Quantum Mechanics
Review of classical mechanics• Lagrangian L(x, x) = 1
2 mx2 −V(x), p = ∂L∂x ,
Action, S =∫
dtL, E-L equation (EOM)
0 = δS⇒ ∂t∂L∂x − ∂xL = 0
• Hamiltonian : H = xp− L,• Hamilton eq., using Poisson bracket A,B = ∂A
∂q∂B∂p −
∂A∂p
∂B∂q
q =∂H∂p = q,H , p = −∂H
∂q = p,H
Quantum Mechanics
Quantum mechanics: Canonical quantization• Poisson bracket ., .→ − i
ℏ [., .], Commutation relation: [x, p] = iℏ. in xrepresentation, p→ −iℏ∂x,
H = − ℏ2
2m∂2
∂x2 + V(x).
• E→ iℏ∂t, Schrödinger equation from E = T + V:
iℏ∂tψ = Hψ =(− ℏ2
2m∂2
∂x2 + V(x))ψ
• Probability conservation:
∂tρ−∇ · j = 0, ρ(x) = ψ∗ψ, j = − iℏ2m (ψ∗∇ψ − ψ∇ψ∗)
Relativistic QM
• Special relativity: E2 = p2c2 + m2c4, relativistic QM, microscopic highvelocity particles, (xµ = (ct, x))• Klein-Gordon eq,
(− 1c2 ∂
2t + ∂2
x −m2c2
ℏ2 )ϕ(x) = 0
However, current conservation:iℏ2m∂µ(ψ∗∂µψ − ψ∂µψ∗) = 0
Charge iℏ2mc2
∫dx3(ψ∗∂0ψ − ψ∂0ψ
∗) not positive definite— negativeprobability.
Relativistic QM
• Dirac Eq. ψ, 4-component spinor
iℏ∂tψ = Hψ, H = −iℏcα · ∇+ βmc2, αi =
(0 σi
σi 0
), β =
(I 00 −I
)Current conservation:
∂tρ+∇ · j = 0 , ρ = ψ∗ψ , j = cψ∗αψ
• Negative energy states: E = ±√
p2 + m2
• For fermions: Pauli principle, Dirac sea, anti-particle. However for bosons,this does not work.
• QM, not consistent with relativity: treating t and x differently, t is not anoperator.• Quantum field: demote the x to variable, φ be viewed as Quantum
operators, labeled by x, infinite degree of freedom, Heisenberg pictureoperator φ(t, x).• We have the Lagrangian density of fields.
L[x, x]→ L[ϕ, ∂µϕ]
• Canonical variables: ϕ(x, t), Π(x, t) = ∂L∂ϕ
,• Canonical Quantization:
[Π(x, t),Π(y, t)] = [ϕ(x, t), ϕ(y, t)] = 0, [ϕ(x, t),Π(y, t)] = iδ3(x− y)
Free scalar field• Klein-Gordon (Real scalar field):
L[ϕ, ∂µϕ] =12∂µϕ∂
µϕ− 12m2ϕ2
• EOM: E-L eq.(∂2 + m2)ϕ(x) = 0
• Canonical momentum,
Π(x) = ∂L[ϕ, ∂µϕ]∂∂tϕ
= ϕ
• Canonical commutation relation:
[Π(x, t),Π(y, t)] = [ϕ(x, t), ϕ(y, t)] = 0, [ϕ(x, t),Π(y, t)] = iδ(3)(x− y)
• Fourier trans: Ladder operators
ϕ(x) =∫
d3k(2π)3
√2ωk
(ake−ik·x + a†keik·x) =
∫d3k(2π)3 (fk(k)ak + f∗k(k)a
†k),
ϕ(x) = i∫
d3k(2π)3ωk
(− fk(x)ak + f∗k(x)a
†k
), fk(x) =
e−ik·x√2ωk
ak = i∫
d3x f∗k(x)←→∂0ϕ(x)⇒ [ak, a
†k′] = (2π)3δ(3)(k− k′)
• Hamiltonian:
