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!?

Size: px
Start display at page:

Download "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!?"

Transcription

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

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) = :

More information

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 $\Vert A\Vert^{2}$ $c*$ - $*:\mathcal{x}\ni A\mapsto A^{*}\in \mathcal{x}$ $\Vert A^{*}A\Vert=$ $\Vert\cdot\Vert$

More information

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)

More information

Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2

Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2 hara@math.kyushu-u.ac.jp 1 1 1.1............................................... 2 1.2............................................. 3 2 3 3 5 3.1............................................. 6 3.2...................................

More information

* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *

* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) * * 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *1 2004 1 1 ( ) ( ) 1.1 140 MeV 1.2 ( ) ( ) 1.3 2.6 10 8 s 7.6 10 17 s? Λ 2.5 10 10 s 6 10 24 s 1.4 ( m

More information

July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i

July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac

More information

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann

More information

数理解析研究所講究録 第1921巻

数理解析研究所講究録 第1921巻 1921 2014 108-121 108 Local state, sector theory and measurement in AQFT 1 1 () $($local state) (quantum operation) ( RIMS ) () [25] ( [22] ) [5, 35, 36] 2 : $c*$ - $E_{\mathcal{X}}$ $\omega(a^{*}a)\geq

More information

Introduction SFT Tachyon condensation in SFT SFT ( ) at 1 / 38

Introduction SFT Tachyon condensation in SFT SFT ( ) at 1 / 38 ( ) 2011 5 14 at 1 / 38 Introduction? = String Field Theory = SFT 2 / 38 String Field : ϕ(x, t) x ϕ x / ( ) X ( σ) (string field): Φ[X(σ), t] X(σ) Φ (Φ X(σ) ) X(σ) & / 3 / 38 SFT with Lorentz & Gauge Invariance

More information

Kaluza-Klein(KK) SO(11) KK 1 2 1

Kaluza-Klein(KK) SO(11) KK 1 2 1 Maskawa Institute, Kyoto Sangyo University Naoki Yamatsu 2016 4 12 ( ) @ Kaluza-Klein(KK) SO(11) KK 1 2 1 1. 2. 3. 4. 2 1. 標準理論 物質場 ( フェルミオン ) スカラー ゲージ場 クォーク ヒッグス u d s b ν c レプトン ν t ν e μ τ e μ τ e h

More information

( ) ) AGD 2) 7) 1

( ) ) AGD 2) 7) 1 ( 9 5 6 ) ) AGD ) 7) S. ψ (r, t) ψ(r, t) (r, t) Ĥ ψ(r, t) = e iĥt/ħ ψ(r, )e iĥt/ħ ˆn(r, t) = ψ (r, t)ψ(r, t) () : ψ(r, t)ψ (r, t) ψ (r, t)ψ(r, t) = δ(r r ) () ψ(r, t)ψ(r, t) ψ(r, t)ψ(r, t) = (3) ψ (r,

More information

Anderson ( ) Anderson / 14

Anderson ( ) Anderson / 14 Anderson 2008 12 ( ) Anderson 2008 12 1 / 14 Anderson ( ) Anderson 2008 12 2 / 14 Anderson P.W.Anderson 1958 ( ) Anderson 2008 12 3 / 14 Anderson tight binding Anderson tight binding Z d u (x) = V i u

More information

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i 1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,

More information

1

1 1 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), 20 82. Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification

More information

[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)

[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization) . D............................................... : E = κ ............................................ 3.................................................

More information

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h 2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h

More information

Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ

More information

L. S. Abstract. Date: last revised on 9 Feb translated to Japanese by Kazumoto Iguchi. Original papers: Received May 13, L. Onsager and S.

