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Квантовая теория поля

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<<< < Предыдущая 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 / 2017 18 19 20 > Следующая >>>

	q
p + q	p

iqµϕ(q)

Λ(2)µ (p +

k, p)

					k
				p + q	p + k + q
		− ieΛµν(2)(p + k, p) =			q
				p	q
				p	p + k
− ieΛµ(2)(p + k, p) = Z	d4q	(−ieγλ)(iS(p + k + q))(−ieγµ)(iS(p + q))(−ieγσ)(−iDλσ(q)).
− ieΛµ(2)(p + k, p) = Z	(2π)4	(−ieγλ)(iS(p + k + q))(−ieγµ)(iS(p + q))(−ieγσ)(−iDλσ(q)).
Λµ(2)(p + k, p) = −4πα Z			d4q	λ(p + k, p + k + q)S(p + k + q) µ(p + k + q, p + q)
Λµ(2)(p + k, p) = −4πα Z			(2π)4	λ(p + k, p + k + q)S(p + k + q) µ(p + k + q, p + q)

× S(p + q) σ(p + q, p)Dλσ(q).

Λ(4)µ (p+k, p)

k2 = 0

S−1(p) = pˆ − m, (0)µ = γµ.

∂S−1(p) = (0). ∂pµ µ

			∂S−1(p)			= µ(p, p).
				∂pµ
					\|eϕ\| 1
ψ(x) → e−ieϕ(x)ψ(x) ≈ (1 − ieϕ(x))ψ(x) = ψ(x) + δψ(x),
¯	¯	ieϕ(x)		¯			¯	¯
ψ(x)	→ ψ(x)e			≈ ψ(x)(1 + ieϕ(x)) = ψ(x) + δψ(x),
Aµ(x) → Aµ(x) + ∂µϕ(x).
				¯				− x2),
δS(x1, x2) = δ(−iψ(x1)ψ(x2)) = ie(ϕ(x2) − ϕ(x1))S(x1								− x2),

δS(p + q, p) = d4x1ei(p+q)x1 d4x2e−ipx2 δS(x1, x2) = ieϕ(q)(S(p + q) − S(p)).

−iqµϕ(q)

qµ

iδS(p + q, p) = (iS(p + q))(−ie µ(p + q, p))(iS(p))(−iqµϕ(q))

δS(p + q, p) = −ieϕ(q)S(p + q)qµ µ(p + q, p)S(p).

S(p) − S(p + q) = S(p + q)qµ µ(p + q, p)S(p).

S−1(p + q) S−1(p)

S−1(p + q) − S−1(p) = qµ µ(p + q, p),

qµ → 0

∂Σ(p) = −Λµ(p, p). ∂pµ

λϕ4

L = 12(∂µϕ0)(∂µϕ0) − 12m20ϕ20 − 4!1 λ0ϕ40,

1/4!

−iλ0

ϕ4

e−iH t|0i = e−iH t

G(n)(x1, x2 . . . , xn) = hΩ|T (ϕH1 (x1)ϕH2 (x2) . . . ϕHn (xn))|Ωi,

|Ωi

H = H0 + V

H |Ωi = E0|Ωi.

H0|0i = 0.

	\|Ωi	\|0i
X	X	X
\|NihN\|0i =	e−iEN t\|NihN\|0i = e−iE0t\|ΩihΩ\|0i +	e−iEN t\|NihN\|0i,
N	N	N6=0
	{\|Ni}

t → ∞

lim e−iH t|0i ≈ lim e−iE0t|ΩihΩ|0i,

t→∞

EN > E0

t → ∞(1 − i0)

e−(EN −E0)t

λϕ4

	k		k1
			k1
		p	p
p	p		k2
	Σ(p2)	λ0	λ02

lim

e−iE0t −1 e−iH t

lim

e−iE0t

−1 e−iH teiH0t

|Ωi = t→∞

hΩ|0i

|0i

t→∞

hΩ|0i

e−

iE0t

−1

|0i =

lim

iH0t

t→∞

hΩ|0i

−

)|0i

|0i = 1

hΩ| = lim h0|S(t, 0) h0|Ωie−iE0t −1 .

