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12
ψ
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S l(k0)
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U (1)
GR/A p (z ) GR/A
p (ω)
G
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c ≈ 3 · 108 m
s
t (x,y,z )
(x0
, x1
, x2
, x3
)
x0 ≡ ct, x1 ≡ x, x2 ≡ y, x3 ≡ z.
µ, ν
xµ = (x0, x) = (x0, x1, x2, x3).
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gµν =
1 0 0 00 −1 0 0
0 0 −1 0
0 0 0 −1
.
xµ xν
xµ =3
ν =0
gµν xν xµ =3
ν =0
gµν xν
xµ
xν
xµ = gµν xν
µ, ν = 0, 1, 2, 3
a, b = 1, 2, 3
aµ
bµ
aµbµ = aµbµ = a0b0 − a · b,
a · b
aµ
aµaµ =
a0
2 − |a|2
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aµaµ =
< 0 aµ
= 0 aµ
> 0 aµ
s2 = aµaµ = aµgµν aν = (ct)2 − |a|2
τ
τ =s
c=
(ct)2 − |x|2
x = const.≡
0
τ
τ
S, S ′
= s
aµaµ = 0
g
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x
ct
world line of (accelerated)
light cone
(world line of a photon)
x=ct
world line of a free, massive
particle: v=x/t<c
massive particle: |slope|>1 always
L
xµ
S ′ −v v xL
s2 = x′0
x′1 x2 x3
γ =1
1 − vc
2, β =
v
c
L
(Lµν ) =
γ βγ 0 0
βγ γ 0 0
0 0 1 0
0 0 0 1
(L) = ±1.
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(L) = +1 L00 ≥ 1 →
(L) = −1 L00 ≥ 1 →
(L) = +1 L00 ≤ −1 →
(L) = −1 L00 ≤ −1 →
i ∂
∂tψ(x, t) =
− 2
2m∆ + V (x)
ψ(x, t)
x0 xa, a = 1, 2, 3
→
→→ →
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→
ψ(xµ)
E =
p2c2 + m2c4.
i ∂
∂tψ =
− 2c2 ∇2 + m2c4ψ,
p = i
∇.
− 2 ∂ 2
∂t2ψ =
−
2c2 ∇2 + m2c4
ψ
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∂ µ =
∂
∂xµ ∂
µ
=
∂
∂xµ ,
xµ = (ct, x1, x2, x3) xµ = (ct, x1, x2, x3) = (ct, −x1, −x2, −x3)
i ∂ ∂ (ct)
= i ∂ ∂x0 = i ∂ 0
i ∂ ∂xa
= −i ∂ a, a = 1, 2, 3
pµ −→ i ∂ µ = i
∂
∂ (ct)
− ∇
− 2 ∂ 2
∂ (ct)2ψ(x, t) = (−
2 ∇2 + m2c2)ψ(x, t)
⇔
∂ µ∂ µ +mc
2
ψ(xµ) = 0
2
:= ∂ µ∂
µ
= ∂
2
0 − ∇
2
mc
ψ(x, t) = exp
− i
(Et − px)
= exp
− i
pµxµ
,
E = ±
p2c2 + (mc2)2.
p
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kµ
= pµ
ψ(xµ) =
d4k√
2π4 δ 4
kµkµ −
mc
2
A(kµ)e−ikµxµ.
A(kµ) k0 < 0
E = +
p2c2 + (mc2)2 > 0
ψ(+)(x, t) = e−i
(Et− p· x)
ψ(−)(x, t) = e−i
(−Et− p·x) = e−i
(E (−t)− p·x)
E < 0
∂/∂x
∼ O( p2) p
∂/∂ (ct)
ρ = ψ∗ψ
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=
2mi ψ∗
∇ψ
−ψ
∇ψ∗ .
∂
∂tρ + ∇ · = 0.
ψ∗
ψ∗
∂ µ∂ µ +mc
2
ψ = 0.
ψ
∂ µ∂ µ +
mc
2
ψ∗ = 0.
∂ µ (ψ∗∂ µψ − ψ∂ µψ∗) = 0,
∂
∂t i
2mc2 ψ∗∂
∂t ψ − ψ
∂
∂t ψ∗ ρ
+ ∇ ·
2mi ψ∗ ∇ψ − ψ ∇ψ∗
= 0.
ρ
ρ
ρ(x, t)
ρ(ω, x) = ω
mc2ψ∗ψ
ρ
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12
12
E = p2c2 + (mc2)2
E 2 = c2 p2 + (mc2)2 != (cα · p + βmc2)2,
α = (αx, αy, αz), β α, β
c2( p2x + p2
y + p2z) + m2c4 = c2(α2
x p2x + α2
y p2y + α2
z p2z) + β 2m2c4
+c2 px py(αxαy + αyαx) + c2 py pz(αyαz + αzαy)
+c2 pz px(αzαx + αxαz) + mc3[ px(αxβ + βαx)
+ py(αyβ + βαy) + pz(αzβ + βαz)]
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12
αx, αy, αz, β
α2i = β 2 = 1
αiα j + α jαi =: [αi, α j]+ = 0
αiβ + βαi =: [αi, β ]+ = 0
i = x,y,z
i, j = x,y,z
i = x,y,z
α, β
[M µ, M ν ]+ = 2δ µν 1
M µ = β, αxαyαz.
