α : G X (s, A) α s (A) X α s (c 1 A 1 + c 2 A 2 ) = c 1 α s (A 1 )+c 2 α s (A 2 ), α st (A) = α s (α t (A)) G X α 1.1 G α X (IO) 5W1H A A B A B 1.2!?
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- あいぞう ひでやま
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1 62 Abstract 0 Coulomb Coulomb Coulomb BCS Higg Cooper 1 G α X, G G = R or Z X X X (A, B) A + B, AB X, K X (c, A) ca X K K = R C K = R or C 1
2 α : G X (s, A) α s (A) X α s (c 1 A 1 + c 2 A 2 ) = c 1 α s (A 1 )+c 2 α s (A 2 ), α st (A) = α s (α t (A)) G X α 1.1 G α X (IO) 5W1H A A B A B 1.2!? 2
3 Duhem-Quine [MicMac] = [ ] 4 [MicMac, Unif03] = 2 5W1H = 4 = Spec ( ) ( ). 5W1H 4(+1) [ & = ], ( ) [ = ], ( ) [ = ], (() ) [= = ], [ = = ]. 2.1 G α X 4 : 3
4 dual 4 : Spec ( ) : = Spec ( ) : dual ( ) 2.2 Spec ) Spec quantum fields Spec Alg : Spec : ( ) Alg : 2.3 ( ) Spec & Spec Fourier ( ) : Spec V I ( ) ( L 1 ) ( L 2 ) ( L ) Galois (event localization) Spec= [ ] phase separation 4
5 semantic space forcing emergence ( ) = ( ) ( ( )) = ( ) Spec co-emergence Spec ( ) : = local net Spec Alg Spec ( ) Galois 2.4 = = Dyn Galois Galois : Galois. Spec center Spec States Alg Dyn Spec Spec States Alg Dyn Spec States Alg Dyn 2.5 Spec. 5
6 sectors (C*-) X (: Alg) Hilbert von Neumann π(x ) (: ) Z π (X ) = π(x ) π(x ) (pure phase) Z πγ (X ) := π γ (X ) π γ (X ) = C1 () *) Warning!: sector = Quantum physics etc. 2.6 vs. Spec(Z π (X )) : as Spec = γ N γ γ 2 γ 1 Spec(Z)..... π γn π γ π γ2 π γ Cf. = 6
7 3 == = [IO03] 3.1 Stone-vN thm QM paradoxes = == = = short cut (e.g., Josephson Josephson, etc.) = Einstein = universality index 7
8 3.2 : Planck = Weyl Stone-von Neumann Schrödinger Dirac 3.3 vs. Hilbert = field operators G 1 (γ i, V i ) = Fourier duality (= quantum theory of fields) (= field operators) 1 G (γ, V ) V End(V ) G γ : G g γ(g) End(V ) γ(e) = I, γ(g 1 g 2 ) = γ(g 1 )γ(g 2 ), γ(g 1 ) = γ(g) 1 8
9 essence Hilbert (= quantum fields) (= QFT) 3.4 vs. = vs. = 3.5 vs. : Gibbs Hilbert 1950 Z(β) := T r(exp( βh)) 9
10 - thermo field dynamics (1975 ) Green Green expert non-expert family - thermo field dynamics = ϕ T =0 ϕ T ϕ T =0 = ϕ T G (γ i, V i ) Fourier duality 3.6 GNS T = 0 ϕ T =0 ϕ T at T 0 ϕ T =0 = ϕ T ϕ T =0 ϕ ϕ ω T =0 Hilbert H T =0 π T =0 (ϕ) A ω ω A E A E A := {ω A s.t. ω(a A) 0 (for A A), ω(1) = 1}. Hilbert H = H ω G(el fand-)n (aimark-)s(egal) GNS ω : A C Hilbert H ω A A H ω π ω (A) π ω : A B(H ω ) ω ω(1) = 1 H ω 1 Ω ω ω(a) = Ω ω π ω (A)Ω ω, π ω (X )Ω ω = H ω 10
11 (π ω, H ω, Ω ω ) unitary A ω E A Mod (π ω, H ω ) GNS ω GNS (π ω, H ω, Ω ω ) ω the Hilbert space [ CCR ] GNS ω (π ω, H ω, Ω ω ) ω ϕ ω = π ω (ϕ) ω T =0 ω T ω T =0 = ω T ϕ T =0 = ϕ T ω T ϕ T = π T (ϕ) π T ϕ ω T =0 ϕ T =0 = π T =0 (ϕ) π T =0 ω T =0 {ω T } T >0 ω T =0 = ω T 3.7 Gibbs = vs = β = (k B T ) 1 Gibbs ω β ω vac = ω β = + Gibbs Gibbs Gibbs ω β ω = lim ω β β + 11
