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Квантовая теория поля

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∂µT µν = 0.

				T µν
	T µν = (∂µϕ)(∂ν ϕ).
ϕ(x) =	X		1	Cpe−ipx + Cp† eipx
					,
		p			,

p 2Ep

Ep = p2 + m2

P µ = Z d3xT µ0 =	p	1
		2pµ(CpCp + CpCp).
	X

[CpCp†′ ] = δpp′ , [CpCp′ ] = [Cp† Cp†′ ] = 0.

Pµ =	p	pµ	Cp† Cp + 2 .
	X		1

1/(p2 − m2)

G(p) = p2 − m2 + i0,

iG(p)

	L =	1	(∂µϕ)(∂µϕ) − U(ϕ),

		2
								T		U(ϕ)
	Z d3x 2ϕ˙2					L = T − U
H =	Z d3x 2ϕ˙2					+ 2( ϕ)2 + U(ϕ) .
				1			1
	U(ϕ) = −				1	µ2ϕ2 +			1	λϕ4

					2				4

µ2 λ

µ2 < 0 λ > 0

ϕ = 0

		6
		5
		4
		3
		2
		1
-2	-1	1	2
	µ2 < 0	λ > 0

µ2 λ

		0.2
		0.1
-1.5	-1.0	-0.5	0.5	1.0	1.5
		-0.1
	µ2 < 0		λ < 0
				U(ϕ)

-1.0	-0.5	0.5	1.0
	-0.1
	-0.2
	-0.3
	-0.4
	-0.5
	-0.6
	-0.7
	µ2 > 0	λ < 0

		0.1
-1.5	-1.0	-0.5	0.5	1.0	1.5
		-0.1
		-0.2
	µ2 > 0		λ > 0

µ2 < 0 λ < 0			ϕ = 0
µ2	> 0	λ < 0
			ϕ = 0
µ2	> 0	λ > 0	ϕ ≡ ±v0 6= 0
			ϕ ≡ ±v0 6= 0
U(ϕ)		ϕ = ±v0	U(−ϕ) = U(ϕ)
η		ϕ = ±v0	v0
η			v0

ϕ = v0 + η, v0 = √ .

+ u(η), u(η) = −

µ4 √

∂µϕ = ∂µη, U(η) = µ

+ µ λη

λη

4λ

L = 2(∂µη)(∂µη) −

2m2η2

− u(η), u(η) = mr

η3

4λη4,

m = µ√

η3

η4

L ′

+ η).

= gψψϕ = gψψ(v0

mf = gv0

e−ipx.

ϕ(x) =

(fpCp + fpCp† ),

fp =

2Ep

Z d3xfp ϕ(x) =

Cp + C−† pe2iEpt

2Ep

Z d3xfp ϕ˙(x) = −

Cp − C−† pe2iEpt ,

Cp = i Z d

xfp

←→0

( )

= − Z

xfp

←→0

( )

∂ ϕ x , C†

∂ ϕ x ,

←→

≡

f˙

−

f˙

∂

|A, piin

|Biout

t → ∞

outh

iin = outh

p |

iin = outh

|(− ) Z

p←→0

in( )|

iin

A, p

†

i d3xf

∂ ϕ

|Aiin = outhB|(−i) Z

0 = outhB|Cp

xfp

←→0

out

( )| iin

†

∂ ϕ

outh

iin

t→+∞

− t→−∞

d3xf

←→0

outh

(

)| iin

A, p

= i( lim

lim )

∂

ϕ x

limt→±∞ ϕ(x)

−∞

∞

(f ←→

) =

−

∂

∂02fp(x) = (Δ − m2)fp(x).

outhB|A, piin = i d3xfp( + m2) outhB|ϕ(x)|Aiin,

iT ({A, p} → B)/ 2Ep

1/ 2Ep

T ({A, p} → B) = d3xe−ipx( + m2)hB|ϕ(x)|Ai,

ϕ1

ϕ2

ϕ ϕ†

(ϕ1 + iϕ2), ϕ†

ϕ =

√

(ϕ1 − iϕ2).