αx, αy, αz, β
H = c α · p + βmc2
M µ λ = ±1
µ = ν
(M µ)2 = 1
M µ ⇒ λ = ±1
2
M µ
(M µ) = 0, µ = 0, 1, 2, 3
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µ = ν
M µM ν = −M ν M µ
⇔ M µM µ 1
M ν = −M µM ν M µ
⇒ (M ν ) = − (M µM ν M µ)
= − (M ν M µM µ =1
)
= − (M ν )
= 0
2
αx, αy, αz, β
0 = (M µ) =d
i=1
λi =d
i=1
(±1) ⇔
2
αx, αy, αz, β d ≥ 4 d = 2
d = 4
α =
0 σ
σ 0
, β =
1 0
0 −1
σ = (σx, σy, σz)
σx =
0 1
1 0
, σy =
0 −i
i 0
, σz =
1 0
0 −1
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α, β
i ∂
∂tψ(xµ) = (c α · p+βmc2)ψ(xµ)
p = −i ∇.
αx, αy, αz, β ψ(xµ)
• ψ(xµ)
• ψ(xµ)
2×2 = 4 = d
d = 2 · (2S + 1)
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i ∂
∂tψ = (ca · p + βmc2)ψ
1c
β
(−i β∂ 0ψ + i βαi∂ iψ + mc)ψ = 0
γ 0 = β
γ i = βαi, i = 1, 2, 3.
α, β
−iγ µ∂ µ +
mc
ψ = 0
mc
γ µ∂ µ = γ µ∂
∂xµ
γ µ =
β
β α
γ ′µ = Lµ
ν γ ν
γ µuµ = γ 0u0 − γ · u =: /u
−i/∂ +
mc
ψ = 0
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γ
γ 0 = β (γ 0)2 = 1
γ i, i = 1, 2, 3 (γ i)2 = −1
(γ i)† = (βαi)† = αiβ = −βαi = −γ i
(γ i)2 = βαiβαi = −ββαiαi = −1
2
γ
[γ µ, γ ν ]+ = 2gµν 1
a, β γ
γ 0 = 1 0
0 1
, γ i =
0 σi
−σi 0
, i = 1, 2, 3
γ µ = Aγ µA−1,
ψ
ψ =
ψ1
ψ4
, ψ† = (ψ∗1 , . . . , ψ∗
4)
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ρ = ψ† · ψ =
4
i=1
ψ∗i ψi,
∂
∂t(ψ†ψ) =
∂
∂tψ†
ψ + ψ†
∂
∂tψ
.
i
∂
∂tψ
= (−i c α · ∇ + βmc2)ψ
ψ†
i ψ†
∂
∂tψ
= (−i cψ† α · ∇ + βmc2ψ†)ψ.
ψ
−i
∂
∂tψ†
ψ = i
∇ψ†
· c αψ + mc2ψ†βψ
i ∂ ∂tψ†ψ = −i ψ†(ca) · ( ∇ψ) + ( ∇ψ†) · (c α)ψ
= − ∇ · ψ†(ca)ψ
∂
∂tρ + ∇ · = 0 ∂ µ j
µ = 0,
( j
µ
) = cρ
, (∂ µ) = ∂
∂ (ct)
∂ ∂x .
ρ = ψ†ψ
= ψ†(c α)ψ = vρ
v := c α
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12
jµ
j′µ = Lµν j
ν .
• −→• −→
•• −→
ψ
E 2 = p2c2 + (mc2)2
E/c =
i ∂/∂ (ct) p = −i ∇ E − p
pµ pµ =
E
c
2
− p2 = (mc)2,
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ψ
ψ
γ ψ
ψ
xµ′ = Lµν x
ν x′ = L x
ψ L
ψ′(x′) = S (L)ψ(x)
= S (L)ψ(L−1(x′))
4 × 4
−iγ µ∂ µ +
mc
ψ(x) = 0 I
−iγ µ∂ ′µ +
mc
ψ′(x′) = 0 I ′
∂ µ =∂
∂xµ∂ ′µ =
∂
∂xµ′ .
γ
γ µ
γ µ = Aγ µA−1 .
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12
γ µ
γ µ′ = T (L)γ µT −1(L).
γ ′
γ ′ = γ µ ⇒ A = T −1(L)
2
∂ µ = ∂ ∂xµ
= ∂xν ′
∂xµ∂
∂xν ′ = Lν µ∂ ′ν
xν ′ = Lν µxµ
∂xν ′
∂xµ= Lν
µ
ψ ψ′
S −1ψ′(x′) = ψ(x).
−iγ µLν
µ∂ ′ν +mc
S −1ψ′(x′) = 0.
S
−iSLν µγ µS −1∂ ′ν ψ
′(x′) +mc
ψ′(x′) = 0.
