Теории / Садовский М.В. Диаграмматика (2005)

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Кузбасский государственный технический университет

Предмет:

Физика

Файл:

exp(−ipRl) = ρ dRe−ipR

= (2π)3ρδ(p)

exp[−i(p1 + p2)Rl] +

exp[−(p1Rl + p2Rm)]

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F C 2 C C

. & I&, W ( $ % ! &

. & I&,,W M 6 % %* 8 !

L	. F - >JK<= L G<=<==< M/0 <1 N J 0
5 I&L I&LQ	I&LL	! * !W
				Ω2
D(qλ, ω) =				qλ				I&LO
D(qλ, ω) =			g2(qλ)Ωqλ2		1 −	g2(qλ)		I&LO
		ω2 −		− Ωqλ2			+ iδ
		ω2 −	Vq e(q0)	− Ωqλ2		Vq	+ iδ

( ! ! ! #) e(qω) # *

! ω % ( ! # * &

! # I&L I&L

W		(qλ)
	g2	(qλ)		= 1			I&O
	Vq			= 1			I&O
	Vq
( ! * I&LO * W
					Ω2
D(qλ, ω) =					qλ		I&O,
D(qλ, ω) =					Ω2		I&O,
			ω2	−	qλ	+ iδ
			ω2	−	e(q0)	+ iδ

G ) ( ) ! % s&v<aw x&@E[yC= ,O W

ω2(qλ) =	Ωqλ2		Ωqλ2		=	mZ		I&O
		≈					vF2 q2
	e(q0)		1 +	κD2		3M
				q2

* & L & O # ( * # Zn

( # & ! ! % !

! ) * #) * c = mZ 1/2 vF &

( ! ! * # ! ! ! # *

Vei(q) * # * ! q * ( # * Vei(q) Ze2/q2&

! ! ! ! # (%% ! ( ! * !

! & ! !! !

! . & I&, W

Vef f (qω) =

4πe2

(qλ)

Ωqλ2

q2 e(qω)

(q0) ω

− ω2(qλ)

I&OI

ω2(qλ) $ ! % *) I&LQ W

ω2(qλ) = Ωqλ2

1 −

e(q0)

I&OJ

g2(qλ)

! Vef f (qω) W

Vef f (qω) =

4πe2

q2 ef f (qω)

I&O

ef f (qω) $ 1 ( ! #W

Ω2

ef f (qω) = e(qω) −

qλ

ω2

I&O

F P G A B @

. & I&, W %% ! ( ! &

! * ! ! # ! ( !

* I&O ef f (qω) ! W

ef f (qω) = e(qω)		g2(qλ)	Ωqλ2			I&OQ
ef f (qω) = e(qω)
	− Vq		ω2 − Ωqλ2	1 −	Vq
	− Vq		ω2 − Ωqλ2	1 −	Vq
					g2(qλ)

M * ) ω2(qλ) > 0 I&OJ

* !W

1 +	g2(qλ)	1 − e(q0)	> 0	I&OL
	Vq	e(q0)

H ( % !* ! # (%% ! !! # 1 23 q 2pF !

% ! +,J-& . *! 5 !

* ! # ! bcd&

> ( ,$+ ' ! =

ω2(qλ) = Ωqλ2 1 + g2(qλ)χ(q, ω(qλ))

( ,,+

! % ! % 1 =

χe(qω) =	1	1	− 1	( )**+

	Vq e(qω)

! 1 ( ,,+ I 8

' χe(qω) 8

' " Mm

& -0 " '

%''

% 1 % 6 1

" " " " ( " d = 2, 3+ 1 e(qω) T = 0

' ! q = 2pF > % '

' 3 %'' q = 2pF !

. ! ! * +I- *) %* ) ! ! W


CT (r) =	< uα(r)uβ (0) >	I&, ,
	αβ

* ! ! * # r → ∞& * I&, , !

# *) *) ! * *) %* ) 8 % & G

LJ	. F - >JK<= L G<=<==< M/0 <1 N J 0

( ! * * # % %* 8 +,- %

! * *) #

! # ( $ % ! & N* 8 ! !

* ! ( %* 8 * ρωk2 ρ $ #

! * ! & 5 ! !W

CT (r) = ρ

(2π)d ωm2 + ωk2

I&,

ddk

eikr

ωk

$ % & '*!! ! ! * ! !

