Добавил:

fench Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Кузбасский государственный технический университет

Предмет:

Физика

Файл:

Теории / Садовский М.В. Диаграмматика (2005)

.pdf

Скачиваний:

Добавлен:

15.08.2013

Размер:

5.49 Mб

Скачать

☆

<<< < Предыдущая 5 6 7 8 9 10 11 12 13 14 15 16 17 1819 / 2919 20 21 22 23 24 25 26 27 28 29 > Следующая >>>

O K . L -

,L,

(Rloc 3 1 ($7),++=

C =

DE (2|εn|)

T νF

ε2

εn

νF DE

2πT (2n + 1)2

+ (2n + 1)3DE ω2

/2πT v2

νF v2

3DE ω2

- − ψ

+ψ ,

3ω02

4πT vF2

= νF Rloc +ψ ,

- − ψ

4πT Rloc2

1.78DE

≈ νF Rloc ln

(? 7*,+

πT R2

loc

DE R−loc2 4πT

> % 8

( (? 7*@++=

DE (ωm) ≈ D0(ωmτ )1/3 ≈ (D0l)2/3ωm1/3

C =

T νF

DE (2|εn|) (D0l)2/3T νF

|εn|

5/3

εn

D0l

2/3

νF

(2n + 1)5/3

n≥0

D0l

2/3

νF

(? 7)*+

(? 7))+

> (? 7))+ (? 7*B+

DE /Tc ≈ D0l/ξlocTc ≤ D02/3(l/Tc )2/3

(? 7)7+

> ! %'' 1 C 8

% (; I 5 o > 2 ),H$+=

2 1/3

ξloc(E) < (ξ0l )

E > Ec

8Tc

C ≡ νF ξ

≈ νF $#"

D0l

2/3

2 2/3

2 1/3

E Ec

≈ (ξ0l )

ξloc(E) > (ξ0l )

(? 7) +

1.78DE

2 1/3

Rloc

(E)ln

Rloc(E) < (ξ0l )

E < Ec

πTcRloc2 (E)

ξ0 =%

0.18vF /Tc l 3

>' % < 8

Ec 1 1 ($7$H+ ξloc %'' 1 C

! %'' 1 '' DE

& "

σ → 0 %'' 1 C !

E < Ec % 2 E 8

" ( Ec + %'' 1 C

1 Rloc E 1

L .; %'' 1 C " % 8

" " ! ( + 2 1

T < Tc % " !

(; I 5 o > 2 ),H$+ A 8

.; 5L

1 %

" >

Tc (-0 + >

" 8

> " %''

,L	. KV K< <10 < J
%	! " )@ %

" " " " Tc 8

" 6 "

( " Tc + " 8

" Tc " (

% ' + )@ 9 "

' 1 L " " 1

% " " /

% " )@

q (? 7) + ! %'' 1 8

'' ( ($)B@++ ' ($7$+ ($7 B+ 6

" ξ (? 7) + !

! ξ(T ) 7) " =

ξ2(T ) =	Tc	ξ0l	σ	σ > σ	(E > Ec)
			σ+σc			(? 7)$+
				σ < σ	(E Ec)
	Tc − T (ξ0l2)2/3

($7$+ σc = e2pF /(π3 2) " σ 8

=			1/3
σ ≈ σc(pF ξ0)−1/3		Tc
	≈ σc		(? 7)?+
		E

6 " σ < σ
ξ(T )	ξ	√
			ξ0l	-"0 "
					2 1/3	2 1/3 " 8
" ξ (ξ0l					)	(ξ0/pF )

" " " ! "

(; I 5 o > 2 ),H$+

" ! " % ns .; 7) =

ns(T ) = 8mC∆2(T ) = 8mC(−A)/B	(? 7)@+
> " % 1 =
ns mN (E)ξ2∆2 mpF (ξ0/pF2 )2/3 ∆2 n(Tc1/2/EF2 )2/3(Tc − T )	(? 7)B+

n p3F 3 % M T 0.5 Tc

1 =

4/3
ns n	Tc	(? 7)H+
	EF
T = 0 &

% " " ! 6

% " 8

" 3 % > σ (? 7)?+ " 8

%'' 1 " ! 8

M σc - 0 o 77 σ

! & " (

" " " " +

σ σc %

( +

! -"0 "

' 1 ξ2(T )

" (? )B7+ > (? 7) +

(? 7)$+

(? 7*$+

(; I 5

o > 2 ),H$+=

dHc2

φ0

σ > σ

−

φ0

σ < σ

(? 7),+

≈ φ0

≈

νF

νF (ξ0l2)2/3Tc

[νF Tc ]1/3

> σ < σ " 3 % 8

(? 7*$+ (

νF Tc + (dHc2/dT )Tc ( Hc2(T )+

" -! 0 % !