L. S. Abstract. Date: last revised on 9 Feb translated to Japanese by Kazumoto Iguchi. Original papers: Received May 13, L. Onsager and S. L. S. Abstract. Date: last revised on 9 Feb 01. translated to Japanese by Kazumoto Iguchi. Original papers: Received May 13, 1953. L. Onsager and S. Machlup, Fluctuations and Irreversibel Processes, Physical

More information

Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W

Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W 003 7 14 Black-Scholes [1] Nelson [] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-Wu Nelson e-mail: takatoshi-tasaki@nifty.com kabutaro@mocha.freemail.ne.jp

More information

φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)

φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1) φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x

More information

1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji

1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji 8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi

More information

1. Introduction Palatini formalism vierbein e a µ spin connection ω ab µ Lgrav = e (R + Λ). 16πG R µνab µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb, e =

1. Introduction Palatini formalism vierbein e a µ spin connection ω ab µ Lgrav = e (R + Λ). 16πG R µνab µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb, e = Chiral Fermion in AdS(dS) Gravity Fermions in (Anti) de Sitter Gravity in Four Dimensions, N.I, Takeshi Fukuyama, arxiv:0904.1936. Prog. Theor. Phys. 122 (2009) 339-353. 1. Introduction Palatini formalism

More information

1 : ( ) ( ) ( ) ( ) ( ) etc (SCA)

1 : ( ) ( ) ( ) ( ) ( ) etc (SCA) START: 17th Symp. Auto. Decentr. Sys., Jan. 28, 2005 Symplectic cellular automata as a test-bed for research on the emergence of natural systems 1 : ( ) ( ) ( ) ( ) ( ) etc (SCA) 2 SCA 2.0 CA ( ) E.g.

More information

[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo

[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo [1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin + 8 5 Clifford Spin 10 A 12 B 17 1 Clifford Spin D Euclid Clifford Γ µ, µ = 1,, D {Γ µ, Γ ν

More information

Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P

Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo

More information

a L = Ψ éiγ c pa qaa mc ù êë ( - )- úû Ψ 1 Ψ 4 γ a a 0, 1,, 3 {γ a, γ b } η ab æi O ö æo ö β, σ = ço I α = è - ø çèσ O ø γ 0 x iβ γ i x iβα i

a L = Ψ éiγ c pa qaa mc ù êë ( - )- úû Ψ 1 Ψ 4 γ a a 0, 1,, 3 {γ a, γ b } η ab æi O ö æo ö β, σ = ço I α = è - ø çèσ O ø γ 0 x iβ γ i x iβα i 解説 4 matsuo.mamoru jaea.go.jp 4 eizi imr.tohoku.ac.jp 4 maekawa.sadamichi jaea.go.jp i ii iii i Gd Tb Dy g khz Pt ii iii Keywords vierbein 3 dreibein 4 vielbein torsion JST-ERATO 1 017 1. 1..1 a L = Ψ

More information

,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising

,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising ,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising Model 1 Ising 1 Ising Model N Ising (σ i = ±1) (Free

More information

ADM-Hamiltonian Cheeger-Gromov 3. Penrose

ADM-Hamiltonian Cheeger-Gromov 3. Penrose ADM-Hamiltonian 1. 2. Cheeger-Gromov 3. Penrose 0. ADM-Hamiltonian (M 4, h) Einstein-Hilbert M 4 R h hdx L h = R h h δl h = 0 (Ric h ) αβ 1 2 R hg αβ = 0 (Σ 3, g ij ) (M 4, h ij ) g ij, k ij Σ π ij = g(k

More information

2016 Course Description of Undergraduate Seminars (2015 12 16 ) 2016 12 16 ( ) 13:00 15:00 12 16 ( ) 1 21 ( ) 1 13 ( ) 17:00 1 14 ( ) 12:00 1 21 ( ) 15:00 1 27 ( ) 13:00 14:00 2 1 ( ) 17:00 2 3 ( ) 12

More information

1. 1.1....................... 1.2............................ 1.3.................... 1.4.................. 2. 2.1.................... 2.2..................... 2.3.................... 3. 3.1.....................

