Теории / Садовский М.В. Диаграмматика (2005)

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Кузбасский государственный технический университет

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Физика

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k <k0 ω − dm M(ω)

M (ω) = 2iγ 1 +

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O P > A H H C 2 2 H D / , ,

3 % " !

q ($)?)+ 6 =

pˆq

(pˆq)2

(pˆq)

Φpp (qω)

−

Φpp (qω) −

Upp ∆Gp Φpp (qω) =

1 p2 2E

pˆq ∆Gp

−

pˆq

ΦpRAp (qω)

∆Gp Upp

($)@?+

2πi

4 K

pˆq

Φpp (qω) +

∆Gp Upp Φp p (qω) =

(I)

pˆq

Upp ∆Gp ΦppRA (qω) +q

ΦppRA (qω)

(II)

A (I) (II) 2 ($)@)+ =

(I) =

pˆq

∆GpUpp ΦppRA

pp m

pˆq

∆Gp

= −

∆GpUpp

1 + d(p qˆ)(p qˆ)/pF

Φp p

2πiν(E)

p p

pˆq

ΦpRA p +

∆GpUpp ∆Gp

2πν(E)

p p

(III)

(pˆq)∆G

∆G

(p qˆ)

p qˆ

ΦRA

2πν(E)pF2

p p

pˆq

(II) =

Upp ∆Gp ΦppRA =

−

pˆq

∆G

∆Gp

1 + d(pˆq)(p qˆ)/p2

ΦRA

p 2πiν(E)

p p

(pˆq)2

p qˆ

= (III) +

∆Gp Upp ∆Gp

Φp p =

2πν(E) p2

p p

1/d

p qˆ

= (III) −

= (III) + Upp ∆Gp

Φp p

2iγ

Φpp

p p

($)@@+

($)@B+

($)@H+

" q ($)?)+

3 % " ($7?+

ΣA − ΣR = 2iImΣA = 2iγ 6 ($)@B+ ($)@H+ =

(I)

−

(II) = 2iγ

pˆq

ΦRA

(pˆq)∆GpU

∆G

(p qˆ)

pˆq

ΦRA

2πν(E)pF2

m pp

pˆq

= M (qω)

Φpp ($)@,+

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

M (qω) ($)@ + > ($)@@+ ($)@7+ K

>1 % 1 - 0 I !

3 %

($7?+ ($)@H+ M !

( ($)@ ++ 1 =

M (qω) =	1		∆Gp ΣR(E + ωp+) − ΣA(Ep−)	+	id	(pˆq)∆GpUpp ∆Gp (p qˆ)
M (qω) =	2πiν(E)		∆Gp ΣR(E + ωp+) − ΣA(Ep−)	+	2πν(E)p2	(pˆq)∆GpUpp ∆Gp (p qˆ)
		p			F pp
						($)B*+

/ 8

Upp 8

= γ E 6 1 8

% " "

. # !* * J&, I J&, !W
ΦRA(qω) = −N (E)	ω + M (qω)

	ω2 + ωM (qω) − d1 vF2 q2	J&,Q,

H # * J&, ! * ! %* ) # $

# *) ! W

1 v2 q2

d F

χ(qω) = ωΦ

(qω) + N (EF ) = vF qΦ1

(qω) = −N (E)

ω2 + ωM (qω) − d1 vF2 q2

J&,Q

G ! ω ω2 ! J&,Q,

J&,Q ! ! #W

ΦRA(qω) = −N (E)

ω + iDE (qω)q2

J&,QI

χ(qω) = N (E)

iDE (qω)q2

ω + iDE (qω)q2

J&,QJ

) 3 5- 8.. ../W

J&,Q

DE (qω) = i dm M (qω) = d M (qω)

) &

H # * J&, ! * !W


σ(ω) =		ne2				i	→ e2DE (00)N (E) ω → 0	J&,Q
			m		ω + M (0ω)
*	n			= 2E		n $ # ( &
	N (E)
					d

5 ! ! ! ! !

) J&, I & # ! !

