( 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) = i ħ θ(t) B(t) A() A()B(t) 3 955 4
A B Φ R 5 j x = lim ω σ xx (ω)e x e iωt σ xx 6 V σ xx (ω) = [ Φ R iω xx (ω) Φ R xx() ] Φ R xx(t) = i ħv θ(t) J x(t)j x () J x ()J x (t) J(t) - J x (t) B(t) J x () Ȧ() -- ) J = (e/m) p ψ p ψ - 5 X, Y X(x)Y (x ) X Y x, x G (, ) = T τ {ψ() ψ( )} 6 σ µν σ xx, σ yy ˆx ẑ ŷσ yx : Φ xx (iω λ ) = 2ħ2 e 2 m 2 2 V β xg (, iε n )G (, iε n iω λ ),n σ xx = ne2 τ m n e m τ 3 3. 7 B ρ B = Tr (ρb) () 7 ) 2
H ρ = e βh /Tr ( e βh) (2) H H = H µn m ħ iħ B(t) t = [B(t), H] (5) 9 m, t ρ = m m ρ m m (3) m, t = U(t) m, (6) 8 U(t) = exp( iht/ħ) B(t) = U (t)b()u(t) (4) (3) ρ(t) = m = m m, t ρ m m, t U(t) m, ρ m m, U (t) = U(t)ρ()U (t) (7) (5) U(t) U (t) iħ ρ(t) [ U(t) = iħ ρ()u (t) + U(t)ρ() U ] (t) t t t = HU(t)ρ()U (t) U(t)ρ()U (t)h = [ρ(t), H] (8) ) [ρ, H] = (9) 8 (3) B = Tr (ρb) = n ρb n = m m B n n mn n m ρ = ρ m m B n n m = ρ m m B m mn m n n n = m B m m B ρ m m Tr(ρB) B 3.2 9 3
A B ) 3.2. ρ t = ρ(t = ) = ρ () ρ = e β(h Ω) Ω e βω = Tr e βh ω Ee iωt H (t) = e i r i Ee iωt () e r i H (t) = Ae i(ω+iδ)t (2) (2) ω + iδ δ H (t ) = Ae iωt e = (3) t = E = ϕ A q e i(q r ωt) q = Ae iωt e iδt ω iδ e iωt e i(ω+iδ)t iħ ρ(t) t = [H + H (t), ρ(t)] (4) ρ(t) = ρ + ρ (t) (5) ρ (t) (4) iħ ρ (t) t = [H, ρ (t)] + [H (t), ρ ] (6) H (t) [H (t), ρ (t)] (9) [H, ρ ] = ρ (t) = U(t)g(t)U (t) (7) 4
(6) iħ ρ (t) t = HU(t)g(t)U (t) + iħu(t) g(t) U (t) t U(t)g(t)U (t)h = iħu(t) g(t) U (t) + [H, ρ (t)] (8) t (6) (6) g(t) t = i ħ U (t) [H (t), ρ ] U(t) (9) g(t) = i ħ t dt U (t ) [H (t ), ρ ] U(t ) (2) (7) g(t) ρ (t) (6) ρ (t) = i ħ t dt U(t t ) [H (t ), ρ ] U (t t ) 3.2.2 (2) () j µ j µ (t) = Tr [{ρ + ρ (t)} j µ ] = Tr [ρ (t)j µ ] (22) Tr[ρ j µ ] = B B(t) = Tr [ρ (t)b] (23) (23) (2) (2) B(ω) = i ħ B(t) = B(ω)e iωt (24) dt Tr {[A, ρ ] B(t)} e iωt (25) Tr{XY Z} = Tr{Y ZX}B(t) = U (t)bu(t) t t t 3.2.3 A B B(ω) σ(ω) (25) B(ω) = i ħ iω = i ħ iω X(t) = Tr {[A, ρ ] B(t)} (26) (25) [ X() + dt dx(t) dt dt ( e iωt) dx(t) dt e iωt ] (27) ω ω + iδ t = 2 X( ) 3 X() = dt dx(t) dt (28) B(t) (5) ρ H dx(t) dt = i ħ Tr { [A, H]ρ B(t) [A, H]B(t)ρ } = B(t)Ȧ() A()B(t) (29) iħȧ = [A, H] () (29) 4 2 t t t t t = t = + 3 4 G R (t) = iθ(t) ψ pσ (t)ψ pσ () ψ pσ ()ψ pσ(t) Ȧ, B ψ ψ Φ 5
