ФИАН / Nefedyev_Ktp
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wh |
h H |
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wh |
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h2 · h1 H |
wh |
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wh2 · wh1 W |
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W |
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wh(λa) = λwh(a), wh(a1 + a2) = wh(a1) + wh(a2),
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λ |
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n × n |
wh W |
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w′ |
= SwhS−1 = |
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· · |
· · · |
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(m × m) |
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(m×m) |
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a |
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a′ = Sa |
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a′ |
H |
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H |
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H
{h1, h2, . . . hk} H
wh W
HW
O(n)
U(n)
SO(3) |
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[JiJj ] = iεijkJk, |
i, j, k = 1, 2, 3, |
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SO(3) |
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J |
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2J + 1 |
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−2 |
J |
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z J |
z = −2 − 2 2 |
J |
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J, J +1 . . . J |
1, J |
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J |
= Jx +Jy +Jz |
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Ji
J(J + 1)
SO(3)
J = 1/2
n
d(1/2)(θ) = e2i (σn)θ ,
σ |
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i |
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0 −1 |
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0 |
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σ1 = |
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1 |
, σ2 = |
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−i |
, σ3 = |
1 0 |
, |
θ |
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d(1/2)(0) = 1, d(1/2)(2π) = −1, |
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2π |
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SO(3) |
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J1 |
J2 |
J1 6= J2 |
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(2J1 + 1)(2J2 + 1) |
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(J2, J1) (J1, J2)
d(J1,J2)
M2 N2
d(J1,J2) d(J1,J2) d(J2,J1)
(1 + iMdη + iNdη ) = 1 + i(−N )dη + i(−M )dη ,
M′ = −N N′ = −M M′2 = N 2 = J2(J2 + 1) N′2 = M 2 = J1(J1 + 1)
(J2, J1)
• d(0,0)
• d(1/2,0) d(0,1/2)
1
2
• d(1/2,1/2)
P : x → −x, |
P = −1, |
P JP −1 = J, P KP −1 = −K.
P MP −1 = N, |
P NP −1 = M, |
P M2P −1 = N2, |
P N2P −1 = M2. |
J1 6= J2
d(J1,J2) |
d(J2,J1) |
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d(J1,J2) |
0 |
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D(J1,J2) = |
0 |
d(J2,J1) |
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P
d(J1,J2) J1 = J2
D(J1,J2)
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P = λp I |
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I |
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|λp| = 1 I |
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(2J1 +1)(2J2 +1) |
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D(1/2,0) = |
d(1/2,0) |
0 |
, P = λpγ0, γ0 = |
0 1 |
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d(0,1/2) |
1 0 |
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ξ |
η |
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d(1/2,0) d(0,1/2) |
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D(1/2,0) |
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η . |
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ξ η |
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ψ = |
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ξ |
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P |
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P |
= λpγ0ψ, |
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ψ → ψ′ |
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ξ → λpη, η → λpξ.
•
•
d(J2,J1) d(J1,J2) d(0,1/2) d(1/2,0)
ξ Λ ξ′ = aξ,
→
SL(2, C)
ξ η
σ2
η
2π
ξ1
ξ = ξ2 , ξ′ =
a
d(0,1/2) d(1/2,0)
S
λp = i,
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J1 6= J2 |
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Λ |
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ξ1′ |
, a = |
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α β |
ξ2′ |
γ δ |
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2 × 2 |
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Sa S−1 = b.
a = b = e2i (σn)ω .
S
Sσ S−1 = −σ.
S
S = iσ2,
b = Sa S−1 = σ2a σ2.
P 2 = −1
ξ |
2 × 2 |
a |
,a = αδ − βγ = 1.