H =Πϕ− L =
∫d3k(2π)3
ωk2 (a†
kak + aka†
k)
=
∫d3k(2π)3ωk(
12 [ak, a
†k] + a†
kak)
Divergence of the vacuum energy : can not detect.• Fock space: vacuum ak|0⟩ = 0, for all k. All
(∏
i√
2Ekia†
ki)|0⟩ = |k1, . . . , kn⟩, form the fock space for free scalar
particles, ⟨k1 |k2⟩ = 2Ek1(2π)3δ(3)(k1 − k2)
• Propagator:
⟨0|Tϕ(x)ϕ(y)|0⟩ =∫
d4k(2π)4
ie−ik·(x−y)
k2 −m2 + iϵ = DF(x− y)
Free Fermion field
L = ψ(i∂/−m)ψ, H =
∫d3xψ(−iγ · ∇+ m)ψ
Euler Lagrangian EOM : (i∂/−m)ψ = 0
• Canonical fields:ψα(x), iψ†α(x)
• Ladder operators:
ψ(x) =∫
d3k(2π)3
√2ωk
∑s(us,kas,ke−ik·x + vs,kb†
s,keik·x)
• Cannonical anti-commutation:ψα (x, t), ψβ (y, t) = ψ†
α (x, t), ψ†β (y, t) = 0, ψα (x, t), ψ†
β (y, t) = δ(3) (x − y)δαβ ,
Hw. as,k, a†r,k′
= (2π)3δ(3) (k − k′)δrs , bs,k, b†r,k′
= (2π)3δ(3) (k − k′)δrs,
• In Ladder operators: H =∫ d3k
(2π)3ωk∑
s(a†s,k
as,k + b†s,k
bs,k)
• Fock space: Vacuum, ak|0⟩ = bk|0⟩ = 0,√
2Eka†k
√2Epb†p · · · |0⟩.
• Propagator:
⟨0|Tψα(x)ψβ(y)|0⟩ =∫
d4k(2π)4
ie−ik·(x−y)
k/−m + iϵ = SF(x− y)
Massless vector field
L = −14FµνFµν , Fµν = ∂µAν(x)− (µ↔ ν)
• Gauge invariance: Aµ → Aµ + ∂µα(x), not all Aµ components aredynamical canonical variables.• A0 has no canonical conjugate momentum.• Choose gauge, Lorenz gauge ∂µAµ = 0, or Coulomb gauge ∇ · A = 0, in
momentum space ki Ai(k) = 0, only (δij − kikj/k2)Aj are independent.• Conjugate momentum Πi =
∂L∂Ai
= Ai, and satisfy ∇ · Π = 0• Canonical commutator:
[Ai(x, t),Πj(y, t)] = i(δij −
∇i∇j
∇2
)δ(3)(x− y) = i
∫d3k(2π)3 eik·(x−y)
(δij −
kikj
k2
)• Ladder operators:A(x) =
∫ d3k(2π)3√2Ek
∑λ=±(ε
∗λ,kaλ,ke−ik·x + ελ,ka†
λ,keik·x)
• Commutation relation for Ladder:
[aλ,k, a†λ′ ,k′
] = (2π)3δ(3)(k− k′)δλλ′ , [aλ,k, aλ′ ,k′ ] = 0
• Propagator: ⟨0|TAµ(x)Aν(y)|0⟩ =∫ d4k
(2π)4−igµνe−ik·(x−y)
k2+iϵ + · · ·
Interacting Fields: Klein-Gordon
H = H0 + Hint = HKlein−Gordon +
∫d3x λ4!ϕ
4
• Heisenberg picture:ϕ(t, x) = eiH(t−t0)ϕ(t0, x)e−iH(t−t0),Interaction picture: ϕI(t, x) = eiH0(t−t0)ϕ(t0, x)e−iH0(t−t0).
ϕI(t, x) =∫
d3p(2π)3
1√2Ep
(ape−ip·x + a†
peip·x)∣∣∣
x0=t−t0
• Relating the Heisenberg & Interaction pictures:U(t, t0) = eiH0(t−t0)e−iH(t−t0), U(t0, t0) = 1,
ϕ(t, x) = U†(t, t0)ϕI(t, x)U(t, t0),
• U satisfies:
i∂tU(t, t0) = HI(t)U(t, t0), HI = eiH0(t−t0)Hinte−iH0(t−t0).