! #] X Upp (qω) J&, I #

J&, Q J&, I (

] ! ! W

	i	!		p 2	"
M (00) =		= 2πiρ δ E			v(p	p ) 2(1	pˆ )	J&,QQ
	τtr		− 2m		\|	− \|	−
		p

> ∆Gp γ → 0

O P > A H H C 2 2 H D / , I

$ ! τtr

! ! ) * !
! * E = EF pˆ = cos θpp	J&,Q			% !*
6 * W		ne2
σ =		ne2	τtr
σ =		m	τtr		J&,QL
		m
ω = 0	ne2
σ(ω) =	ne2		τtr		J&,QO
σ(ω) =					J&,QO
	m 1 − iωτtr

&
#! ! # Upp (qω) * J&,JQ & 5
%%* ) !
M (0ω) ω → 0 # * ω → 0W
	kd−1	1		d = 1
		√
			ω
dk	ω + iD0k2	ln ω		d = 2	J&,L
1 % 2 ! # d ≤ 2 & G ( !* *

! * ! ! ω → 0 ! ! * # *

1 % !2 ω → 0 M (0ω) #

! ! : & 4 #

* ! &

G J&,JQ

J&, I

! ! q → 0, ω → 0 W

M (0ω) = 2iγ

2id

(pˆq)ImGR(Ep)Upp (0ω)ImGR(Ep )(p qˆ) =

− πν(E)pF2

4dU0γ

= 2iγ + πν(E)pF2

J&,L,

(pˆq)ImGR(Ep) ω + iD0(p + p )2 ImGR(Ep )(p qˆ)

' ! # ! * ! W k = p + p ; p = k

−

p& 5 J&,L,

4dU0γ

M (0ω) = 2iγ +

(pˆq)ImGR(Ep)

ImGR(Ep − k)(kˆq) −

πν(E)p2

ω + iD0k2

4dU0γ

1 p2

−πν(E)p2

(pˆq)2 ImGR(Ep) ω + iD0k2 ImGR(Ep − k)

*d+,F -

J&,L

/ ! ! # ! # ω → 0 ! #

1 ! * 2 2iγ # ! * ω → 0

# *

k → 0 J&,L & !W

			1
M (0ω) ≈ 2iγ − 4U02	(ImGR(Ep))2
M (0ω) ≈ 2iγ − 4U02	(ImGR(Ep))2	k	ω + iD0k2	J&,LI
p		k

/ d ≥ 2 ($)H*+ ! " k l−1 vF−1γ (

'' ( ($) )++

, J	. O - >JK<= = E <K 1<U == V 0 J /V
5 # * ! # * ! ! W					= 2γ ν(E)
p	(ImGR(Ep))2 ≈ ν(E) −∞ dξ [(E			ξ)2 + γ2]2	= 2γ ν(E)		J&,LJ
		∞		γ2		π
			−
W
	2U0	(ImGR(Ep))2 = πU0ν(E)γ−1			= 1		J&,L

' J&,LI W

M (0ω) = 2iγ − 2U0

ω + iD0k2

. ! ! # * *! ! (d = 2)&

I =

Ω2

I =

dkk

ω + iD0k

2π 0

ω + iD0k

k ω + iD0k

(2π)

Λ2

+ i D0

4π

ω + iD0x

4πiD0

x +

4πiD0 0

x2 +

ω2

iD0

> =

Λ2

ReI =

→ 0

ω → 0

4πD0 D0 0

x2 +

ω2

D02

" 2 =

Λ2

ImI = −

4πD0

ω2

+ D02

3 ' " ω → 0=

Λ4

ImI = −

= −

ln 1 +

8πD0

z +

ω2

8πD0

ω2

D02

2 ω → 0 =

D02Λ4

ImI ≈ −

= −

D0Λ2

8πD0

ω2

4πD0

= −

−

4πD0

ωτ

2πEτ

ωτ

Λ =

D0Λ2

= 2γ

2m 1

= 2

γ l−1

l = vF τ

vF Λ

= 2γ

D0 =

Eτ

d = 2 2

−2U0I ≈ imU0

2πEτ

ωτ

I ! % K

J&,L

($)HB+

($)HH+

($)H,+

($),*+

($),)+

($),7+

($), +

($),$+

($),?+

O P > A H H C 2 2 H D /								,
q ! d = 2 ν(E) = m		mU0 =			m		= 1
					2πν(E)τ
2π							τ
J&,L W
M (0ω) =	i	+	i		1	ln	1	J&,O

	τ		τ	2πEτ			ωτ

H # * ( J&,Q ! (%% %%* *! ! W

D(ω) = D0

i 1

≈

M (ω)

1 +

2πEτ

≈ D0

ωτ

J&,OQ

1 − 2πEτ ln ωτ

! * * )

! # * 2πEτ 1& ! J&,OQ * %%*

! *! ! ( ! ! R&G&8 # :&H&R 6&X&U! # ,OQO & G # * ( !

! * # # *! # )

(%% %%* ) ! 1 * !2 !

D0 1 2 &

! # T = 0& G ! *

* * * ( !

