数理解析研究所講究録第1908巻

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$1908 2014 78-85 78 1 D3 1 [20] Born [18, 21] () () RIMS ( 1834) [19] ( [16] ) [1, 23, 24] 2 $\Vert A\Vert^{2}$ $c*$ - $*:\mathcal{x}\ni A\mapsto A^{*}\in \mathcal{x}$ $\Vert A^{*}A\Vert=$$

1 D3 1 [20] Born [18, 21] () () RIMS ( 1834) [19] ( [16] ) [1, 23, 24] 2 $\Vert A\Vert^{2}$ $c*$ - $*:\mathcal{x}\ni A\mapsto A^{*}\in \mathcal{x}$ $\Vert A^{*}A\Vert=$ $\Vert\cdot\Vert$ 1 ) $\omega(1)=1$ Banach ( $E_{\mathcal{X}}$ $\omega(a^{*}a)\geq 0$ () $C^{*}-$ $(\mathcal{x}, \omega)$ $c*$ - () 2-3 $c*$ - (continuous functional calculus) - () : 1( [20,19-1 kazuqi@kurims kyoto-u ac.$ip$ 2 3 $*:A\mapsto A^{*}$

2 ) ) $\hat{\mathcal{x}}$ 79 $c*$ - $E_{\mathcal{X}}$ $A_{i}\in \mathcal{x},$ $\epsilon_{i}>0(i=1,2, \cdots, n)$ : $O_{\omega}(\{A_{i}, \epsilon_{i}\}_{i=1}^{n})=\{\omega \in E_{\mathcal{X}} \omega(a_{i})-\omega (A_{i}) <\epsilon_{i}, i=1, 2, \cdots, n\}.$ 1(GNS ). $\omega\in $B(\mathcal{H}_{\omega})$ E_{\mathcal{X}}$ Hilbert $\mathcal{h}_{\omega}$ () $\pi_{\omega}$ $\Omega\omega$, $\in \mathcal{h}_{\omega}$ $\omega(a)=\langle\omega_{\omega} \pi_{\omega}(a)\omega_{\omega}\rangle$ $\mathcal{h}\omega$ $=\pi_{\omega}(\mathcal{x})\omega_{\omega}$ $\{\pi_{\omega}, \mathcal{h}_{\omega}, \Omega_{\omega}\}$ 3 GNS GNS $S\subset B(\mathcal{H})$ $S =\{A\in B(\mathcal{H}) AB =BA, B\in S\}$ (1) $S$ $S $ $:=(S ) $ $S$ $\mathcal{m}$ $B(\mathcal{H})$ - $\mathcal{m}"=\mathcal{m}$ ( $\mathcal{h}$ ( $\mathcal{h}_{\omega}$ $\pi_{\omega}(\mathcal{x})"$ von Neumann von Neumann 2(). $F_{\mathcal{X}}/\approx$ 3(). $\omega\in $\mathfrak{z}_{\omega}(\mathcal{x})$ $F_{\mathcal{X}}$ E_{\mathcal{X}}$ $\pi_{\omega}(\mathcal{x})"$ von Neumann $:=\pi_{\omega}(\mathcal{x})"\cap\pi_{\omega}(\mathcal{x}) =\mathbb{c}1$ $\mathcal{m}$ 4 ( $\pi$-). (1) von Neumann $A_{\alpha}\nearrow A$ $\lim_{\alpha}\omega(a_{\alpha})=\omega(a)$ $\mathcal{m}_{*,1}$ $\mathcal{m}$ (2) $\rho$ $c*$ - $\pi$ $\omega\in E_{\mathcal{X}}$ $\pi$- $\pi(\mathcal{x})"$ $\omega(x)=\rho(\pi(x))$ $X\in \mathcal{x}$, 5(). (1) 2 $\pi_{1},$ $\pi_{2}$ $\pi_{1}$ $\pi_{1}\approx\pi_{2}$ (2) 2 $\pi_{1},$ $\pi_{2}$ $\pi_{1}d\pi_{2}$ $\pi_{1}$ - $\pi_{2}$ $GNS$ - $\pi_{2}$

