A L Fetter and J D Walecka Quantum Theory of Many

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1 1 Green Green Green 3,1 2 Green 4,1 3 Hartree Fock 7. 1 4 Feynman Fermion 8,1 5 Fermi 9,1 6 19,1 7 22,1 8 32. 1 9 35,1 10 Feynman 38,1 11 Feynman 39,1 12 Dyson 40. 1 13 Green 42,1 14 Green 46,1 15 Hartree Fock 47,1 16 Fermi 51. 1 17 Ring 55,1 18 Green 60,1 19 63,1 20 64,1 21 67.
1 22 Bogoliubov Bose 69,1 23 Bogoliubov Fermi 75,1 24 Phonon 86. 1 25 88,1 26 Phonon Green 92,1 27 Migdal 95, 1 1 Green Green Green . Green Lehmann , , A k B k , G k d 1 , 0 i i , A k B k Green. Green , , R A k B k , G k d 2 , 0 i i , , A A k B k . G k d 3 , 0 i i , GR , GR k ReG k isgn ImG k 4 , G GR .
P i 5 , i , , i A k , ImG k 6 , i B k , , , R A k B k . G k P d , 0 , i A k B k , , A k , P d , i d A k , 0 0. , B k , P d , i d B k , 0 0, , A k , d P i A k , 0 . , B k , d P i B k , 0 , , A k B k , d 7 , 0 i i , . 1 2 Green , Green Fourier , d i t, G k t e G k , 2 .
, d i t d i t, e G k e G k 8 , 2 2 , , GR k G k 9 . G k G k 10 , , d i t d i t A, e G k e G k 11 , 2 2 . GA , C1 h , 1 GA , I , 0, C1 C1 C1 C1 , , 12 , C1 C1 . G GA GR 1 Jordan , , , d i t d i t A, e G k e G k 13 . 2 i 2 , 8 , , d i t d i t R, e G k e G k 14 , 2 2 .
GR GR k i k, a k k k 0 , , I , 0 15 , C2 C2 C2 C2 . h C2, 2 GR , , d i t d i t R 1, e G k e G k 2 i e i k i k t a. 2 i 2 2 , , d i t R k, e G k iae i t e k t 16 , i 2 . t , k damping constant k , 1 t k 17 , 2 t k 1 18 . G k t iae i t e k t 19 , , 16 Im , , GR k t 20 , k i k.
GA k t GR k t 21 , k i k, 16 , , d i t A, e G k GR k . i 2 , , d i t a a, e , i 2 k i k k i k, , d i t 2i k. a e, i 2 k 2 k2, idu i iu t 1, 2ia k e u i 1 , 2 k. 2 k iu 2, 2 2 1, k a i t i x, dxe x k2 , e 1 x ut 22 . t 0 k t, t k t k 1 k 1 k , t k , 1 1 23 , t k u , 2.
d i t A k a i t , e G k GR k e dx e x, i 2 t 0 k, t k a 2 k 2 e i t. iae i t e k t 24 , 2, t2 , k, 2, k, k t 1 25 , k t. , 1 3 Hartree Fock , 1 10 17 , , U x k x1 dx2 x1 x2 k x2 k k x1 26 . U x 0 k x V 1 2 eik x , k x1 x1 , dx1 dx2 k x1 x1 x2 k x2 k 27 . 0k r x1 x2 R x1 x2 2 , , , dr dR x1 x2 e ik x1 x2 k 28 .
, k 0k k 29 , Self Energy Fourier , , k d x x e ik x x x x 30 . 1 10 16 , , , x x 2s 1 x x dx1 V x x1 k x1 2 kF k . V x x k x k x kF k , , 2s 1 x x dx1 V x x1 kF k , dk ik x x . V x x e kF k 31 , n n 2s 1 kF k 30 , , ik x x , k n d x x e x x dx1 V x x1 . , ik x x dk ik x x , d x x e V x x e F k , , dk , n dx1 V x x 3.