S (L)
SLν µγ µS −1 = γ ν
S −1(L)γ ν S (L) = Lν µγ µ
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S (L)
Lν µ = δ ν
µ + ∆ων µ,
∆ων µ
L00 = 1
Lab = cos(φ)
Lν ν = cosh(φ)
⇒ Lν ν = 1 + O(φ2)
Lν
µ ∼ sin(φ)
sinh(φ) = O(φ), ν = µ
⇒ ∆ων ν = O + O(φ2)
(∆ων µ) φ
S (L) ∆ων µ
S = 1 + τ S −1 = 1− τ τ
(1− τ )γ µ(1 + τ ) = γ µ + γ µτ − τ γ µ + O(τ 2)
= γ µ + ∆ωµν γ ν
[γ µ, τ ] = ∆ωµν γ ν
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τ 1
τ 1, τ 2
[γ µ, τ 1 − τ 2] = 0.
τ 1 − τ 2 = α · 1 α ∈ R
2
τ
• S (L)
ψ†(x)ψ(x) =4
α=1
ψ∗α(x)ψα(x)
ρ =
ψ†(x)ψ(x)x ρ
(S ) = 1,
1 = (S ) = (1 + τ ) = (1) + (τ )
= 1 + (τ ) + O(τ 2)
(τ ) = 0
τ =1
8∆ωµ
ν ′gν ′ν (γ µγ ν − γ ν γ µ) =
1
8∆ωµ
ν ′gν ′ν [γ µ, γ ν ]
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Lν µ = δ ν
µ + ∆ων µ
S (L) = 1 +1
8∆ωµ
ν ′gν ′ν [γ µ, γ ν ]
•
(ων µ)
δ ν
µ +
η
N ων
µ, N → ∞
η
Lν µ =
lim
N →∞
1 +
η
N ωN ν
µ
= (eηω )ν µ
−β = −v/c
ω =
0 1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
=: (τ 01)x 101 :=
1 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
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(Lν µ) = 1 +
∞k=1
1
k! (ητ x1)k
= 1 − 101 +∞
k=0
1
(2k)!η2k101
+∞
k=0
1
(2k + 1)!η2ki+1(τ 01)x
(Lν µ) = 1 − 101 + cosh(η)101 + sinh(η)(τ 01)x
= cosh(η) sinh(η) 0 0
sinh(η) cosh(η) 0 00 0 1 0
0 0 0 1
η
v/c
tanh(η) =v
c= β,
cosh(η) = γ =1
1 − vc2
,
sinh(η) = βγ.
•
S (L) = limN →∞
[1 +η
N
1
8(ωµ
ν ′gν ′ν [γ µ, γ ν ]
(∗)
)]N
= expη · 18
(ωµν ′g
ν ′ν [γ µ, γ ν ])∗ 4×4
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−v/c x
ωµν ′g
ν ′ν [γ µ, γ ν ] =
0 1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
µν
× [γ µ, γ ν ]
= 4α
0 1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
=
0 −1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
[γ µ, γ ν ] = [γ µ, γ ν ]+− 2γ ν γ µ
= 2gµν 1
−2γ
ν γ
µ
[γ 0, γ 1] = −2γ 1γ 0
= −2
0 −σ1
σ1 0
1 0
0 −1
= −2
0 σ1
σ1 0
= −2α1
S (L)
S (L) = expη
2α1
=∞
k=0
1
(2k)!
η
2
2k
+∞
k=0
1
(2k + 1)!
η
2
2k+1
α1
S (L) = cosh(η/2)1+ sinh(η/2)α1
S (L) =
cosh(η/2) 0 0 sinh(η/2)
0 cosh(η/2) sinh(η/2) 0
sinh(η/2) 0 0 cosh(η/2)
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x tanh(η) = v/c
•
jµ = c ψ†γ 0 ψ
γ µψ = cψ†
1
α
ψ
S †
S −1
S †γ 0
= bγ 0
S −1
b =
+1, L00 ≥ 1
−1, L00 ≤ −1
ψ := ψ†γ 0,
jµ
ψ′ = Sψ
ψ′ = ψ†S †γ 0 = bψ†γ 0S −1 = bψS −1
jµ = cψγ µψ
jµ′ = cbψ S −1γµS
=Lµ
νγ ν
ψ = cbLµν ψγ ν ψ = bLµ
ν jν
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ψψ = ψ†γ 0ψ
ψ′ψ′ = bψS −1Sψ = bψψ
i
∂
∂t ψ = [c α · p + βmc
2
]ψ
ψ(x) −→ ψ′(x) = e−iθ(x)
ψ(x)∂
∂ (ct)−→ ∂
∂ (ct)+ i
∂θ
∂ (ct)=: Dt
∂
∂x−→ ∂
∂x+ i
∂θ
∂x=: Dx
pµ = i ∂
∂xµ−→ pµ −
∂θ
∂xµ= Πµ
x = ctx .
(∂θ/∂xµ)
Πµ = pµ − q
cAµ
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q
cAµ =
∂θ
∂xµ
q
Πµ = gµν Πν = pµ − q
cAµ
cΠ0 = i ∂
∂t− q Φ
Π = −i
∂
∂x −q
c A
i ∂
∂tψ = [c α ·
p − q
c A
Π
+βmc2 + q Φ]ψ
v ≪ c Φ = 0
Eψ = c α · Π + βmc2ψ.