L =

(∂µϕa)(∂µϕa) − m2ϕa2

= (∂µϕ)(∂µϕ†) − m2ϕϕ†.

=1,2

P µ =

Z d3x

∂µϕ† ∂(∂0ϕ†) + ∂(∂0

ϕ)∂µ

= Z

d3x (∂µϕ†)(∂0ϕ) + (∂0ϕ†)(∂µϕ) .

∂L

ϕ → eieaϕ

jµ = ie ϕ† ∂(∂µϕ†)

+ ∂(∂µϕ)ϕ

= ie(ϕ†(∂µϕ) − (∂µϕ†)ϕ).

∂L

a b

ape−ipx + bp† eipx

ϕ(x) =

2Ep

[apa† ′ ] = [bpb† ′ ] = δpp′ .

P µ =

pµ(ap† ap + bp† bp + 1),

Q = e (a†pap − b†pbp).

p1 p2

e2Q2AµAµ|ϕ|2

jµ

ϕ2A2

ieQ(p1+p2)µ

ψp

π±

∂µ → ∂µ − ieQAµ,

L ′ = −Aµjµ + e2Q2AµAµ|ϕ|2,

ϕ2A

ϕ2A2

2ie2Q2gµν

ψn

±e π0

¯	ψnπ	+	¯	ψpπ	−	¯	0	¯	0	,
ψpγ5	ψnπ		, ψnγ5	ψpπ		, ψpγ5	ψpπ	, ψnγ5	ψnπ	,

γ5

ψγ5ψ

¯	ψπ	P
ψγ5

ψψϕ

ψ =

ψn ,

ψp

{πa} a = 1, 2, 3

π1

(π+

+ π−), π2

(π+ − π−), π3 = π0

√

−1

τ1 =

, τ2 =

−i

, τ3 =

	τa	α
L ′ = gψ¯αγ5	τa	β ψβπa,
L ′ = gψ¯αγ5	2	β ψβπa,
a		α β

τ+ = 2(τ1		+ iτ2) =	0	0	,	τ− = 2(τ1		− iτ2) =	1	0	,
1			0	1		1			0	0

ˆ ˆ		1	ˆ		τ3
Q = T3	+	2	Y, T3	=	2	,
Y = 1					1		0
Y = 1					0		1

ψ → ψ′ = Uψ, U = e− 2i τaωa ,

SU(2)

τaπa → UτaπaU†.

SU(N)

mp ≈ 938.3

mn ≈ 939.6

iδab/(p2 − µ2 + i0)			µ
a	α
−igγ5 τ2	β	α	β

		SU(N)
N	ψ
ψ → ψ′ = Uψ	N × N		U
SU(N)
	U = eiωata , a =			,
	U = eiωata , a =		1, N2 − 1	,
{ωa}	U		ta			U UU† = U†U = 1
	SU(N)					U UU† = U†U = 1
	U detU = 1					ta
(ta)† = ta, Sp(ta) = 0, Sp(tatb) =					1	δab,
(ta)† = ta, Sp(ta) = 0, Sp(tatb) =					2	δab,
					2
N2 −1	ta

a =

√

, ta

Sp( a b) =

21 δab

N × N

M = Ca a = 2Sp(M a) a,

M b

Sp(M b) = CaSp( a b) =

Caδab

Cb.

Mδ δαδγ

−

2( a)α( a)γ = 0,

δ β

δδαδβγ − N δβαδδγ .

