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

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1 (Emmy Noether 1) [1] [2] [ (Paul Gordan Clebsch-Gordan ] 1915 habilitation habilitation außerordentlicher Professor Außerordentlich(=extraordinary) 1

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

3 2 (Gell=Mann-Brueckner) (Baym-Kadanoff) [6] (Luttinger-Ward) Ω 2 Σ Ω Σ 2: (Luttinger-Ward) Ω 3

4 3 (Maksym) [7] (yrast spectrum) [8] [V = m 2 ω2 (x 2 + y 2 )] 1920 (Fock-Darwin) 3(a) [V = m 2 (ω2 xx 2 + ωyy 2 2 )] 3(b) ( 3(c) (1 + cos2θ) 0, ±2 1,3 4

5 3: (a) 2 2 [ (Fock-Darwin) ] (b) ( hω x = 17 mev, hω y = 10 mev) [8] 4 1/r L Runge-Lenz 5

6 A = p L r/r 1920 A = 1 (p L L p) r/r 2 L A L SU(2) SU(2) SO(4) [9] [10] [11] [ (Yang-Baxter) ] (rapidity 2 (Sutherland) [12] non-diffractive 6

7 (rapidity) 4: 1 tj AB J/t [14] SU(2)( ) [11, 13] 4 [14] 7

8 anticrossing 4 anticrossing 4 tj t J [15] J = J = 2t J J = 0, 2t 1 AB rapidity [14] (Hans Bethe) (!) habilitation PhD 8

5 C P T Bloch 5 1929 PhD [16] 5: Felix Bloch;

9 5 C P T Bloch PhD [16] 5: Felix Bloch; ) Nobel Foundation Ψ Ψ l H ˆT l [H, ˆT l ] = 0 9

10 H ˆT l Ψ ˆT l Ψ(r) Ψ(r + l) = λψ(r) ˆT l Ψ [ λ = 1 Ψ ] Ψ α (r) = e if α(r) u α (r) α f α (r) u α ˆT l1 ˆT l2 ˆT ˆTl1 l2 = ˆT l1 +l 2 f(r) r (f(l 1 ) + f(l 2 ) = f(l 1 + l 2 ) k Ψ Ψ sk (r) = eik r u sk (r) f k s( ) 1929 k e ik l k a 1, a 2, a 3 (a i b j = 2πδ ij ) G = n 1 b 1 + n 2 b 2 + n 3 b 3 (n i k k + G e ik l u (u k = u k+g ) k k + G ˆT 1, ˆT 2, ˆT 3 ˆT 1 ˆT 2 ˆT 3 ˆT i e ik l E s (k) E s (k + G) = E s (k) k Ψ Ψ E s ( k) = E s (k) p k k G hk 10

11 crystal momentum k k + G (G 0 ) (Peierls) (Umklapp) (Bragg) k Ψ(r 1, r 2 ) e ik l Ψ(r 1, r 2 ) k G k G [17] [18] k 6 [19] 11

12 6: 6 (Kramers) GaAs E(k, ) = E( k, ) 7 GaAs 12

13 7: (Rashba) BCS (B 1g ) (E u ) 7 (Gutzwiller) 13

14 [20] L z 19 (Hadamard) (genus) (trace formula) [21] [22, 23] 8: [ (Schwarz) P ] 8 (Schwarz) [24] 14

15 (Bonnet) 8 U(1) (Thouless) [25] (nonintegrable phase) (Berry) [26] [27] 9 15

16 [1] Emmy Noether, Nachr. Gesellsch. Wiss. Göttingen 2, 235 (1918). [2] : p.28 (1997) [ 1999) p.80 ] Constance Reid: Hilbert (Springer, New York, 1996) Sonya Kovalevskaya Joan Spicci: Beyond the limit The dream of Sonya Kovalevskaya (Tom Doherty Associates, New York, 2002) [3] Jagdish Mehra, The Beat of a Different Drum The life and science of Richard Feynman (Oxford Univ. Press, 1996). [4] 2001 [5] M. Suzuki, Physica 51, 277 (1971). [6] 1999 [7] Peter A. Maksym 53, 36 (1998) [8] P.A. Maksym, Physica B , 233 (1998). [9] Brian G. Wybourne, Classical Groups for Physicists (Wiley, 1974). [10] K. Asano and T. Ando, Phys. Rev. B 65, (2002) [11] [12] Bill Sutherland in Exactly Solvable Problems in Condensed Matter and Relativistic Field Theory ed. by B.S. Shastry et al (Springer, 1985), p.1. [13] 1990 [14] R. Arita, K. Kusakabe, K. Kuroki and H. Aoki, J. Phys. Soc. Jpn 66, 2086 (1997). [15]

17 [16] 57, 118 (2002) [17] K. Yamada and K. Yosida, Prog. Thoer. Phys. 76, 621 (1986). [18] R. Arita, K. Kuroki and H. Aoki, Phys. Rev. B 61, 3207 (2000). [19] Johannes Kepler: Gesammelte Werke (Frankfurt, 1634; Beck sche Verlag, München 1988 [20] Martin C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, 1990). [21] [22] :, 1997 [23] E.B. Bogomolny et al, Phys. Rep. 291, 219 (1997). [24] H. Aoki, M. Koshino, H. Morise, D. Takeda, and K. Kuroki, Phys. Rev. B 65, (2001). [25] David J. Thouless: Topological Quantum Numbers in Nonrelativistic Physics (World Scientific, Singapore, 1998). [26] 29, No.11, p.11 (1991) [ 1999 p.107 ] [27] [25] Y. Hatsugai, J. Phys. Condens. Matt. 9, 2507 (1997) 17

(extended state) L (2 L 1, O(1), d O(V), V = L d V V e 2 /h 1980 Klitzing

1 2 2.1 [1] [2] 2.1 STM [3, 4, 5, 6] 2.1: 2 ( 3 [1] ) [7, 8] [9]( 2.2) 2 2 2.1.1 (extended state) L (2 L 1, O(1), d O(V), V = L d V V 2.1.2 1985 2 e 2 /h 1980 Klitzing 2.1. 3 [7, 8] 2.2 [10] [8] 2.2: (a)