Теории / Садовский М.В. Диаграмматика (2005)

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Кузбасский государственный технический университет

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Физика

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. KV K< <10 < J

K G %* F F + τ = τ1 −τ2

F +(τ ) = −F +

F (τ ) = −F

.pdf

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$ *

E 2 .					, ,
i ∂t −		2m ( + ieA)2 − µ F +(x, x ) + iλF −(x, x)G(x, x ) = 0			&O
	∂	1
G ! W
			A → A + ϕ		&OI
%* 8 G F F + *) W
			G(x, x ) → G(x, x )eie[ϕ(r)−ϕ(r )]		&OJ
			F (x, x ) → F (x, x )e+ie[ϕ(r)+ϕ(r )]		&O
			F +(x, x ) → F +(x, x )e−ie[ϕ(r)+ϕ(r )]		&O
# * 1 5 ( *) ( ! W
			ψ(x) → ψeieϕ(r)	ψ+(x) → ψ+e−ieϕ(r)	&OQ

5 1 2 ∆(x) |λ|F (x, x) ∆ (x) |λ|F +(x, x) !

) %* ! x *) *W

F (x, x) → F (x, x)e2ieϕ(r) F +(x, x) → F +(x, x)e−2ieϕ(r) &OL

! ! 1 #2 ∆ 1 5

! ! 2e ! * !* * ( *

]

G ! ! ) * ! * & ! *

* 1 ! # 2 %* 8 ( *

1 ! # *)2W

Fαβ (τ1, r1; τ2, r2) = Sp e	Ω+ T	−	H	Tτ (ψα(τ1r1)ψβ (τ2r2))	&OO
Fαβ (τ1, r1; τ2, r2) = Sp e	µN		H	Tτ (ψ¯α(τ1r1)ψ¯β (τ2r2))
Fαβ (τ1, r1; τ2, r2) = Sp e	T			Tτ (ψ¯α(τ1r1)ψ¯β (τ2r2))	&,
+	Ω+µN	−H

ψα(τ r) = eτ (H−µN )ψα(r)e−τ (H−µN )	ψ¯β (τ r) = eτ (H−µN )ψβ+(r)e−τ (H−µN )				&, ,

* &OO &, # !* !* ! ) 8Ω $ *) ! ! Tτ

! ! !* ! * !* ! &

' ! # ( %* & I W


Fαβ = gαβ F		Fαβ+ = −gαβ F +		&, I

/ ' 1 . ) =
Gαβ (τ1, r1; τ2, r2) = −Sp	e	Ω+µN −H	¯	(? )*7+
		T
			Tτ (ψα(τ1r1)ψβ (τ2 r2))

& (? ? + " (? ,,+ (? )**+

i " T = 0

N* # τ ( %* # εn = πT (2n + 1)& 3 * ψ $ τ = 0 ) !

t = 0 ( !* &OO &, &J &J & J

F (0, r; 0, r) = Ξ(r)

F +(0, r; 0, r) = Ξ (r)

Ξ ! ! # %*

* 8 *&

M 1 2 ! * %* 8 G F F + * & & ! ! %%

t %% * ! τ ! &IJ # * ! !

! it → τ & * # * ! * W

−∂τ

−

∂

+ µ

G(τ, r; τ , r ) + λΞF +(τ, r; τ , r ) = δ(τ

τ )δ(r

r ) &,

∂τ

−

∂

+ µ F +(τ, r; τ , r )

λΞ G(τ, r; τ , r ) = 0

&, Q

G %* # $ ! ! ( * ! ) W

(iεn − ξ(p))G(εnp) + ∆F +(εnp) = 1

&, L

−(iεn + ξ(p))F +(εnp) − ∆ G(εnp) = 0

&, O

∆ = λΞ = λF (0r; 0r) ∆ = λΞ = λF +(0r; 0r)

&,,

. ! * &, L

&, O

! W

G(εnp) = −

iεn + ξ(p)

= −

iεn + ξ(p)

ε2 + ξ2(p) + ∆ 2

ε2 + ε2

(p)

&,,,

+(εnp) =

∆

εn2

+ ξ2(p) + |∆|2

εn2 + ε2

(p)

&,,

ε(p) ! & L W

ε(p) =

ξ2(p) + |∆|2

&,,I

* T = 0 # (

%* 8 * &

M ) ( *) # ! W

∞	d3p	&,,J
Ξ = F +(τ = 0, r = 0) = T n=	(2π)3 F +(εnp)

−∞

E 2 .

