omega_{\xi}$ $\mathcal{m}\ni B\mapsto\omega_{\xi}(B)=\langle\xi B\xi\rangle$ $A$ $\gamma\in\overline{\mathcal{u}(a})$ $\Delta$ Borel $p(\delta \omega_{\xi})=2(\delta \omega_{\xi})(1)$ $3(\Delta \omega_{\xi}.)/p(\delta \omega_{\xi})$ type I $=$ 4 $\mathcal{m}\mathrm{x}_{\alpha}\mathcal{u}$ dynamics coupling $E_{*}(V)$ \Phi $\text{ }$ $(V\eta)(\gamma_{1} \gamma_{2})=\eta(\gamma_{1} \gamma_{1}^{-1}\gamma_{2})(\eta\in L^{2}(\hat{\mathcal{U}}\mathrm{x}\hat{\mathcal{U}}))$ K-T $V$ Fourier $W=(F\otimes F)^{-1}V(\mathcal{F}\otimes F)$ $(W\xi)(u_{1} u_{2})=\xi(u_{2}u_{1} u_{2})$ (for $u_{1}$ $u_{2}\in \mathcal{u})$ $(F \xi)(\gamma)=\int\overline{\gamma(g)}\xi(g)dg(\xi\in L^{2}(\mathcal{U}))$ $\xi\in L^{2}(\mathcal{U}\cross \mathcal{u})$ $\mathrm{w}^{r}$ l 5 $W_{12}W_{13}W_{23}=W_{23}W_{12}$ intertwining relation $W(\lambda\otimes\lambda)=(\iota\otimes\lambda)W$ K-T $E$ $\mathcal{u}arrow Aarrow \mathcal{m}$ $EW=(E\otimes id)(w)$ 5 $(EW)_{12}(EW)_{13}W_{23}=W_{23}(EW)_{12}$ intertwining relation $EW(u\otimes\lambda_{u})=$ (I\otimes \mbox{\boldmath $\lambda$} EW $\alpha=ad$

10 $\hat{\alpha}$ 114 $\mathcal{m}arrow$ $L^{\infty}(\mathcal{U} \mathcal{m})=\mathcal{m}\otimes L^{\infty}(\mathcal{U})$ $\pi_{\alpha}$ $\mathcal{m}\otimes L^{\infty}(\mathcal{U})$ $(\pi_{\alpha}(x)\xi)(u)=\alpha_{u}^{-1}(x)(\xi(u))=(u^{-1}xu)(\xi(u))$ (3) for $\xi\in L^{2}(\mathcal{M})\otimes L^{2}(\mathcal{U})u\in \mathcal{u}$ $EW$ unitary implementer $\pi_{\alpha}(x)=(ew)(x\otimes I)(EW)^{*}$ for $X\in \mathcal{m}$ $\pi_{\alpha}(\mathcal{m})$ $\mathbb{c}i\otimes\lambda(\mathcal{u}) $ $\alpha$ $\mathcal{m}\aleph_{o}\mathcal{u}$ $\mathcal{m}\nu_{\alpha}\mathcal{u}=\pi_{\alpha}(\mathcal{m})\vee(\mathbb{c}\otimes\lambda(\mathcal{u}) )$. von Neumann [10] $\mathcal{m}=\mathbb{c}1$ $(\lambda L^{2}(\mathcal{U}))$ $\mathcal{m}\aleph_{\alpha}\mathcal{u}$ convolution $(X*\mathrm{Y})(u)=$ $X(v)\alpha_{v}(\mathrm{Y}(v^{-1}u))dv$ ck $X\#(u)=$ $\alpha_{\mathrm{u}}(x(u^{-1}))^{*}$ $L^{1}(\mathcal{U}\mathcal{M})=\mathcal{M}\otimes L^{1}(\mathcal{U})$ *- operator-valued Fourier (X)=(Xdu\otimes id)(\mbox{\boldmath $\sigma$}(ew \mbox{\boldmath $\sigma$}) $= \int_{\mathcal{u}}x(u)udu$ for $X\in L^{1}(\mathcal{U}\mathcal{M})=\mathcal{M}\otimes L^{1}(\mathcal{U})$ ; $\text{ }(X*Y)=$ (X) (Y) and (X ) $=$ (X)* $\alpha$ ( ) $A$ coupled dynamics switch-on off ( ) $\iotaarrow\alphaarrow\iota$ $(\mathcal{m}\otimes A\supset)\mathcal{M}\otimes L^{\infty}(\hat{\mathcal{U}})=$ $=$ $\mathcal{m}\aleph_{\alpha}\mathcal{u}$ initial $\mathcal{m}\cross_{\iota}\mathcal{u}arrow \mathcal{m}\rangle \mathrm{t}_{\alpha}\mathcal{u}arrow \mathcal{m}\otimes L^{\infty}(\hat{\mathcal{U}})$final Fourier [15] $\mathcal{m}\simeq \mathcal{m}\otimes B(L^{2}(\mathcal{U}))$ $\mathrm{o}\mathrm{k}_{0}\hat{\alpha}$ $Y\in \mathcal{m}\aleph_{\alpha}\mathcal{u})$ $(\mathcal{m}\mathrm{x}_{\alpha}\mathcal{u})*_{\hat{\alpha}}\hat{\mathcal{u}}\simeq \mathcal{m}\otimes B(L^{2}(\mathcal{U}))\simeq \mathcal{m}$. \mbox{\boldmath $\pi$} (for $\pi_{\overline{\alpha}}(y)=ad(1\otimes\sigma W^{*}\sigma)(Y\otimes 1)$ $\mathcal{m}\aleph_{\alpha}\mathcal{u}$ $\hat{\mathcal{u}}$ dual co action [10] $\mathcal{m}x_{\alpha}\mathcal{u}$ $\hat{\mathcal{u}}$ Fourier dual co-action MASA $A$ Spec $(A)\subset\hat{\mathcal{U}}$ $\mathcal{m}\aleph_{\alpha}\mathcal{u}$ $[16]_{\text{ }}$ M Fourier-Galois [5]