Φ R (t) = i ħ θ(t) B(t) A() A()B(t) (3) B(ω) = iω dt ( e iωt) Φ R (t) (3) Φ R (t) Φ R (t) = 5 dω 2π ΦR (ω)e iωt (32) B(ω) = [ Φ R (ω) Φ R () ] (33) iω (24), (3), (33) 3.3 () (33) A = e i r i E (34) A = i ħ [A, H] = i e [ ri E, p 2 ] i ħ 2m i = e p i E (35) m i 5 ω ω ω B(ω) = iω = iω = iω = iω dt ( e iωt) dω dt dω 2π 2π ΦR (ω )e iω t { } e iω t e i(ω ω )t Φ R (ω ) dω { 2πδ(ω ) 2πδ(ω ω ) } Φ R (ω ) 2π [ ] Φ R () Φ R (ω) [x, p x ] = iħ, [x, p 2 x] = 2iħp x J = e p i (36) m i A = J E (37) µ j µ = J µ /V B = j µ j µ = lim σ µν (ω)e ν e iωt ω (38) σ µν (ω) = [ Φ R iω µν (ω) Φ R µν() ] (39) Φ R µν(t) = i ħv θ(t) J µ(t)j ν () J ν ()J µ (t) (4) σ µν σ xx σ xy 6 (38) - ) - - 6 σ xy = ), 2) 6
E m m Φ R (t) = i ħ θ(t)eβω m,n e βe n [ n B(t) m m A() n n A() m m B(t) n ] (4) 4 (38)- (4) 7 4. H m = 7 ) n B(t) m = n e iht/ħ B()e iht/ħ m = n B() m e i(e n E m )t/ħ Φ R (t) = i ħ θ(t)eβω m,n e βe n [ e i(e n E m )t/ħ n B m m A n e i(e n E m )t/ħ n A m m B n = i ħ θ(t)eβω m,n ] ( e βe n e βe m ) (42) e i(en Em)t/ħ n B m m A n (43) m n Φ R βω (ω) = e m,n n B m m Ȧ n 8 e βen e βem ħω + E n E m + iδ (44) 8 dω e iωt θ(t) = lim δ 2πi ω iδ F (t) = iθ(t)e ixt { } F (ω) = i dt e iωt θ(t)e ixt { } dω e i(ω+ω +X)t = i dt 2πi ω { } iδ = dω δ(ω + ω + X) ω { } iδ = ω + X + iδ { } 7
Φ(τ) = T τ { B(τ) Ȧ() } (45) Φ(iω λ ) = β dτ e iω λτ Φ(τ) (46) B, Ȧ 4 ω λ = 2λπ B T (47) λ τ > Φ(τ) = e βω βh B(τ) m m Ȧ() n m,n n e = e βω e βen e (En Em)τ m,n n B m m A n (48) Φ(iω λ ) = β dτ e iω λτ Φ(τ) = e βω m,n n B m m A n β dτ e iω λτ e βe n e (E n E m )τ βω = e m,n n B m m Ȧ n e βe n e βe m iω λ + E n E m (49) e iω λβ = (45) (46) iω λ ħω + iδ (5) (44) ω λ ħω 4.2 4.2. x σ xx = [ Φ R iω xx (ω) Φ R xx() ] (5) Φ xx (iω λ ) = V β dτe iω λτ T τ {J x (τ)j x ()} (52) e J = e ħψ σ m ψ σ (53) (52) Φ xx (iω λ ) = e2 ħ 2 m 2 x V xk (iω λ ) (54) K (iω λ ) = β dτe iω λτ K (τ) (55) K (τ) = T τ { ψ (τ)ψ (τ) ψ ()ψ ()} (56) K K (τ) ψ (τ)ψ () ψ (τ) ψ () = T τ {ψ () ψ (τ)} T τ { ψ (τ) ψ ()} = G (, τ)g (, τ) (57) 8