, I

&,,

1 = |λ|T

∞

εn2

d3p

|λ|T

∞

d3p

&,,

(2π)3

+ ξ2(p) + |∆|2

(2π)3

εn2 + ε2(p)

−∞

'*!! ! # ! # ! #) &OO

* TK * ! * W

1 =	\|	2\|	(2π)3	ξ2(p) + ∆2	(T ) th			2T		&,,
		λ	d3p	1			ξ2	(p) + ∆2	(T )

' ( * + ,, ,- ( !* ! * !

! # ! ! &,, ! ! *)

! * *) ! # ∆(T ) TK & #

* # T = Tc ! &I, & * ∆ = 0 * &,, W

1 =	\|λ2\| π2 F 0	ξ	th 2T		&,,Q
	mp	ωD dξ		ξ

% * ! ) ! )

* 1 2 &I & G T = 0 &,,

&QI

* ! &Q &QQ

∆0

= π Tc&

M 8 # &, L &, O

! # W

G(εnp) =

∆F +(εnp)

&,,L

−

iεn − ξ(p)

F +(εnp) =

∆ G(εnp) = −

∆ G(εnp)

&,,O

−

iε

n −

ξ(

−

iε

+ ξ(p)

! &L &L, ! !! ! . & &J

# * ! # p = (εn, p)& H * ! &,,O

&,,L * ! ! # %* 8 !*

∆W

G(εnp) =

|∆|2

iεn

− ξ(p)

iεn − ξ(p) (iε + ξ(p))(iε − ξ(p))

|∆|4

· · ·

iεn − ξ(p) (iεn + ξ(p))2(iεn − ξ(p))2

iεn + ξ(p)

(iεn)2 − ξ2(p) − |∆|2

% . & & [ & . * # *!! (

&,,, & : ! ! ! * &,,L

# &,,O 1 #2 !! F +(εnp)

. & & e ! % * * !

1 !2 ∆ *!! * &,, & S ! !

! # %* 8 * * *

! ) ^ 1 2 ! # %* * * & / ! !

! # %* 8 F + . 5 1 2 !

∆&

& - 0 "

" " )) 7? -0 ∆

, J	. KV K< <10 < J

. & & W 6 !! ! # [ ! # e %* 8

! ∆&

E 2 .

. & &QW 6 !! * & G*

%* ) 8 % &

: &L &,, &,,J ! # W

∞	d3p	&, ,
∆ = λΞ = λT n=	(2π)3 F (εnp)

−∞

# % ! ! . & & [ * 8 #

!* % . & & e &

o "

5L 3 . (? + -8

0 % 3 ' ( < " +

' 1 . " 8

' A ? B 2 % " (. o ),@*+

" A % " 8

' 6

! (u i WcW\hhCU ),@H+ 1 " " 8

% 3 '

' 1 .

" 7@

> " ' I

(? ,,+ (? )**+ (? )*7+ ! =

; τ2, r2) = −Sp

Ω+µN −H

Gαβ (τ1, r1

Tτ (ψα(τ1r1)ψβ (τ2 r2))

1 ' 1 . =

Gˆ(εnp) =

G(εnp)

−F (εnp)

−F

(εnp)

G(εnp)

iεn1ˆ − σˆz ξ(p) −

)∆ˆσ+ + ∆ σˆ−* Gˆ(p) = 1ˆ

(? )*H+ (? )*,+ ˜

(iεn − ξ(p))F (εnp) + ∆G(εnp) = 0

(iεn + ξ(p))G˜(εnp) + ∆ F (εnp) = 1

o ' 1 . (? )7 + (x = (τ, r)+=

− < T ψα(x1)ψβ (x2) > = < T ψˆ(x1)ψˆ+(x2) >

Gˆ(x1, x2) = − < T ψα(x1)ψβ (x2) >

−

< T ψ¯α(x1)ψ¯β (x2) >

−

< T ψ¯α(x1)ψβ (x2) > −

. I

ψα(x1)

ˆ+

ψ(x1) = ¯

(x2) = (ψα(x2), ψβ (x2))

ψβ (x1)

(? )77+

(? )7 +

(? )7$+

(? )7?+

(? )7@+

(? )7B+

(? )7H+

> " ' 1 . I ! " -0 ' 8 1 " ( ! " ' 8

" xx + 7$ q

-0 % '

% ' 1 .

" -"0 -"0 ' 1

,	. KV K< <10 < J

. & &LW M 8 # ! ! &

( ) ' #

. ! ! ! ! ) *

ρ ! %* # $

# v(p)& ( * * * ! #

! # ! ! # ! ! & S !