ψ(t) = ψe−i
Et
ψ(x) =
χ(x)
Φ(x)
,
χ, ΦE − mc2 −cσ · Π
−cσ · Π E + mc2
χ
Φ
= 0
(E − mc2) E s
χ − cσ · Π = 0
(E + mc2)
E s+2mc2
Φ − cσ · Πχ = 0
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E s mc2
E
v ≪ c q
c A ≪ | p|
E s ≈ p2
2m≪ mc2
| Π| ≈ m|v| ≪ mc
Φ =
cσ · Π
E + mc2
χ
E + mc2 = E s + 2mc2
≈2mc2
c · | Π| ≈ mc|v|Φi
χi
∼= 1
2
v
c≪ 1, i = 1, 2
Φ ∼= 1
2mc(σ · Π)χ
χ Φ
E sχ =1
2m(σ · P i)(σ · P i)χ
E s
−→i
∂
∂t
i ∂
∂tψ =
1
2m(σ · Π)(σ · Π)ψ
(σ · u)(σ · v) = u · v + iσ · [u × v]
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u v
[ Π × Π] = p −q
c A× p −q
c A= [ p × p]
=0
−q
c[ A × p] − q
c[ p × A] +
q 2
c2[ A × A]
=0
= −q
c[ A × p] − q
c[ p × A] +
q
c[ A × p]
=i q
c[ ∇ × A]
=i q
c B
i ∂
∂tχ =
1
2m( p − q
c A)2 − q
mc =2µB
·1
2σ · B
χ
12
σqmc
= 2µB = gµB
1
2 g = 2
µB =q
mc
= c = 1
(−
iγ µ∂ µ + m)ψ = 0
m
, c
m −→ mc
=
1
λ
−→ mc2 = E 0
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pψ = 0
(−iγ 0∂ 0 + m)ψ = 0
γ
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1
(−i∂ 0) +
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
m
ψ = 0
ψ(+)(x) = ur(E = m, p = 0)e−imt r = 1, 2
ψ(−)(x) = vr(E = m, p = 0)e+imt m =
u1(m, 0) =
1
0
0
0
, u2(m, 0) =
0
1
0
0
v1(m, 0) =
0
0
1
0
, v2(m, 0) =
0
0
0
1
( pµ) =
m 0
E > 0(u1, u2) E < 0(v1, v2)
p
−v
x p px = p
p v
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E p = γ βγ
βγ γ m
0 = γm
βγm
γ =1
1 − β 2
β =v
c= v
px = βγ m
γm
L µ
x ν
=
cosh η sinh(η) 0 0
sinh(η) cosh(η) 0 0
0 0 1 0
0 0 0 1
.
S (Lx
) = cosh(η/2)1 + sinh(η/2)α1,
u′1(E, p) =
cosh(η/2)
0
0
sinh(η/2)
, u′2(E, p) =
0
cosh(η/2)
sinh(η/2)
0
v′1(E, p) = 0
sinh(η/2)cosh(η/2)
0
, v′2(E, p) = sinh(η/2)
00
cosh(η/2)
u′i, v′i, i = 1, 2 E
p
cosh(η) = γ =E
msinh(η) = βγ =
px
m(E > 0)
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1/2
cosh(η/2) = 12
(cosh(η) + 1) = 12m
(E + m)
sinh(η/2) = (η)
1
2(cosh(η) − 1) = ( px)
1
2m(E m)
= ( px)
1
2m
E 2 − m2
E + m= px
1
2m(E + m)
u1(E, px) =
1
2m(E + m)χ1
px
1
2m(E +m)χ2
u2(E, px) =
12m
(E + m)χ2
px
1
2m(E +m)χ1
v1(E, px) =
px
1
2m(E +m)χ2
12m
(E + m)χ1
v2(E, px) =
px
1
2m(E +m)χ1
1
2m(E + m)χ2
χ1 =
1
0
χ2 =
0
1
.
v ≪ c
u1, u2 ≈ 1,
u1, u2
≈v
2c ≪1,
v1, v2 ≈ v
2c≪ 1,
v1, v2 ≈ 1.
p
pxχ2 −→ σ · pχ1
pxχ1 −→ σ · pχ2.
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12
px
e−imt′ = e−ip0′x0′ = e−ipµxµ
= e−i(Et− px)
e+imt′ = e+ip0
′x0′ = e+ipµxµ
= e−i(−Et+ px)
p
ψ(+) p,r (x) = ur(E, p)e−i(Et− px)
ψ(−) p,r (x) = vr(E, p)e−i(−Et+ px)
E = +
p2 + m2
H 2D = −i ∂ ∂t
H D
ur, vr
ur(k)us(k) = δ rs r, s = 1, 2
vr(k)vs(k) = −δ rs
ur(k)vs(k) = 0
vr(k)us(k) = 0
Lµν = δ µν + ∆ωµ
ν
S (L) = 1 +1
8∆ωµν [γ µ, γ ν ]
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∆ ϕ, |∆ ϕ| =
∆ωij = −εijk ∆ϕk (∆ω0µ = ∆ωµ
0 = 0)
σµν =i
2[γ µ, γ ν ]
σij = σij = εijk Σk
Σk = σk 0
0 σ
k σk
S (L) = 1− i
4∆ωµν σµν
ψ′(x′) = L {ψ(x)} = Sψ(x) = Sψ
L−1x′
β ±1
E < 0
E > 0
E < 0
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12
ψ(x) =
d4 p
(2π)4δ ( p2
0 − E 2)
s=1,2
(2π)2mb( p, s)W s( p) e−ipµxµ
d4 p(2π)4
δ ( p20 − E 2) = d3 p
E
ψ
2m
b( p, s) E =
| p|2 + m2
W s( p) =
us( p), p0 > 0
vs( p), p0 < 0.
