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1 ( ) ) AGD ) 7)

2 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, t)ψ (r, t) ψ (r, t)ψ (r, t) = (4) ψ()ψ ( ) G(, ) = i T {ψ()ψ ( )}. (5) (r, t ; r, t ) (, ) T { }. ψ ψ ψ (r) = i ψ(r) = i φ i (r)a i, φ i (r)a i φ i (r) i a i a i 5 t > t (r, t ) (r, t )

3 .3 T = T T = T 8) T T = Matsubara (955) AGD (959) Fradkin (959) : G R,A G A(ε) 9) ) 3 3 T = S T 3 3 time ordered G(, ) = i T {ψ()ψ ( )} retarded G R (, ) = iθ(t t ) {ψ(), ψ ( )} advanced G A (, ) = iθ(t t) {ψ(), ψ ( )} greater G > (, ) = i ψ()ψ ( ) lesser G < (, ) = i ψ( ) ψ() contour ordered G(, ) = i T C {ψ()ψ ( )}

4 ) 4) T ˆX [ ] ˆX Tr e β(ĥ µ ˆN) ˆX = [ ] (6) Tr e β(ĥ µ ˆN) 4 µ β = /k B T µ β Tr Tr[ ] = n n n ˆX ψψ [ ] Tr e β(ĥ µ ˆN) ψ(t)ψ (t ) [ ]. (7) Tr e β(ĥ µ ˆN) ˆX(t) = e iĥt/ħ ˆX()e iĥt/ħ 5 exp(±iĥt/ħ) exp( βĥ) 6 Time ordered ( Causal) T Contour ordered ( Keldysh) T C Retarded, Advanced {A, B} = AB + BA 4 Ω = Tr[ e β(ĥ Ω) ] Z = Tre βĥ = e βω Ĥ = Ĥ µ ˆN 5 e iĥt/ħ +iĥt/ħ+ +(/n!) (iĥt/ħ ) n+ 6 e iĥt S 8) exp(±βĥ) 3. G αβ (r, τ ; r, τ ) = T τ {ψ α (r, τ ) ψ β (r, τ )} (8) T τ { } T P ( ) P { T τ {ψ()ψ( ψ()ψ( ) for τ > τ )} = ±ψ( )ψ() for τ < τ (9) ψ(r, τ) = e τĥψ(r)e τĥ ψ(r, τ) = e τĥψ (r)e τĥ () () ψ ψ ψ ψ 4

5 3. T = t τ ) 9) 7 Tr{ABC} = Tr{BCA} τ > τ G αβ (r, τ ; r, τ ) = Tr [ e β(ĥ Ω) e (τ τ )Ĥψ α (r )e (τ τ )Ĥψ β (r ) ] () 8 τ τ τ < τ G αβ (r, τ; r, ) = T τ {ψ α (r, τ) ψ β (r, )} (3) β < τ < 9 G αβ (τ) = ±G αβ (τ + β). (4) 7 9) (4) 8 e τ Ĥe β(ĥ Ω) = e β(ĥ Ω) e τ Ĥ Ĥ Ω 9 G αβ (r, τ) = (π) 3 dp e ip r iεnτ G αβ (p, iε n ) β G αβ (p, iε n ) = dr β n dτe ip r+iε nτ G αβ (r, τ) (5) (6) r = r r 56 4 ε n π/β { nπk B T ε n = (n + )πk B T (7) τ < G αβ (r, τ; r, ) = ψ β (r, )ψ α(r, τ) = Tr [e β(ĥ Ω) ψβ (r )e τĥψ α(r )e τĥ] [ = Tr e βω e τĥψ ] α(r )e (τ+β)ĥ ψβ (r ) [ = Tr e β(ĥ Ω) e (τ+β)ĥψ ] α(r )e (τ+β)ĥ ψβ (r ) = ±G αβ (r, τ + β; r, ) β τ β f(τ) f(τ) = β e inπτ/β f(iε n ) n= f(iε n ) = β dτ f(τ)e inπτ/β β β < τ < f(τ) = ±f(τ + β) f(iε n ) = [ β ] dτ f(τ)e inπτ/β + dτ f(τ)e inπτ/β β = β ( ± e inπ ) dτ f(τ)e inπτ/β n ( ± e inπ ) β f(iε n ) = dτ e iεnτ f(τ) { nπk ε n = B T (n + )πk B T 5