! ! 1 ! # 2 1 ! # 2 %* 8 & ' * # !* * ( %* *

! % * ! ! & 1G ! 2 !! !

# # % % * *) 1 * 2 ! !*

! !* ) %* 8 G F +& 5* #

# ^ ! ! ) & & !

! ( ! ∆(r) ∆ (r)&

# ! # * # !! *) * # * *)

∆ * * # # * *

∆(r) = λF (x, x)& G ( !* / 1 *

# ! W < ∆(r) >= ∆(0)

) * ! ! < ∆2(r) > − < ∆(r) >2= 0

! * ) & # !

! ! * # ! *

F (x, x) ! ) ! ! & G ( !*

!! ! ! ^ * ! $ !* ! # ρv2(q)

( ! ! &

) # ! * ) & M

* G(p) F +(p) ! ) % . & &L

# & * !* * # * # !

! %* 8 1 2 G(0) F +(0) !

< pF l 1, EF τ 1

F DH C H 2 ? C C										, Q
* ) ! #									!* * +,- W
					¯	¯		+	(p) = 1	&,I
	(iεn − ξ(p) − Gε)G(p) − (∆ + Fε)F								(p) = 1	&,I
−	(iε	n	+ ξ(p)	−	G¯	)F +(p) + (∆ + F¯+)G(p) = 0				&,I,
−		n		−	−ε		ε

W		(2π)3 \|v(p − p )\|2G(p )
G¯ε = ρ
		d3p
F¯ε+ = ρ		(2π)3 \|v(p − p )\|2F +(p )
		d3p

*! p = (p, εn)&

. ( ! * ! ! * ! ¯

Gε

&,I

&,II

= − ¯

G−ε W

			¯	+ ξ(p)
G(p) =	−	iεn − Gε		+ ξ(p)
G(p) =		iεn − Gε				+	2
		¯ 2	2			+	2
	−(iεn − Gε) + ξ			(p) + \|∆ + Fε \|
					¯+
F +(p) = −			∆ + F
					ε
		¯ 2	2			+	2
		−(iεn − Gε) + ξ			(p) + \|∆ + Fε		\|

&,IJ

&,I

G ( &,I &,II *
¯	¯+

Gε Fε & K # ! ! ! # ! !

$ ( # ¯

Gε *) * *) !

! # ! *&

! * ! $ d3p

N ! & G ( !* ( % !

! # ! ! W

(2π)3 |v(p − p )|2

ξ(p )

&,I

δµ ≈ ρ

d3p

G ( !* % &,I &,II #

!* * N ! &,I &,I,

(iε˜n − ξ(p))G(p)

(p) = 1

&,IQ

− ∆F

(iε˜n + ξ(p))F

˜ G(p) = 0

&,IL

(p) − ∆

−∞ dξ −(iε˜n)2 + ξ2

+ ∆˜ 2

&,IO

iε˜n = iεn − G¯ε = iεn + ρv2νF

∞

iε˜n + ξ

∞

∆˜

∆˜ = ∆ + F¯ε = ∆ + ρv2νF −∞ dξ

&,J

−(iε˜n)2 + ξ2 + ∆˜ 2

! * ! ! & / ! !

! ! &,IO * ) $ W

¯			¯
−	Gε	=	Fε
−	iεn	=	∆	&,J,

ρv(p)2 → 0 (? ) *+ (? ) )+ " (? )*H+ (? )*,+ I

) (? ))*+ (? )7)+

=	+(x, x)
∆ = \|λ\|F (x, x) ∆ = \|λ\|F	+(x, x)	(? )7,+

" ( (? )*H+ (? )*,++

, L

. KV K< <10 < J

5 # ! #W

= ∆ηε

∆ = ∆ + Fε

&,J

iε˜n = iεn − Gε = iεnηε

&,JI

ηε & &,IO

&,J

* !W

εn2

ηε = 1 + 2πτ

−∞ ξ2

+ (εn2 + ∆2)ηε2

= 1 + 2πτ

ηε

+ ∆2

&,JJ

ηε

∞

dξ

ηε

* ρv2νF =

2πτ & S # ! !W

ηε = 1 +

&,J

2τ

εn2 + ∆2

5 ! ! * ! ! %* 8

G(p) F +(p) * ) *) # %* 1 2

&,,, &,, ! W

{εn, ∆} → {εnηε, ∆ηε}

&,J

5 * &,,J &,, &,, ! * #

ηε * ∆(T )

# * ! ξ ! # ! *

! ξ → ξ/η ! & G ! ! ( *

& / ! &,J &,,J &,, * *)