12 4 Gibbs generic von Neumann Gibbs trace ω β (A) = Tr(e βĥa) Tr(e βĥ) = Tr(eβ(F Ĥ) A), (1) Tr(e βĥa), Tr(e βĥ) operator e βĥ trace-class OK Hamiltonian Ĥ A trace (Tr) (1) 4.1 KMS (1) Haag-Hugenholtz-Winnink [HHW] Gibbs [ ω β (Aα t (B)) = Tr(e β(f H) Ae ith Be ith ) ] = Tr(e β(f H) e i(t iβ)h Be i(t iβ)h A) = ω β (α t iβ (B)A), (2) K(ubo)-M(artin)-S(chwinger) ω β KMS Definition 1 (KMS ) A ω β E A KMS (β-)kms : A, B A D β {z C; 0 Im(z) β} D β F AB (z) F AB (t) = ω β (Aα t (B)) F AB (t + iβ) = ω β (α t (B)A) t R 12
13 4.2 KMS Gibbs α t (B) := e itĥbe itĥ 2 t B KMS trace trace A trace M n (C) Gibbs (1) KMS Gibbs Gibbs KMS 4.3 KMS KMS ω β GNS (π β, H β, Ω β ) s.t. ω β (A) = Ω β, π β (A)Ω β, H β π β (A) Ω β π β (A) (communtant) : π β (A) C1 Hβ M π β (A) w = π β (A) M = π β (A) H β J modular Je βh β/2 AΩ β = A Ω β (A M), (3) JΩ β = Ω β, e ith β Ω β = Ω β, (4) J 2 = 1, JΦ JΨ = Φ Ψ, (5) JMJ = M, e ith β Me ith β = M, (6) JH β J = H β. (7) : Je βh β/2 AΩ β = A Ω β (3) trace (2) 4.4 Modular β = e βh β modular J A) M J A1) M M OK A, B A B A B & B C A C 2 13
14 () A2) Modular J M M : JMJ = M & JM J = M. Hamiltonian H β GNS (π β, H β, Ω β ) JH β J = H β Hamiltonian H β KMS 4.5 modular B) observation e βh trace class Hamiltonian H J JH β J = H β H β = H JHJ (8) H β (8) Gibbs H β JHJ 4.6 vs. C) Gibbs (1) C1) ( von Neumann ) M trace III 14
15 KMS ( ) (1) C2) coupling scale up t micro = λt macro, λ 1 KMS T micro = T macro /λ 0 () T = 0 K λ T 0 K (3) couple M M T = 0 K decouple C3) T 0 K ω vac (A) = Ω, π vac (A)Ω Feynman ω β= (A B) = ω vac (A)ω vac (B) = ω vac (A)ω vac (B ), (9) scale up bulk matter M M 5 focus 15
16 branch sector Spec Spec 5.1 [Unif03] Definition 2 F (π, H) Z π (F) = Z(π(F) ) Spec(Z π (F)) G- (G, τ) unbroken = G = = G-unbroken G- and/or device stabilized Macro level dynamical Micro system
17 4 a) A A A A A ω : A A ω(a) C A ω(a) Micro-Macro interface GNS : ω(a) = Ω ω, π ω (A)Ω ω, Ω ω H ω Hilbert H ω A π ω : A A π ω (A) B(H ω ) A virtual Hilbert A π ω (A) = disjoint = Cf. Schrödinger [MicMac, Unif03, IO13, IOOk13] Z π (A) := π ω (A) π ω (A) Sp(Z) := Sp(Z π (A)) =: Spec [IO10] b) [BOR] D(oplicher) H(aag)R(oberts) [DHR, DR89] = [Unif03, IO04, IO13] F A = F G A = (π, H) s.t. Z π (A) = C1 (= unitary ) Z π (A) C1 Z π (A) π(a) Sp(Z) = Spec 17
18 π(a) = χ Spec π χ(a) dµ(χ) 5.3 due to disjointness π 1, π 2 unitary (disjointness) i.e., T π 1 (A) = π 2 (A)T ( A A) T = 0 = Z π (A) = χ Spec = = = vs. duality γ N γ γ 2 γ 1 Spec.... π γn. π γ. π γ2. π γ DHR c) F F G F τ F G- A := F G G- G F G- G F A {ω α} ω α (A), A A τ gap A G G- F DHR O O A π 0 DHR : π A(O ) = π 0 A(O ) A π= Ĝ F G A F = A Ĝ Gal(F/A) = G [DR89](π 0 A A(O ) O O C*- ) DHR Ĝ G F G Ĝ τ 18