9 τ > K (iω λ ) = G (τ) = e iεnτ G (iε n ) (58) β β n dτe iω λτ β 2 n,n e i(ε n εn)τ G (, iε n)g (, iε n ) = G (, iε n iω λ )G (, iε n ) (59) β n 2 Φ xx (iω λ ) = 2ħ2 e 2 m 2 V 2 β xg (, iε n )G (, iε n iω λ ),n (6) 2 9 = T τ {ψ (τ) ψ ()} = dr dr 2 T τ {ψ (τ) ψ 2 ()} e i r e i r 2 = dr dr 2 = dr dr 2 = d (2π) 3 G (, τ)e i (r r 2 ) e i r +i r 2 d (2π) 3 G (, τ)e i( ) r e i( ) r 2 d (2π) 3 G (, τ)(2π) 6 δ( )δ( ) = G (, τ)(2π) 3 δ( ) δ( ) (54) 2 ε n ω λ ω n + ε n ε n 8. β dτe i(εn ε n ) = βδ εn,ε n 4.2.2 K (iω λ ) = G (iε n )G (iε n iω λ ) β n n ε n = (2n + )π/β (6) dz 2πi n F(z)F (z) (62) C F (z) n F (z) n F (z) = e βz + (63) n F (z) z = (2n + )πi/β iε n 2 β n dz F (iε n ) = C 2πi n F(z)F (z) (64) 22 ε n z iε n z C 2 2 n B (ε) = /(e βε ) 22 e βz z = (2n + )iπ/β e βz + β z (2n + )iπ/β n F (z) /β (65) 9
Im z (a) Im z Ca Im z = iω λ Re z Cb Re z Cc (b) Im z 2: n F (z) C C2 C3 C4 Re z 4.2.3 F (z) G (z)g (z iω λ ) Im z = Im z = iω λ 23 Im z = Im z = iω λ C 3(a) C a (Im z > iω λ ); C b ( < Im z < iω λ ); C c (Im z < ) ω λ > G (z), G (z iω λ ) 3(b) Im z =, iω λ C C 4 + x C, C 2 z = x + iω λ ± iδ, C 3, C 4 z = x ± iδ 23 G (p, iε n ) iεn ε+iδ G R (p, ε) for ε n > (66) G (p, iε n) iε n ε iδ G A (p, ε) for ε n < (67) 3: K (iω λ ) = + + dx 2πi n F(x)G (x + iω λ )G (x + iδ) (C ) dx 2πi n F(x)G (x + iω λ )G (x iδ) (C 2 ) dx 2πi n F(x)G (x + iδ)g (x iω λ ) (C 3 ) dx 2πi n F(x)G (x iδ)g (x iω λ ) (C 4 ) G iω λ iω λ iδ C, C 2 e iβω λ = n(x + iω λ ) = n(x) iω λ ħω + iδ
24 dx K(ω) = 2πi n F(x) [ G R (x + ħω)g R (x) G R (x + ħω)g A (x) ] + G R (x)g A (x ħω) G A (x)g A (x ħω) 25 (68) iω λ < x < C C 4 G R G R, G R G A, G A G A C 2 + C 3 G R G A C +C 4 G R G R G A G A G R G A 26 4.2.4 ω (39) ω Φ R µν(ω) = Φ R() µν + 24 iε n iω λ 25 iω λ ħω + iδ K (ħω + iδ) dx = 2πi n F(x)G (x + ħω + iδ)g (x + iδ) dx + 2πi n F(x)G (x + ħω + iδ)g (x iδ) dx 2πi n F(x)G (x + iδ)g (x ħω iδ) dx + 2πi n F(x)G (x iδ)g (x ħω iδ) G ( ± iδ) ( + iδ) G R ( iδ) G A (68) 26 G R G R G A G A Φ R() µν ω + Φ R(2) µν ω 2 + ω (39) ω σ µν (ω) = iφ R() µν (69) ω G R G A G R G R, G A G A G R G A ω C 2 + C 3 dx 2πi n F(x) [ G R (x)g A (x ħω) G R (x + ħω)g A (x) ] x x + ħω dx 2πi [n F(x + ħω) n F (x)] (7) G R (x + ħω)g A (x) (7) n F (x + ħω) ħω ħω ( dx dn ) F(x) G R (x)g A (x) (72) 2πi dx dn F (x)/dx T dn F (x)/dx (Fermi surface term) C 2 + C 3 27 (Fermi sea term) 27 C 2 +C 3 ω (7) ω = G R (x)g A (x) G R (x)g A (x) = ω C + C 4 Φ R µν()
