ФИАН / Nefedyev_Ktp
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C
z = x − y
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Δ(z)|z0=0 Z−∞ dp0ε(p0)δ(p02 − Ep2) = 0, |
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∂Δ(z) |
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= δ(3)(z). |
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∂z0 |
|z0=0 |
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(3) |
(x − y) |
{ψα(x)ψβ (y)}|x0=y0 = (γ0)αβ δ |
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{ψα(x)ψβ† (y)}|x0=y0 |
= δαβ δ(3)(x − y). |
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C
UC
UC apλU† |
= λcbpλ, UC bpλU† |
= λ apλ, |
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C |
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C |
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a b |
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ψ |
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= λcψ |
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¯T |
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UC ψUC |
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= λcCψ |
= −λcγ2ψ |
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uλ(p)
ucλ(p)
C
UC2 = 1 = |λc|2 = 1.
λc = 1
Cψ(t, x) = −γ2ψ (t, x).
P
λc = ±1
λc = ±1 |
λc |
CAµ(t, x) = −Aµ(t, x).
P
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P : t → t, |
x → −x, |
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UP |
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λ |
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U a U† |
= λ a |
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U b U† |
= λ b |
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P pλ P |
p −pλ |
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P pλ P |
p −pλ |
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uλ(−p) = γ0uλ(p)
UP ψ(t, x)UP† = λpγ0ψ(t, −x).
UP2 = 1 λp = ±1 |
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λp |
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λp |
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2π |
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Rn(θ) = e |
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(σn)θ , |
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z |
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2π |
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Rz(0) = 1, Rz(2π) = |
eiπ |
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e−iπ |
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λp |
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ψ(t, x) → UC |
UP ψ(t, x)UP† UC† |
= UC [λpγ0ψ(t, −x)] UC† = −γ2 |
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= UP [ψc(t, x)] UP† |
c |
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(t, |
ψ(t, x) → UP hUC ψ(t, x)UC† i UP† |
= λpγ0ψ |
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= −1.
[λpγ0ψ(t, −x)] = λpγ0γ2ψ (t, −x), −x) = −λcpγ0γ2ψ (t, −x).
T
λp = −λcp.
λp
P
ˆ |
ˆ |
† |
UP (i∂ |
−eA |
−m)ψ(x)UP |
ψc = ψ |
λpc = λp |
λp = ±i.
P ψ(t, x) ≡ UP ψ(t, x)UP† = iγ0ψ(t, −x).
P A0(t, x) |
≡ UP A0(t, x)UP† = A0(t, −x), |
P A(t, x) |
≡ UP A(t, x)UP† = −A(t, −x). |
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ˆ |
ˆ |
− m)ψ(x) = 0 |
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(i∂ |
− eA |
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UP |
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U† |
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P |
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ˆ |
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ˆ |
ˆ † |
= UP (i∂ |
−eA |
−m)UP UP ψ(x)UP |
= (i∂ |
−eUP AUP |
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λp = −λp
−m)UP ψ(x)UP† = 0.
− − ˆ − − −
λpγ0 iγ0∂0 iγ∂ eA(t, x) m ψ(t, x) = 0.
x → −x
T
T : t → −t, x → x,
i∂t∂ |Φi = H|Φi
CP T
T
t
T
¯ |
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|Φi → |Φi = Ut|Φi |
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t → −t Ut
H → UtHT Ut†,
H |
H → HT |
T
O
T T †
O → UtO Ut ,
T
T
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T |
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T ψ(t, |
x |
) ≡ Utψ |
T |
(t, |
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¯T |
(−t, |
x |
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(−t, |
x |
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)Ut |
= iγ3γ1γ0ψ |
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) = iγ3γ1ψ |
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T A0(t, x) ≡ UtAT0 (t, x)Ut† = A0(−t, x), T A(t, x) ≡ UtAT (t, x)Ut† = −A(−t, x),
CP T
CP T |
S = R d4xL |
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CP T |
CP T
CP T : |
ψ(t, x) → iγ5ψ(−t, −x), |
CP T : |
Aµ(t, x) → −Aµ(−t, −x), |
CP T : |
ϕ(t, x) → ϕ(−t, −x). |
CP T
d(1/2,0) d(0,1/2)
ξ
ψ = η ,
CP T : ξ(t, x) → −iξ(−t, −x),
CP T : η(t, x) → iη(−t, −x).
±i |
4CP T |
±1 |
d x |
x → −x |
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CP T |
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CP T |
(−1)
ab
(ψaψb)CP T = ψbCP T ψaCP T = −ψaCP T ψbCP T .