δ ( p20 − E 2) = 1
2 p0[δ ( p0 − E ) + δ ( p0 + E )]
ψ(x) = d3 p(2π)3
mE
s=1,2
b( p, s)us( p)e−ipµxµ + d∗( p, s)vs( p)e+ipµxµ
b( p, s) = 2πb(E, p, s)
d∗( p, s) = 2π
b(−E, p, s).
E > 0
E > 0
d∗ ≡ 0 p0 = E
ψ(+)(x) =
d3 p
(2π)32
m
E
s=1,2
b( p, s)us( p)e−ipµxµ
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J (+) =
d3x
(2π)3 (+)(x)
= c
d3x
(2π)32
ψ(+)†(x) αψ(+)(x)
= c
s
d3 p
(2π)3
m
E
b( p, s)2 p
E
= p
E =
vG
vG = ∂E ∂ p
=∂ √ | p|2+m2
∂ p= p
E .
u v
E > 0 E < 0
E > 0 E < 0
t = 0
E > 0 d
4d
ψ(t = 0, x) =1
(2πd2)34
eix k− x2
(2d)2 w
w E > 0 w =
χ1
0
=
1
0
0
0
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12
4d
k
eix k− x2
(2d2) =
d3 p
(2π)3
4πd2
2π
32
e−d2( p− k)2
p k
b( p, s) = 232 d3e−d2( p− k)2u†s(vp)w = 0
d∗( p, s) = 232 d3e−d2( p− k)
2
v†s( p)w = 0
ψ(t, x) E > 0 E < 0
E < 0 E > 0
χC
d ≪ χC =
mc
,c=1=
1
m.
w =
χ1
0
us vs
d∗( p)
b( p)=
| p − k|E + m
| k| ≪ 1
.
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• d ≫ 1m
| p − k| d−1 ≪ m ⇐⇒ d∗
b≪ 1.
• d ≪ 1m
| p − k| ≈ d−1 ≫ m =⇒ | p − k| ≈ E ⇐⇒ d∗
b≈ 1
x
E > 0
x =
d3x ψ†(x)xψ(x)
d
dtx =
d
dt
d3x ψ†(0, x)e+iHtxe−iHt ψ(0, x)
=
d3x ψ†(t, x)i [H, x]
−ic α
ψ(t, x)
=
d3x ψ†(x)c αψ(x) = J (t)
J i(t) = d3 p
(2π)3
m
E pi
E s |b( p, s)
|2 +
|d( p, s)
|2
+ is,s′
b∗( p, s)d∗( p, s)e2iEtus( p)σi0vs′( p)
− b( p, s)d( p, s)e−2iEtvs′( p)σi0us( p)
2E > 2mc2
= 2 × 1021s−1
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12
I II
x1=x
V(x1)=q (x1)φ
V0
V (x1) = const
E
E +m
2m
E > 0
ψin(x) = e−iEteipx
1
0
0 p
E +m
ψrefl(x) = e−iEtae−ipx
1
0
0− pE +m
u1
+be−ipx
0
1
− p
E +m
0
u2
E = +
p2 + m2 > m, p = +√
E 2 − m2.
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ψrefl
ψtrans(x) = eiEt
ceiqx
1
0
0q
E −V 0+m
+ deiqx
0
1q
E −V 0+m
0
q =
+
(E − V 0)2 − m2, |E − V 0| ≥ m
i
m2 − (E − V 0)2, |E − V 0| ≤ m.
E − V 0 + m
q = 0
E
q
Re(q)Re(q)
Im(q)
m+V0m−V0 V0
•
E ≥ m + V 0 > 0 E ≤ −m + V 0 ( E ≥ m : V 0 ≤ 2m)
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12
•
−m + V
0< E < m + V
0
∂ ∂xµ
ψI (0)!
= ψII (0)
ψ
1 + a = c
b = d
−b pE +m
= d qE −V 0+m
2.⇐⇒ b = d = 0
V 0 = 0 , p = q
(1 − a) = rc , r = q p
E +mE −V 0+m
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propagating
solutions
m0
E>m
E
x
E−V0<−mE<−m
(II)(I)
(2)
−m
E−V0>m
V0+mV +m
V0−m
m
(1)
0
m
• |E | > m
• |E
−V 0
|> m
E > m
V 0 > 0
1) E − V 0 ≥ m V 0 > 0 E − V 0 > 0
2)
E
−V 0
≤ −m
E ≥ m m
≤E
≤V 0
−m
E 0 > 2m E − V 0 < 0.