3 : $\mathfrak{b}$ ( ) 80 : $\Rightarrow$ $\Rightarrow$ 3 -, $E_{\mathcal{X}}$ Choquet $\omega\in$ $(E_{\mathcal{X}}, \mathcal{b}(e_{\mathcal{x}}))$ Borel E 4 7( ). $=$ $(E_{\mathcal{X}}, \mathcal{b}(e_{\mathcal{x}}))$ Borel $\mu$ $\triangle\in \mathcal{b}(e_{\mathcal{x}})$ : $\int_{\delta}d\mu(\rho)\rho d \int_{e_{\mathcal{x}\backslash \triangle}}d\mu(\rho)\rho$. (2) : 8 ( [1, Theorem ] ). (1) $\mathfrak{b}$ von Neumann $\mu,$ $L^{\infty}(E_{\mathcal{X}}, \mu)$ $\kappa_{\mu}$ $L^{\infty}(\mu)arrow \mathfrak{b}$ $\mathfrak{b}$ $\mu$ $\mathfrak{z}_{\omega}(\mathcal{x})$ $L^{\infty}(\mu):=$ - : $\langle\omega_{\omega} \kappa_{\mu}(f)\pi_{\omega}(x)\omega_{\omega}\rangle=\int d\mu(\rho)f(\rho)\rho(x)$. (3) (2) $\mathfrak{z}_{\omega}(\mathcal{x})$ $\mu_{\omega}$ $F_{\mathcal{X}}$ $\mu_{\omega}$ F von Neumann $F_{\mathcal{X}}$ $\mathfrak{z}_{\omega}(\mathcal{x})$ $(\pi_{\omega}(\mathcal{x})"$ $)$ ( GNS ) () 2( [20,19 $\omega\in E_{\mathcal{X}}$ $\Delta\in \mathcal{b}(e_{\mathcal{x}})$ $\mu_{\omega}(\delta)$ 4

4 ) 81 () () () $c*$ - $c*$ - : 9 (Gel fand-naimark ). Hausdorff $S$ - $0$ $C_{0}(S)$ - Hausdorfff $S$ : Spec $(\mathcal{x}$ $)$ Spec $(\mathcal{x})=\{\chi\in E_{\mathcal{X}} \chi(ab)=\chi(a)\chi(b), A, B\in \mathcal{x}\}$. (4) $E_{\mathcal{X}}$ () $\mathcal{e}(e_{\mathcal{x}})$ Spec $(\mathcal{x})=\mathcal{e}(e_{\mathcal{x}})=f_{\mathcal{x}}$ (5) Gel fand-n\ aimark $c*$ - Spec $(\mathcal{x}$ $)$ Borel : 10 (Riesz-Markov-Kakutani ). Hausdorff $S$ $0$ - $C_{0}(S)$ $S$ Borel $c*$ - $c*$ $(\mathcal{x}, \omega)$ - (Spec ( ), $\mathcal{b}$(spec ( ), $\mu$) ( $\mu$ Borel ) $C(Spec(\mathcal{X}))$ Riesz-Markov-Kakutani () Riesz-Markov-Kakutani : 11 ([23, Chapter III, $S$ Theorem 1.2]). Hausdorff $\mu$ $S$ Borel ( $\mu$ ) (i) $C_{0}(S)$ $\mu$ $GNS$ $(\pi_{\mu}, \mathcal{h}_{\mu}, \Omega_{\mu})$ Hilbert $L^{2}(S, \mu)$, $L^{2}(S, \mu)$ $M_{f},$ $f\in C_{0}(S)$ $S$, 1 : $\mathcal{h}_{\mu}=l^{2}(s, \mu), \pi_{\mu} =M., \Omega_{\mu}=1$ ; (ii) von Neumann $\pi_{\mu}(c_{0}(s))"$ $L^{2}(S, \mu)$ $\pi_{\mu}(c_{0}(s))"=\pi_{\mu}(c_{0}(s)) $ ; (6) (iii) $\pi_{\mu}(c_{0}(s))"$ $\pi(f)$, $f\in L^{2}(S, \mu)$ $(\pi(f)\xi)(s)=f(\mathcal{s})\xi(s), \xi\in L^{2}(S, \mu), s\in S$. (7)

$\mathcal{o}_{1}$ 82 $c*$ $(\mathcal{x}, \omega)$ - 5, ( ) 4 ( ) von Neumann (von Neumann ), $\{\mathcal{a}(\mathcal{o}) \mathcal{o}\in \mathcal{k}\}$ Minkowski $M_{4}$