G R G R, G A G A ω C + C 4 dx 2πi n F(x) [ G R (x + ħω)g R (x) G A (x)g A (x ħω) ] (73) G R (x + ħω) = G R (x) + ħω x G R (x) [ ] ω G R x G R = 2 x { G R } 2 [ ] ħω G R (x) GR (x) + G A (x) GA (x) x x [ { = ħωre G R (x) } ] 2 x G A = [G R ] ħω = ħω dx 2πi n F(x) [ {G x Re R (x) } ] 2 ( dx dn F(x) 2πi dx ) [ {G Re R (x) } ] 2 lim ε ± G R (ε) = C 2 + C 3 C + C 4 dn F /dx ) σ xx σ yy (dissipative) C + C 4 dn F /dx σ xy (dispersive) C + C 4 n F 4.2.5 σ xx = ħe2 ħ 2 2 x πv m 2 dε ( dn F(ε) dε [ G R (, ε)g A (, ε) Re { G R (, ε) } 2 ] (74) dn F /dε G R G A Re { G R} 2 = 2 [ ImG R ] 2 ) (75) T µ σ xx = 2ħe2 πv ħ 2 2 x m 2 [ ImG R (, ) ] 2 (76) G R (, ε) = ε ξ Σ (, x) + iσ (, x) (77) ) Σ Σ Σ > Σ Σ (, ε) ξ x Σ ξ ε G R (, ε) = a ε b ξ + iσ (78) a, b F a = ε Σ ( F, ε) ε= (79) b = + ξ Σ (, ) =F (8) - a 3), 4) ρ(ε) = 2π A(, ε) = Im G R (, ε) π (8) 2
ρ = bρ (82) ρ (= ρ()) b 28 G R G A G R (, )G A (, ) = b bσ Σ ξ 2 + (bσ ) 2 = πb Σ δ(ξ ) (83) bσ πδ(x) = lim a a/(x 2 + a 2 ) G R G A Σ (84) G R G A Σ C + C 4 Re{G R } 2 [ {G Re R (, ) } ] 2 = b 2 ξ2 b2 Σ 2 (ξ 2 + b2 Σ 2 ) 2 (85) ξ 29 Re [ {G R } 2 ] (86) 28 (8) G R = /(ε ξ + iδ) ρ(ε) = π δ (ε ξ ) 2 + δ 2 = δ(ε ξ ) (78) ρ(ε) = π Σ ρ = Σ (a ε b ξ ) 2 + Σ 2 δ(b ξ ) = b 29 E F B T = ρ 2 E E dξ δ(ξ ) = bρ E G R G A σ xx = ħe2 V ħ 2 2 x m 2 = ħe 2 d ħ 2 6π 2 b Σ δ(ξ ) b m 2 4 Σ m ħ 2 F δ( F ) = ħe2 F 3 b 6π 2 m Σ (87) 3 dξ /d = ħ 2 /m τ = ħ 2Σ (88) n = 3 F /3π2 σ xx = ne2 τ m ρ ρ (89) σ xx = ne 2 τ/m 5 (85) ξ E dx x2 a 2 [ E (x 2 + a 2 ) 2 = x ] E x 2 + a 2 E = 2E E 2 + a2 2 E E F E E F τ/ħ Re{G R } 2 3 x 2 d 3 (2π) 3 2 x... = 2 d 2π dφ π sin θ dθ 2π 2π 2π 2 sin θ 2 cos φ 2... = 6π 2 4 d... 3
5. 3 R i U(r R i ) N i N i U(r) = U(r R i ) (9) i= dr ψ(r)u(r)ψ(r) (9) (85) G (, ) = G (, ) + d2 G (, 2)U(2)G (2, ) + d2d3 G (, 2)U(2)G (2, 3)U(3)G (3, ) + (92) (92) G U(r) U(r) U(r) = i,q u(q)e iq (r R i) (93) 3 U(r) ave = i,q = i,q = i,q u(q)e iq r e iq R i ave u(q)e iq r dri V e iq R i u(q)e iq r δ q, = N i u() (94) U(r) dr U(r) 32 u() = (92) U()U(2) ave = i,j q,q 2 u(q )u(q 2 )e i(q r+q2 r2) e i(q R i +q 2 R j ) ave (95) u() = R i = R j q = q 2 U()U(2) ave = N i u(q)u( q)e iq (r r 2 ) 33 5.2 q (96) 4(a) u(q) N i N i n i = N i /V u(q) q = u() = 32 4(a) 33 u( q) = u (q) u(q)u( q) = u(q) 2 4