More information

Yang-Mills Yang-Mills Yang-Mills 50 operator formalism operator formalism 1 I The Dawning of Gauge T

Yang-Mills Yang-Mills Yang-Mills 50 operator formalism operator formalism 1 I The Dawning of Gauge T Yang-Mills 50 E-mail: kugo@yukawa.kyoto-u.ac.jp 2004 Yang-Mills 50 2004 Yang-Mills 50 operator formalism operator formalism 1 I The Dawning of Gauge Theory O Raifeartaigh [1] I, II, III O Raifeartaigh

More information

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R

More information

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp

More information

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

$)\triangleleft\hat{g}$ $\mathcal{t}\mathcal{h}$ 106 ( ) - Einstein ( ) ( ) $R_{\mu\nu}- \frac{1}{2}g_{\mu\nu}r=\kappa T_{\mu\nu}$ bottom-up feedback duality 1532 2007 105-117 105 - $-*$ (Izumi Ojima) Research Institllte for Mathematical Sciences Kyoto University 1? 3 ( 2-4 ) 1507 RIMS. ( ) (2006 6 28 30 ). $+\mathrm{f}_{\mathrm{o}\mathrm{l}1\gamma}\mathrm{i}\mathrm{e}\mathrm{r}$

More information

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes ) ( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)

More information

Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q

Chern-Simons   Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0

More information

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................

More information

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple

More information

Einstein ( ) YITP

Einstein ( ) YITP Einstein ( ) 2013 8 21 YITP 0. massivegravity Massive spin 2 field theory Fierz-Pauli (FP ) Kinetic term L (2) EH = 1 2 [ λh µν λ h µν λ h λ h 2 µ h µλ ν h νλ + 2 µ h µλ λ h], (1) Mass term FP L mass =

More information

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+ 1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z)

More information

/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat

/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat / Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiation and the Continuing Failure of the Bilinear Formalism,

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................

More information

0406_total.pdf

0406_total.pdf 59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ

More information

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................

More information

MS#sugaku(ver.2).dvi

MS#sugaku(ver.2).dvi 1 1,,,.,,,,,.,.,,,.,, Computer-Aided Design).,,, Boltzmann,, [1]., Anderson, []., Anderson, Schrödinger [3],[4]., nm,,.,,,.,, Schrödinger.,, [5],[6].,,.,,, 1 .,, -. -, -., -,,,. Wigner-Boltzmann Schrödinger,

More information

講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K

講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K 2 2 T c µ T c 1 1.1 1911 Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ 58 213 email:takada@issp.u-tokyo.ac.jp 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B =

More information

( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =

( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) = 1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =

More information

xia2.dvi

xia2.dvi Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,

More information

Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull

Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and

More information

K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X

K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X 2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ

More information

反D中間子と核子のエキゾチックな 束縛状態と散乱状態の解析

反D中間子と核子のエキゾチックな   束縛状態と散乱状態の解析 .... D 1 in collaboration with 1, 2, 1 RCNP 1, KEK 2 . Exotic hadron qqq q q Θ + Λ(1405) etc. uudd s? KN quasi-bound state? . D(B)-N bound state { { D D0 ( cu) B = D ( cd), B = + ( bu) B 0 ( bd) D(B)-N

More information

25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3

More information

D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane

D-brane K 1, 2   ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane K 1, 2 E-mail: sugimoto@yukawa.kyoto-u.ac.jp (2004 12 16 ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3]

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................

More information

ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ)

ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý  (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ) (2016 ) Dept. of Phys., Kyushu Univ. 2017 8 10 1 / 59 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER 2 / 59 ( ) ( ) (Dirac, t Hooft-Polyakov)

More information

0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t

0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t e-mail: koyama@math.keio.ac.jp 0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo type: diffeo universal Teichmuller modular I. I-. Weyl

More information

006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................

More information

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論 email: takahash@sci.u-hyogo.ac.jp April 30, 2009 Outline 1. 2. 3. 4. 5. 6. 2 / 260 Today s Lecture: Itinerant Magnetism 60 / 260 Multiplets of Single Atom System HC HSO : L = i l i, S = i s i, J = L +

More information

4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n 2 n = n +,n +2, n = Lyman n =2 Balmer n =3 Paschen R Rydberg R = cm 896 Zeeman Zeeman Zeeman Lorentz

4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n 2 n = n +,n +2, n = Lyman n =2 Balmer n =3 Paschen R Rydberg R = cm 896 Zeeman Zeeman Zeeman Lorentz 2 Rutherford 2. Rutherford N. Bohr Rutherford 859 Kirchhoff Bunsen 86 Maxwell Maxwell 885 Balmer λ Balmer λ = 364.56 n 2 n 2 4 Lyman, Paschen 3 nm, n =3, 4, 5, 4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n

More information

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi 1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys

More information

橡超弦理論はブラックホールの謎を解けるか?