5 $\mathcal{o}_{1}$ 82 $c*$ $(\mathcal{x}, \omega)$ - 5, ( ) 4 ( ) von Neumann (von Neumann ), $\{\mathcal{a}(\mathcal{o}) \mathcal{o}\in \mathcal{k}\}$ Minkowski $M_{4}$ $\mathcal{k}=\{\mathcal{o}=(a+$ $V_{+})\cap(b-V_{+}) a,$ $b\in M_{4}\}$ $(V;= \{x\in M_{4} x^{2}=x_{0}^{2}-\sum_{j=1}^{3}x_{i}^{2}>0, x_{0}>0\}$ $M_{4}$ ) $c*$ - $\mathcal{o}\mapsto () 3 : \mathcal{a}(\mathcal{o})$ 1) $\mathcal{o}_{1}\subset \mathcal{o}_{2}$ $\mathcal{a}(\mathcal{o}_{1})\subset \mathcal{a}(\mathcal{o}_{2})$ ; 2) $\mathcal{k}$ $\mathcal{o}_{1}$ $\mathcal{o}_{2}$ $\mathcal{a}(\mathcal{o}_{1})$ 2 $\mathcal{o}_{1}$ $\mathcal{o}_{1} \supset ; $\mathcal{a}(\mathcal{o}_{2})$ $\mathcal{o}i=\{x\in M_{4} (x-y)^{2}<0, y\in \mathcal{o}_{1}\}$ $\mathcal{o}_{2}$ \mathcal{o}_{2}$ $\mathcal{a}:=\bigcup_{\mathcal{o}\in \mathcal{k}}\mathcal{a}(\mathcal{o})$ $c*$ - $Aut(\mathcal{A})$ - $\alpha_{9}$ : $\mathcal{p}_{+}^{\uparrow}$ 3) Poincar\ e $(^{*}$-) $\alpha_{g}(\mathcal{a}(\mathcal{o}))=\mathcal{a}(g\mathcal{o})$ $\mathcal{o}\in $\mathcal{p}_{+}^{\uparrow}arrow Aut(\mathcal{A})$, \mathcal{k}$ $g\in $g\in \mathcal{p}_{+}^{\uparrow}$, \mathcal{p}_{+}^{\uparrow}$ () $\omega_{0}$ 2 : $\omega_{0}$ 5

6 $\tilde{\mathcal{o}}$ \mathcal{o}^{-},\mathcal{o}\in \mathcal{k}}\mathcal{a}(\mathcal{o})}$ 83 A) $\omega_{0}$ $\mathcal{p}_{+}^{\uparrow}$- A $\in \mathcal{a}$ $9\in \mathcal{p}_{+}^{\uparrow}$ $\omega_{0}(\alpha_{g}(a))=\omega_{0}(a)$ ; (8) $\omega_{0}$ A $\in GNS $(\pi_{0}, \mathcal{a}$ $g\in \mathcal{p}_{+}^{\uparrow}$ \mathcal{h}_{0}, \Omega_{0})$ $\alpha_{g}$ : $\pi_{0}(\alpha_{g}(a))=u_{g}\pi_{0}(a)u_{9}^{*}$. (9) $U_{g}\Omega=\Omega$ $U_{g}$ : $\mathcal{p}_{+}^{\uparrow}$ $\mathbb{r}^{4}$ B) Poincar\ e $U_{g}$ $P=(P_{\mu})_{\mu=0,1,2,3}$ $\overline{v_{+}}=\{x\in M_{4} x^{2}=x_{0}^{2}-\sum_{j}^{3_{=1}}x_{i}^{2}\geq 0, x_{0}\geq 0\}$ ; A) B) $\{\mathcal{a}(\mathcal{o}) \mathcal{o}\in \mathcal{k}\}$ $DHR$(Doplicher-Haag-Roberts) $\omega_{0}$ $\mathcal{h}_{0}$ GNS Hilbert $\{0\})$ 2 : 1(Haag ). $\mathcal{o}\in \mathcal{k}$ $\{\mathcal{a}(\mathcal{o}) \mathcal{o}\in \mathcal{k}\}$ 2( B). $\mathcal{o}_{1}$ $\mathcal{o}$2 $\mathcal{a}(\tilde{\mathcal{o}})=\overline{\bigcup_{\mathcal{o}\subset $(ker(\pi_{0})=$ $\pi_{0}(\mathcal{a}(\mathcal{o}))"=\pi_{0}(\mathcal{a}(\mathcal{o} )) $ Haag $E\in $W^{*}W=E,$ $WW^{*}=1$ 2 DHR [3, 4, 5, 6, 7] $W\in \mathcal{a}(\mathcal{o}_{2})$ \mathcal{a}(\mathcal{o}_{1})$ $=$ (10) : DHR $\mathcal{o} $ $\pi_{0}$ () $\mathcal{o}\in : \mathcal{k}$ $\pi _{\mathcal{a}(\mathcal{o} )}\cong\pi_{0} _{\mathcal{a}(\mathcal{o} )}$. (11)

$$\gamma$ 84 DHR $B$ $\pi$ $\pi$- DHR $\mathcal{o}$ 12. $DHR$ $\pi$ $\rho$ : (1) $\pi=\pi_{0}\circ\rho,$ (2) $\rho(a)=a,$ $A\in \mathcal{a}(\mathcal{o} )$ - 1(Haag ).$