19 d) DHR focus G unitary F G F F ( τ ) (π, H) : π(τ g (F )) = U(g)π(F )U(g) ( F F) G unitary (U, H) G Lie Lie F (π, H) Z π (F) := π(f) π(f) = C1G- (U, H) [Unif03] F (π, H) G unitary G unitary (U, H) G F (π, H) Z π (F) C1 disjoint [IO04] [IO13] = dynamics 4 [IO13, IOOk13] Spec= States = GNS RepMod= Galois Alg = Dyn= 4 19
20 Spec Alg(ebra) map ϕ : Spec Alg ϕ : States Dyn(amics) RepMod GNS GNS : States RepMod Galois RepMod Dyn ϕ = Gal GNS 5.6 Goldstone : a) H Ĥ, b) G/H = M [IO03, 04], c) 3 Γ/G = R [IO10]. Γ G H R P Γ P G P H soldering R ρ P G /H, P G /H σ P Γ /H, R τ P Γ /G Goldstone ρ, σ, τ R P Γ /G Γ/G G/H P Γ /H H P Γ Γ τ σ R G/H P G /H H P G G ρ R H P H H. 6 = Spec Stone-von Neumann = broken symmetries Doplicher-Haag- Roberts augmented algebra 3 epigenetic 20
21 [IO03] 6.1 gap 20 particle physics 4 No Go thm p-space support on-shell p µ (p 2 = m 2 ) [ Einstein E = mc 2 ] support p-space support intertwiner 0 = Haag 4(+1) events Spec (= Schrödinger (IO, &, in preparation) Schrödinger intertwiner + ρ = ψ ψ with ψ = ( ± )/ 2 + σ = [ + ]/2 (IO 96): ρ(a) = σ(a) for all observables A, 1/2 4 21
22 : GNS Hilbert EPR entangled states [ ψ (A) 1 ϕ (B) 1 + ψ (A) 2 ϕ (B) 2 /] 2 state vector (A) (B) EPR EPR 6.2 : G G- G H G/H CT scan Radon Helgason Hecke dual (augmented algebra) X = X Ĥ\G = X H Ĝ [IO03] : K\G/H : Hecke K\G Helgason G/H G H K H K dual X G = X H G/H H X H G X H G/H X Ĥ\G Ĝ Ĥ 22
23 6.3 X G = X H G/H H X H X H G/H X Ĥ\G Ĝ Ĥ : X = X H Ĝ X H Galois G/H Galois G = Gal( X /X H ) = ) U(1) Cooper Lie g Lie G Lie H & Lie h M = G/H Lie m = g/h M [m, m] h (É. Cartan) [IO, unpublished] [m, m] = M loop M [m, m] h i.e., [m, m] h Chiral symmetry : [V, V ] = V, [V, A] = A, [A, A] = V (V h: vector currents, A m: axial currents) 6.4 G Lorentz L +, H SO(3) Lorentz boosts Lorentz M = G/H = R 3 h := {M ij ; i, j = 1, 2, 3, i < j}, m := {M 0i ; i = 1, 2, 3} [h, h] = h, [h, m] = m, [m, m] h Lorentz Lie [im µν, im ρσ ] = (η νρ im µσ η νσ im µρ η µρ im νσ + η µσ im νρ ). Lorentz boosts Lorentz boosts ) Borchers-Arveson-Araki T = 0K 23
24 T 0 Lorentz boosts (IO 86), Lorentz Lorentz M = R 3 [boost, boost] = Q q E = Q + W p W, i.e., Im(q) = ker(p), 5 Lie h g m = g/h M [m, m] [m, m] h E = Q+ W = 0 W = [m, m] = Q > 0 [m, m] h in Kelvin s version 6.5 Helgason Hecke Hecke K\G/H Radon Helgason K\G G/H: K\G/H K\G G/H G K\G/H : Hecke K\G Helgason G/H G H K H K dual X G = X H G/H H X H G X H G/H X Ĥ\G Ĝ Ĥ Helgason duality Radon CT scan 5 Im(q) = ker(p) energy 0 (ker(p)) (Im(q)) 24
25 6.6 [IO03] : G/H X H = X G H X H X H X G/H Ĥ\G Ĝ Ĥ H R O ρ = Od H A(R) O d H X (R) R R Γ Ĥ : [ ] O d, O ρ = Od H Cuntz 7 [H\G X H = X G Ĥ X H X [H\G Ĥ X Ĥ\G Ĥ Ĝ [H\G X H = X G Ĥ X H G X Doplicher, Roberts Ĥ X [H\G Galois X (R) = A(R) O ρ O d H R O ρ = Od H A(R) O d X (R) R DHR-DR focus R 7.1 Galois (H, G), (G, Γ), (H, Γ) 25