d x x V x x ei k k x x kF k , nV 0 3, dk V k k kF k 32 . , 1 4 Feynman Fermion , 1 n 2n 1 , 2 4 V p G 0 p . 3 , i 1 nl, 4 nl , 2 4n, 5 n , 6 ei , Green , i d4 k1 0 . k G k V 0 G 0 k1 ei 1 G 0 k , 2 4, i d4 k1 0 , G k V k k1 G 0 k1 ei 1 G 0 k 33 . 2 4, Self Energy , k k k k k1 k, 3 Feynman , i d4 k1.
k V 0 G 0 k1 ei 1 , 2 4, i d4 k1, V k k1 G 0 k1 ei 1 34 . 2 4, , 1 5 Fermi , Fermi Fermi , E 2 kF2 , A B kF a C kF a 2 35 . kF 0 a 0 , Hatree Fock , 2 1 , 10 22 , , 2V 1 , E dk k k kF k . 2 3 2, 32 , , E 3 2 kF2 2 1, dk kF k dknV 0 kF k . V 5 2m 2 3 2 3, , dk dk , 3 3, V k k kF k kF k , 2 2 .
, 3 kF 2 4 kF3, dk dk , 2V 0 kF k kF k , 5 2m 2 3 3 2 3 2 3. , dk dk , V k k kF k kF k , 2 3 2 3, kF kF, 3 2 kF2 N dk dk . 2V 0 V k k 37 , 5 2m V 2 3 2 3, , N k3, n 2 kF k F2 38 . V 0 , 1 , dye ik y v y k y , f k k 39 , v y 40 , Born k y eik y . V 0 dxV x f k k 41 , kF 0 , f k k a 42 , k k , V k k V 0 43 .
37 , 3 2 kF2 N2, E N V 0 , 5 2m 4V, 3 2 kF2 N 2 a. N N, 5 2m V m, 3 2 kF2 k 3 2 a, N N F2, 5 2m 3 m . E 2 kF2 3 2, kF a 44 , N 2m 5 3 2, , dk 1 4 kF3 N. 45 , 2 2 3 2V, , 42 s , , f 2l 1 ei l sin l Pl cos 46 . l 0 , f e sin 0 47 , , r a , V r 48 , Schro dinger .
, k 0 r 0 49 , 0 r A sin k r a 50 , 0 a 0 , 0 k ka 51 . k 0 , f 1 i 0 0 a O 02 52 , k k, p p q, 0 q, k q q q. k q p, p p p p, p k p p q p k q p, 4 , 42 , 4 , Self energy . , d4 k d4 k, 1 p 2i U0 0 G 0 k eik0 i U0 p k G 0 k eik0 53 . 2 4 2 4, , , 2 d4 k d4 q, 2 p U0 q G 0 k G 0 k q G 0 p q U0 q .
2 4 2 4, , 1 d4 k d4 q, U0 q G 0 k q G 0 k G 0 p q U0 k p q 54 . 2 4 2 4, , p p p, 5 vertex ,proper Self energy , . d4 k 0 d4 k 0 , p 2i, G k pk pk i G k kp pk 55 , 2 4 2 4. vertex , i d4 q, p1 p2 p3 p4 U0 p1 p3 U0 q G 0 p1 q G 0 p2 q U0 p1 q p3 . 2 4, 56 , 7 , p1 q, p1 p2, p1p2 p3p4 p2 q, p1 p3, p3 p4 p1 q p3.
p3 p4, 6 vertex , p1 p2, p1 q p2 q, , p3 p4, 7 vertex . i d4 q, p1 p2 p3 p4 U0 p1 p3 U0 q G 0 p1 q G 0 p2 q p1 q p2 q p3 p4 . 2 4, Bethe Salpeter , , , p1 p2 p3 p4 U0 q Q p1 q p2 q p3 p4 58 . i 0 d4 q, Q p1 p2 p3 p4 2 p1 p3 G p1 G p2 , 4 0 , U0 q Q p1 q p2 q p3 p4 . 2 4,Q , , d4 q i 0 , U 0 q 2 4, p 1 q p 3 G p1 q G 0 p2 q .
2 4 , , d4 q , U0 q Q p1 q q p2 q q p3 p4 , , i d4 q d4 q . U0 p1 p3 U0 q G 0 p1 q G 0 p2 q U0 q , 2 4 2 4, 2 4 p1 q q p3 . i d4 q, U0 p1 p3 U0 q G 0 p1 q G 0 p2 q U0 p1 q p3 60 . 2 4, 56 , P , P p1 p2 p3 p4 61 , , 1 1, p p1 p2 p p3 p4 62 . 2 2, U0 q U0 q 59 , P 59 , , dp0 1 1 1 1, p p P Q P p P p P p P p .
