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Квантовая теория поля

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2 × 2

Spρ = 1.

hθi = Sp(θρ).

ραβ = wαwβ, Spρ = wαwα = w†w = |w1|2 + |w2|2 = 1.

hθi = Sp(θρ) = θαβ ρβα = wαθαβ wβ = w†θw.

P = Sp(σρ),

hSi = 12P ,

{I, σ}

2 × 2

ρ = C0I + Cσ.

σi

C0 C

ρ= 12ISpρ + 12Sp(σρ)σ,

ρ= 12(1 + P σ).

ραβ = uαu¯β, Spρ = uu¯ = 2m.

ρ = u u¯

a(0)µ = (0, P ),

a0(0) + vak(0)

a(0) =

|p|

(pP )

√1 − v2

√1 − v2 k

a(0) + va0(0)

(0)

E (pP )

ak =

√

m |p|

1 − v2

a = a(0),

a0 =

(pP )

a = P +

p(pP )

m(m + Ep)

aµpµ = 0.

a2µ = −P 2.

(ˆp − m)ρ = ρ(ˆp − m) = 0.

			ρ = N(ˆp + m)(1 + Aγ5aˆ)(ˆp + m),
	aµ
γ5			aµ
			Spρ = 2m,
	N = 1/(4m)						A
hSi =	Z d3xψ†	2	Σ ψ = 4Ep uγ¯ 0Σu =	4Ep uγ¯	5γu =	4Ep Sp[(uu¯)γ5γ].
		1	1	1		1

uu¯ → ρ

hSi = 4Ep Sp[ργ5γ].

hSi = −12AP .

A = −1

ρ = 4m(ˆp + m)(1 − γ5aˆ)(ˆp + m),

(ap) = 0

1 1

ρ = 2(1 − γ5aˆ)(ˆp + m) = 2(ˆp + m)(1 − γ5aˆ).

ρc

ρcαβ = ucαu¯cβ = (Cu¯)α(Cu)β = Cασu¯σCβρuρ = CασρTσρCρβT = −(CρT C†)αβ,

ρc = −CρT C†,

CT = −C

ρc =

[(ˆp − m)(1

− γ5aˆ)] .

λ uλu¯λ

λ uλc u¯λc

uλ uλc

uu¯ → ρ

u¯

→ ρ

P = 0

huu¯i hucu¯ci

aµ = 0

huu¯i =

uλu¯λ = ρ(aµ = 0) =

(ˆp + m),

hucu¯ci =

uλc u¯λc = ρc(aµ = 0) =

(ˆp − m),

uλc (p)¯uλc (p) = pˆ − m.

uλ(p)¯uλ(p) = pˆ + m,

(ˆp − m)u = u¯(ˆp − m) = (ˆp + m)uc = u¯c(ˆp + m) = 0,

p0 = Ep = p2 + m2

aµ = 0

pˆ m

(ˆp − m)ρ = 0, Spρ = 2m,

ρc

ρ2 = (u u¯)(u u¯) = u (¯uu) u¯ = 2m(u u¯) = 2mρ,

ρ2 = ρ

a2µ = −1

|P | = 1

ρ2 = ρ

γ0	=	0	−1	,	γi =	−σi	0i	, i = 1, 2, 3.
		1	0			0	σ

u(p) = √

σp

= √

p m √

0 ,

Ep+m

→

| |

ψ → ψ′ = Uψ, UU† = 1.

γµ → γµ′ = UγµU†.

i∂ψ′

= (α′p + β′m)ψ′

∂t

∂ξ

− (σp)ξ = mη

∂t

∂η

+ (σp)η = mξ,

∂t

uλ(p) → u˜λ(p) =

2Ep

χλ

ϕλ

u˜λ(p) =

√2

(σn)ϕλ ≡

χ˜λ .

ϕλ

ϕ˜λ

ξλ = √2(ϕ˜λ

+ χ˜λ) =

2(1 + (nσ))ϕλ =

(1 + 2λ)ϕλ

ηλ = √2(ϕ˜λ − χ˜λ) =

2(1 − (nσ))ϕλ =

2(1 − 2λ)ϕλ.