CP T |
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(LQED(x))CP T |
= −ψ¯CP T (t, x)(i∂ˆ − eAˆCP T (t, x) −1m)2ψCP T (t, x) − |
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FµνCP T (t, x) |
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¯ |
ˆ |
ˆ |
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(−t, −x) = LQED(−x), |
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= ψ(−t, −x)γ5 |
(i∂ + eA(−t, −x) − m)γ5 |
ψ(t, x) − |
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Fµν |
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T
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QED) |
Z |
QED(− ) x→−x Z |
QED( |
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QED |
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(S |
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CP T = d4xL |
x = |
d4xL x |
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S |
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CP T
CP T
CP T
CP T
CP T
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f¯ |
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L |
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S |
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(−1)L |
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f |
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(−1) |
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P = (−1)L+1. |
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SM |
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C |
X |
SM |
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ψ = |
C21 µ1 21 µ2 |f(p, µ1)f(−p, µ2)i → |
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C21 µ1 21 µ2 |f(p, µ1)f(−p, µ2)i = |
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µ1µ2 |
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µ1µ2 |
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SM |
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SM |
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= − C |
21 µ1 21 µ2 |
|f(−p, µ2)f(p, µ1)i |
= −(−1) |
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C |
21 µ2 21 µ1 |
|f(p, µ1)f(−p, µ2)i = |
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µ1µ2 |
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µ1µ2 |
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SM |
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= −(−1) |
L |
(−1) |
S− |
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− 2 |
C |
p |
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p |
, µ2)i |
= (−1) |
L+S |
ψ, |
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µ1 2 µ2 |f( |
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µ1µ2 |
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p → −p |
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(−1)S− |
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(−1)L |
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2 − 2 |
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J L |
S |
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C = (−1)L+S . |
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2S+1LJ , |
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L |
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S P D F |
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L = 0, 1, 2, 3, . . . |
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L = S = J = 0 |
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2S+1LJ = 1S0, JP C = 0−+, |
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S = 1 |
L = 0 |
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J = 1 |
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JP C = 1−−. |
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0++ |
L = S = |
1 J = 0 |
3P0 |
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JP C |
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0−− 0+− 1−+ 2+− |
J L S |
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{J, L, S} |
JP C |
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S = 1 |
J |
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1−− |
3S1 |
3D1 |
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•
•
ˆ
hΦ|O|Φi = 0
DS
JP
ψ |
c |
¯T |
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= Cψ |
(iγµ∂µ − m)ψ = eAµγµψ, (iγµ∂µ − m)ψc = −eAµγµψc.
¯
Aµ = eN(ψγµψ).
ψ ψ† Aµ
O = 0
′ |
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˙ |
′ |
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=x0 |
= gµν δ |
(3) |
( |
x |
− |
x′ |
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[Aµ(x)Aν (x )]|x0=x0 |
= 0, [Aµ(x)Aν |
(x )]|x0 |
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{ψα(x)ψβ (x′)}|x0′ =x0 |
= 0, {ψα(x)ψβ† (x′)}|x0′ =x0 |
= δαβδ(3)(x − x′), |
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′ |
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[ψα(x)Aµ(x )] = 0, [ψα(x)Aµ(x )] = 0. |
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L = − |
1 |
2 |
¯ ˆ |
ˆ |
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Fµν |
+ ψ(i∂ |
− eA |
− m)ψ = Lγ + Le + LI , |
− ¯
LI = Aµ(x)jµ(x), jµ(x) = eN(ψ(x)γµψ(x)).
Z Z
H = H0 + V, V = − d3xLI = d3x jµ(x)Aµ(x),
H0
•
OS = |
, ψS (t) = e−iH tψ(0), |
e−iH t
i∂ψS (t) = HSψS (t). ∂t
•
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ψH = |
= ψ(0), |
OH (t) = eiH tOS e−iH t, |
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∂OH |
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∂ |
eiH t |
OS |
e−iH t = iH eiH t |
OS |
e−iH t |
− |
ieiH t |
OS |
e−iH tH = i[H |
OH |
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∂t |
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hOiS = hψS |OS |ψS i = hψH |eiH tOS e−iH t|ψH i = hψH |OH |ψH i = hOiH .
Z
∂µOH = i[PµOH ], Pµ = d3xTµ0.
∂ν Aµ = i[Pν Aµ], ∂ν ψ = i[Pν ψ],
T S
•
OI = eiH0tOS e−iH0t, ψI (t) = eiH0tψS (t),
hOiI = hOiS ,
∂tψI (t) = |
∂t |
eiH0tψS (t) = iH0eiH0tψS (t) + eiH0t |
∂tψS (t) |
= |
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∂ |
∂ |
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∂ |
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=iH0eiH0tψS (t) − ieiH0t(H0 + VS )ψS (t) = −ieiH0tVSψS (t) =
=−ieiH0tVS e−iH0teiH0tψS (t) = −iVI ψI (t).
OH = eiH te−iH0tOI eiH0te−iH t ≡ S†(t, 0)OI S(t, 0),
S(t, 0) = eiH0te−iH t, S†(t, 0) = S−1(t, 0) = S(0, t),
I
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S |
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∂ψ |
= V ψ, V = |
Z d3xjµ(x)Aµ(x). |
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∂t |
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t
t0
ψ(t) = S(t, t0)ψ(t0).
S(t, t′)
• S(t, t) = 1
t0 = 0 |
t0 6= 0 |
t → t − t0
T S
•S(t, t′)S(t′, t0) = S(t, t0)
•S†(t, t′) = S(t′, t) = S−1(t, t′)
e−iEn(t−t′)
•
i∂t∂ S(t, t′) = V S(t, t′).
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∂ψ |
= i |
∂S |
ψ0 = V Sψ0 = V ψ, |
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∂t |
∂t |
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V |
C |
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t t′ |
V (τ)dτ! . |
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Snaive(t, t′) = exp |
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Z |
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V
[V (t)V (t′)] 6= 0.
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S(t0 + t0, t0) − S(t0, t0) |
= V (t0)S(t0, t0), |
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t0 |
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S(t0 + t0, t0) = S(t0, t0) − i |
t0V (t0)S(t0, t0) = 1 − i |
t0V (t0) ≈ exp −i Zt0t0+Δt0 |
V (τ)dτ . |
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t0 + t0 = t1 |
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S(t1, t0) = exp −i Zt0t1 |
V (τ)dτ . |
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S(t2, t0) = exp −i Zt1t2 |
V (τ)dτ S(t1, t0) = exp −i Zt1t2 |
V (τ)dτ exp −i Zt0t1 |
V (τ)dτ , |
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t2 = t1 + t1 |
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Ztn |
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( 0) N→∞ n=0 exp − |
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n+1 = |
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N |
tn+1 |
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Y |
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S t, t |
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V τ dτ , t |
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t. |
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