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Квантовая теория поля

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A = 0, A0 = A0(r), U = eA0 = U(r).
i	∂ψ	= (αp + βm + U)ψ.
i	∂t	= (αp + βm + U)ψ.
	∂t
		ψ = e−iEt			χ ,
					ϕ
(E + m − U)χ = (σp)ϕ.
	(E	− m − U)ϕ = (σp)χ
									j
m
	l	0			1 σ	0		j 0
0 l					0	21 σ		0 j
J = L + S =				+	2		=		,
J = L + S =				+			=		,
j2ϕjm = j(j + 1)ϕjm, j3ϕjm = mϕjm
j2χjm = j(j + 1)χjm, j3χjm = mχjm.
Ωjlm							Ylm		ξµ
				l
		X X			jm
Ωjlm(n) =		µ=			C21 µlm′ Ylm′ (n)ξµ
		µ=	±	21 m′=−l
			±

l = l′ ± 1

Ωjl′m

(

ϕjm

χjm

j2 j3 l2 s2

l′−l+2

(n) = (−1) 2 (σn)Ωjlm(n).

(r) = fjl(r)Ωjlm(n)

l−l′+1

(r) = (−1) 2 gjl′ (r)Ωjl′m(n),

l(l′) = j ±	1	, l = l′ ± 1.
	2

ψjm

U)f = g′ +

κ − 1

−

(E + m

′

+ 1

−

U)g = f

(σp)(σr) = −i 3 + r ∂r + (σl)

∂

σr

σp

∂

σl

(

)(

) =

− ∂r

−

(

)

(σl) = j(j + 1) − l(l + 1) −

≡ −κ − 1,

= (

1, j = l + 21

κ = −j(j + 1) + l(l + 1) −

− − l, j = l − 21

= j +

, l = j ±

F (r) = rf(r) G(r) = rg(r)

F ′

−

(E + m

−

U)G = 0

G′

G + (E

−

U)F = 0.

− r

U(r)

U(r) = −

Zα

En = m + εn, εn = −

m(Zα)2

2n2

εn m

(Zα) 1

m + E ≡ A m − E ≡ B √

= √

≡ D

Dr ≡ ρ

m2 − E2

|E| < m

D +

G = 0

dρ +

ρ F −

Zα

= 0

−

D −

dρ −

∞

			X
	F = e−ρ		aν ργ+ν
		ρ	∞
		ρ	γ+ν
			bν ρ ,
G = e−			bν ρ ,
			ν=0

ν=0

∞	− aν +		A		ρν + ((γ + ν + κ)aν − Zαbν )ρν−1
ν=0	− aν +		D bν		ρν + ((γ + ν + κ)aν − Zαbν )ρν−1
X
∞
		B			ν		κ	ν 1
	bν +			aν	ρ + ((γ + ν		κ	)bν + Zαaν )ρ −
ν=0	bν +	D		aν	ρ + ((γ + ν	−		)bν + Zαaν )ρ −
ν=0	−	D				−
X

κ → −κ, aν ↔ bν , A ↔ B, Z → −Z.

ν = 0

((γ +

κ)a0 − Zαb0)

∞

+ (γ + ν + κ)aν − Zαbν

+ ν=1

−aν−1 − D bν−1

∞

((γ −

)b0

+ Zαa0)ρ

+ ν=1

−bν−1 − D aν−1 + (γ + ν −

)bν + Zαaν

ρ−1

((γ + κ)a0 − Zαb0 = 0 (γ − κ)b0 + Zαa0 = 0.

γ > 0

γ = p

κ2 − (Zα)2

−aν−1 −

+ (γ + ν + κ)aν − Zαbν = 0

bν−1

bν

aν

+ (γ + ν

)bν + Zαaν = 0.

−

− D −

aν−1

=0,

ν → ν − 1

ρν−1 = 0

ρν−1 = 0.

bν−1

	Zα − D (γ + ν + κ)				aν bν
aν	Zα − D (γ + ν + κ)		= bν	−D Zα − γ − ν + κ .
		B			B

aν bν

nr + 1

anr +1 = bnr +1 = 0.

ν = nr + 1

anr = −

bnr .

ν = nr

A − B

Zα = 2(γ + nr)

Enrj = m 1 +

−1/2

Zα

+ κ2

(Zα)2

−

Zα 1

Enrj = m −

m(Zα)2

2(nr + |κ|)2

n = nr + |κ| = nr + j + 21

a0 = −

b0,

a0 b0

a0 =

Zα

b0,

κ2

pb0

− (Zα)

κ < 0

nr = 0

κ > 0

nr = 0, 1, 2, 3, . . . ,

κ < 0

nr = 1, 2, 3, . . . ,

κ > 0,

Zα 1

|κ| = j + 21

nr 6= 0

nr + pκ2 − (Zα)2 ≈ nr + |κ| = nr + j +

≡ n,

nr j

nr = 0

l = 0

j = l + 1 =

E0 = m 1 − (Zα)2.

nr = 0

l = j 12

κ = −1

(Zα) 6 1	Z . 137	r0
		E0	Z
		E0 = 0
E0 = −m		Zc(r0)

Z − 2

			C
C†C = CC† = 1, ψc = Cψ¯T				ψαc = Cαβψ¯β !			,
					X
					β
T								C
C = γ0γ2.
CC† = 1, C† = C, CγµT C† = −γµ,
							µ = 0 µ = 2 µ = 1, 2
ψc = −γ2ψ .
								C
Cψ(t, x) = −γ2ψ (t, x).
	T			¯T	(x) = 0,
(−i∂µ − eAµ)γµ		− m ψ			(x) = 0,
C						1 = C†C		γµT ψ¯T
T †		¯T	)		¯T	) = 0.
(−i∂µ − eAµ)(Cγµ C	)(Cψ		)	− m(Cψ		) = 0.