	t→∞
\|Ωi	hΩ\|Ωi = 1
tlim \|hΩ\|0i\|2e−2iE0t = tlim h0\|S(t, −t)\|0i,
→∞	→∞

ϕHi (xi) = eiH ti e−iH0ti ϕi(xi)eiH0ti e−iH ti = S−1(ti, 0)ϕ(xi)S(ti, 0),

ϕi(xi) ≡ ϕIi (xi)

G(n)(x1, x2 . . . , xn) = hΩ|T (S−1(t1, 0)ϕ1(x1)S(t1, t2)ϕ2(x2)S(t2, t3) . . . S(tn−1, tn)ϕn(xn)S(tn, 0))|Ωi =

= tlim |hΩ|0i|2e−2iE0t

−1 h0|S(t, 0)T (S(0, t1)ϕ1(x1)S(t1, t2) . . . S(tn−1, tn)ϕn(xn)S(tn, 0))S(0, −t)|0i =

→∞

1 1

= h

1 1

n n

= lim h

− |

T (ϕ (x ) . . . ϕ (x )S(t, t))

T (ϕ (x ) . . . ϕ (x )S)

t→∞

−

S(t, t)

= h0|T (ϕ1(x1) . . . ϕn(xn)S)|0ic,

G0(p2) = 1/(p2 −

m2 + i0)	h0\|T (ϕ1(x1)ϕ2(x2))\|0i
	(x1, x2) = h0\|T (ϕ1(x1)ϕ2(x2)S)\|0i = Z	d4p	d4p
G(2)		1	2	ei(p1x1	+p2x2)G(p1	, p2).
		(2π)4	(2π)4

λϕ4

G(2)

(x1, x2) = G

(2)

(x1

− x2)

G(p1, p2) =

(2π)4δ(4)(p1 + p2)G(p1)

G(p) = G(p2)

Σ(p2)

iG(p2) = iG0(p2)+iG0(p2)(−iΣ(p2))iG0(p2)+. . . = iG0(p2)

1 + G0(p2)Σ(p2) + . . . =

G0−1(p2) − Σ(p2)

G(p2) =

) + i0

− m0

− Σ(p

Σ(p2)

λ0

λ02

d4k

Σ1(p2) =

λ0 Z

(2π)4

k2 − m02 + i0

Σ(p2)

p2 = m2

′

˜ 2

Σ(p

) = Σ(m ) + (p

− m )Σ (m ) + Σ(p

Σ(m2) Σ′(m2) Σ(˜ p2)

Σ(m ) = 0

m2 = m02 + δm2, δm2 = Σ(m2),

G(p2) =

Zϕ

, Zϕ−1 = 1 − Σ′(m2).

)

− m −

Σ(p

G(p2)

p2 = m2

ϕ = Zϕ−1/2ϕ0,

Zϕ

GR(p2) = Z

d4x

eipxh0|T (ϕ(x)ϕ(0)S)|0i =

G(p2) =

(2π)4

Zϕ

p2 − m2 − Σ(˜ p2)

δm2

= Σ(m2) =

λ0Λ2

, Zϕ ≈ 1 + Σ′(m2) = 1 + O λ02 .

32π2

0(s, t, u) = −iλ0 + (s) + (t) + (u).

(3 · 4)/4!

ϕ0

λ0

) ≈

˜ 2

)

ZϕΣ(p

Σ(p

λϕ4

p1	p3
p2	p4

p2	p4	p2
(s, t, u)		λ2 (s) (t) (u)

s = (p1 +p2)2 t = (p1 −p3)2

s = t = u = 4m2/3

u = (p1 −p4)2

2	˜	˜	˜		−1	λ0	˜	˜	˜
0(s, t, u) = −iλ0 + 3 (4m /3) + (s) + (t) + (u) = −iZλ						λ0	+ (s) + (t) + (u),
									λ0
	Zλ = 1 +	3λ0	ln	Λ2	+ O λ02 .
	Zλ = 1 +	32π2	ln	m2	+ O λ02 .

