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Квантовая теория поля

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S = 1

		e+e−
¯	1	×	1
=	2	×	2

0 1

								1
t 4m2								2
t 4m2
t			2πα2			t		−
t			t			m		−
σ2γ	4m2				ln		2		1 ,
									t → 4m2
			√	πα2					πre2
t								=		,
t									v
σ2γ	→	4m2				2			+
	→		m t − 4m						+

ψ(0) ≡ ψ(r = 0)			ψ(0)						S
	L = 0	L
		S = 0
(2S + 1)\|S=0 0 + (2S + 1)\|S=1 1			=	1	0	+	3	1,

				4			4
1	(−1)L+S	S						3	S1
	S0								S1

= 2γ + 3γ + 4γ + . . . ,

0 = 2γ

+ 4γ + . . . ,

= 3γ + 5γ + . . . ,

0 ≈ 4 2γ ,

1 ≈

3γ.

e+e−

τ0 =

, 0

≈ 4 2γ = 4|ψ(0)|2(v+σ2γ )v+→0 =

mα5,

v = (p+ − p−)/m = 2p/m

1−−

|p| αm

3D1

		p		k1						p
		p		k1


	P							P
		p − P		k2					p − P
		p − P							p − P
							α
									σ2γ
j = v+\|ψ(0)\|2								j =2 v+/V = ρ+v+
				ρ+ = \|ψ(0)\|
						1	2
			ψ(r) =		√		e−r/a, a =				.
										mα
						3				mα
						πa

τ0 1.23 · 10−10

τ1 =	~	,	1 ≈	4	3γ =	4	\|ψ(0)\|2(vσ3γ )v→0	mα6,

	1			3		3

		α
T = Z	d4p
T = Z	(2π)4 Sp S(p) (p, P )S(p − P )W ,

W = (−ieγν )S(p − k1)(−ieγµ)εµ(k1)εν (k2) + (k1 ↔ k2),

e+e−

iT (e+e− → γγ) = u¯(−p+)W u(p−) = v¯(p+)W u(p−).

(p, P )

G = G0 + G0V G0 + . . . = G0 + GV G0 = G0 + G0V G,

e+e−

τ1 = 1.4 · 10−7

→ γγ

Φ + . . . ,

G = Φ

P 2 − M2

P 2 = M2

Φ = ΦV G0

= ΦG0−1, ¯ = G0−1Φ¯.

= G0V

(p, P ) = −i Z

d4k

S(k) (k, P )S(k − P )V (p − k),

(2π)4

V (p−k)

P = 0

p− = (M/2, p) p− = (M/2, −p)

(p) = Nu(p)ψ(p)¯v(−p),

ψ(p)

(p, P )

(G01 − V )G = 1,

∂P 2 (G0−1 − V ) P 2

=M2 Φ¯

= 1,

∂

−1

G0−1 − V

(G0

− V )|P

Φ = 0

Pµ

Φ Φ

∂Pµ

¯ |P 2=M2

= 2Pµ

∂G0

		d4p		∂
i	Z		βγ (p, P )¯σα(P, p − P )		Sαβ(p)Sγσ(p − P ) P 2	=M2	= 2Pµ,
		(2π)4		∂Pµ
				\|

d3p

|ψ(p) 2

, κ = √

mε

Z (2π)3

(p2 + κ|2)2

N = (p2 + κ2)/m3/2

ε = 2m − M

i Z

≈

2π

(p0 − Ep + i0)(p0 − M + Ep − i0)

2Ep − M

(2m + p2/m) − (2m − ε)

p2 + κ2

uu¯ = −vv¯ = 2m

T = i

d4p

u(p)¯u(p)

p2 + κ2

u(p)ψ(p)¯v( p)

v( p)¯v(−p)

W ,

(2π)4

2Ep(p0

i0)

−

Ep + i0) m3/2

−

2Ep(p0

− M + Ep

−

Wp0

p2 + κ2

T = −√1m Z	d3p	ψ(p) v¯(−p)W u(p) = −√im
T = −√1m Z	(2π)3	ψ(p) v¯(−p)W u(p) = −√im


				T (e+e− → γγ)
	\|p\| m		√
		\|p\| κ	√	mε	αm

Z	d3p	ψ(p)T (e+e− → γγ),
	(2π)3

ψ(p)

T (e+e− → γγ)

ψ(0)

T |2

dτ

ψ(0)

T (e+e−

γγ)

2dτ

ψ(0)

2σ

= 4v

ψ(0)

2σ

4m2|

Z |

→

2γ

|2M

2γ

I = p(p+p−)2 − m4

≈ 2m|p| = m2v,

v =

ψ(0) = 0

T (e+e− → γγ)

ψ′(0)

ψ(0)

T (e+e− → γγ)

T (e+e− → γγ) p2 + m2

ψ(0)

p1 p2

p3 p4

p1 + p2 = p3 + p4.

		s = (p1 + p2)2, u = (p1 − p4)2, t = (p1 − p3)2,
		s + t + u = 4m2.
			t u
Tif = e2	t	(¯u(p3)γµu(p1))(¯u(p4)γµu(p2)) − u(¯u(p4)γµu(p1))(¯u(p3)γµu(p2)) .
	1	1
		−

m2)(s

6m2)

(s2 + u2) + 8m2(t

m2)

(s2 + t2) + 8m2(u

m2)

|Tif |2 = 64π2α2

− 2

−

2t2

2u2

E1 = E2 → m

dσ

e−e−

2α

3 cos2 θ)