6 T = 3.3 Ĥ = ε (p)ψ pσψ pσ (8) pσ ˆN = ψ pσψ pσ (9) pσ Baker-Hausdorff e A Ce A G (p, iε n ) = ( ± n p ) β dτ e (iε n ξ p )τ = ( ± n p ) eβ(iε n ξ p ) iε n ξ p = ( ± n p ) e βξp iε n ξ p. (5) G (p, iε n ) = iε n ξ p (6) iε n ξ p = C + [A, C] +! [A, [A, C]] + [A, [A, [A, C]]] + 3! () ψ pσ (τ) = e τ(ĥ µ ˆN) ψ pσ e τ(ĥ µ ˆN) = e τξ p ψ pσ () ψ pσ (τ) = e τ(ĥ µ ˆN) ψ pσe τ(ĥ µ ˆN) = e τξp ψ pσ () ξ p = ε (p) µ θ(τ) G (p, τ) = θ(τ)e ξpτ ψ pσ ψ pσ + θ( τ)e ξpτ ψ pσψ pσ = e ξpτ [θ(τ) ( ± n p ) θ( τ)n p ] iε n ε±iδ (3) n p = ψ pσψ pσ n p = e βξ p e βξ p + (4) 4 (retarded and advanced Green s function) G (p, iε n ) iε n ε+iδ G R (p, ε) for ε n > (7) G (p, iε n ) iεn ε iδ G A (p, ε) for ε n < (8) δ 6

7 4. G R (p, t t ) = iθ(t t ) ψ pσ (t)ψ pσ(t ) + ψ pσ(t )ψ pσ (t) (9) t i θ(t t ) t t G A (p, t t ) = iθ(t t) ψ pσ (t)ψ pσ(t ) + ψ pσ(t )ψ pσ (t). (3) Ĥ m = E m m G R (t t ) = iθ(t t )e βω n e βĥ { ψ(t)ψ (t ) + ψ (t )ψ(t) } n n n ψ(t) m = n e iĥt ψ()e iĥt m = n ψ() m e i(e n E m )t (3) 3 G R (t t ) = iθ(t t )e βω m,n e βe n { e i(e n E m )(t t ) n ψ m + e i(e n E m )(t t ) m ψ n } = iθ(t t )e βω m,n ( e βe n + e βe m ) e i(en Em)(t t ) n ψ m. (33) 4. G R (r, t) = (π) 4 dpdε e ip r iεt G R (p, ε) (34) G R (p, ε) = drdt e ip r+iεt G R (r, t) (35) (33) dε θ(t) = lim δ πi e iε t ε iδ (36) = iθ(t t )e βω m,n e βen { } n ψ(t) m m ψ (t ) n + n ψ (t ) m m ψ(t) n (3) m m m = ψ p,σ 3 { } m n 7

8 δ > (33) (35) 4 G R dε (ε) = i dt πi ε iδ e βω ( e βe n + e βe ) m n ψ m m,n = e βω m,n e i(ε+ε +E n E m)t dε δ (ε + ε + E n E m ) ε iδ ( e βe n + e βe ) m n ψ m = e βω n ψ m e βe n + e βe m ε + E m,n n E m + iδ (37) G A G R G A (ε) = [ G R (ε) ] (38) A(p, ε) G R (p, ε) = A(p, ε) = πe βω m,n dε A(p, ε ) π ε ε + iδ (39) n ψ p m ( e βe n + e βe m ) δ(ε + E n E m ). (4) (39) A(p, ε) p ε p ε (39) A(p, ε) = ImG R (p, ε) (4) 4 ε ε dt e iεt = πδ(ε) π x x ± iδ = P iπδ(x) (4) x P A(p, ε) p, ε 5 dε A(p, ε) = (43) π A(ε) A(p, ε) n p = ψ pσψ pσ (44) 5 (3) dε A(p, ε) π = e ( βω n ψ p m e βe n + e βe m m,n = e ) βω n (ψ pψ p + ψ pψ p n e βen n = e βω n e βe n. Z = e βω = Tre βĥ = n e βe n ) 8