1 % 2 %* ! ! ! τφ = AT −p

# p * * & G ( !* J&,OQ !

D(ω) = D0 1

J&,OL

−

2πEτ

M ax ω,

τφ

* ! *) ! #

= 0

τφ = σ0

σ = σ0

1 − 2πEτ ln

1 − 2πEτ ln T0

J&,OO

T0 $ ! ! * ( & H # *

= 2π

= E

N (E)

ne2

d = 2 * ! σ0

τ =

Eτ J&,OO

2π

σ = 2π Eτ 1 −

2πEτ ln

τφ

/ ! !

= e2

e2 =

2π

h ! ! W

2.5 · 10−4!−1&

R %! ! * ! # ! J&,OO J&

# ) ( ! # ! *

) *! ! ! *!

( 3SG $ * * & & +,L ,O-&

M ' ($),B+ ω → 0 ' " 8

" 8 ! T = 0 ( " 3 %

" + A % " "

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

. & J& ,W N ! ) A

* B [ ! ! ! e &

>-> -. 4 . @ > 2-. =, 4 4 2:

A ' " " ( 9 ;

/ M v 1 ),H*+ o ( K+

l p ( = 1+ pF l 1 Eτ 1 2 -0

! - 1 0=

σ = σ0 + δσ; |δσ| σ0 ($7*)+

& ( +

σ0 e22 Eτ > % ! 8

( + % % ( + ''

1 " ! t = 0 r0

t > 0 r '' d 3 =

e−

|r−r0|2

P (rt) =

4D0t

(4πD0t)d/2

($7*7+

%'' 1 '' D0 = v2

τ /d !

" Vdif f |r − r0|2 4D0t =

P (rt)

Vdif f

(D0t)d/2

($7* +

> % A ' 8

" 1 7*

! A B %

A $7) (C+ L -0 !

( K+=

λF =

($7*$+

mvF

% -0 λd−1

→

0 λ = 0

λF /l = pF l 1 -0

1 " τφ =

τφ τ

O P > A H H C 2 2 H D /			, Q
> " A → B < =
2
W = Ai!	= \|Ai\|2 + AiAj		($7*?+
! i	i	i=j

Ai 3 " A → B i8 > 1 8

' 1 ( K+ ($7*?+ >

% J ( + Ai

( K+ ' & A B

A $7) (D+ 6

" " " " 1 2 1 2

' A1 A2 ($7*?+ K

2 ! " A1 = A2 = A Wcl = 2|A|2 ( ($7*?++ ! ' 1

( ($7*?++ =

Wqm = 2 A	2 + A A	2	+ A	A = 4 A		2	($7*@+
\| \|	1	2	1	2	\| \|

	Wqm = 2Wcl						($7*B+

6 1

J '' -0 σ

3 1 1 K

&1 δσ/σ0 9 " δσ/σ0 1 8

1 ( " '' 8

K+ A (-0 λdF−1 + d 3 8

4 dt % " dl = vF dt !

-0 dV = vF dtλd−1

F 2 '' 8

! 1 Vdif f ($7* + 6 %

1 % " " =

τin	dV	τφ	dt
W =		= vF λd−1
W =	Vdif f	= vF λd−1	(D0t)d/2
τ	Vdif f	F τ	(D0t)d/2	($7*H+

I '' 8

" 3

1 > σ0 e22 Eτ

1 =

" τφ

1/2

d = 1

δσ

$# ln

d = 2

($7*,+

σ0

Eτ

τφ

−1/2

M τφ T

d = 3

($7*,+ =

δσ

T −p/2

d = 1

($7)*+

Eτ

%$" p ln T

d = 2

T p/2

d = 3

9 A1 = A2 % % 8

p −p ! t → −t

" > H %

" ' =

A1 → A1eiϕ		A2 → A2e−iϕ		($7))+
			2πφ
ϕ =	e	dlA =		($7)7+
	c &		hc/e

A 1 " (DCcr_cC^^[Y\Ua+

1 " >

" J %" K

& " 8

! " " ! " 8

! " . wxxx 7)

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

φ = HS 3 S ( +

% '' S D0t φ HD0t >

($7*@+ =

WH = 2\|A\|2 1 + cos 2π		φ		($7) +

		φ0
φ0 = ch			!	2eK q
2e 3 " ))
'	% p −p ( 8
"	%

+ H = 0 WH=0 = 4|A|2 2 8

" =

	∆σ(H) = δσ(H) − δσ(0) WH − WH=0
	τφ	dt				φ
v	λd−1		1	−	cos 2π		> 0	($7)$+
F	F τ	(D0t)d/2		−		φ0

" ( 1 "+ 8

∆σ(H) ' 1	HDoτφ
∆σ(H) ' 1	φ0 -L 0 !
	φ0 -L 0 !
1 "		HcD0	τφ	1 "
1 "		φ0		1 "

D0 τφ Hc 100 ÷ 500 '' 1 " "

" " " 8

" " τφ )H ),

! * & S ! ! # *)

!* ! ) *) ! # # #

& 6 ( ! ) J&,L & S 6

- ;&}{<EqC ,OQO s&z<>>a[=?E c&;{<>|C ,OL

! (%% %%* D0 J&,L

(%% %%* J&,Q !