$5 $=$ decoherence 115 coupling $\lambda^{\otimes n}=\lambda\otimes\cdots\otimes\lambda$ $\lambda^{\otimes m}\approx\lambda^{\otimes n}(\forall m n\in \mathrm{n})$ K-T $V$ $V_{nn+1} \cdots$

11 5 $=$ decoherence 115 coupling $\lambda^{\otimes n}=\lambda\otimes\cdots\otimes\lambda$ $\lambda^{\otimes m}\approx\lambda^{\otimes n}(\forall m n\in \mathrm{n})$ K-T $V$ $V_{nn+1} \cdots V_{23}E_{*}(V)_{12}(\xi\otimes)\frac{ \iota\rangle\otimes \iota\rangle\cdots\otimes \iota\rangle}{n}$ $= \sum_{\gamma\in\hat{g}}c_{\gamma}v_{nn+1}\cdots V_{34}V_{23}(\xi_{\gamma}\otimes \gamma\rangle\otimes \iota\rangle\cdots\otimes \iota\rangle)$ $= \sum_{\gamma\in\hat{g}}c_{\gamma}v_{nn+1}\cdots V_{34}(\xi_{\gamma}\otimes \gamma\rangle\otimes \gamma\rangle\cdots\otimes \iota\rangle)=\cdots$ $n arrow\inftyarrow\sum_{\gamma\in\hat{g}}\mathrm{r}_{\gamma}\xi_{\gamma}\otimes[ \gamma\rangle^{\otimes\infty}]$ Heisenberg $A\otimes f_{2}\otimes\cdots\otimes f_{n+1}$ $\mapsto E_{*}(V)_{12}^{*}V_{23}^{*}\cdots V_{nn+1}^{*}(A\otimes f_{2}\otimes\cdots\otimes f_{n+1})v_{nn+1}\cdots V_{23}E_{*}(V)_{12}$ $=Ad(E_{*}(V)_{12}^{*})\circ Ad(V_{23}^{*})\cdots Ad(V_{nn+1}^{*})(A\otimes f_{2}\otimes\cdots\otimes f_{n+1})$ $=Ad(E_{*}(V)^{*})(A\otimes Ad(V^{*})(f_{2^{-}}\otimes Ad(V^{*})(\cdots\otimes Ad(V^{*})(f_{n}\otimes f_{n+1})))\cdots)$ time-ordered Dyson matrix Accardi quantum Markov chain [17] Ising Heisenberg spin $ \gamma\rangle^{\otimes\infty}$ $ +\rangle^{\otimes\infty}$ $\Pi\overline{\mathrm{p}}$ $\iotaarrow\gamma$ decoherence $\lambda^{\otimes n}=\lambda\otimes\cdots\otimes\lambda$ repeatablity hypothesis ( ) $f(x+y)=f(x)+f(y)$ $f$ $f(\lambda x+\mu y)=\lambda f(x)+\mu f(y)(\forall\lambda \mu>0)$