% L

" 1 " "8

C A DH

,LI

( (

( ! ! ! ! ! *) )

( ! #) # #

! # ! & 4 ! +-& . ! !

7 ! ! ! ( ! !

! ! ! ! ! A(rt) ϕ(rt) ! *) !

J (x = (r, t))W

Aµ(x) =	cϕ(x)		µ = 0	&
	Ai(x)		µ = i	= 1, 2, 3

! 1 2 # c& ! Aµ !

( ( ! ! ! *) ! !W

	Hp = − c d3rjµp(x)Aµ(x) = − c d3r[jp(x)A(x) − ρe(x)cϕ(x)]						& ,
		1	1
! # ! 1 ! !2 J !
! ) ! W					µ = 0
µ		ρe(x) = −eψ{+(x)ψ(x) = −ρ(x),					&
jp	(x) =	jp(x) =	e	i ψ+(x) ψ(x) − [ ψ+(x)]ψ(x)},		µ = i = 1, 2, 3
			2m

G # jµ(x) * $ A ( !

! * ! *!! ! 1 !

2 W				jµ(x) = jp		(x) + jd	(x)	& I
				jµ(x) = jp		(x) + jd	(x)	& I
					µ	µ
# ! !W
d	(x) =		e	ρe(x)A(x)				µ = i = 1, 2, 3
d			mc	ρe(x)A(x)				µ = i = 1, 2, 3
jµ		0			µ = 0			& J

' ! ( ! ( ! !

! ! W
Hint = Hp + Hd				&
! # ! !W
Hd = −2mc2		d3rρe(x)A2	(x)	&
	e

5 # ! ! ! # # ! ! Hint

Aµ → 0 t → −∞ *) !

! ! ! +,-W

\|Φ(t) >= T exp −i −∞ dt Hint(t ) \|0 >≡ U (t, −∞)\|0	>	& Q
t
5 \|Φ(t) > !W
Jµ(x) =< Φ(t)\|jµ(rt)\|Φ(t) >=< 0\|U +(t, −∞)jµ(rt)U (t, −∞)\|0	>	& L

& I

)

,LJ . KV K< <10 < J

4 *) Jµ(x) Aµ !

! * !W		(rt),	−∞ dt Hint(t )]\|0 > & O
Jµ(x) = mc < 0\|ρe(x)\|0 > Aµ(x)[1 − δµ0] − i < 0\|[jµp
	e		∞

* Jµ(x) * ) ! < j0(x) > $

* !* * & H # * & , ^ &

! ! Jµ Aµ

*) ! # ! ! ! #) # Kµν W

Jµ(x) = −4π Kµν (rt; r t )Aν (r t )d3r dt

& I

! W

(x, x ) =

4πi

< 0 [jp

(x), jp

(x )] 0 > θ(t

t )

4πe

< 0 ρ

(x) 0 > δ(x x )δ

]

− c2

−

µν

| µ

−mc2

−

µν

−

ν0

& I,

X ! Kµν #

* %* # $ )W

Kµν (q, t − t ) =

Kµν (x, x )e−iq(r−r )d3rd3 r =

4πi

< 0 [jp(qt), jp(

qt )] 0 > θ(t

−

t ) +

4πne2

δ(t

−

t )δ

−

)

& I

mc2

− c2

−

µν

ν0

n $ # ( & G # * & I !

% * # # !

