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

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1 1 vs. 90 mescoscopic physics 1

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

3 (: ) [1]: 1. Newton =[( ) vs. ] (a) =0 x v ( p = mv) [ a), b), c)] (b) = : [dp/dt = F = ] [ b), c), d)] F Newton (c) = [ c), d): dynamics] 3

4 2. (a) & = Lagrange ( ) vs. Euler ( ) [ a), c)] Newton ( x, t) v( x, t) [ a), b)] ( x, t) vs. [ d)] (b) [ a), d)] Newton 4

5 F µν = 0 E 1 E 2 E 3 E 1 0 B 3 B 2 E 3 B 3 0 B 1 E 3 B 2 B 1 0 ( E:, B: ) source J µ = (ρ, J) p µ p µ ea µ Maxwell ν F νµ = J µ J µ =Minkowski J µ Maxwell F µν Lorentz T µν [2] (c) duality J µ F µν T µν g µν duality T µν g µν Einstein : T µν Einstein g µν. T µν T µν g µν Maxwell Einstein R µν 1 2 g µνr = κt µν g µν 5

6 Γ λ µν = 1 2 gλρ ( µ g ρν + ν g ρµ ρ g µν ) Newton [ a), b), d)] 3. vs. (a) [ d W = ] vs. [ d Q = ] = = = : de = d Q+ d W = = ds = d Q/T 0 (b) e.g., Joule (c) [ vs. ] (= ) ( ) + (,,...) + + [= ] 5W1H a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] 6

7 b) c) = dynamics d) a) 3 M M Hilbert 1. M M M C *- c 1 A 1 + c 2 A 2 A i M, c i C A 1 A 2 A i M (c 1 A 1 + c 2 A 2 )B = c 1 A 1 B + c 2 A 2 B, A(c 1 B 1 + c 2 B 2 ) = c 1 AB 1 + c 2 AB 2, B i M M (involution) 1 (c 1 A 1 + c 2 A 2 ) = c 1 A 1 + c 2 A 2, A = A, A1 = 1A = A, 1 = 1. M : AB BA (A, B M) M ω 7

8 : ω : M C : ω(c 1 A 1 + c 2 A 2 ) = c 1 ω(a 1 ) + c 2 ω(a 2 ) : ω(a A) 0 (A M); : ω(1) = 1. (A i M, c i C); ω = ω(1) = 1 1 ω(a A) 0 M A B ω := ω(a B) M Hilbert A ω = ω(a A) = 0 N ω := {A M; ω(a A) = 0} H ω := M/N ω [A] := A + N ω ω Hilbert H ω M π ω (A)[B] := [AB], [1] =: Ψ ω ω(a) = Ψ ω π ω (A)Ψ ω H ω = π ω (M)Ψ ω Ψ ω Gel fand, Naimark, Segal GNS (π ω, H ω, Ψ ω ) Hilbert Hilbert Hilbert ( [3] ) parallel M A M ω(a) [4] 2. vs. M AB = BA for A, B M 8

9 M x p f i = f i (x, p) f 1 f 2 = f 1 (x, p)f 2 (x, p) M c 1 ψ 1 + c 2 ψ 2 ψ 1 ψ 2 Schrödinger 3. A : ω A = A M ω(a) π ω (A) Spec(A) := {λ R; (A λ1) : non-invertible} π ω (A) = λdâω(λ) Spec(A) Hilbert H ω operator A a A = 0 a = a i Â i ; i 0 0 a d Â i = (δ pi δ qi ) d p,q=1 : i- f(x) = k f kx k k f ka k = k f k( i a iâi) k = i k f ka k i Âi = i f(a i)âi, f(a) = i f(a i)âi Gel fand [5] A M(= Spec(A)) 9