橡超弦理論はブラックホールの謎を解けるか? 1999 3 (Can String Theory Solve the Puzzles of Black Holes?) 305-0801 1-1 makoto.natsuume@kek.jp D-brane 1 Schwarzschild 60 80 2 [1] 1 1 1 2 2 [2] 25 2.2 2 2.1 [7,8] Schwarzschild 2GM/c 2 Schwarzschild

More information

1: (Emmy Noether; ) (Feynman) [3] [4] {C i } A {C i } (A A )C i = 0 [5] 2

1: (Emmy Noether; ) (Feynman) [3] [4] {C i } A {C i } (A A )C i = 0 [5] 2 2003 1 1 (Emmy Noether 1) [1] [2] [ (Paul Gordan Clebsch-Gordan ] 1915 habilitation habilitation außerordentlicher Professor Außerordentlich(=extraordinary) 1 1: (Emmy Noether; 1882-1935) (Feynman) [3]

More information

SUSY DWs

SUSY DWs @ 2013 1 25 Supersymmetric Domain Walls Eric A. Bergshoeff, Axel Kleinschmidt, and Fabio Riccioni Phys. Rev. D86 (2012) 085043 (arxiv:1206.5697) ( ) Contents 1 2 SUSY Domain Walls Wess-Zumino Embedding

More information

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,

More information

163 KdV KP Lax pair L, B L L L 1/2 W 1 LW = ( / x W t 1, t 2, t 3, ψ t n ψ/ t n = B nψ (KdV B n = L n/2 KP B n = L n KdV KP Lax W Lax τ KP L ψ τ τ Cha

163 KdV KP Lax pair L, B L L L 1/2 W 1 LW = ( / x W t 1, t 2, t 3, ψ t n ψ/ t n = B nψ (KdV B n = L n/2 KP B n = L n KdV KP Lax W Lax τ KP L ψ τ τ Cha 63 KdV KP Lax pair L, B L L L / W LW / x W t, t, t 3, ψ t n / B nψ KdV B n L n/ KP B n L n KdV KP Lax W Lax τ KP L ψ τ τ Chapter 7 An Introduction to the Sato Theory Masayui OIKAWA, Faculty of Engneering,

More information

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977

More information

Gelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ,

Gelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ, A spectral theory of linear operators on Gelfand triplets MI (Institute of Mathematics for Industry, Kyushu University) (Hayato CHIBA) chiba@imi.kyushu-u.ac.jp Dec 2, 20 du dt = Tu. (.) u X T X X T 0 X

More information

1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ +

1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ + 1 1.1 21 11 22 10 33 cm 10 29 cm 60 6 8 10 12 cm 1cm 1 1.2 2 1 1 1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr

More information

42 3 u = (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = (4) MeV/c 2 = (13) u m n = (4) MeV/c 2 =

42 3 u = (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = (4) MeV/c 2 = (13) u m n = (4) MeV/c 2 = 3 3.1 3.1.1 kg m s J = kg m 2 s 2 MeV MeV [1] 1MeV=1 6 ev = 1.62 176 462 (63) 1 13 J (3.1) [1] 1MeV/c 2 =1.782 661 731 (7) 1 3 kg (3.2) c =1 MeV (atomic mass unit) 12 C u = 1 12 M(12 C) (3.3) 41 42 3 u

More information

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i

More information

第5章 偏微分方程式の境界値問題

第5章 偏微分方程式の境界値問題 October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ

More information

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

More information

Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha

Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha Euler, Yang-ills Clebsch variable Helicity Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity i) Yang-ills 3 A T T A) Poisson Hamilton ii) Clebsch parametrization iii) Y- Y-iv) Euler,v)