! W

Rµν (qt) = −i < 0\|[jµp(qt), jνp(−q0)]\|0 > θ(t)	& II
* * ! ! #) T $ * %* 8 W
Pµν (qt) = −i < 0\|T jµp(qt)jνp(−q0)\|0 >	& IJ

' # ! %* # $ & II

& IJ +,- * * # # Rµν Pµν ! ! ) ! ω < 0W

RePµν (qω) = ReRµν (qω)

ImPµν (qω) = signωImRµν (qω)

* ! Rµν Pµν & G # * Kµν

# ! * $ Aµ

j	jp					p&
µ	µ ) ! * !
	! $
Aµ(q) = [A(q), cϕ(q)] q = [q, ω]			! W
	Jµ(q) = −4π Kµν (q)Aν (q) = −4π			0	Kµi(q)Ai(q) − Kµ0(q)cϕ(q)1		& IQ
		c	c		3
					i
					=1

' 1 . ( 1 + " ($)*H+

C A DH									,L
W				4π		1
				4π		1
		Kµν (q) =			Rµν (q) +			δµν [1 − δν0]	& IL
		Kµν (q) =		c2	Rµν (q) +		λL2	δµν [1 − δν0]	& IL
! # λ2 =	2
	mc		$ * !
	4πne	2	$ * !
L	4πne
* T = 0& 6 ! & IL						) !

! &

# ! ! * ! # # *) W

Qαβ (q) = −	c
Qαβ (q) = −	4π Kαβ (q)	& IO

. ! ! # % ! $ ! # ! +I-& 4 !

j ) * # $

$ A& ! * # !

! * j A W

ˆ	& J
jq = Q(q)Aq

* ! * ! $

) ( ! # ! ! # !

% ! $ ! & ˆ(q = 0) = 0

q = 0 ! !

) & IL & ! q → 0 ω = 0 !

) ! ! R *& . ! ! ( * * )

& ! * * ! * # * #

! * & H ! *

J&, L ^ J&, ! ! #

% ! $ ! W

αβ

− τ →0

m2c

(iεn

ξ(p))(iεn

ξ(p)) (2π)3

& J,

Qp (q = 0) =

lim 2

iεn τ

pαpβ

d3p

−

# * ! ! ! ! %*

8 W
pG02(εnp) = m pG0(εnp)					& J
! & J, W			pα pβ G0(εnp)
Qαβp (0) = −mc 2T		n	pα pβ G0(εnp)	(2π3)	& JI
	e2			d3p

H d3p ! # ! pβ pα W

	e2	n	G0(εnp)	d3p	=	ne2	& JJ
Qαβp (0) = mc δαβ 2T				(2π)3		mc δαβ

#) ! *) # & IL &

= divA = 0,			ϕ = 0

/ 8 " " ' 1 . " >

" !

,L	. KV K< <10 < J

G ! ! # * q = 0& . ˆ(q) q

W Q(q) = aq2 + O(q4) ! ! * #

!! . & &,J& 5 ! !W

ˆ	2	A	& J
j = QA = −a

' * ( ! ! #

! #) ! !W

j = c rotM = c rotχ0 B = χ0c rot rotA & J

# * # rot rotA = grad divA− 2A ! * ! divA = 0

! #W
j = −χ0c 2A	& JQ

χ0 $ ! ! # ( & 5 & J

& JQ !W χ0

= a

a ! # !! . & &,J&

c *

! ! ( !! ! !W

Qxx = − τ →0

n e

(iεn

ξ(p + q))(iεn

ξ(p)) (2π)3

& JL

lim 2

iεn τ

vx(p + q)vx(p)

d3p

−

vx(p + q) = ∂ξ(p + q)/∂px vx(p) = ∂ξ(p)/∂px

$ *)

( & G !

q z& ! #

! # # #) N ! !

! # * R *

E < EF ! χ0 $

*!! ( !* *&

S ! * ! ( ξ(p) = 2pm − EF & . ! Qxx

q = qz * ! vx(p + q) = vx(p) = px/m !W

Qxx =

lim

e2q2

∂2

d3p

−

eiεn τ

iεn

−

ξ(p) ∂pz2

iεn

−

ξ(p) (2π)3

τ →0

H! ! #W

∂2

+ 2

∂pz2

iεn − ξ(p)