12 116 $\lambda\approx\lambda^{n}(\forall n\in \mathrm{n})$ $\lambda\approx\lambda^{n/m}$ $(\forall m n\in \mathrm{n})$ $(AdV^{*})^{t+S}\approx(AdV^{*})^{t}(AdV^{*})^{s}(t s>0)$ ( L\ evy ) 1 1 [18]? 6 $\mathcal{m}\rangle\triangleleft_{\alpha}\mathcal{u}\simeq A\otimes B(L^{2}(\mathcal{U}))$ semi-duality canonical References [1] Doplicher S. Haag R. and Roberts J.E. Fields observables and gauge transformations I&II Comm. Math. Phys. 13 (1969) 1-23; 15 (1969) ; Local observables and particle statistics I& II 23 (1971) ; 35 (1974) [2] Doplicher S. and Roberts J.E. Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics Comm. Math. Phys. 131 (1990) ; Endomorphism of $\mathrm{c}^{*}$-algebras cross products and duality for compact groups Ann. Math. 130 (1989) ; A new duality theory for compact groups Inventiones Math. 98 (1989) [3] Dixmier J. $C^{*}$-Algebras North-Holland 1977; Pedersen G. $C^{*}-$ $Algebra.9$ and Their Automorphism Groups Academic Press [4] Ojima I. A unified scheme for generalized sectors based on selection criteria-order parameters of symmetries and of thermality and physical meanings of adjunctions- Open Systems and Information Dynamics 10 (2003) ; Temparature as order parameter of broken scale invariance Publ. RIMS (2004). [5] Ojima I. Micro-macro duality in quantum physics pp in Proc. Intern. Conf. on Stochastic Analysis Classical and Quantum World Scientific 2005.

13 $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{a}\mathrm{k}\mathrm{a}-\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}$ -Tatsuuma 117 [6] Ojima I. How to formulate non-equilibrium local states in QFT?- General characterization and extension to curved spacetime- pp in ( $ \mathrm{a}$ Garden of Quanta World Scientific (2003); -print cond- $\mathrm{m}\mathrm{a}\mathrm{t}/ $. [7] Buchholz D. Ojima I. and Roos H. Thermodynamic properties of non-equilibrium states in quantum field theory Ann. Phys. (N.Y.) 297 (2002) [8] Takesaki M. Theo $\gamma\eta/of$ Operator Algebras $I$ Springer-Verlag [9] Takesaki M. A characterization of group algebras as a converse of duality theorem Amer. J. Math. 91 (1969) [10] Nakagami Y. and Takesaki M. Lec. Notes in Math. 731 Springer [11] Enock M. and Schwartz J.-M. $Kac$ Algebras and Duality of Locally Compact Groups Springer [12] N. Tatsuuma A duality theory for locally compact groups J. Math. Kyoto Univ. 6 (1967) ; ( 1994) [13] Ozawa M. Perfect correlations between noncommuting observables $\mathrm{a}$ $33\bm{5}$ Phys. Lett (2005). [14] Ozawa M. Quantum measuring processes of continuous observables. J. Math. Phys (1984); Publ. RIMS Kyoto Univ (1985); Ann. Phys. (N.Y.) (1997). [15] Takesaki M. Duality for crossed products and the structure of von Neumann algebras of type III Acta Math (1973); Theory of Operator Algebras II Springer-Verlag $\mathrm{m}$ [16] Ojima I. and Takeori How to observe quantum fields and recover them from observational data? Takesaki duality as a Micro-Macro $\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}-\mathrm{p}\mathrm{h}/ $ duality- (2006). [17] Accardi L. Noncommutative Markov chains in Intern. School of Math. Phys. Camerino pp (1974); Topics in quantum probability Phys. Rep. 77 (1981) [18]. ( III 2 pp j 1996)

2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2

1 vs. 90 mescoscopic physics 1 2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2 (: ) [1]: 1. Newton =[( ) vs. ] (a) =0 x v ( p = mv) [ a), b), c)] (b) = :