9 n p = e βω m,n m e βĥψ pσ n n ψ pσ m = e βω m,n e βe m n ψ pσ m (45) (4) ( e βe n + e βe m ) δ(ε + E n E m ) = e βe m ( e βε + ) δ(ε + E n E m ) (46) p p e βε + n p = dε A(p, ε) π e βε + (47) n F (ε) = (e βε + ) n p = dε π n F(ε)A(p, ε) (48) p 6 (4) 4.3 (9) G R (p, t t ) = iθ(t t )e i(t t )ξ p ψ pσ ψ pσ + ψ pσψ pσ = iθ(t t )e i(t t )ξ p (49) 6 n p n F (48) 3: A (ε) A(ε) 8. (43) A (ε) A(ε) Baker-Hausdorff ψ pσ (t) = e itξ p ψ pσ (37) G R (p, ε) = ε ξ p + iδ (5) (6) iε n ε + iδ (4) A (p, ε) = πδ(ε ξ p ) (5) p ξ p p ξ p 3 9

10 5 5. V (r r ) Ĥ = dr ψ(r) ( ) m ψ(r) + drdr ψ(r) ψ(r )V (r r )ψ(r )ψ(r). (5) V (r r ) ψ(r) ψ σ (r) r σ ħ = X(t) i X(t) t ] = [X(t), Ĥ(t) (53) ψ(τ) τ = it ψ(τ) τ ] = [ψ(τ), Ĥ(τ) (54) (5) Ĥ Ĥ [ ] ψ(τ), Ĥ = ψ(r, τ) dr ψ(r, τ) m ψ(r, τ) + dr ψ(r, τ) m ψ(r, τ)ψ(r, τ) (55) i r i ψ(r, τ) m ψ(r, τ)ψ(r, τ) = ψ(r, τ)ψ(r, τ) m ψ(r, τ) = { δ(r r ) ψ(r ) ψ(r ) } m ψ(r, τ) (56) (55) r [ ] ψ(τ), Ĥ = ψ(r, τ) (57) m ψ ψ ψ ψ ψ ψ ψ [ψ, ψ ] + = δ( ) [ψ(τ), Ĥ ] [ ψ(τ), Ĥ ] = dr dr 3 [ψ ψ ψ3 V ( 3)ψ 3 ψ ψ ] ψ3 V ( 3)ψ 3 ψ ψ (58) ψ ψ3 V ( 3)ψ 3 ψ ψ = ψ ψ3 V ( 3)ψ ψ 3 ψ = ψ { δ(3 ) ψ ψ3 } V ( 3)ψ3 ψ = ψ V ( )ψ ψ + { δ( ) ψ ψ } ψ3 V ( 3)ψ 3 ψ = ψ V ( )ψ ψ + ψ 3 V ( 3)ψ 3 ψ ψ ψ ψ3 V ( 3)ψ 3 ψ (59) (58) [ ψ(τ), Ĥ ] = dr ψ(r )V (r r )ψ(r )ψ(r) (6)