! J&,L & * # * ! *) /

1 M (0ω) W

J& ,

( * J&,Q

* (%% %

%* W

|k|

−

= 1 +

DE (ω)

πν(E)

iω + DE (ω)k2

J& ,

! * # k0 * * # &

%''

∆σ(H)

−(ωH τ )2

ωH = mceH > 8

σ0

! " " ($7)$+ K

4 % = M (qω) →

(0ω)		DE (qω) → DE (0ω)
M
	A " % -8

0 " " -"0 > !

($7)?+ ($7)@+ (g wVhh]CYZ^ F uyVhz[ ),H7+ 8

! " ( ω → 0+

% & "

" '

O P > A H H C 2 2 H D /

, O

G ! ! # # ! ! J& , J& , !

! ! # 1 2& 6 ( ! )

J&,QI ΦRA(qω)& 3 ! M (ω) ! # * !

ω → 0& * * W

2	2E	lim	1
2		lim
Rloc(E) = −		ω 0		J& ,Q
Rloc(E) = −	dm	ω 0	ωM (ω)	J& ,Q
		→

) *) * Rloc J&,QI *

*! q → 0 W

ΦRA(qω) = −

N (E)

1 + R2

J& ,L

loc

( * * *) * ω0(E)W

lim ωM (ω) =

J& ,O

(E) = − ω 0

> 0

ω0

dm R2

(E)

→

loc

Rloc(E) =

dm ω0(E)

6 Rloc ! ! / ( *

! +,-& * * *

( * N ! E ) !

! 8& G :

( * !

* ! * ! +-& 4 ! * ! ! ! * ( &

4 ( :

! * ! J& ,Q & & #)

ω0(E) ! J& ,O & H ! ! ( *

							2
) ! W ReM (0ω) = −								ω0 (E)	ω → 0&
) ! W ReM (0ω) = −								ω	ω → 0&
q → 0 ω → 0 *) !
W	M (0ω) =			i		ω02(E)	( :			J& ,
				i		!
				τE		!
				τE
					−
				τE	−	ω
# ! ! $ * J& , \
G * ! J& , J& ,									! !
J&, I		* ]								∆Gp
J&, I		2								∆Gp
! #		0 2pF & ' * ! Upp (qω)
ImGR(Ep) δ	E −	2pm	!! ! # k = p + p Upp (qω) !

! * )

J&,JQ 1 * 2 # * ! ! # |p + p | ≤ l−1

! ! %%* & S ) ! * #

J& , J& , * ! # W

k0 M in{pF , l−1} J&

I ΦRA(qω) ! 1 %

1 " % )@

√k

2mE

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

N : pF l 1 !

H %% $ . +- #W

k0 = x0pF = x0	√
		2mE	J& I

x0 = const 1 ÷ 2&

G ! # ) * ! J& , :& &3 3& &' ,OL & ! ! *) ! *) y = x0

! J& , W

y2 − 4x02E2 M (ω)

M (ω) = 2iγ + dλxd−2M (ω)

dydd−1

dω

J& J

λ ^ ! !* * W

λ =

d/2

E d2 −2

ρv2

2πEτ

πE

2π

* γ = πρv2ν(E) (

! dW

. 6 2 > A :

G J& J ω = 0 ! ! ! J& ,

ReM (ω = 0) = 0 ImM (ω = 0) = 1/τE * * ! J& J W

τE = 2γ	1 − d − 2 λx0d−2			J& Q
1		d

5 J&,Q ! * ! *) ! # ! W

σ = m 2γ 1 − Ec

4−d

;

2 < d < 4

J& L

ne2 1

Ec = d d

2 Γ 0 d

(2π)− 2

Esc

J& O

xd−2

4−d

( W−

Esc = m

(ρv2)

4−d

J& I

* ! J&QJ ) # 1 # 2 #

( ( * ! ! )

γ ρv2ν(E) E * ! J& *)

) H %% $ . * ! ! # * ! !

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