(1 +

dΩ

mv2

sin4 θ

p2)/m = 2p/m

= p

= (

1 −

+ m

2 v

E1 = E2 = E

dΩ e−e−

sin4 θ

)

dσ

+ cos2

p′− p′+

p1 → −p′+, p2 → p−, p3 → −p+, p4 → p−,

s → (p− − p′+)2, t → (p+ − p′+)2, u → (p− + p+)2.

u → 4m2

e+e−

2θ ,

dΩ e+e−

mv2

sin4

dσ

u 4m2

dσ

e+e−

(3 + cos2 θ)2

dΩ

sin4

2θ

p1	p2
m	M

s = (p1 + P1)2 = (p2 + P2)2,

|Tif |2

dσ dΩ

t = (p1 − p2)2 = (P2 − P1)2,	u = (p1 − P2)2 = (P1 − p2)2,
s + t + u = 2m2 + 2M2.

eµ

= 8πα 2 t2 + st + (s − m2 − M2)2 , t 2

eµ

|Tif |2

α2

+ t +

−

M2)2 .

64π2s

p1 = (E, Ev),				p2 = (E, Ev′), \|v\| = \|v′\| ≈ 1,						(vv′) = cos θ,
		= (√					= (√			, Ev′),
P	1	= (√	E2	+ M2, Ev), P		2	= (√	E2	+ M2	, Ev′),
	1				−	2				−

p− p+

P1 P2

t = −4E2 sin2

− M2)2 = 4E2.

→ ∞

/M2

s ≈ M

O(1 2

)

t /(2s)

dσ

Mott 1 + 2η tg2

η =

Q2 ≡ −(p1 − p2)2,

dΩ

4M2

dΩ

Mott =

2E sin22 θ ! .

dσ

α cos θ

M = ∞ η = 0

Z = 1

p1 = (E1, E1v1),

p2 = (E2, E2v2),

|v1| = |v2| ≈ 1,

(v1v2) = cos θ,

P1 = (M, 0), P2 = (M + E1 − E2, E1v1 − E2v2),

P22 = M2

E1 E2

sin2

s = (p1 + P1)2 ≈ M2 + 2ME1, t = (p1 − p2)2 = −2(p1p2) = −4E1E2 sin2 θ2.

dσ	=	dw/ t	=		\|Tif \|2	dτ2,
dσ	=	j	=	(2M)(2E1)v1		dτ2,
		j		(2M)(2E1)v1
j = v1		V

dτ2 =	1 d3p2	δ((p1	− p2	+ P1)2 − M2).
	(2π)2 2E2

dτ2 =	\|p2\|E2	dΩ =	v2E22	dΩ.
			16π2ME1
	16π2ME1

v1 ≈ v2 ≈ 1

eµ

dΩ

64π2M2

dσ

|Tif |2

dt =

dΩ.

dΩ

2E1 sin2 θ !

4M2E1E2

2 + st + (s − M2)2

dσ

. . . ≈ cos2	θ		2E2		θ		θ	1 + 2η tg2	θ	.
		+	1	sin4		≈ cos2
	2	+	M2	sin4	2	≈ cos2	2		2

dΩ		dΩ Mott 1 + 2η tg2		2	E1 .
	dσ		dσ	θ		E2
		η = 0 E2 = E1

eµ

	p1 → p−, p2 → −p+, P1 → −P+, P2 → P−,
p− P−	p− P−
	s → (p− − P+)2, t → (p− + p+)2,
	e+e− → µ+µ−

σ(e+e− → µ+µ−) = 2πα2 v 1 − v2 , ε2 3

	v	ε	t = (p− + p+)2 =
(E	+ E )2	= ε2
−	+		v 1
			v 1
			S

εM v 1

σ(e+e− → µ+µ−) 4πα2 .

ε M 3ε2

					1 m2A2
					2	µ
					F 2
					µν
	1	2	1	2	2
L = −		Bµν +		m Bµ, Bµν = ∂µBν − ∂ν Bµ,
L = −	4	Bµν +	2	m Bµ, Bµν = ∂µBν − ∂ν Bµ,

∂µBµν + m2Bν = 0

∂ν

m2∂ν Bν = 0,

m 6= 0

(( + m2)gµν − ∂µ∂ν )Gνλ(x) = −gµλδ(4)(x)

Gµν (k) = k2 − m2 gµν −		m2	,
1		kµkν

ρµν =	1 3		(k, σ) = −	1	gµν −	kµk	.
	3 σ=1 eµ(k, σ)eν			3		m2ν
		X

X eµ(k, σ)eν (k, σ) = −gµν + kmµk2ν ,

σ=1

kµ

kσ

kσkσ′

, σ, σ′ = 1, 3,

eµ(k, 0) =

, e0(k, σ) =

, eσ′ (k, σ) = δσσ′ +

√

m(m +

+ m )

σ=1

−iGµν

	kµ
nµeµ(k, 0) = 1, nµeµ(k, σ) = 0, σ = 1, 3, nµ =	kµ
		.
	m	.
3			kµkν
X			kµkν
eµ(k, σ)eν (k, σ) = − eµ(k, σ)eν (k, σ) + eµ(k, 0)eν (k, 0) = −gµν +				,
			m2	,
σ=0

eµ(k, 0)

		1	µ	ϕ) −	1 2 2
L =			(∂µϕ)(∂				m ϕ	,
		2			2
		( + m2)ϕ = 0.
T µν = −L gµν + (∂µϕ)							∂L		.

						∂(∂ν ϕ)
1
T µν = −		∂λ(ϕ∂λϕ)gµν + (∂µϕ)(∂ν ϕ),
	2

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