2 mx2 −V(x), p = ∂L ∂x ,
Action, S = ∫
0 = δS⇒ ∂t ∂L ∂x − ∂xL = 0
• Hamiltonian : H = xp− L, • Hamilton eq., using Poisson bracket {A,B} = ∂A
∂q ∂B ∂p −
∂q = {p,H}
Quantum Mechanics
Quantum mechanics: Canonical quantization • Poisson bracket {., .}→ − i
[., .], Commutation relation: [x, p] = i. in x representation, p→ −i∂x,
H = − 2
2m ∂2
∂x2 + V(x).
i∂tψ = Hψ = ( − 2
• Probability conservation:
∂tρ−∇ · j = 0, ρ(x) = ψ∗ψ, j = − i 2m (ψ∗∇ψ − ψ∇ψ∗)
Relativistic QM
• Special relativity: E2 = p2c2 + m2c4, relativistic QM, microscopic high velocity particles, (xµ = (ct, x)) • Klein-Gordon eq,
(− 1 c2 ∂
However, current conservation: i 2m∂µ(ψ∗∂µψ − ψ∂µψ∗) = 0
Charge i 2mc2
Relativistic QM
i∂tψ = Hψ, H = −icα · ∇+ βmc2, αi =
( 0 σi
σi 0
• Negative energy states: E = ± √
• 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 an operator. • Quantum field: demote the x to variable, φ be viewed as Quantum
operators, labeled by x, infinite degree of freedom, Heisenberg picture operator φ(t, x). • We have the Lagrangian density of fields.
L[x, x]→ L[, ∂µ]
• Canonical variables: (x, t), Π(x, t) = ∂L ∂
, • Canonical Quantization:
[Π(x, t),Π(y, t)] = [(x, t), (y, t)] = 0, [(x, t),Π(y, t)] = iδ3(x− y)
Free scalar field • Klein-Gordon (Real scalar field):
L[, ∂µ] = 1 2∂µ∂
µ− 1 2m22
• Canonical momentum,
=
• Canonical commutation relation:
[Π(x, t),Π(y, t)] = [(x, t), (y, t)] = 0, [(x, t),Π(y, t)] = iδ(3)(x− y)
• Fourier trans: Ladder operators
∫ d3k (2π)3 (fk(k)ak + f∗k(k)a
† k ),
† k′ ] = (2π)3δ(3)(k− k′)
† k ] + a†
k ak)
( ∏
2Eki a†
ki )|0 = |k1, . . . , kn, form the fock space for free scalar
particles, k1 |k2 = 2Ek1 (2π)3δ(3)(k1 − k2)
• Propagator:
Free Fermion field
• Canonical fields:ψα(x), iψ† α(x)
s,k eik·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,
(2π)3ωk ∑
bs,k)
2Eka† k
Massless vector field
L = −1 4FµνFµν , Fµν = ∂µAν(x)− (µ↔ ν)
• Gauge invariance: Aµ → Aµ + ∂µα(x), not all Aµ components are dynamical 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
∇i∇j
( δij −
kikj
k2
λ,k eik·x)
• Commutation relation for Ladder:
] = (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+i + · · ·
∫ d3x λ4!
• Heisenberg picture:(t, x) = eiH(t−t0)(t0, x)e−iH(t−t0), Interaction picture: I(t, x) = eiH0(t−t0)(t0, x)e−iH0(t−t0).
I(t, x) = ∫
d3p (2π)3
1√ 2Ep
x0=t−t0
• Relating the Heisenberg & Interaction pictures: U(t, t0) = eiH0(t−t0)e−iH(t−t0), U(t0, t0) = 1,
(t, x) = U†(t, t0)I(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′) = T { exp[−i
∫ t
• Correlation functions:
0|T{I(x1) · · ·I(xn) exp[−i ∫ T −T dtHI(t)]}|0
0|T{exp[−i ∫ T −T dtHI(t)]}|0
• Wick’s theorem:
• Vacuum matrix element: 0|T{I(x1) . . . }|0 = Fully contracted terms.
Feynman diagrams
−T dtHI(t)]}|0
)2 + . . .
} |0
Use the Wick’s theorem, we can get Feynman diagrams: for example
Feynman rules in coordinate space:
Feynman rules in momentum space:
Scattering Matrix • In states and out states:
|k1, · · · knin t→−∞−−−−−→ |k1, · · · knfree , |k1, · · · knout
t→+∞−−−−−→ |k1, · · · knfree
• T-matrix: S = 1 + iT. • Invariant matrix element M (scattering amplitude)
p1p2 · · · |iT|k1k2 · · · = (2π)4δ(4)( ∑
pj)·iM(k1, · · · · · · → p1, · · · )
• For 2→ many, physical observable: differential cross section, and total cross section. Peskin (4.79) • For 1→ many: decay rate. Peskin (4.86). • LSZ reduction formula:∫ n∏
1 d4xieipi·xi
m∏ j=1
−−−−−→ p0
( n∏ i=1
p1, · · · pn|iT|pApB
= lim T→∞(1−i)
( 0p1, · · · , pn|T
QED: Dirac + Maxwell field interaction between charged fermions and photons • Lagrangian density:
L = iψ∂/ψ −mψψ + 1 4FµνFµν + eψA/ψ = iψD/ψ −mψψ +
1 4FµνFµν
Dµ = ∂µ + ieAµ, • Gauge transformation: ψ → ψ′ = eiα(x)ψ, Aµ → A′
µ = Aµ − 1 e∂µα.
L → L[ψ′,A′] = L[ψ,A]
• Note : m2AµAµ is not invariant under local gauge trans. Photons are massless.
Feynman rules:
• Momentum conservation at each vertex. Integrate over each undetermined loop 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
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.