26 b R= d G\Γ X G = X H R= b G\Γ d X G = X H [H\G Ĥ H\G [ Ĥ X H Ĥ Ĥ b R= d G\Γ X H Ĥ X G Ĥ X br= G\Γ d [H\G Ĥ [ H\G X br= d G\Γ br= d G\Γ X or=o b G\Γ d 7.2 Lagrangian Maxwell : Einstein revise 7.3 Doplicher-Roberts Galois Doplicher & Roberts (DR) DR T Galois T Obj(T ) : DHR π 0 ρ A(O ) = π 0 A(O ) A ρ End(A), Mor(T ): T ρ, σ ρ(a)t = T σ(a) for A A intertwiners T T (ρ σ) A. DR T ρ End(A) V ρ V ρ Hilbert Hilb V : T Hilb Galois 26
27 DR T V V unitary u = (u ρ ) ρ T End (V ) = {u : V V ; u ρ = u 1 ρ, u ρ1 ρ 2 = u ρ1 u ρ2 for ρ ρ 1 ρ 2 T } H H = End (V ). 7.4 Galois V W v : W V : i.e., W (T )v σ = v ρ V (T ) T T (ρ σ) V τ v (V ) τ v : τ v (V ) V v ρ W (ρ) V (ρ) W (T ) V (T ) W (σ) vσ V (σ), τ v (V )(T ) := v ρ V (T )v 1 σ for T T (ρ σ), τ v gauge link, u End (V ) u ρ V (T ) = V (T )u σ, u H = End (V ) τ u Galois V τ u (V ) = V 7.5 Maxwell ρ T End(A) Doplicher-Haag-Roberts 6 Galois V τ u (V )(T ) := u ρ V (T )u 1 σ for T T (ρ σ) u ρ u σ u = (u ρ ) ρ T ρ- τ u (V )(T ) = u ρ V (T )u 1 σ = V (T ) 6 sectors End(A)/Inn(A) 27
28 u : T ρ u ρ U(V ρ ) V Čech DR unbroken symmetry H Lie, Ĥ DR T End(A) = End(X H ) factor (augmented algebra) X = X H R T = End( X H ) Γ unbroken symmetry H G Γ/G = R(: ) X H H DR T Γ = End (Ṽ : T Hilb) Γ Maxwell Noether References [1] Bratteli, O. and Robinson, D.W., Operator Algebras and Quantum Statistical Mechanics (2nd ed.), Vol.1, Springer-Verlag, [2] Ojima, I., A unified scheme for generalized sectors based on selection criteria Order parameters of symmetries and of thermal situations and physical meanings of classifying categorical adjunctions, Open Sys. Info. Dyn. 10, (2003); Micro-macro duality in quantum physics, , Proc. Intern. Conf. Stochastic Analysis: Classical and Quantum, World Sci., 2005, arxiv:math-ph/ [3] Ojima, I., Lévy Process and Innovation Theory in the context of Micro- Macro Duality, 15 December 2006 at The 5th Lévy Seminar in Nagoya, Japan. [4] Ojima, I. and Okamura, K., Large deviation strategy for inverse problem I & II, Open Sys. Inf. Dyn., 19, & (2012) [5] Ojima, I., Okamura, K. and Saigo, H., Derivation of Born rule from algebraic and statistical axioms. 21, (2014). [6] Ojima, I., Temperature as order parameter of broken scale invariance, Publ. RIMS (Kyoto Univ.) 40, (2004) (math-ph ). 28
29 [7] Ojima, I., Lorentz invariance vs. temperature in QFT, Lett. Math. Phys. 11, (1986). [8] Ojima, I., Space(-Time) Emergence as Symmetry Breaking Effect, Quantum Bio-Informatics IV, (2011). (arxiv:mathph/ (2011)); Micro-Macro Duality and Space-Time Emergence, Proc. Intern. Conf. Advances in Quantum Theory, (2011); New interpretation of equivalence principle in General Relativity from the viewpoint of Micro-Macro duality (arxiv:genph/ ), Foundations of Probability and Physics 6, Sweden, (invited talk). [9] Ojima, I., Micro-macro duality in quantum physics, , Proc. Intern. Conf. Stochastic Analysis: Classical and Quantum Perspectives of White Noise Theory ed. by T. Hida, World Scientific (2005), arxiv:math-ph/ [10] Haag, R., On quantum field theories, Kgl. Danske Videnskab. Selskab. Mat.