10 (C(M) or L (M)) Â ω : L (M) f Âω(f) = M f(λ)dâω(λ) = f(a) ω B(H ω ) : all bounded operators on H ω f 1, f 2 operator Âω(f 1 f 2 ) = Â ω (f 1 )Âω(f 2 ) [6, 4] Âω { = 1 (x ) χ, χ (x) = f Âω( ) = = 0 (x / ) Â ω (χ ) M dâω f(λ) 4. M α R with R t α t Aut(M) M Hamiltonian, ω (disjoint) = Hilbert Hamiltonian = parallel Hamiltonian ω : ω α t = ω ω GNS U t π ω (A)Ψ ω := π ω (α t (A))Ψ ω U t U t π ω (α t (A)) = U t π ω (A)U t, U t Ψ ω = Ψ ω U t = exp(ih ω t) = Hamiltonian H ω (Stone ) GNS ω 10

11 Hilbert H ω Hamiltonian H ω ω ω 5. Born = A(= A ) a Spec(A) random ω A a Spec(A) P (a ω) P (a ω) = ω(âω( )) 4 = [7] Hilbert 11

12 Hilbert ψ Hilbert H ω GNS ω Hilbert H ω interface [4] 1) (or ) M π 1 = π2 U : unitary s.t. π 2 (A) = Uπ 1 (A)U 1 ( A M), π 1 = π π 2 = π π [ ] quasiequivalence π 1 π 2 m, n : cardinality s.t. mπ 1 = nπ2 {T rρπ 1 ( ) ; ρ: } = {T rρπ 2 ( ) ; ρ: } π 1 (M) = π 2 (M) (as von Neumann algebras) 12

13 π Z π (M) := π(m) π(m) = C1, factor [π] [6] [ T M AT = T A for A M] [π 1 ], [π 2 ] i.e., π 1 π 2 (disjoint) T π 1 (A) = π 2 (A)T ( A M) intertwiner T 0 2) π Z π (M) Spec(Z π (M)) π Z π (M) Spec(Z π (M)) phase diagram [8, 9] 3) 1) 2) 4) [x i, p j ] = i δ ij (Stone-von Neumann ) 5) selection criterion - [8, 9] 13

14 i) [ q : ( ) ] = ii) [ c : ( ) ] iii) i) ii) [ ] selection criterion: matching [ ii) i) : iv) c-q channel ii) = i) adjunction q-c channel i) = ii) ] [10, 8] Example 1 {(U λ, ϕ λ : U λ R n )} M: i) = U λ, ii)= R n, iii)= ϕ λ : U λ R n, iv)=,, K-, M Example 2 [11, 10] A) B) DHR [12] [8] C) (SSB) [8] D) [10, 8] E) [13] F) [15] A) : D) - : B) : C) : & F) : [ E) ] 14

15 5 Fourier-Pontryagin [15] 3. 2) 1. [] [ = 0] A 1, A 2,, A n A k = a (k) = A ia j = A j A i 0 0 a (k) d a (k) (i, j, k = 1, 2,, n). Sec.2 3. Gel fand Gel fand [5] : C*- A M := Spec(A) := {χ; χ : A C: } compact Hausdorff A M A C(M) A : repeatablity hypothesis Born repeatablity Born Born coupling 15

16 2. coupling 2) M A = M A 3. Fourier-Pontryagin : A U U A = U U U γ : U C, γ(u 1 u 2 ) = γ(u 1 )γ(u 2 ) (u 1, u 2 U), γ(e) = 1 Û Û U Û Û Fourier- Pontryagin U U Û Fourier Û U; FL p q (U, du) = L (Û, dγ), du, dγ U Û Haar p, q 1/p+1/q = 1 Fourier Fourier (Ff)(γ) := γ(u)f(u)du; U (F 1 ϕ)(u) := γ(u)ϕ(γ)dγ Gel fand Spec(A) = {χ; χ : A C: } U A = U χ Spec(A) χ U Û Spec(A) Û 16 bu