(iεn − ξ(p))2

(iεn − ξ(p))3

! * !W

χ0

= − τ →0 c2 T

m (iεn

ξ(p))3 + 2

(iεn

ξ(p))4

(2π)3

lim

v2 v2

d3p

−

iεn τ

& JO

& ,

5 # * * * ! < v2								>= v2/3 <
							x
v2 v2	>= v4/15 ! # * ! ! !
x z
) ! ! ! & , W							+ EF − p2/2m
	0	(iεn + EF − p2/2m)3 dp =	3	2	0	iεn	+ EF − p2/2m	&
	∞	p4		m2		∞	dp

C A DH

,LQ

. & &,JW 6 !! ) # % ! $

$ &

! ! ! !

dp (iεn + EF − p2/2m)4 = −5

iεn + EF − p2/2m

& I

∞

5 *!! n & ,

%* ) N ! +,-

e2p

! * ! −

! & ,

4π2mc2

6π2mc2

! *!! # W

χ0 = −12π2mc2

dpn(ξ(p)) = −12π2mc2

& J

∞

e2pF

^ * # ! ! ( +,,-& X

! # * # * ! W

χ	p	= 2µ2 ν	F	=		e2	mpF
χ		= 2µ2 ν		=	4m2c2			&
		B					π2	&
							π2
* ! * # ! +,,-W
		χ0 = −			1	χp
								&
					3			&

! # ! ) & . !

$ ( * & ! # !

% ! $ ! * ! # * *

! ! %* & G ( !*

q = 0 & J ! * ! W

nse2
j(r) = − mc A(r)	& Q

& & * R ^ * ( ! &

ns ) #)

* ( & M & Q 1 #)2 * *)

# & 6 ! % *

! ! Q(0) )

! # * !

# /& H! ( !

,LL	. KV K< <10 < J

!! ! ! #

!* !* )& ! # ! ! * Q(0)

% ! M ! !

#) ( ! +-&

( 1 2 $ % !

* *) ! TK * #

! &

G ! !* * Q(0) & 6 !

#) ! # W

(2)	(0) = −	ne2
Qαβ			δαβ	& L
		mc

: ! ! # ! ** # Q(0) ! # ! # %* 8 &

! ! ! Q(1)αβ (ω, q) !*

! ) * ! * ! * *

! * *) ! ! σ, σ $

Qαβ (ωm, q) = 2c σσ

−β eiωmτ d3reiqr < Tτ ψσ+(rτ )ˆjαψσ (rτ )ψσ+ (00)ˆjβ ψσ (00) >

(1)

& O

& 5 # * #

$ *

= −i m r

! # ! # & 5 ! ! * !W

< Tτ ψσ (rτ )jαψσ (rτ )ψσ (00)jβ ψσ (00) >= −jαGσσ (rτ )jβ Gσ σ (−r, −τ ) −

−jαFσσ (rτ )jβ Fσ σ (−r, −τ )

5 Qαβ (ωmq) ! W

d3reiqr{2ˆjαG(rτ )ˆjβ G(−r, −τ ) +

Qαβ (ωm, q) = 2c σσ

−β eiωmτ

(1)

+2ˆjαF (rτ )ˆjβ F (−r, −τ )}

& ,

/ ! ! # %* G F # ! & G & , ! * # ! ! !W

Qαβ (ωm, q) = −2 c

(2π)3 vα

p − 2

vβ p +

d3p

× G εnp −

F εn

G εn + ωmp +

+ F εnp −

+ ωmp +

5 ωm = 0 q = 0 ! !W

(2π)3 p2{G2

(εnp) + |F (εnp)|2}

Qαα(1)(0) = −3m2c T

& I

2e2

d3p

> ! % -0


C A DH									,LO
S 7 ! & L			& I * ns *) & Q
%* 8 # &,,, &,, * !W
n	−	ns =		−	3m	n	−∞	(εn2 + ξ2 + ∆2)2	& J
n		ns =			2νF pF2 T		∞ dξ	∆2 + ξ2 − εn2

# * # (

! ! ! N ! & G ( !* ) !* * ξ

& J !& ! #

* ( ns ! #) N !