ξ1 = ϕ1 , η

1 = ϕ

−

1 , ξ

−

1 = η1 = 0,

−

U = √2

−1 .

0 1

= −γi, i = 1, 2, 3,

γ0′

1 0 = −γ5, γi′ =

σi

−0 i

γ5′ α′

γ′

= i(γ0)′(γ1)′(γ2)′(γ3)′ = γ0

−1 0

, α′ = γ′ γ′ = γ5γ =

σ 0 .

−

0 1

0 −σ

≡

ψ′ = U χ

= √2

+ χ

η .

−

γ0′

m = 0

γ5

ψR =

− γ5)ψ, ψL

(1 + γ5)ψ, ψ = ψR + ψL.

, ψ =

γ5 = − 1 0

ϕ + χ

ψR =

ϕ + χ

, ψL =

χ −

−

γ′

−1

, ψ′ =

, ψL′ =

ψR′

= 0

η .

ψγµψ = ψRγµψR + ψLγµψL, ψψ = ψRψL + ψLψR,

	¯		¯	¯
mDψψ = mD(ψRψL + ψLψR)
(1)	¯C	¯	C	(2)	¯C	¯	C	(1)	(2)	χ¯RχR,
mM	(ψL	ψL + ψLψL ) + mM			(ψR	ψR + ψRψR ) = mM			χ¯LχL + mM	χ¯RχR,
			χL = ψL		+ ψC , χR = ψR + ψC ,
						L		R
									χLC = χL	χRC = χR

γ5

•

apλuλ(p)e−ipx + b

uc (p)eipx

ψ(x) =

pλ

p,λ

2Ep

uλ(p) =

p m (σn)ϕλ ! , u¯λuλ = 2m, u¯λγµuλ = 2pµ

E + m ϕλ

−

uλc (p) = Cu¯λT (p), u¯λc uλc = −2m, u¯λc γµuλc = 2pµ.

•

T µ0

ψ†(∂µψ) − (∂µψ†)ψ .

•

jµ = eψγµψ.

Z	Z	X
Q = d3xj0	= e d3xψ†ψ = e		(a	apλ + bpλb	),
			pλ	pλ
		p,λ
P µ = Z d3xT		µ0 = p,λ pµ(ap	λapλ − bpλbpλ).
		X

u†λ(p)uλ′ (p) = ucλ†(p)ucλ′ (p) = 2Epδλλ′ , u†λ(p)ucλ′ (−p) = 0,

	u†(p)uc(−p) = u¯(p)γ0uc(−p) =				Ep u¯(p)(γp + m) uc(−p) =
						1
=	1	u¯(p) [(γp + m)uc(−p)] =	1	u¯(p)[−Epγ0uc(−p)] =			−u†(p)uc(−p).

	Ep		Ep

{. . .}

{apλ {apλ {apλ

[. . .]+

a† λ } = {bpλb† λ } = δpp′ δλλ′ ,

p′ ′ p′ ′

ap′λ′ } = {a† λa† λ } = {bpλbp′λ′ } = {b† λb† λ } = 0,

p p′ ′ p p′ ′

bp′λ′ } = {a† λb† λ } = {apλb† λ } = {a† λb† λ } = 0,

p p′ ′ p′ ′ p p′ ′

2 × 2				n					2n × 2n
a =	1	0	,	a† =	0	0	,	\|0i =	1	, \|1i =	0 .
	0	0			0	1			0		1

a2 = (a†)2 = 0, {aa†} = 1, {aa} = {a†a†} = 0, h0|0i = h1|1i = 1, h0|1i = 0,

a|0i = a†|1i = 0, a†|0i = |1i, a|1i = |0i. N = a†a

N|0i = 0|0i, N|1i = 1|1i, [Na] = −a, [Na†] = a†.