I&,

! #) W

∞

= π cthπa

I&, I

m=−∞

m2 + a2

* # * !W

(2π)d ωk cth

2T eikr

CT (r) = 2ρ

ddk

ωk

I&, J

S ) ! # # * (T = 0) % * * ! #) % !* W

nB (ω) = ω1

e T −1

1	cth	ω	=	1	+ nB (ω)	I&,

2		2T		2

$ %* T & S nB → 0 T → 0

*) % * * &

H ! ! W

C0(r) = CT =0(r) = 2ρ

(2π)d ωk

I&,

ddk eikr

∆C(r, T ) = ρ

(2π)d

Bωk k

eikr

I&, Q

ddk n

(ω

)

4 * (

→ ∞& G ( !* ∆C(r, T ) )

# ! ! k r ωk

T & G ( !* $ %* )

! ! # nB (ωk) ≈

ωk

≈ ρ

(2π)d

ωk2

∆C(r, T )

ddk

eikr

I&, L

( % !* ! !

+,,- &

6 # ! ! & '

! ! C0(r)&

> % ' (B ,+ (B )*+ (B ) + )

4 % ' 1 ' 1 .

3 ! 1

F P G A B @ L

5 I&,W : ! (r → ∞) %*

% * * ! ! ! &


d	C0(r)		∆CT (r)
3	1			T
	1			L
		r2		r
2	rL		T ln r
1	rL		T ln r
1	ln r		T L

6 d = 3 ! !W	(r) = (2π)3		ρ 0			4π2ρcr2
C0	(r) = (2π)3		ρ 0	ck	kr	4π2ρcr2	I&, O
(3)		4π		∞ dkk2	sin kr	1

% ! # !

k 1r # * # k #

* * & 6 d = 2W

C0 (r) = 4πρc 0

2π 0

dkeikr cos θ =

4πρc

dkJ0

(kr) = 4πρcr

(2)

2π dθ

∞

J0(r) $ *) %* T &

4 d = 1 ! !W

2πρc 0

2πρc ln r

(r) = 4πρc −∞ |k|eikr =

cos kr =

(1)

∞ dk

G I&, O

%! ! !

! k

L $ ! ! &

. ! ! # % * * &

6 d = 3 ! !W

2π2ρc2r 0

sin kr = 4πρc2r

∆CT (r) =

(3)

∞ dk

6 d = 2W

∆CT (r) =

(2π)2ρc2

dθ

k eikr cos θ

(2)

2π

∞ dk

I&,,

I&,,,

k 1r

I&,,

I&,,I

* # !* (

# C1(r)& * !W

	∆C(2)(r) =				T	ln	L	I&,,J
	∆C(2)(r) =			2πρc2		ln		I&,,J
		T					r
							r
6 d = 1 ! !W			0		k2 cos kr = ConstL
∆CT	(r) = πρc2		0		k2 cos kr = ConstL			I&,,
(1)		T	∞ dk

L	. F - >JK<= L G<=<==< M/0 <1 N J 0

( ! * ! k ! !

(%% ! ! $

* ! ) !

) !&

H 5 & * #

) * # * # ! *

] # ! ! % * * ! ! &

! ! ( ! # ! C(r)

r → ∞& X ( ! C(r) → 0 #

% * * ! * $ # u(0)

# !* ! ) u(r)& 4 C(r) → ∞ (

# & 5 * ! !

% * * d = 1 ^ d = 1, 2]

& 8 #

. ! ! ( * ! ! Ni

! 1 ! ! 2 * ! !

% #) ρi = Ni V $ 7 ! ! &

'*!! * ] ! ! ! #W

	Ni
	j
V (r) =	v(r − Rj )	J&,
	=1

v(r − Rj ) $ # * ]

Rj & : ) *

* %* W

P{Rj} = V −Ni

6 % * %* 8 ( ** )W

∂

v(r

Rj ) G(rr t

) = δ(r

r )δ(t)

J&I

∂t

− j=1

−

{

}

%* # Rj & S * !

+,- % ! ! ( !

) ! ) ! ! ! !

1 ! 2 * ! ! & 5 ! ! *)

* %* 8 W

G(r − r , t) =< G(rr t) >= V Ni	...	Ni	J&J
		j=1 dRj G(rr t{Rj })
1

LL	. O - >JK<= = E <K 1<U == V 0 J /V

. & J&,W . ( ! ! % %

* &

' ! ! # ) !*

$ ! # ! ( !

J&, W


	Hint =	drψ+(r)V (r)ψ(r)		J&
5 !* 1 !*2 ) +,-			# & ' *)
!* %* 8 J&I			! W
G(1, 1 ) = G0(1, 1 ) +		d2G0(1, 2)V (2)G0(2, 1 ) +
+	d2d3G0(1, 2)V (2)G0(2, 3)V (3)G0(3, 1 ) + ...			J&

1 = (r, t) 1 = (r , t ) & & 8 % (

!! ! ! . & J&,& 4 * * %* 8 < G(rr t) > J&J & ' * J&

J& ) W

< V (2) >,

< V (2)V (3) >,

< V (2)V (3)V (4) >, ...