(2π)3

∞

εn2

ηε2 + ξ2(p) + |∆|2ηε2

≈ | |

∞

ωD

εn2 ηε2

+ ξ2 + |∆|2ηε2

−ωD

1 = |λ|T

d3p

ηε

λ νF T

dξ

ηε

−∞

&,JQ

M Tc * ! &,JQ ∆ = 0W

(2π)3

∞

εn2

ηε2 + ξ2

≈ | |

∞

ωD

εn2

ηε2 + ξ2

−ωD

&,JL

1 = |λ|T

d3p

ηε

λ νF T

dξ

ηε

−∞

ηε = 1 +

&,JO

2|εn|τ

5 # ! ! # !

# *

ε2 +ξ2

&OO ! * Tc W

2ξ th 2Tc +

1 = |λ|νF −ωD

ωD

dξ

| |

∞

ωD

ηε

−

−ωD

εn2 ηε2 + ξ2

εn2 + ξ2

+ λ νF

−∞

dξ

! * !

! * ! # ωD → ∞&

5 ! ! ξ → ξ/ηε ! ! !

! !* ! *) * * ]

F DH C H 2 ? C C	, O
* # &, * ) &,,Q ) !* Tc	1 2

&I, &

: ! ! ! ! # 1 2 * &,JQ

) ∆(T ) ! ! & * # &,,

1 2 * &

5 ! ! Tc ∆(T ) 1 ! # ! 2 ! !

! ! ( ! ! :&:&:

R&G&8 # ,O O & 3 * ( * #

pF l 1, EF τ 1 ! # 1 ! 2 % ! !

& N ( * ! *) #

! ! +,- ! ! ! * ! #

! & ( * ( ! #

- * #

! ) ! # ! !

* &

A ! 5L 8

I %''

( 1 )) 7) + % ! "

! Tc )@ " 5L

" " % ! " <

( " " 8

d 3 " " "

+ > % " 8

)@ & "

( + ( ! 8

S+ ( ; . ),@*+ A

M " 1

% =

V (r) = v(r) + J(r)(S · s) (? )?)+

J(r) 3 ! S 3 s = 1	σ 3 % > %
2

" . "

A ? H G F + > Gαβ = Gδαβ Fαβ = gαβ F Fαβ+ = −gαβ F + gαβ = iσαβy ( (? ?7++ >

(? )?)+ 3 % " A ? H " " ' 1 " . 6

iεn ∆ (? )$7+ (? )$+ (? )$?+

iε˜n = iεn +

iε˜n

(? )?7+

˜ 2

2τ1 ε˜n + ∆

∆

(? )? +

∆ = ∆ +

˜ 2

2τ2 ε˜n + ∆

= 2πρνF v2 +

S(S + 1)J2

(? )?$+

τ1

= 2πρνF v2 −

S(S

+ 1)J2

(? )??+

τ2

1 " < S2 >= 13 S(S + 1) L

% s2 = 41 σ2 = 43

A ! " (? )?$+ (? )??+=

−

πρJ2νF S(S + 1)

(? )?@+

τ1

τ2

τs

9 %

σαγ δγδ σδβ = σαδ σδβ = σ2 δαβ σαγ gγδ σδβ = −σ2gαβ σ2 = 3

,Q	. KV K< <10 < J

. & &OW / ! # ! *

! ! # * 1 2

! &

( + 1 8

v ' τ1s τ1 τ 3 1

τs ! " 8

" ! !

! (? )$+ (? ) ?+ (? ) B+

(? ) H+ (? )?7+ (? )? +

! A 8

" "

" ! "

3 . =

Tc0

= ψ

− ψ

(? )?B+

2πTcτs

Tc0 3 " ! "

Γ (z)

∞

ψ(z) =

= − ln γ

−

+ n=1

−

(? )?H+

Γ(z)

n + z

3 ' Γ 3 ' 1 ( ' 1 + ln γ = C

= 0.577... 3 8

! " Tc

A ? , 2 !

( ! 1 1

" +=

πTc0

∆0

Tc0

(? )?,+

τ c

2γ

( 1 1 " + " 8

(Tc ! + ( 1 1 + τs → ∞

(? )?B+ =	π
Tc ≈ Tc0 −	π	(? )@*+

	4τs

" Tc & Tc 1 1 8

" (? )?B+ % / !

-' 0 A ? ,

(? ) B+ (? ) H+ (? )?7+ (? )? + ! " 8

> % Tc ∆ (∆ !

T= Tc+ % ! % " (

! .

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