More information

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345

More information

G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2

G (n) (x 1, x 2,..., x n ) = 1 Dφe is φ(x 1 )φ(x 2 ) φ(x n ) (5) N N = Dφe is (6) G (n) (generating functional) 1 Z[J] d 4 x 1 d 4 x n G (n) (x 1, x 2 6 Feynman (Green ) Feynman 6.1 Green generating functional Z[J] φ 4 L = 1 2 µφ µ φ m 2 φ2 λ 4! φ4 (1) ( 1 S[φ] = d 4 x 2 φkφ λ ) 4! φ4 (2) K = ( 2 + m 2 ) (3) n G (n) (x 1, x 2,..., x n ) = φ(x 1 )φ(x

More information

1/2 ( ) 1 * 1 2/3 *2 up charm top -1/3 down strange bottom 6 (ν e, ν µ, ν τ ) -1 (e) (µ) (τ) 6 ( 2 ) 6 6 I II III u d ν e e c s ν µ µ t b ν τ τ (2a) (

1/2 ( ) 1 * 1 2/3 *2 up charm top -1/3 down strange bottom 6 (ν e, ν µ, ν τ ) -1 (e) (µ) (τ) 6 ( 2 ) 6 6 I II III u d ν e e c s ν µ µ t b ν τ τ (2a) ( August 26, 2005 1 1 1.1...................................... 1 1.2......................... 4 1.3....................... 5 1.4.............. 7 1.5.................... 8 1.6 GIM..........................

More information

sakigake1.dvi

sakigake1.dvi (Zin ARAI) arai@cris.hokudai.ac.jp http://www.cris.hokudai.ac.jp/arai/ 1 dynamical systems ( mechanics ) dynamical systems 3 G X Ψ:G X X, (g, x) Ψ(g, x) =:Ψ g (x) Ψ id (x) =x, Ψ gh (x) =Ψ h (Ψ g (x)) (

More information

2016 ǯ¥Î¡¼¥Ù¥ëʪÍý³Ø¾Þ²òÀ⥻¥ß¥Ê¡¼ Kosterlitz-Thouless ž°Ü¤È Haldane ͽÁÛ

2016 ǯ¥Î¡¼¥Ù¥ëʪÍý³Ø¾Þ²òÀ⥻¥ß¥Ê¡¼  Kosterlitz-Thouless ž°Ü¤È Haldane ͽÁÛ 2016 Kosterlitz-Thouless Haldane Dept. of Phys., Kyushu Univ. 2016 11 29 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER ( ) ( ) (Dirac,

More information

Aharonov-Bohm(AB) S 0 1/ 2 1/ 2 S t = 1/ 2 1/2 1/2 1/, (12.1) 2 1/2 1/2 *1 AB ( ) 0 e iθ AB S AB = e iθ, AB 0 θ 2π ϕ = e ϕ (ϕ ) ϕ

Aharonov-Bohm(AB) S 0 1/ 2 1/ 2 S t = 1/ 2 1/2 1/2 1/, (12.1) 2 1/2 1/2 *1 AB ( ) 0 e iθ AB S AB = e iθ, AB 0 θ 2π ϕ = e ϕ (ϕ ) ϕ 1 13 6 8 3.6.3 - Aharonov-BohmAB) S 1/ 1/ S t = 1/ 1/ 1/ 1/, 1.1) 1/ 1/ *1 AB ) e iθ AB S AB = e iθ, AB θ π ϕ = e ϕ ϕ ) ϕ 1.) S S ) e iθ S w = e iθ 1.3) θ θ AB??) S t = 4 sin θ 1 + e iθ AB e iθ AB + e

More information

2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W

2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W SGC -83 2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ 1 3 4 Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W [14] c = 1 CFT [8] Rational CFT [15], [56]

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

YITP50.dvi

YITP50.dvi 1 70 80 90 50 2 3 3 84 first revolution 4 94 second revolution 5 6 2 1: 1 3 consistent 1-loop Feynman 1-loop Feynman loop loop loop Feynman 2 3 2: 1-loop Feynman loop 3 cycle 4 = 3: 4: 4 cycle loop Feynman

More information

main.dvi

main.dvi SGC - 48 208X Y Z Z 2006 1930 β Z 2006! 1 2 3 Z 1930 SGC -12, 2001 5 6 http://www.saiensu.co.jp/support.htm http://www.shinshu-u.ac.jp/ haru/ xy.z :-P 3 4 2006 3 ii 1 1 1.1... 1 1.2 1930... 1 1.3 1930...