J&Q

6 * ! * J&,

J& ( ! # ! & 6 (

! %* # $ W

V (r) = v(p)eip(r−Rj )

J&L

v(p) ^ %* # $ # v(−p) = v (p)& 6

* ! # # * ( !

v(p) = v = const& # &

' * ! J&L J&Q )

Ms(p1, p2, ..., ps) =< ρ(p1)ρ(p2)...ρ(ps) >≡ l1	l2	... ls	exp(−i j	pj Rlj )	J&O
M ! ) # # 1
2] ! & G* # Ni		!
* ! ! N * ! * *

& 5 ! ! ρi ! !

*) ) ! ρ =	Ni	0
	N ! ! #

O 9 1 CC 2 D 2 2 2 WH C 2DX LO

1& 5 * # * # *!! ! !

	... →	i	... = ρ	l	... = ρi	dRl...	J&,
	... →	N	... = ρ	a3	... = ρi	dRl...	J&,
li		N	l	dR

!! ! * ! a
) & G ( ! ! !
# ρi			= Ni =		Ni	= ρa−3& G !
					3
			V		N a
1 2 ! * a → 0 %
ρ *	*	ρi	→ ∞	& ! % #			ρi
	3

a → 0 ρ = ρia → 0& X ( !

# 7 ! ! V = 1 N = a−3

) & G ( !* # ! ! # * !

ρ&

G *) # * #

# * *!! !

J&O & G ! ! W

=(2π)3ρδ(p1 + p2) + ρ2[(2π)3δ(p1)(2π)3δ(p2) − (2π)3δ(p1 + p2)] =

=(2π)6ρ2δ(p1)δ(p2) + (2π)3(ρ − ρ2)δ(p1 + p2) ≡

≡< ρ(p1) >c< ρ(p2) >c + < ρ(p1)ρ(p2)) >c J&,

) / / 1 5 < ... >c & : ! ! * !W

M3(p1, p2, p3) =< ρ(p1) >c< ρ(p2) >c< ρ(p3) >c + < ρ(p1) >c< ρ(p2)ρ(p3) >c + + < ρ(p2) >c< ρ(p1)ρ(p3) >c + < ρ(p3) >c< ρ(p1)ρ(p2) >c + < ρ(p1)ρ(p2)ρ(p3) >c

J&,J

* J& ) *) ( !

* !* W

v < ρ(p1) >c= (2π)3ρvδ(p1) (a)	J&,
v2 < ρ(p1)ρ(p2) >c= (2π)3(ρ − ρ2)v2δ(p1 + p2) (b)	J&,
v3 < ρ(p1)ρ(p2)ρ(p3) >c= (2π)3v3(ρ − 3ρ2 + 2ρ3)δ(p1 + p2 + p3) (c)	J&,Q
v4 < ρ(p1)ρ(p2)ρ(p3)ρ(p4) >c= (2π)3v4(ρ−7ρ2 + 12ρ3 −6ρ4)δ(p1 + p2 + p3 + p4)	(d)
	J&,L

! ) !! . & J& [ ? & K*!*

! ) ! &

< =

exp αj ρ(pj ) = exp exp αj ρ(pj ) − 1

($) +

j	j	c
		c

O	. O - >JK<= = E <K 1<U == V 0 J /V

. & J& W 8 % ! ! *!* ! * !

!* &

& : %

H * * !* !!

( %* 8 * * ! % * !

! ! . & J&I& H

! ! ! ) 1 * 2 7 ! 1

! 2 1 2 !! @&k&o?`[=?r ,O L &

G ! (ρ → 0) ! 1 2 ! !

! ! # # ! ! ρ& 5 * !

! ! (v → 0) ! # # ! J&,

J&, % ! *!* !

! . & J& [ e & G ( ! J&,

* ( ( * ρv

*) ! ! & *!* . & J& e

( ! (2π)3ρv2δ(p1 +p2)& 5 * %* 8 *!! % . & J&J& !* * )

* . 1 * J&Q 1 *2W

< V (1)V (2) >= 0 < V (1) >= 0 < V (1)V (2)V (3) >= 0

$ & &

< V (1)V (2)V (3)V (4) >=< V (1)V (2) >< V (3)V (4) > + < V (1)V (4) >< V (2)V (3) >

$ & & &

J&,O

% ($B+ < V (2) >= 0 %

> 3 % % 8

A $7 (C+ Σ = ρv(0) = ρ drv(r) ' 1

. = G(εnp) =	1
	iεn −ξ(p)−ρv(0)

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