7 $\gamma$ 84 DHR $B$ $\pi$ $\pi$- DHR $\mathcal{o}$ 12. $DHR$ $\pi$ $\rho$ : (1) $\pi=\pi_{0}\circ\rho,$ (2) $\rho(a)=a,$ $A\in \mathcal{a}(\mathcal{o} )$ - 1(Haag ). - $DR(\mathcal{A}) :=\{\rho\in End(\mathcal{A}) \exists \mathcal{o}\in \mathcal{k}s.t. \rho(a)=a, A\in \mathcal{a}(\mathcal{o} )\}$ (12) $\{\mathcal{a}(\mathcal{o}) \mathcal{o}\in \mathcal{k}\}$ $DR(\mathcal{A})$ $DR(\mathcal{A})$ $DR(\mathcal{A})$ $c*$- Doplicher-Roberts [6] $DR(\mathcal{A})$ $G$ Rep (G) $G$ () $G$ DHR Haag Haag (essential duality) DHR DHR [13, 14] 5 () () [2, 8, 9] () Sanov 1 () Sanov [11] :

8 $\mu,$ $v$ $\psi$ $\varphi,$ $m$ $\mu,$ $v\ll m$ $S(\varphi\Vert\psi)=D(\mu\Vert\nu)$. (13) $S(\varphi\Vert\psi)$ [22, 25] $D(\mu\Vert v)$ : $D(\mu\Vert\nu)=\{\begin{array}{ll}\int d\mu(\rho)\log\frac{d\mu}{dv}(\rho), (\mu\ll\nu),+\infty, (otherwise).\end{array}$ (14) [1] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics (vol. 1), Springer-Verlag (1979). [2] I. Csisz\ ar, Bull. Brazilian Math. Soc. 37, (2006) [3] S. Doplicher, R. Haag and J.E. Roberts, I & II, Comm. Math. Phys. 13, 1-23 (1969); 15, (1969). [4] S. Doplicher, R. Haag and J.E. Roberts, I& II, 23, (1971) & $35,$ (1974). [5] S. Doplicher and J.E. Roberts, Ann. Math. 130, (1989). [6] S. Doplicher and J.E. Roberts, Invent. Math. 98, S (1989). [7] S. Doplicher and J.E. Roberts, Comm. Math. Phys. 131, (1990). [8] A. Dembo and O. Zeitouni, Large deviations techniques and applications (2nd ed (Springer, 2002). [9] R.S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, (Springer, 1985). [10] R. Harada and I. Ojima, Open Sys. $Inf$. Dyn. 16, (2009). [11] F. Hiai, M. Ohya and M. Tsukada, Pacific J. Math. 107, (1983). [12] I. Ojima, Order Parameters in QFT and Large Deviation, RIMS Kokyuroku (1998), (in Japanese). [13] I. Ojima, Open Sys. $Inf$. Dyn. 10, (2003). [14] I. Ojima, Publ. RIMS 40, (2004). [15] I. Ojima, Micro-Macro Duality in Quantum Physics, pp in Proc. Intern. Conf. on Stochastic Analysis, Classical and Quantum (World Scientific, 2005), arxiv:math-ph/ [16] (2013). [17] I. Ojima and K. Okamura, Open Syst. $Inf$. Dyn. 19, (2012). [18] I. Ojima and K. Okamura, Open Syst. $Inf$. Dyn. 19, (2012). [19] (2013). [20] I. Ojima, K. Okamura and H. Saigo, Derivation of Born Rule from Algebraic and Statistical Axioms, (2013), $arxiv: $, to appear in Open Sys. $Inf$. Dyn. [21] K. Okamura, Quant. $Inf$. Process. 12, , (2013). [22] M. Ohya and D. Petz, Qunatum Entropy and Its Use, (Springer, Berlin, 1993). [23] M. Takesaki, Theory of Operator Algebras $I$, (Springer, 1979). [24] M. Takesaki, Theory of 0perator Algebras II, (Springer, 2002). [25] $(1983,1984)$.

数理解析研究所講究録第1921巻

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数理解析研究所講究録第1908巻