(a) (b) (c) -q (d) (e) (f) -q (g) - - - - q -q (h) q -q q q -q-q -q -q-q q -q q -q -q -q 4: (a) (b), (d), (e) u() = 4(a) 4(b) (c) (b) (a) (c) R i = R j q + q 2 = 4(f) 34 4(g) (h) n i n2 i 5.3 4(c) G (c) = G Σ (c) G (97) 34 e i(q +q 2 +q 3 ) R i q + q 2 + q 3 = 4(h) G (h) = G Σ (c) G Σ (c) G (98) Σ (c) Σ (c) G (c) = G + G Σ (c) G + G Σ (c) G Σ (c) [ G + ] = G + G Σ (c) G + G Σ (c) G + = G + G Σ (c) G (c) (99) G (, iε n ) ave = G (, iε n ) + G (, iε n )Σ(, iε n ) G (, iε n ) ave () 35 5 G 4(h) G (c) (c) Σ Σ 4(c) 35 (98) G = G + G ΣG G ( ) G Σ G = (98) G G = G Σ 5
36 = + = + + + + +... 5: Σ Σ(, iε n ) = dξ ħ iε n + ξ 2πτ ε 2 n + ξ 2 = ħ iε n 2τ ε n (3) 37 τ (88) τ (2) G = [ G Σ ] G (, iε n ) = iε n ξ + i sgn(ε n )ħ/2τ (4) Σ(, iε n ) = N i u(q) 2 G ( q, iε n ) q u( ) 2 = N i iε n ξ = N i u( ) 2 iε n + ξ ε 2 n + ξ 2 () u( ) 2 ħ = τ 2 N iρ dω u( ) 2 (2) 36 dξ dω = 2V (2π) 3 2 d sin θdθdφ = V dω π 2 2 d 4π,σ dω = sin θdθdφ dω/4π = 2 ξ = ħ 2 2 /2m 2 d = ( ) 2m 3/2 ξdξ 2 ħ 2 ρ(ϵ) =,σ V 2π 2 ( ) 2m 3/2 ϵ ħ 2 dω = dξρ(ξ) 4π dn F (x)/dx ρ(ξ) ρ ρ 37 ξ ξ iε n ε n ε n > ±iε n +iε n dξ ξ 2 + ε 2 = dξ n (ξ iε n )(ξ + iε n ) = π ε n ε n < iε n dξ (ξ iε n )(ξ + iε n ) = π ε n dξ ξ 2 + ε 2 n = π ε n 6
(a) (b) (c) G R (, ε) = ε ξ + iħ/2τ (5) G A (, ε) = ε ξ iħ/2τ (6) (4) ħ/τ A(, ε) = (ε ξ ) 2 + ħ 2 /4τ 2 (7) (d) (e) (f) 6: (a)-(c) (d)-(e) ħ/2τ x x x 6 4 (6) 6 (a)-(c) (d)-(f) 38 7: (8) x 6(d) 6(e) (6) = + 8: (9) 38 vertex. 7
Φ xx (iω λ ) = 2ħ2 e 2 m 2 x Γ x () V β,n G (, iε n )G (, iε n iω λ ) (8) Γ () = + N i u( ) 2 Γ ( ) G (, iε n )G (, iε n iω λ ) (9) 7, 8 Γ 39 Γ () = γ () Γ (9) γ γ = + γ ħ 2π τ dξ G (, iε n )G (, iε n iω λ ) () ħ τ = 2 N iρ dω u( ) 2 2 (2) 4 () ξ η(ε n ) = sgn(ε n )ħ/2τ dξ iε n ξ + iη(ε n ) i(ε n ω λ ) ξ + iη(ε n ω λ ) (3) ε n iε n + 39 Γ (; iε n, iε n iω λ ) 4 (9) Γ () = γ γ = + N i γ v 2 G G 2 = 2 γ = + N i γ 2 v 2 G G iη(ε n ), i(ε n ω λ ) + iη(ε n ω λ ) ω λ > ε n > ε n ω λ < (3) iω λ ω + iδ ω 2π/(ħ/τ ) γ () γ = γ = + γ τ τ (4) τ /τ = τ tr τ (5) ħ = ħ ħ τ tr τ τ = ( 2 N iρ dω u( ) 2 ) 2 (6) Φ xx (iω λ ) = 2ħ2 e 2 [ ( τtr ) ] m 2 + V β τ θ(ε n )θ(ω λ ε n ),n 2 xg (, iε n )G (, iε n iω λ ) (7) θ(x) ε n > ε n ω λ < 4 C 2 + C 3 G R G A (89) C 2 + C 3 τ tr /τ σ xx = ne2 τ tr m ρ ρ (8) 8