# * # ! (

! & * % N ! & &

ξ EF & G ( !*

*) !& ! & J ( ! # ! & & ∆ = 0& 5 * & G ( ! *

ns = 0 ∆ → 0& * # * !W

ns =

−

−∞

(ξ2

+ εn2 )2

− (εn2

+ ξ2 + ∆2)2

2νF pF2

∞ dξ

ξ2

− εn2

∆2 + ξ2 − εn2

! ξ # * # ! ! W

−∞ (x2

+ a2)2 = 2a3 ,

−∞ (x2 + a2)2

= 2a

∞

dxx2

5 * !W

ns =

2νF pF2 T

π∆2

(εn2 + ∆2)3/2

& Q

5 # ! # # * &

• ' * T → 0& / ! ! *!!* ! * !W

ns =

2π

−∞ (ε2 + ∆2)3/2 =

3π2 = n

& L

2νF pF2 π∆2

∞

dε

2νF pF2

pF2

! T = 0 * # (

# ! &

•' * T → Tc& ( ! * ! # ∆2 ! & Q & S ) *!! ! % ! * #

ζ $ %* ) W

n		4π2Tc2			− Tc
ns	=	7ζ(3)∆2	= 2	1		T	& O

T → Tc * # ! * )&

. ! *! # * *) # ns

& Z # ( ! ! Q(0) *

! ! & G ( ! # !

% ! * ! ! %*

,O	. KV K< <10 < J

8 # * ) *) %* 1 2

&,,, &,, ! #) ! &,J & !!

! ) %* 8 ! *

# # * ) * # $ * !

! ! ( ! ! ! & G ( !* % !* * ns ! * # & & J W

n		−		−∞ (ξ2			+ ∆˜ 2		+ ε˜n2 )2
				∞	ξ	2	˜	2	2
n − ns	=		T						− ε˜n	& Q
	=		T							& Q

ε˜n ∆ &,J &,J & ' ! (

∆ = 0 * ! ) # * * * % N ! & 5 ξ * !

& Q W

˜ 2

ns = πT

& Q,

(ε˜n

+ ∆ )

2 3/2

* ! &,J &,J W

∆2

n = πT

(εn + ∆ )

1 +

2 2

2τ

√εn +∆

& Q

3/2

G ! ! ! ∆0τ 1 (

& Q 1 !2 ∆0τ 1 W

ns = 2πτ T			∆2	= πτ ∆th ∆		& QI

n		(εn2	+ ∆2)2		2T
	n

5 ! ! 1 !2 * T → 0 ! !W

ns(T → 0)	= πτ ∆	0	1	& QJ
n

& & # # ( &

G ! *) * ) * # ! ! & I ^ & IL c = 1& ! !

) ( ! ! ! Kµν = Kδµν & H *

! % !* ! ! ! * !

! W

lim lim	lim lim K(qω) = K(0, 0) = 0		& QQ
q 0 ω 0 K(qω) =	ω 0 q 0
→ →	→ →
9 Kµν (qω)= Kµνp		Kµνd	/

1 %

> 3 1

( ; + %

= Kµνn (q0) = Kµνnp(q0) + Kµνd ≈ 0 Kµνnp(q0) ≈ −Kµνd (

q → 0+ " =

Jµ(qω) = −	1	{Kµνsp (qω) − Kµνnp(q0)}Aqνω
	4π		(? 7B?+

% A

" ! =

Jµs (qω) − Jµn(qω) = {Kµνs (qω) − Kµνn (q0)}Aνqω =

= {Ksp (qω) − Knp(qω)}Aν

µν µν qω (? 7B@+

<<< < Предыдущая 5 6 7 8 9 10 11 12 13 14 15 16 17 1819 / 2919 20 21 22 23 24 25 26 27 28 29 > Следующая >>>

Соседние файлы в папке Теории

#
15.08.201330.48 Mб482Роуч П. Вычислительная гидродинамика.pdf
#
15.08.201310.55 Mб137Руденко О.В. Солуян С.И. Теоретические основы нелинейной акустики.pdf
#
15.08.20135.49 Mб36Садовский М.В. Диаграмматика (2005).pdf
#
15.08.20131.78 Mб15Садовский М.В. Лекции по статистической физике (2000).pdf

\|Φ(t) >= T exp −i −∞ dt Hint(t ) \|0 >≡ U (t, −∞)\|0	>	& Q
t
5 \|Φ(t) > !W
Jµ(x) =< Φ(t)\|jµ(rt)\|Φ(t) >=< 0\|U +(t, −∞)jµ(rt)U (t, −∞)\|0	>	& L