More information

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2 2014 6 30. 2014 3 1 6 (Hopf algebra) (group) Andruskiewitsch-Santos [AFS09] 1980 Drinfeld (quantum group) Lie Lie (ribbon Hopf algebra) (ribbon category) Turaev [Tur94] Kassel [Kas95] (PD) x12005i@math.nagoya-u.ac.jp

More information

The Physics of Atmospheres CAPTER :

The Physics of Atmospheres CAPTER : The Physics of Atmospheres CAPTER 4 1 4 2 41 : 2 42 14 43 17 44 25 45 27 46 3 47 31 48 32 49 34 41 35 411 36 maintex 23/11/28 The Physics of Atmospheres CAPTER 4 2 4 41 : 2 1 σ 2 (21) (22) k I = I exp(

More information

V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H

V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H 199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)

More information

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe 3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)

More information

1 1.1,,,.. (, ),..,. (Fig. 1.1). Macro theory (e.g. Continuum mechanics) Consideration under the simple concept (e.g. ionic radius, bond valence) Stru

1 1.1,,,.. (, ),..,. (Fig. 1.1). Macro theory (e.g. Continuum mechanics) Consideration under the simple concept (e.g. ionic radius, bond valence) Stru 1. 1-1. 1-. 1-3.. MD -1. -. -3. MD 1 1 1.1,,,.. (, ),..,. (Fig. 1.1). Macro theory (e.g. Continuum mechanics) Consideration under the simple concept (e.g. ionic radius, bond valence) Structural relaxation

More information

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib ( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson

More information

19 /

19 / 19 / 1 1.1............................... 1. Andreev............................... 3 1.3..................................... 3 1.4..................................... 4 / 5.1......................................

More information

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0 1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45

More information

QMI13a.dvi

QMI13a.dvi I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1 (d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c)

More information

1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25

1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25 .. IV 2012 10 4 ( ) 2012 10 4 1 / 25 1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) 2012 10 4 2 / 25 1. Ω ε B ε t

More information

Seiberg Witten 1994 N = 2 SU(2) Yang-Mills 1 1 3 2 5 2.1..................... 5 2.2.............. 8 2.3................................. 9 3 N = 2 Yang-Mills 11 3.1............................... 11 3.2

More information

スケーリング理論とはなにか? - --尺度を変えて見えること--

スケーリング理論とはなにか?  - --尺度を変えて見えること-- ? URL: http://maildbs.c.u-tokyo.ac.jp/ fukushima mailto:hukusima@phys.c.u-tokyo.ac.jp DEX-SMI @ 2006 12 17 ( ) What is scaling theory? DEX-SMI 1 / 40 Outline Outline 1 2 3 4 ( ) What is scaling theory?

More information

Dynkin Serre Weyl

Dynkin Serre Weyl Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................

More information

References: 3 June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima,

References: 3 June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima, References: 3 mailto:hukusima@issp.u-tokyo.ac.jp June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima, and H. Takayama, cond-mat/0204225. Typeset by FoilTEX

More information

note4.dvi

note4.dvi 10 016 6 0 4 (quantum wire) 4.1 4.1.1.6.1, 4.1(a) V Q N dep ( ) 4.1(b) w σ E z (d) E z (d) = σ [ ( ) ( )] x w/ x+w/ π+arctan arctan πǫǫ 0 d d (4.1) à ƒq [ƒg w ó R w d V( x) QŽŸŒ³ džq x (a) (b) 4.1 (a)

More information

4/15 No.

4/15 No. 4/15 No. 1 4/15 No. 4/15 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = m ψ(r,t)+v(r)ψ(r,t) ψ(r,t) = ϕ(r)e iωt ψ(r,t) Wave function steady state m ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem

More information

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler

More information