G(4)

R(s, t, u)

R(s, t, u) = Zϕ4 G4(s, t, u) = Zϕ4 (Zϕ−1/2)4 0(s, t, u) = Zϕ2 0(s, t, u),

Zϕ4

Zϕ−1/2

λ = Zλ−1Zϕ2 λ0

− ˜ ˜ ˜

R(s, t, u) = iλ + (s) + (t) + (u),

2 ˜	˜	˜	˜
Zϕ (s) ≈ (s)		(t) (u)

+ m2

λϕ4

ϕ20

G0(n)({pi}, λ0	n	pi) =	Z
	, m0)(2π)4δ(4)( i=1
	X

n = 4

λϕ4

ϕ40

(0n)({pi}, λ0, m0) i = 1, n

d4xieipixi h0|T (ϕ0(x1) . . . ϕ0(xn)S)|0i

i=1

(Rn)({pi}, λ, m, µ) = Zϕn/2(µ) (0n)({pi}, λ0, m0),

(n)

({pi

}, λ0, m0) = µ

Zϕ−n/2

(µ) R

({pi}, λ, m, µ) = 0.

dµ

d/dµ

Zϕn/2

∂λ

∂

µ ∂m

β(λ) = µ

, γ(λ) = µ

ln pZϕ, γm(λ) =

∂µ

µ∂µ + β(λ)

∂λ − nγ(λ) + mγm(λ)

∂m

({pi}, λ, m, µ) = 0

∂

(n)

R(n)({pi}, λ, m, µ)

ξpi

(n)

({pi}, λ, m/ξ, µ/ξ).

({ξpi}, λ, m, µ) = ξ

λϕ4

							ξ
ξ ∂ξ R(n)		({ξpi}, λ, m, µ) =	d − m∂m		− µ∂µ		ξd R(n)({pi}, λ, m/ξ, µ/ξ).
	∂			∂		∂
							ξd R(n)({pi}, λ, m/ξ, µ/ξ)

(Rn)({ξpi}, λ, m, µ)

ξ ∂ξ

+ m∂m

+ µ∂µ

− d R

({ξpi}, λ, m, µ) = 0,

∂

(n)

−ξ ∂ξ

+ β(λ)∂λ − nγ(λ) + m(γm(λ) − 1)∂m

+ d R

({ξpi}, λ, m, µ) = 0.

∂

(n)

β γ γm

λm

(Rn)({ξpi}, λ, m, µ) = f(ξ) (Rn)({pi}, λ(ξ), m(ξ), µ),

ξ	∂g(ξ)	= β(λ), ξ	∂m(ξ)	= m(γm(λ) − 1),

	∂ξ		∂ξ

ξ df(ξ) = d − nγ(λ). f dξ

β(λ)

λ0

λ = 1 + 3λ0 ln Λ2 . 32π2 m2

λϕ4

λ0

λ(µ2) =

µ	∂λ	=	∂λ	≈	∂

	∂µ		∂ ln µ		∂ ln µ

λϕ4

β(λ) =

λ = 0

λ0

λ < λ0 β

λ > λ0 β

λ = λ0

λ0

1 + 3λ0 ln Λ2 .

32π2 µ2

λ0 1 −	3λ		Λ2
	0	ln
	32π2		µ2

3λ2 + O(λ3), 16π2

ξ → ∞

λ = 0

= 3λ20 , 16π2

µ2

ξ → 0

λ = 0

λ0

λ(∞) = λ0

λϕ4

β(λ) λ0

	λ0(Λ) = Zλ(λ, Λ/µ)λ,
λ	µ
Zϕ = 1
λ0 → λ	Zλ
Λ µ
λ

β(λ) = Λ∂λ0 ∂Λ

U= 12m20ϕ20

(2n) = i (2n)! Z

2n(2n)

2n ϕ

= λ	∂Zλ	=	3λ2	+ O(λ3).
	∂ ln Λ		16π2

								~
						λϕ4
	λ0			∞	1
+		ϕ04 −		X			(2n)ϕ02n,
+	4!	ϕ04 −		n=1	(2n)!		(2n)ϕ02n,
				n=1
d4k							i	n
d4k			(−iλ0)				i	,
(2π)4			(−iλ0)			k2 − m02 + i0		,
						(2n)!		2n

U =

1 2

λ0

+ i Z

d4k ∞ 1

λ0ϕ02/2

ϕ0

(2π)4 n=1 2n

−

m02

+ i0

<<< < Предыдущая 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 / 2017 18 19 20 > Следующая >>>

	\|Ωi	\|0i
X	X	X
\|NihN\|0i =	e−iEN t\|NihN\|0i = e−iE0t\|ΩihΩ\|0i +	e−iEN t\|NihN\|0i,
N	N	N6=0
	{\|Ni}