	Q = e p,λ	Np(+)λ	− e p,λ Np(−λ ) + Q0,			P µ = p,λ		pµ Np(+)λ		+ Np(−λ ) + P0µ,
	X		X				X
N	(±)
	pλ
			N(+)	= a†	apλ,	N(−)	= b†		bpλ,
			pλ	pλ		pλ	pλ
			X			X			b	b†
			X			X			X
			Q0 = e	1, P0µ		= pµ = 2gµ0 Ep.
			p,λ			p,λ			p
						Q0	P00

{ ¯ }

ψα(x)ψβ (y)

•


		jµ =		1		e ψγ¯ µψ − ψ¯cγµψc .

					2
Q =	1	d3x(ψ†ψ − ψc†ψc) =	e	p,λ			[ap†	λapλ] − [bp† λbpλ]	= e(N(+) − N(−)),
Q =	2e Z	d3x(ψ†ψ − ψc†ψc) =	2	p,λ			[ap†	λapλ] − [bp† λbpλ]	= e(N(+) − N(−)),
				X

[a†a] = a†a − aa† = 2a†a + 1 = 2N + 1,

•

N(a†a) = a†a, N(aa†) = −a†a, N(b†b) = b†b, N(bb†) = −b†b,

Q = e Z d	xN(ψγ0		ψ) = e p,λ	(N(apλapλ) + N(bpλbpλ)) = e p,λ				(Npλ	− Npλ	),
3		¯	X	†	†		X	(+)	(−)
			X				X
P	µ =	1	d3xN ψ†i(∂µψ) − i(∂µψ†)ψ		=	p,λ	pµ Np(+)λ + Np(−λ ) .
P	µ =	2 Z	d3xN ψ†i(∂µψ) − i(∂µψ†)ψ		=	p,λ	pµ Np(+)λ + Np(−λ ) .
						X

: AB := N(AB).

				¯
				{ψα(x)ψβ(y)}
ψ(x) = p,λ		1	apλuλ(p)e−ipx	+ bp† λuλc (p)eipx , px = Epx0 − px,
ψ(x) = p,λ		2Ep	apλuλ(p)e−ipx	+ bp† λuλc (p)eipx , px = Epx0 − px,
X
	p	¯
		ψ
			¯	¯
		ψ(x) = a(x) + b(x), ψ(x) = a¯(x) + b(x).
			¯	¯

{ψα(x)ψβ (y)} = {aα(x)¯aβ(y)} + {bα(x)bβ (y)}.

{ψα(x)ψβ (y)}

(+)

(x − y),

{aα(x)¯aβ (y)} = −i(i∂ + m)αβ

(+)(z) = i

d3p

p 2Ep e−iEpz0+ipz

= i Z (2π)3 2Ep e−iEpz0+ipz,

pµ

i∂/∂xµ = i∂µ

{bα(x)bβ (y)}

(−)

− y),

{bα(x)bβ (y)} = −i(i∂ + m)αβ

(−)(z) = −i

= −i Z

d3p

+ipz,

2Ep eiEpz0−ipz

(2π)3 2Ep eiEpz0

p → −p

(±)(z)

(±)(z) = (±2πi) Z

d4p

θ(±p0)δ(p2 − m2)e−ipz,

(2π)4

θ(p0)

= 2π Z

e iEpz0

θ(±p0)δ(p2 − m2)e−ip0z0 ,

2Ep

2π

(pz)

(±)(z)

{ψα(x)ψβ (y)} = −i(i∂ + m)αβ Δ(x − y),

Δ(z) =

(+)(z) +

(−)(z) = 2πi Z

d4p

ε(p0)δ(p2 − m2)e−ipz,

(2π)4

ε(p0)

+1,

p0 > 0

ε(p0) = −1, p0 < 0.

{ψα(x)ψβ (y)}|x0=y0

{ψα(x)ψ¯β (y)}|x0=y0 = −i iγ0

∂Δ(z)

|z0=0 + (iγ + m)Δ(z)|z0=0 αβ ,

∂z0

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