11 τ ψ(r, τ) = ψ(r, τ) m + dr ψ(r, τ)v (r r )ψ(r, τ)ψ(r, τ). 5. (6) (8) G (, ) = θ(τ τ ) ψ() ψ( ) + θ(τ τ ) ψ( )ψ() (6) τ τ G (, ) = δ(τ τ ) [ ψ(), ψ( ) ] T τ { τ ψ() ψ( ) } = δ( ) m T { τ ψ() ψ( ) } { + dr V (r r ) T τ ψ()ψ()ψ() ψ( ) } τ=τ (63) δ( ) = δ(τ τ )δ(r r ) G (; ) = T τ { ψ()ψ() ψ( ) ψ( ) } (64) G [ ] G (, ) = δ( ) τ m dr V (r r )G (; + ) τ =τ. (65) + τ G (6) (65) G G 5.3 G [ ] G (, ) = δ( ) (66) τ m exp[ ip (r r ) + iε n (τ τ )] r, τ 7 β dτ e iε n(τ τ ) G (, ) = iε n G (r r τ, iε n ) (67) dr e ip (r r ) G (, ) = p G (p, τ τ ) G (p, iε n ) = iε n p /m (68) (69) (6) τ iε n (7) p (7) 7 (67) (68) τ (4) r G (, ) r r ± = (6)

12 (63) G (, ) = iε n Ĥ (7) Ĥ /(iε n Ĥ) (iε n Ĥ) Ĥ p /m (69) (7) iε n G δ, iε n (73) 6 G (r, τ ) (r, τ ) G (, ) G (, )G (, ) (74) (65) [ ] τ m + dr V (r r )G (, + ) τ =τ G (, ) = δ( ) (75) G (, + ) n(r ) (75) G (, + ) n nv = dr V (r r )G (, + ) (75) (69) G (p, iε n ) = iε n p /m nv (76) iε n ε + iδ G R (p, ε) = ε p /m nv + iδ (77) A(p, ε) = πδ(ε p /m nv ) (78)

13 (, (, ) ( ; ) ( ; ) E(p) = p m + nv (π) 3 dp V (p p )G (p, iε β n) n (83) A(p, ε) = πδ(ε E(p)) (84) E(p) (83) G (p, iε n ) (8) E(p) E(p) (8) (83) G (, ) G (, )G (, ) G (, )G (, ) (79) 8 [ ] G (, ) + dr U(r, r )G (, ) τ m = δ( ) (8) U(r, r ) = δ(r r ) dr 3 V (r r 3 )G (3, 3 + ) τ =τ 7 (8) G (, ) = G (, ) + d d G (, )Ṽ ( ) [ G (, + )G (, ) G (, + )G (, ) ] (85) 9 Ṽ ( ) = δ( τ τ )V ( r r ) 4 V (r r )G (, + ) (8) U(r, r ) G (p, iε n ) = iε n E(p) (8) 8 (64) G (; ) = ±G (; ) Hartree Fock 4: ( 9 (8) τ m ) [ ] G (, ) = δ( ) τ m 3

14 Hartree Fock 5: 7: 6 8: 7 6: G G G G G V 5G G 4 G m. m 7 i j ) 4

15 . G (i, j) i j 3. i j V (i, j) 4. β dr i dτ i (86) 5. ( ) m+f (s + ) F (87) F s s = / m 5. 8 m+f (s + )F ( ) (π) 3m (89) p E(p) p (, ) τ r r (, ) 9. m. G (p i, iε ni ) 3. V (p) 4. β m n i dp i (88) 9: G (, ) G (, )G (, ) G (, )G (, ) β dτ dr dr V (r r ) [G (, )G (, )G (, )G (, ) G (, )G (, )G (, )G (, )] τ =τ (9) 5

16 (65) ( ) G (, ) + τ m β dτ dr Σ(, )G (, ) = δ( ). (9) Σ Σ(, ) = Σ HF (, ) + Σ B (, ) (9) Σ HF (, ) = δ(τ τ ) [ δ(r r ) dr V (r r )G (, + ) ] V (r r )G (, + ) (93) Σ B (, ) = dr dr V (r r )V (r r ) [ G (, )G (, )G (, ) G (, )G (, )G (, ) ] τ =τ 8. τ =τ (94) (66) e ip (r r ) (9) Σ(r, r ) dp Σ(r) = Σ(p)eip r (π) 3 (95) G (9) dr Σ(r, r )G (r, r ) = dp dp (π) 3 (π) 3 dr eip (r r (π) 3 ) ip (r e r ) Σ(p)G (p ) (96) r r r r p p r (π)3 δ(p p) *4 p p = p dp eip (r r (π) 3 ) Σ(p)G (p) (97) dp(π) 3 e ip (r r ) (9) ) (iε n p m Σ(p, iε n) G (p, iε n ) = (98) (95) dpe ip (r r ) Σ B (94) β dτe iωnτ = βδ ωn AGD (4.5) ω n ε n ε n ( ) πt 6