E ≥ m V 0 ≥ 2m
•V 0
•
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12
jtrans ⊥ jrefl
ψ x = 0
c =2
1 + ra =
1 − r
1 + r.
j = cψ† αψ α = 0 σ
σ 0 c
j ψ
jy = jz, jx = 0.
jtrans
jin
=4r
(1 + r)2
,jrefl
jin
= 1 − r
1 + r2
jtrans
jin
+jrefl
jin
= 1.
q,p > 0, m < E < V 0 − m, i.e. V0 > 2m, E − V0 + m < 0, r < 0
jrefl
jin
> 1,jtrans
jin
< 0
vtrans,x =dE
dq =
d
dq
q 2 + m2
=
2q
+
q 2 + m2=
2q
E − V 0= − 2q
|E − V 0|
vtrans
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E > m V 0 > 2m
jtrans
E < 0
E < 0
• E < 0
• E < 0
E < 0E > 0
E < 0
e−
e+
hω
p
EE(p)
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12
E < 0
E < 0, p, q
E > 0, − p, − q ; − σ
ω > 2mc
2
hω
p
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E < 0
-4 -2 0 2 4-3
-2
-1
0
1
x
E < 0
ψ(±) ≡ 0
• ψ(+)(x) E > 0 p σ E > 0 p σ e
• ψ(−)(x) E < 0 p σ −E > 0 − p −σ −e
particle
particle
antiparticle
0
V
x
V
jin(+)
(+)
jrefl>jin
ψ (−)
jtrans<0
V0>2m
m<E<V0−0
ψ
ψ
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12
•
••
∆x
∆ p ≥ ∆x−1
∆E ≈ c ∆ p ≥ c
∆x
=⇒ ∆x <1
2
mc=
1
2χC
∆E ≥ 2mc2
χC
E < 0
E < 0 e = −e0 < 0 E > 0
−e = e0 > 0
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[γ µ(i∂ µ − eAµ) − m] ψ = 0
[γ µ(i∂ µ + eAµ) − m] ψC = 0
ψ ψC ψC = C ψ
i∂ µ eAµ
C [−γ
µ
∗(i∂ µ + eAµ) − m] C−1C ψ∗ = 0.
γ µ −→ γ µ∗ CC γ µ∗ C−1 = −γ µ
γ µ
γ µ
γ 0 = β =
1 0
0 −1
, γ i =
0 σi
−σi 0
, i = 1, 2, 3 ,
C•
•σi
−→σi∗ (i = 1, 2, 3)
C =
0 0 0 1
0 0 −1 0
0 −1 0 0
1 0 0 0
= iγ 2
C−1 =
C .
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12
C ψ = C ψ∗ = iγ 2 ψ∗
γ µγ ν + γ ν γ µ = 2gµν 1
(iγ
2
)γ
µ
∗(iγ
2
) = [(−i
−γ 2
γ
2
∗ ) =iγ 2
γ
µ
(−iγ
2
∗)]∗
= −[γ 2γ µγ 2]∗
= −[γ 2(2gµ21− γ 2γ µ)]∗
= [2δ µ2γ 2 − γ µ]∗
=
−γ µ , µ = 2
2γ 2∗ − γ 2∗ = −γ 2 , µ = 2
(γ 2)2 = −1
• ψ C• E −→ −E ψ ( )∗
• p −→ − p
• (C)
• e −e
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ψ =
1
0 p√
E +m
0
e−i(Et− pz)
C ψ =
0 0 0 1
0 0 −1 0
0
−1 0 0
1 0 0 0
1
0 p√
E +m
0
e+i(Et− pz)
=
0− p√ −E −m
0
1
e−i(−Et−(− p)z)
E > 0 p|| z pz = p
↑E < 0 p || z pz = − p ↓
Aµ
T : t −→ −t
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12
T : p −→ − p ( )
L = [r × p] −→ − L ( )
S −→ − S ( L)
Φ(x, t) −→ Φ(x, −t) = Φ(x, t)
A(x, t) −→ A(x, −t) = − A(x, t)
e −→ e ( )
(1.264),(1.265)=⇒ Aµ(x, t) −→ Aµ(x, t) ( )
[γ µ(i∂ µ − eAµ) − m] ψ = 0
[γ µ(−i∂ µ − egµν Aν ) − m] ψT = 0
ψT = T ψ
ψT = T ψ = iΣ2ψ∗
iΣ2 =
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
= i
σ2 0
0 σ2
.
C ≡ T
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P : x −→ −x t −→ t , E −→ E
p −→ − p
L −→ + L
S −→ + S
Aµ −→ Aµ
Φ −→ Φ A −→ − A
e−ipµxµ
P : p0 −→ p0 , x0 −→ x0
pi −→ − pi , xi −→ −xi , i = 1, 2, 3.
P ψ(x, t) = γ 0 ψ(−x, t) γ 0 =
1 0
0 −1
.