d(1/2,0)
η |
b |
S
ω 1
b = (a†)−1
a
η′ = (σ2a σ2)η = (−iσ2η′) = a (−iσ2η),
η2˙ ! |
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− 2 η2 η1 |
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˙ |
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−η2 |
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η1 |
= |
iσ |
η1 |
= |
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˙ |
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˙ |
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η1 = η2 = η1˙ , η2 |
= −η1 = η2˙ , |
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wα = εαβwβ, α, β = 1, 2, ε12 = −ε21 = 1. |
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ηα˙ = ηα, |
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ψ |
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ξα |
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ψ = |
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ηα˙ |
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ξα |
P |
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P |
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→ |
λpηα˙ , ηα˙ → λpξα, |
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λp
c |
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H = 0, |
E + |
∂H |
= 0 |
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∂t |
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E = ρ, |
H − |
∂E |
= j, |
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∂t |
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ρ j |
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ρ(x, t) = Xi |
eiδ(3)(x − xi(t)), |
j(x, t) = Xi |
eiviδ(3)(x − xi(t)), |
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∂ρ |
j = 0 |
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+ |
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∂t |
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∂µjµ = 0, |
jµ = (ρ, j). |
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Aµ = (A0, A) |
E |
= − |
∂A |
− A0 |
= −∂0A − ∂A0, |
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∂t |
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H |
= |
A = [ A]. |
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Fµν = ∂µAν − ∂ν Aµ,
F0i = ∂0(−Ai) − ∂iA0 = Ei, |
Fij = ∂i(−Aj ) − ∂j (−Ai) = −εijkεklm∂lAm = −εijkHk, |
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Fµν = −E1 |
0 |
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−H3 |
H2 |
, F µν = |
E1 |
0 |
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−H3 |
H2 |
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E1 |
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E2 |
E3 |
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−E1 |
−E2 |
−E3 |
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−E2 |
H |
2 |
H |
1 |
−0 1 |
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E H |
2 |
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1 |
−0 |
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H3 |
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H1 |
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E H3 |
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E2 |
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− |
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− |
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Ei = F0i = −Fi0 = −F 0i = F i0, |
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1 |
εijkF jk. |
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Hi = − |
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εijkFjk = − |
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2 |
2 |
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∂µFλρ + ∂λFρµ + ∂ρFµλ = 0. |
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λ = i ρ = j |
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µ 6= λ 6= ρ |
µ = 0 |
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εijk |
∂Hk |
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+ (∂iEj − ∂j Ei) = 0, |
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∂t |
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εijk |
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H |
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E k |
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∂ |
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= 0, |
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∂t |
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µ = i λ = j ρ = k |
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εijk |
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εµνλρ |
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1 |
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˜ |
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εµνλρF |
λρ |
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Fµν = |
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2 |
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˜µν |
= 0. |
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∂µF |
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∂µF µν = jν ,
jµ
ϕ |
Aµ → Aµ′ = Aµ + ∂µϕ |
Fµν |
↔ ˜
F F
2J + 1 J = 1
Aµ
Aµ
∂µAµ = 0.
ϕ = 0.
∂µA′µ = ∂µ(Aµ + ∂µϕ) = ∂µAµ + ϕ = ∂µAµ = 0.
Aµ = jµ,
∂µjµ = 0
A(0)µ = 0
A(0)µ
jµ
− D(x) = δ(4)(x),
Z
A(1)µ (x) = − d4yD(x − y)jµ(y).
Z
A(1)µ (x) = − d4y D(x − y) jµ(y) = jµ(x).
D(k) = Z |
d4x |
− kx, |
(2π)4 eikxD(x), kx = kµxµ = k0x0 |
1
D(k) = k2 .
k0
k0 = ±|k|
1
DR(k) = (k0 + i0)2 − k2 ,
R
δ(x)
DR(x)
x0 + |x|
x0
Z
Aµ(x) = A(0)µ −
1
DR(x) = −4π|x|δ(x0 − |x|)Θ(x0 Θ(x0)
|x| = x0
DR(x) = −21π δ(x2)Θ(x0).
d4yDR(x − y)jµ(y) = A(0)µ (x) + 41π Z
A(0)µ
),
2|x|
δ
x2
d3y jµ(x0 − |x − y|, y),
|x − y|
j0(x) = ρ(x) = Xi |
eiδ(3)(x − xi), |
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j(x) = 0 |
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e δ(3)(y |
x |
) |
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1 |
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e |
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A0(x) = |
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Z |
d3y |
Pi i x |
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y− |
i |
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= |
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4π |
− |
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4π i |
x |
xi |
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X | |
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− |
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1/(4π)
Z
S = d4xL ,
L
L |
xµ |
L |
Aµ ∂ν Aµ |
L
L |
m4 |
L = Lγ + L ′, |
Lγ = − |
1 |
Fµν F µν = |
1 |
(E2 − H2), L ′ = −jµAµ. |
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4 |
2 |
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Lγ |
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L ′ |
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∂µf |
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S′ = − Z d4xjµAµ → S′ + δS′, |
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δS′ = − Z d4xjµ∂µf = − IΣ→∞ dΣµjµf + Z |
d4xf∂µjµ = 0, |
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∂L |
= ∂ν |
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∂L |
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∂Aµ |
∂(∂ν Aµ) |
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δS = δ Z d4xL (Aµ, ∂ν Aµ) = Z d4x |
∂Aµ δAµ + |
∂(∂ν Aµ)δ(∂ν Aµ) |
= |
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∂L |
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∂L |
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= Z d4x |
∂Aµ |
− ∂ν ∂(∂ν Aµ) |
δAµ = 0, |
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∂L |
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∂L |
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δAµ