• Generalize to initial t′: Time evolution operator,
U(t, t′) = Texp[−i
∫ t
t′dt′′HI(t′′)]
= eiH0(t−t0)e−iH(t−t′)e−iH0(t′−t0), (t ≥ t′)
• Correlation functions:
⟨Ω|T(ϕ(x1) · · ·ϕ(xn))|Ω⟩ = limT→∞(1−iϵ)
⟨0|TϕI(x1) · · ·ϕI(xn) exp[−i∫ T−T dtHI(t)]|0⟩
⟨0|Texp[−i∫ T−T dtHI(t)]|0⟩
• Wick’s theorem:
T(ϕI(x1) · · ·ϕI(xn)) = NϕI(x1) · · ·ϕI(xn) + all possible contractions
• Vacuum matrix element: ⟨0|TϕI(x1) . . . |0⟩ = Fully contracted terms.
Feynman diagrams
⟨0|TϕI(x1) · · ·ϕI(xn) exp[−i∫ T
−TdtHI(t)]|0⟩
=⟨0|TϕI(x1) · · ·ϕI(xn)
1− i∫ T
−TdtHI(t) +
12(− i
∫ T
−TdtHI(t)
)2+ . . .
|0⟩
Use the Wick’s theorem, we can get Feynman diagrams: for example
Feynman rules in coordinate space:
Scattering Matrix• In states and out states:
|k1, · · · kn⟩int→−∞−−−−−→ |k1, · · · kn⟩free , |k1, · · · kn⟩out
t→+∞−−−−−→ |k1, · · · kn⟩free
• S-matrix:out⟨p1p2 · · · |k1k2 · · · ⟩in ≡ ⟨p1p2 · · · |S|k1k2 · · · ⟩
• T-matrix: S = 1 + iT.• Invariant matrix element M (scattering amplitude)
⟨p1p2 · · · |iT|k1k2 · · · ⟩ = (2π)4δ(4)(∑
iki−
∑j
pj)·iM(k1, · · · · · · → p1, · · · )
• For 2→ many, physical observable: differential cross section, and totalcross section. Peskin (4.79)• For 1→ many: decay rate. Peskin (4.86).• LSZ reduction formula:∫ n∏
1d4xieipi·xi
m∏j=1
d4yje−ikj·yj⟨Ω|Tϕ(x1) · · ·ϕ(xn)ϕ(y1) · · ·ϕ(ym)|Ω⟩
−−−−−→p0
0→Epik00→Eki
( n∏i=1
√Zi
p2i −m2 + iϵ
)( m∏j=1
√Zi
k2j −m2 + iϵ
)⟨p1 · · · pn|S|k1 · · · km⟩
Compute S-matrix using Feynman diagrams
⟨p1, · · · pn|iT|pApB⟩
= limT→∞(1−iϵ)
(0⟨p1, · · · , pn|T
(exp[−i
∫ T
−TdtHI(t)]
)|pApB⟩0
)connected,amputated
QED
QED: Dirac + Maxwell field interaction between charged fermions and photons• Lagrangian density:
L = iψ∂/ψ −mψψ +14FµνFµν + eψA/ψ = iψD/ψ −mψψ +
14FµνFµν
Dµ = ∂µ + ieAµ,• Gauge transformation: ψ → ψ′ = eiα(x)ψ, Aµ → A′
µ = Aµ − 1e∂µα.
L → L[ψ′,A′] = L[ψ,A]
• Note : m2AµAµ is not invariant under local gauge trans. Photons aremassless.
Feynman rules:
Fermion propagators and external fermions:
• Momentum conservation at each vertex. Integrate over each undeterminedloop momentum. Overall sign.
QED Elementary process• (1) Two fermion 2→ 2 scattering:
e+e− → µ+µ−, e−µ− → e−µ−.
Bhabha scattering:e+e− → e+e−, Mo/ller scattering, e−e− → e−e−.• (2) Compton scattering.
• (3) the electron - positron annihalation:
QFTII
• Path integral.• Higher order corrections: Renormalization and renormalization group.
Renormalization : use the coupling to obsorb the high energy information.• Non-abelian Gauge Field theory:• Spontaneous symmetry breaking.• Higgs Mechanism: Standard Model, SU(3)× SU(2)×U(1) gauge theory
with SSB.• Anomaly.
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