' ' 9: τ tr τ tr ( / 2 ) 4 cos χ χ cos χ /τ tr cos χ /τ tr 9 4 ρ ρ(t) B(t) = Tr [ρ (t)b] (9) B ρ A B(t) = B(ω)e iωt (2) B(ω) Φ R (ω) B(ω) = [ Φ R (ω) Φ R () ] (2) iω A B Φ R (t) = i ħ θ(t) B(t)Ȧ() A()B(t) (22) A B ψψ ψ(τ)ψ(τ) ψ()ψ() Φ xx (iω λ ) = 2ħ2 e 2 m 2 V 2 β xg (, iε n )G (, iε n iω λ ),n (23) 9
F (iε n ) iε n β n dz F (iε n ) = C 2πi n F(z)F (z) (24) G (iε n )G (iε n iω λ ) C C 4 3(b) iε n ε iω λ ω + iδ K (iω λ ) = G (, iε n )G (, iε n iω λ ) β n (25) dε K(ω) = 2πi n F(ε) [ G R (ε + ħω)g R (ε) G R (ε + ħω)g A (ε) ] + G R (ε)g A (ε ħω) G A (ε)g A (ε ħω) (26) σ xx = ħe2 ħ 2 2 ( x πv m 2 dε dn ) F(ε) dε [G R (, x)g A (, x) Re { G R (, x) } ] 2 (27) G R G A C 2 + C 3 Re [ {G R (, x)} 2] C + C 4 (dissipative) (dispersive) G R (, ε) = a ε b ξ + iσ (28) C 2 + C 3 C + C 4 G R G A Σ (29) Re [ {G R (, )} 2] (3) σ xx = ne2 τ m ρ ρ (3) ρ = bρ τ τ = ħ 2Σ (32) Σ(, iε n ) = ħ 2τ iε n ε n (33) τ N i ħ = τ 2 N iρ dω u( ) 2 (34) G (, iε n ) = ε n ξ + i sgn(ε n )ħ/2τ (35) G R (, iε n ) = ε ξ + iħ/2τ (36) 2
ħ = ( τ tr 2 N iρ dω u( ) 2 ) 2 (37) τ tr σ xx = ne2 τ tr m ρ ρ (38) ( / 2 ) τ 42 42 Green-Kubo Green 43 (24), (3), (33) Φ(ω) = 2ħ2 e 2 m 2 V m,n m x n 2 n F(E m ) n F (E n ) E m E n + ħω + iδ (39) δ δ σ δ δ > 5 43 B(ω) = i e iωτ Tr(e ih τ/ħ [A, ρ]e ih τ/ħ B)dτ ħ ( ) δ M = i t { Tr ρ [H (t ), ˆM(t)] } dt (Ricayzen) ħ σ αβ (q, ω) = t dt e iω(t t ) ψ [j ωv α(q, t), j β (q, t )] ψ + i n e 2 mω δ αβ (Mahan) A(t) = dt f(t ) A(t)B(t ) R (Zagosin) 2
/(ε ξ + iħ/2τ ) /(E m E n + ħω + iħ/2τ ) τ E m,n (39) δ [2] H. Fuuyama: Prog. Theor. Phys. 42 (969) 284. [3] S. Naajima and M. Watabe: Prog. Theor. Phys. 29 (963) 34. [4] J. C. Swihart, D. J. Scalapino, and Y. Wada: Phys. Rev. Lett. 4 (965) 6. http://www.ooai.pc.uec.ac.jp/otaibutsuri.html [] R. Kubo: J. Phys. Soc. Jpn. 2 (957) 57. [2] R. Kubo and K. Tomita: J. Phys. Soc. Jpn. 9 (954) 888. [3] : 84 (955) 25. [4] : 88 (955) 53. [5] H. Naano: Prog. Theor. Phys. 5 (956) 77. [6] : 88 (955) 45. [7] : 89 (955) 72. [8] : 89 (955) 79. [9] : 89 (955) 99. [] J. v. Neumann: Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl. (927) 245. [] H. Fuuyama, H. Ebisawa, and Y. Wada: Prog. Theor. Phys. 42 (969) 494. 22