17 dq ΣB st (r, r ) = dr r dq (π) 3 (π) 3 V (q)v (q ) e iq (r r ) e iq (r r ) dp dp dp (π) 3 (π) 3 (π) 3 G (p )G (p )G (p ) e ip (r r ) e ip (r r ) e ip (r r) (99) r, r (π) 3 δ( q + p p ), (π) 3 δ( q p + p ) q, q ΣB st (r, r ) dp dp dp = (π) 3 (π) 3 (π) 3 ei(p +p p ) (r r ) V (p p )V (p p )G (p )G (p )G (p ) Σ B (r, r ) = dp eip (r r (π) 3 () ) Σ B (p) () {V (p p ) V (p p )} / 8. (98) G (p, iε n ) = iε n ξ p Σ(p, iε n ). (5) iε n ε + iδ Σ(p, iε n ) Σ R (p, ε) (6) ΣB st dp dp dp (p) = (π) 3 (π) 3 (π) 3 {V (p p )} G (p )G (p )G (p ) (π) 3 δ(p + p p p ) () p p [ {V (p p )} {V (p p )} + {V (p p )} ] / ΣB nd dp dp dp (p) = (π) 3 (π) 3 (π) 3 V (p p )V (p p )G (p )G (p )G (p ) (π) 3 δ(p + p p p ) (3) dp dp dp Σ B (p) = (π) 3 (π) 3 (π) 3 {V (p p ) V (p p )} G (p )G (p )G (p ) (π) 3 δ(p + p p p ) (4) G R (p, ε) = ε ξ p Σ R (p, ε) + iδ (7) (4) Γ (p, ε) A(p, ε) = [ε ξ p ReΣ R (p, ε)] + [Γ (p, ε)/] (8) Γ (p, ε) = ImΣ R (p, ε) 3 Γ ε dε A(p, t t ) = A(p, ) ε)e iε(t t π (9) p t t Γ ε e Γ (p)(t t ) Γ 7

18 τ = /Γ τ ξ p G (p, iε n ) = iε n ξ p () iε n τ Ĥ G (, ) = iε n Ĥ () G R, G A iε n ε ± iδ A(p, ε) = ImG R (p, ε) () A (p, ε) = πδ(ε ξ p ) E(p) G (p, iε n ) = iε n E(p) (3) E(p) G A(p, ε) = πδ(ε E(p)) G (p, iε n ) = iε n ξ p Σ(p, iε n ) (4) Σ(p, iε n ) 8

19 [] R. Kubo: J. Phys. Soc. Jpn. (957) 57. [] A. Abrikosov, L. Gorkov, and I. Dzyaloshinskii: Methods of Quantum Field Theory in Statistical Physics (Dover Books on Physics Series. Dover Publications, 975), Dover Books on Physics Series. [3] A. Fetter and J. Walecka: Quantum Theory of Many-Particle Systems (Dover Books on Physics. Dover Publications, ), Dover Books on Physics. [4] G. Mahan: Many-Particle Physics (Physics of Solids and Liquids. Springer, ), Physics of Solids and Liquids. [5] : (, ). [6] : (, 99). [7] : (, 999). [8] T. Matsubara: Prog. Theor. Phys. 4 (955) 35. [9] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii: Sov. Phys. JETP 9 (959) 636. [] E. S. Fradkin: Sov. Phys. JETP 9 (959) 9. [] H. Ezawa, Y. Tomozawa, and H. Umezawa: Nuovo Cim. 5 (957) 8. 9

( ) ) ) ) 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)