Σ =
2
σ 0
0 σ
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12
ψ(+) p=0 , ↑ =
1
0
0
0
e−imt , ψ(+) p=0 , ↓ =
0
1
0
0
e−imt , E = m > 0
ψ(−) p=0 , ↑ =
0
0
1
0
e+imt , ψ(−) p=0 , ↓ =
0
0
0
1
e+imt , E = −m < 0
Σz =
2
1 0 0 0
0 −1 0 0
0 0 1 0
0 0 0 −1
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•
z
ψ(+) p|| b z , ↑(x) =
E + m
2m
1
0 pz
E +m
0
e−ipµxµ
ψ(+)
p|| b z , ↓
(x) = E + m
2m
0
1
0 pz
E +m
e−ipµxµ
ψ(−) p|| b z , ↑(x) =
E + m
2m
pz
E +m
0
1
0
e+ipµxµ
ψ(−) p|| b z , ↓(x) = E + m
2m 0
pz
E +m0
1
e+ipµxµ
z||p^ ^
Σz
Σz s = ±12
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12
• ⊥
z
ψ(+) p|| b x , ↑(x) =
E + m
2m
1
0
0 pz
E +m
e−ipµxµ
ψ(+) p|| b x , ↓(x) =
E + m
2m
0
1 pz
E +m
0
e−ipµxµ
ψ(−) p|| b x , ↑(x) =
E + m
2m
0
pzE +m
1
0
e+ipµxµ
ψ(−) p|| b x , ↓(x) =
E + m
2m
pzE +m
0
0
1
e+ipµxµ
z
x||p^ ^
^
1 i∂
∂tψ =
m + eΦ α · π
ւ
ր α · π −m + eΦ
ψ = H ψ.
γ µ
γ µ = A γ µ A−1
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E < 0
E < 0
ψ
••
Σz σz
σz = ±m
E
2!
|v| → c E =
p2 + m2 → ∞
σz px→∞ = 0 σy px→∞ = 0.
|v| → c
x
v=c
γ µ
ψ(+)
↑ ↓ ↑ ↓
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12
ψ(+)† p|| b x , ↑ · ψ(+)
p|| b x , ↑ =E + m
2m 1 0 0 pxE +m
1
0
0 pxE +m
=E + m
2m
(E 2 + m2 + 2Em) +
E 2−m2 p2
(E + m)2
=E + m
2m
2E 2 + 2Em
(E + m)2=
E
m
d4x
σz =
normalization m
E ψ
(+)† p|| b x , ↑ Σz ψ
(+) p|| b x , ↑
=m
E
E + m
2m
(E 2 + m2 + 2Em) − p2
(E + m)2
=m
E
E + m
2m
2m2 + 2Em
(E + m)2=
m
E .
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ψ(+) p , ↑(x) =
E + m
2m
1
0 pz
E +m px+ipy
E +m
e−ipµxµ
ψ(+) p , ↓(x) = E + m2m 0
1 px+ipyE +m
pzE +m
e−ipµxµ
ψ(−) p , ↑(x) =
E + m
2m
pz
E +m px+ipy
E +m
1
0
e+ipµxµ
ψ(−) p , ↓(x) =
E + m
2m
px+ipy
E +m pzE +m
0
1
e+ipµxµ
E −mm
A−1 =: e−iS e−iS
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12
ψ′ = e+iS ψ
i∂
∂te−iSe+iS
1
ψ = H e−iSe+iS 1
ψ
i∂
∂t
e−iS ψ′
i( ∂ ∂t
e−iS)ψ′+ie−iS ∂ ∂t
ψ′
= H e−iS ψ′ e+iS ·
i∂
∂tψ′ = e+iS
H − i
∂
∂t
e−iS
H′
ψ′.
H′
H = α · p + βm =
1m α · p
α · p −1m
= βm + O
odd
B =
Bx
0
Bz
H = σxBx + σz Bz
Θ
σ
z
x
y
ϑ0
→ ei2
σyϑ0 = e−12
σzσxϑ0
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e±iS = e±
σz
β
σx
( α · p ) ϑ( p) = 1 cos ϑ ± β ( α · p )sin ϑ
= e±β O ϑ| p | ( )
p = p
| p |
H′ = eiS
H e−iS
= eβ ( α· b p )ϑ( α · p + βm)(1 cos ϑ − β ( α · p )sin ϑ)
= eβ ( α· b p )ϑ (1 cos ϑ + β ( α · p )sin ϑ) eβ( α·bp )ϑ
( α · p + βm)
= (cos 2ϑ + β ( α · p )sin2ϑ)( α · p + βm)
= α · p
cos2ϑ − m
| p |
!
= 0
+βm
cos2ϑ +
| p |m
sin2ϑ
=⇒ tan2ϑ = | p|m
sin2ϑ =p
m2 + p2=
p
E , cos2ϑ =
m m2 + p2
=m
E
[α , β ]+ = 0
H′
H′ = βm m
E + | p
|2
mE = β
1
E | p |2
+ m2
E p
| p | ≪ m
iS = β O ϑ
| p | ≈ β O 1
2m
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12
e±β ( α· b p )ϑ =∞
k=0
1
(2k)![β ( α ·
p )]2k +
∞k=0
1
(2k + 1)![β ( α ·
p )]2k+1
( α · p )2 = αi piα j p j = 12{αi, α j}
δij
pi p j = | p |2
H = α ·
p − e A
+ βm + eφ = βm + ε + O
ε = 1eφ βε = ε
O = α( p − e A) β O = −Oβ
βm O(m)1m
iS =β
2mO O = α·
p − e A
.
| p−e A|m
H′ = eiSH − i ∂ ∂t e−iS
∂ ∂t
e−iS
eABe−A = B+[A , B]+1
2[A , [A , B]]+. . .+
1
k![A , [A , . . . , [A , B] . . .]]