-fys. Medd., 29, no.12, 1-37 (1955); Local Quantum Physics Fields, Particles, Algebras, Springer-Verlag (1992). [11] Wightman, A.S., Quantum field theory in terms of vacuum expectation values, Phys. Rev. 101, (1956); Streater, R.F. & Wightman, A.S., PCT, Spin and Statistics and All That, Benjamin (1964). [12] Ojima, I., Covariant Operator Formalism of Gauge Theories and its Extension to Finite Temperature, pp in Lecture Notes in Physics, No.176, Gauge Theory and Gravitation (Proceedings, Nara, Japan 1982), Springer-Verlag, Berlin-Heidelberg-New York, [13] Buchholz, D. and Ojima, I., Spontaneous collapse of supersymmetry, Nucl. Phys. B498, Nos.1,2, (1997). [14] Matsubara, T., Prog. Thcor. Phys. 14, 351 (1955); Takahashi, Y. and Umezawa, H., Thermo field dynamics, Collect. Phenom. 2, (1975). [15] (, 2013). [16] ( SGC98, 2013). [17] Buchholz, D., Ojima, I. and Roos, H., Thermodynamic properties of non-equilibrium states in quantum field theory, Ann. Phys. (N.Y.) 297, (2002). [18] Doplicher, S., Haag, R. and Roberts, J. E., Fields, observables and gauge transformations I & II, Comm. Math. Phys. 13, 1-23 (1969); 15, (1969); Local observables and particle statistics, I & II, 23, (1971) & 35, (1974). 29
30 [19] Doplicher, S. and Roberts, J.E., 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); Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Comm. Math. Phys. 131, (1990). [20] Ojima, I., Dynamical relativity in family of dynamics, 1921, (2014). [21] Ojima, I., Local gauge invariance and Maxwell equation in categorical QFT, 1961, (2015); Algebraic QFT and local gauge invariance, 2010, (2016). [22] Haag, R., Hugenholtz, N.M. & Winnink, M., On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5, (1967). [23] Kubo, R., J. Phys. Soc. Japan 12, (1957); Martin, P.C. & Schwinger, J., Theory of many particle systems I, Phys. Rev. 115, (1959). [24] Bratteli, O. & Robinson, D.W., Operator Algebras and Quantum Statistical Mechanics, Vols.1 & 2, Springer-Verlag (1979, 1981). [25], (1985). [26] Kugo. T. and Ojima, I., Local Covariant Operator Formalism of Non- Abelian Gauge Theories and Quark Confinement Problem, Suppl. Prog. Theor. Phys. No. 66 (1979); Nakanishi, N. and Ojima, I., Covariant Operator Formalism of Gauge Theories and Quantum Gravity, World Scientific Lecture Notes in Physics Vol.27, World Scientific Publishing Company, Singapore-New Jersey-London-Hong Kong (1990). [27] Milnor, J., Morse theory, Princeton Univ. Press (1963), (1968); (1991); II (1993); Morse (2005). 30
2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2
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