2 2 2 2 2, Q p1 p2 p3 p4 63 , p1 P 2 p p2 P 2 p p3 P 2 p p4 P 2 p 63 . Q , , i dp0 d4 q, p p P 2 p p , U0 q Q p1 q p2 q p3 p4 G 0 p1 G 0 p2 . 2 2 4, , i dp0 0 d4 q, 2 p p , G P 2 p G P 2 p , 2 2 4. U0 q Q P 2 p q P 2 p q P 2 p P 2 p , , i dp0 0 dq. 2 p p , G P 2 p G P 2 p , p q p P 64 , 2 2 3, Green Green .
, , , i dp0 0 , G P 2 p G 0 P 2 p , , i dp0 P 2 p kF kF P 2 p . , 2 P0 2 p0 P 2 p i P0 2 p0 P 2 p i , , P 2 p kF kF P 2 p . , P0 2 p0 P 2 p i P0 2 p0 P 2 p i , , i dp0 P 2 p kF P 2 p kF . 2 P0 2 p0 P 2 p i P0 2 p0 P 2 p i , kF P 2 p kF P 2 p . P0 2 p0 P 2 p i P0 2 p0 P 2 p i , , i dp0 P 2 p kF P 2 p kF .
, 2 P0 2 p0 P 2 p i p0 P0 2 P 2 p i , kF P 2 p kF P 2 p . 65 , P0 2 p0 P 2 p i p0 P0 2 P 2 p i , , i dp0 0 P 2 p kF P 2 p kF . G P 2 p G 0 P 2 p , 2 P0 0P p 0P p i , kF P 2 p kF P 2 p . 66 , P0 0P p 0P p i , E, E P0 67 , N P p 1 n0P p n0P p 68 . 2 2, n0p kF p , , 2 P2 P2, P0 P p P p, 0 0, P0 P p p2 P p p2.
2 2 2m 4 4, E 69 , 66 , i dp0 0 N P p , G P 2 p G 0 P 2 p 70 . 2 E p m i N P p , 64 , N P p , dq, p p P 2 p p , U0 q p q p P 71 . E p m i N P p , 2 2 2 3, , p p P P 2 p P 2 p P 2 p P 2 p . U0 q p q p P 72 , 71 72 f , mU0 mE, v 73 , 2 2, 71 N 1 kF 0 . 0 71 N 1 , dp, p i 0 p p P , v q 0 p q p P 2 3 p2 i p p 74 .
Schro dinger , k p i k p , v q k p q 0 75 , 75 0 . 2 dk k p q k p , 0 p p P p i 76 , 2 3 k 2 i , , , 2 dk k p k p dq. p i p i , v q p 2 i , 2 3 k 2 i 2 3, dk k p q k p . 2 3 k 2 i , , 2 2 dk k p k p , p i p i , 2 3 k 2 i . , dq dk k p q k p , v q , 2 3 2 3 k 2 i , , 2 2 dk k p k p .
p i p i , 2 3 k 2 i , , dk k 2 p2 i , k p k p , 2 3 k 2 i . , 2 dk p2 i k 2 p2 i , p i k p k p , 2 3 k 2 i , 2 dk. p i k p k p 2 3 p 2 i p p 77 , , k p 2 3 p k 78 , k 2 p2 i . 76 , , dk p f p k , 0 p p P p 2 i 2 3 p k 2, 2 3 k 2 i k p 2 i . 2 dk k p f p k , p p p i , 2 3 k 2 i k 2 p 2 i , .
2 dk 1 1 f p k , p p p i k p 2, 2 3 k i k p i p 2. 2 2, , dk 1 1, p p k p f p k 79 , 2 3 k 2 i k 2 p 2 i . 0 72 , dq, 0 p p P U0 q 0 p q p P , 3, v q 0 p q p P 80 . A L Fetter and J D Walecka Quantum Theory of Many Particle Systems 2011 9 15 4 9 12 Fetter and Walecka 1

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