+ . . .
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H′ = H + i[S , H] +i2
2 [S , [S , H]]
+i3
6[S , [S , [S , H]]] +
i4
24
O( 1m3 )
[S , [S , [S , [S , H]]]]
− S − i
2[S , S] − i2
6[S , [S , S]] + . . .
β α 1m
i[S , H] = −O + β 2m
[O , ε] + 1m
β O2
O O 1m0
S = −i β 2m
O ≡ 0
H′ = βm +
ε′
β O2
2m − O4
8m2+ ε
− 1
8m2[O , [O , ε]] +
β
2m[O , ε] − O3
3m2 O′
= βm + ε′ + O′
O′ ∼ O 1m
O 1
miS =
β
2mO′ ∼ O
1
m2
.
H′′ = βm + ε′ +β
2m[O′ , ε′] +
O
1
m4
= βm + ε′ + O′′.
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12
iS = β 2m
O′′ ∼ O 1m3
O
1m3
H′′′ = β
m +
O2
2m− O4
8m3
+ ε − 1
8m2
O , [O , ε] + iO
+ O
1
m4
O −→ H′′′ O 1
m3
O H′′′
H′′′ = β m + p
−e A
2
2m −1
8m3 p − e A2
− e Σ · B2 + eφ
− e
2mβ Σ · B − e
8m2 Σ ·
∇ × E
− e
4m2 Σ ·
E ×
p − e A
− e
8m2 ∇ · E .
H′′′
E > 0 ψ′ = χ0
i∂ϕ
∂t=
m + eφ +
1
2m
p − e A
2
− e
2mσ · B
−(| p |2)2
8m3− e
4m2σ ·
E ×
p − e A
− e
8m2 ∇ · E
ϕ
H1 = − (| p |2)2
8m3
| p |2
2
m2 + p2
=| p |2
2m− (| p |2)2
8m3+ . . .
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E = − ∇φ(r) = −1
r
∂φ
∂rr , A = 0
σ ·−1
r
∂φ
∂rr × p
= −1
r
∂φ
∂r
σ · L
H2 =
e
4m2
1
r
∂φ
∂r
σ · L
∂φ∂r
H3 = − e
8m2 ∇ · E
H3
∼ λCompton =
mc
m = 0
γ µ pµ ψ = 0
1 p0 ψ = c α
· p ψ
m = 0
{αi, α j} = 2i εijk αk
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E =
p2c2 + (mc2)2 m = 0
E > 0
v p =∂E ∂ p
m=0=
∂ ∂ p
(| p |c) = c p
| p | ,
| p |E
= 1
p σ⊥ = 0 m = 0
σ2⊥ > 0 σz
σx σy
m = 0 Σ p
h( p ) = Σ · p
| p |
h( p )
Σ · p
| p |2
2
=3
i,j=1
ΣiΣ j pi p j
| p |2
=
3i=1
Σ2i
=1
p2i
| p |2 = 1,
pi p j =
0 , i = j
p2i , i = j
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h( p ) h = ±1
Σ
Σ
p
p helicity h=+1
helicity h=−1
Σ · p
Σ · p , (c α · p + mc2) H
= 0
Σ · p
Σ · p
γ µ pµ ψ = 0
γ · p ψ = γ 0 p0 ψ γ = 0 σ
−σ 0 −→ Σ · p Σ =
σ 0
0 σ
γ Σ
0 −11 0
0 −11 0
= γ 5γ 0 , γ 5 ≡ iγ 0γ 1γ 2γ 3 = 0 1
1 0
m = 0 γ 5γ 0
Σ · p ψ = γ 5 p0 ψ
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Σ · p , γ 5
= 0
γ 5 = iγ 0γ 1γ 2γ 3
ψch = Uψ U =1√
21 + γ 5
γ µ ch = Uψ U −1 ,
p0 − σ · p
ψch
1 = 0 p0 + σ · p
ψch
2 = 0
σ · p| p |
ψch
=1
√ 21 + γ 5
σ · pP−→ −σ · p
h = +1 h = −1
h = −1
h = +1
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γ
• γ 0
• γ k
γ 0 = 0 σ2
σ2 0 ,
γ 1 = i
0 σ1
σ1 0
,
γ 2 = i1 0
0−1
σ3 = i
0 σ3
σ3 0
iγ µ∂ µ − m
ψ = 0 .
ψC = ψ∗ ,
γ µ(i∂ µ − eAµ) − m
ψ = 0 | ∗ − γ µ(−i∂ µ − eAµ) − m
ψ∗ = 0
γ µ(i∂ µ + eAµ) − m
ψ∗ = 0 ψ∗ = ψC
ψ ψ∗ = ψ