((i∂µ + eAµ)γµ − m)ψc(x) = 0,

((i∂µ − eAµ)γµ − m)ψ(x) = 0,

− γ2ψ = ψ

ϕ = −σ2χ

χ = σ2ϕ ,

					1		1 σ
					2		2
	S = 2Σ,				Σ =	0 σ		= −αγ5.
			1			σ	0
S2 =	2Σ			2	4(γ0γiγ5)(γ0γiγ5) = −4γ2					=	4,
S2 =	2Σ			=	4(γ0γiγ5)(γ0γiγ5) = −4γ2					=	4,
		1			1				1		3

S2 = S(S + 1) = 12 1 + 12 = 34

[H Σ] = 2((γp)γ + p)γ5,

p = 0

p 6= 0

p = 0

[H Σ2] = Σ[ΣH ] + [ΣH ]Σ = 0

Σ2 = (2S)2 = 3

n = p/\|p\|	(γn)2 = −1
	±21

2(σn)ϕλ = λϕλ.

n = (sin θ cos φ, sin θ sin φ, cos θ),

(nσ) =

cos θ

sin θe−iφ

21 sin θeiφ

−21 cos θ ,

λ = ±21

eiφ/2 sin 2θ

−

eiφ/2 cos

2θ

ϕ1

e−iφ/2 cos θ

, ϕ

e−iφ/2 sin

−

ϕ†λϕλ′ = δλλ′ .

iσ2ϕλ = (σn)ϕ−λ = −2λϕ−λ.

u(p)

E = ±Ep.

E =

+Ep

m + E = m + Ep > 0

m + E

uλ(p) =

(σp)λ

n =

E +m ϕλ

−

(σn)ϕλ

ϕλ

qEp+m

| |

1 = ZV d3x

(x) = ZV d3xψγ¯ 0ψ =

ZV d3x uλ† uλ =

uλ† uλ,

2EpV

2Ep

uλ† uλ = 2Ep,

N = p

Ep + m

uλ(p) =

p m (σn)ϕλ ! .

E + m ϕλ

p = 0

−

uλ(p = 0) = √2m

0λ ,

ψpλ (x) =		2EpV e−iEpx0+ipx		Ep		m (σn)ϕλ ! .
				p
(+)	1				Ep + m ϕλ
	p		p
	p		p		−

u¯λ(p)uλ(p) = 2m,

Ep = E = p0

u†λuλ = u¯λγ0uλ

u¯λγµuλ = 2pµ.

ucλ(p)

		p
uλc (p) = −γ2uλ(p) =	−i Ep −		m ϕ−λ
uλc (p) = −γ2uλ(p) =			m ϕ−λ
	− p			−	λ
	i	Ep + m(σn) ϕ			λ

E = −Ep < 0			m−E = m+Ep > 0
ϕ			χ
vλ(p) =	−	Ep + m χλ	! ,

Ep − m (σn)χλ

v¯λ(p)vλ(p) = −2m.

ϕλ ≡ −i(σn)χλ = −2iλχλ
		p
vλ(p) =	−i Ep − m ϕλ
	p
i		Ep + m (σn)ϕλ
		Ep > 0

E > 0

Ep < 0


Ep − m	[m, ∞)			Ep = p			Ep −m
Ep				Ep = p		p2 + m2	> m
Ep − m → eiπ(Ep + m) = p				→ ip
Ep − m → eiπ(Ep + m) = p			Ep − m	→ ip	Ep + m.
	p				(−∞, −m]
Ep → −Ep	p	Ep + m			(−∞, −m]

Ep + m → e−iπ(Ep − m) = pEp + m → −ipEp − m.

Ep → −Ep

−i Ep − m ϕ−λ v−λ(−p) = p

−i Ep + m(σn) ϕ−λ

				v−λ(−p) = uλc (p).
	X		1		apλuλ(p)e−ipx + b	uc	(p)eipx
ψ(x) =								,


		p			pλ	λ

p,λ 2Ep

−p −λ

(ˆp − m)u(p) = 0, (ˆp + m)uc(p) = 0.

uc(p) = v(−p) = u|E=−Ep (−p)

u¯(ˆp − m) = 0 = (ˆpT − m)¯uT = 0 = C(ˆpT − m)C†Cu¯T = 0 = (−pˆ − m)uc = 0.

u−p

u(−p)

p p′	puˆ (p) = mu(p)

u¯(p′)ˆp′ = mu¯(p′)

γµu(p) =	1	(pµ + σµν pν )u(p), u¯(p′)γµ =		1	u¯(p′)(pµ′ − σµν pν′ ),

	m			m
γ					u¯(p′)
u(p)
u¯(p′)γµu(p) =		1	u¯(p′) (Pµ − σµν qν ) u(p), P = p + p′, q = p′ − p.

		2m

p′ = p

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