(ta)βα(ta)δγ = 2

SU(N)

tata	a
(tata)α = (ta)α(ta)γ		=	N2 − 1	δα.
β	γ β		2N	β
			2N

{ta}

[tatb] = ifabctc,

fabc

[[tatb]tc] + [[tbtc]ta] + [[tcta]tb] = 0

				fabdfdce + fcadfdbe + fbcdfdae = 0.
					N2 − 1		N × N {λa}					ta =		λa
SU(2)					N2 − 1		N × N {λa}					ta =		2
SU(2)
									εijk					SU(3)
t1 = 1 0 0			, t2 = i		0 0 , t3		= 0 −1 0 , t4			=		0 0 0 ,
	0 1 0			0	−i 0		1 0 0					0 0 1
t5 =	0 0	0		0	0 0	=	0 0	0			√13	1 0		0			,
t5 =	0 0 0		, t6 =	0 0 1 , t7		=	0 0 −i	, t8 =			√13	0 1		0			,
	0 0 −i			0 0 0			0 0 0					1 0 0
	i 0 0			0	1 0		0 i 0					0 0			−2
	i 0 0			0	1 0		0 i 0					0 0			−2
								1					√
	f123 = 1,	f147 = f165 = f246 = f257 = f					345 = f376 =	1	, f458		= f678 =			3		.
	f123 = 1,	f147 = f165 = f246 = f257 = f					345 = f376 =	2	, f458		= f678 =					.
								2						2
			SU(N)				N										{T a}
							N2 − 1										{T a}

(T a)bc = −ifabc.

N	2	ˆ a	}
		{1, T

[T aT b] = ifabcT c.

U(1)	U = eiω		ψ′ = Uψ
	ψ	→	ψ′ = Uψ
		→

Aµ → A′µ = Aµ + 1e ∂µω

Dµψ = (∂µ − ieAµ)ψ → UDµψ

SU(N)

Dµ → UDµU†, Fµν =		1	[DµDν ] → UFµν U† = Fµν .

		e
¯ ˆ	2
ψDψ	Fµν
			SU(N) U = eiωata
Aa
µ

Dµψ = (∂µ − igAaµta)ψ → (∂µ − igA′µata)ψ′ = (∂µ − igA′µata)Uψ.

Dµψ → Dµ′ ψ′ = Dµ′ Uψ = UDµψ,

U†∂µU − U†igAµ′ataU = −igAµa ta,
Aµa ta → UAµa taU† +			i		†,
				U∂µU
			g
−U∂µU†					U	U†∂µU =
		U ≈ 1 + iωata
δAµa =	1	Dµωa,

	g

Dµωa ≡ Dµacωc = (∂µ − igAbµT b)acωc = (∂µδac − gfbacAbµ)ωc = ∂µωa + gfabcAbµωc,


Z	d4xfaDµϕa = Z		d4xfa(∂µϕa + gfabcAµb ϕc) = Z		d4x(−ϕa∂µfa + gfabcfaAµb ϕc) =
= Z		d4x(−ϕa∂µfa + gfcbafcAµb ϕa) = Z		d4x(−ϕa∂µfa − gfabcϕaAµb fc) = − Z		d4xϕaDµfa.

Dµ → UDµU†.

Fµν = Fµνa ta = g1[DµDν ],

Fµνa = ∂µAaν − ∂ν Aaµ + gfabcAbµAcν .

Fµν → UFµν U†,

SU(N)

L = −

¯ ˆ

a a

Sp(Fµν ) + ψ(iD − m)ψ = −

(Fµν )

+ ψ(i∂ + gAµt

− m)ψ.

− m)ψ = 0,

DµFµν

= Jν

, (i∂ + gAµt

Jµ

= ψγµt ψ,

∂µJµa 6= 0

DµJµa = 0.

SU(3)

A3 A4	(F a	)2
	µν

•

−igγµ(ta)αβ

b ν k2

•	−gfabc[(k1 − k2)λgµν +

a µ k1

c λ k3

(k2 − k3)µgνλ + (k3 − k1)ν gλµ]

a µ k1

b ν k2

•

d σ k4 c λ k3

−ig2[fabcfcde(gµλgνσ − gµσgνλ) + facefbde(gµν gλσ − gµσgνλ) + fabcfcde(gµν gλσ − gµλgνσ)]

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