17 A Spec(A) Û 4. coupling = Kac- operator instrument repeatability hypothesis M A M U α u (A) = U(u)AU(u) 1 α adjoint action Ad(u)A := uau 1 dynamics coupling Kac- (K-T) [14] unitary operator Ũ(V ) = de(χ) λ χ. χ Spec(A) [15]dE(χ) U U(u) = χ(u)de(χ) χ Spec(A) λ χ Û (λ χξ)(γ) := ξ(χ 1 γ) for ξ L2 (Û, dγ), γ Û coupling Ũ(V ) A ξ χ ι ι χ Ũ(V )(ξ χ ι ) = ξ χ χ ξ = χ Spec(A) c χξ χ coupling Ũ(V ) ξ ι Ũ(V )(ξ ι ) = c χ ξ χ χ, χ Spec(A) [16] χ Spec(A) Û ξ χ instrument I( ω ξ )(B):=(ω ξ m U )(Ũ(V ) (B χ )Ũ(V )) = ( ξ ι )Ũ(V ) (B χ )Ũ(V )( ξ ι ) de(γ) = dµ(γ) B de(γ) dµ(γ) =: de(γ)b de(γ) dµ(γ) 17

18 ω ξ : M B ω ξ (B) = ξ Bξ M m U (f) = ι f ι A γ Spec(A) Borel p( ω ξ ) = I( ω ξ )(1) M I( ω ξ )/p( ω ξ ) [17] 5. Fourier duality [12] G A = F G A G = Gal(F/A) F G Galois : Ĝ [A = F G ] = [F = A Ĝ] G = Gal(F/A). G RepG [14] Lie [15] U M( M α(u) ˆα Û) (M α U) Û α ˆα References [1] vs. I, , ( Ojima, I., Nature vs. science. I, Acta Inst. Phil. et Aesth. 10, (1992));, ( [2],, 1990). [3] Haag, R., Local Quantum Physics Fields, Particles, Algebras (2nd ed.) (Springer-Verlag, 1996); 21 (1992). [4] ;

19 ;, , pp [5] (, 1985). [6] large deviation 1066 (1998), [7], (, 1982);, ( 28, 1995). [8] 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), (math-ph/ ). [9] ( 2004) pp ; ; ; ; [10] Ojima, I., How to formulate non-equilibrium local states in QFT? General characterization and extension to curved spacetime, pp in A Garden of Quanta, World Scientific (2003) (condmat/ ). [11] Buchholz, D., Ojima, I. and Roos, H., Thermodynamic properties of non-equilibrium states in quantum field theory, Ann. Phys. (N.Y.) 297 (2002), [12] 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), 49-85; 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 C*-algebras, cross products and duality for compact groups, Ann. Math. 130 (1989), ; A new duality theory for compact groups, Inventiones Math. 98 (1989), [13] Ojima, I., Temperature as order parameter of broken scale invariance, Publ. RIMS 40, (2004) (math-ph ). 19

20 [14] Takesaki, M., A characterization of group algebras as a converse of Tannaka-Stinespring-Tatsuuma duality theorem, Amer. J. Math. 91 (1969), ; Enock, M. and Schwartz, J.-M., Kac Algebras and Duality of Locally Compact Groups, Springer, 1992;, [15] Ojima, I., Micro-macro duality in quantum physics, pp in Stochastic Analysis: Classical and Quantum Perspectives of White Noise Theory ed. by T. Hida, World Scientific (2005) (mathph/ ); Ojima, I. and Takeori, M, How to observe quantum fields and recover them from observational data? Takesaki duality as a Micro-Macro duality, Open Sys. & Inf. Dyn. 14, (2007) (math-ph/ (2006));, pp.34-44;, (No.457), pp [16] Ozawa, M., Perfect correlations between noncommuting observables, Phys. Lett. A, 335, (2005). [17] Ozawa, M., Quantum measuring processes of continuous observables. J. Math. Phys. 25, (1984); Publ. RIMS, Kyoto Univ. 21, (1985); Ann. Phys. (N.Y.) 259, (1997). 20

数理解析研究所講究録第1908巻

数理解析研究所講究録第1908巻 1908 2014 78-85 78 1 D3 1 [20] Born [18, 21] () () RIMS ( 1834) [19] ( [16] ) [1, 23, 24] 2 $\Vert A\Vert^{2}$ $c*$ - $*:\mathcal{x}\ni A\mapsto A^{*}\in \mathcal{x}$ $\Vert A^{*}A\Vert=$ $\Vert\cdot\Vert$