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Quantization of Scalar Field
Wei Wang
2017.10.12
Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41
Contents
1 From classical theory to quantum theory
2 Quantization of real scalar field
3 Quantization of complex scalar field
4 Propagator of Klein-Gordon field
5 Homework
Wei Wang(SJTU) Lectures on QFT 2017.10.12 2 / 41
Free classical field
Klein-GordonSpin-0, scalarKlein-Gordon equation
∂µ∂µφ+m2φ = 0
DiracSpin- 12 , spinorDirac equation
i∂/ψ −mψ = 0
MaxwellSpin-1, vectorMaxwell equation
∂µFµν = 0, ∂µF
µν = 0.
Gravitational field
Wei Wang(SJTU) Lectures on QFT 2017.10.12 3 / 41
Klein-Gordon field
scalar field, satisfies Klein-Gordon equation
(∂µ∂µ +m2)φ(x) = 0.
Lagrangian
L =1
2∂µφ∂
µφ−m2φ2
Euler-Lagrange equation
∂µ
(∂L
∂(∂µφ)
)− ∂L∂φ
= 0.
gives Klein-Gordon equation.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 4 / 41
From classical mechanics to quantum mechanics
Mechanics: Newtonian, Lagrangian and Hamiltonian
Newtonian: differential equations in Cartesian coordinate system.
Lagrangian:
Principle of stationary action δS = δ∫dtL = 0.
Lagrangian L = T − V .Euler-Lagrangian equation
d
dt
∂L
∂q− ∂L
∂q= 0.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 5 / 41
Hamiltonian mechanics
Generalized coordinates: qi; conjugate momentum: pj = ∂L∂qj
Hamiltonian:
H =∑i
qi pi − L
Hamilton equations
p = −∂H∂q
,
q =∂H
∂p.
Time evolution
df
dt=∂f
∂t+ {f,H}.
where {...} is the Poisson bracket.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 6 / 41
Quantum Mechanics
Quantum mechanicsHamiltonian: Canonical quantizationLagrangian: Path integral
Canonical quantizationObservables: operatorscommutation relations
[qi, qj ] = [pi, pj ] = 0,
[qi, pj ] = i~δij
Poisson bracket → commutation bracket: {...} → 1i~ [...]
Time evolution (Heisenberg equation)
qi = i[H, qi],
pi = i[H, pi].
For any observable F (q, p),
F (q, p) = i[H,F ].
Wei Wang(SJTU) Lectures on QFT 2017.10.12 7 / 41
1D harmonic oscillator (classical)Lagrangian
L =1
2mq2 − mω2
2q2,
(ω =
√K
m
)
Canonical momentum
p =∂L
∂q= mq.
Hamiltonian
H = pq − L =1
2m(p2 +m2ω2q2)
Hamilton equation
p = −∂H∂q
= −mω2q,
q =∂H
∂p=
p
m.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 8 / 41
1D harmonic oscillator (quantum)
equal-time commutation relation
[q, p] = i, [q, q] = [p, p] = 0
equation of motion
p = i[H, p] = −Kq,
q = i[H, q] =p
m.
raising and lowering operators
a =
√1
2mω(p− imωq), a† =
√1
2mω(p+ imωq).
[a, a†] = 1, [a, a] = 0, [a†, a†] = 0.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 9 / 41
particle-number representation
Hamiltonian:
H = ω(aa† +1
2)
particle-number operator
N = aa†, H = ω(N +1
2),
N |n〉 = n|n〉, H|n〉 = (n+1
2)ω|n〉
Vacuum state: |0〉
N |0〉 = 0, H|0〉 =ω
2|0〉.
Creation and annihilation
a†|n〉 =√n+ 1|n+ 1〉,
a|n〉 =√n|n− 1〉.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 10 / 41
From mechanics to field theory
Mechanics: finite degree of freedom.field: infinite (continuum) degree of freedom.
canonical coordinates: x→ φ(~x, t).
canonical momentum: p→ π(~x, t) =∂L(φ,∂µφ)
∂φ(~x,t)
Hamiltonian: H =∫d3xH(π(~x, t), φ(~x, t)) =
∫d3x(πφ− L),
H: Hamiltonian density.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 11 / 41
Analogy between mechanics and field theory
Discretization:
φi(t) =1
∆Vi
∫(∆Vi)
d3xφ(~x, t)
φi(t) is the average value in ∆Vi.Continuum → denumerable.
Lagrangian:
L =
∫d3xL(φ(x), ∂µφ(x))→
∑i
∆ViLi(φi(t), φi(t), φi±s(t), · · · ).
φi(t) =1
∆Vi
∫(∆Vi)
d3x∂
∂tφ(x, t).
Wei Wang(SJTU) Lectures on QFT 2017.10.12 12 / 41
Analogy between mechanics and field theory
canonical momentum:
pi(t) =∂L
∂φi(t)= ∆Vi
∂Li∂φi(t)
≡ ∆Viπi(t).
Hamiltonian
H =∑i
piφi − L =∑i
∆Vi(πiφi − Li).
Canonical quantization
[φi(t), pi(t)] = iδij , [φi(t), φj(t)] = [pi(t), pj(t)] = 0.
Heisenberg equation
φi(t) = i[H,φi(t)],
pi(t) = i[H, pi(t)].
Wei Wang(SJTU) Lectures on QFT 2017.10.12 13 / 41
Continuum limit
When ∆Vi → 0,δij
∆Vi→ δ3(~x− ~x′)
commutation relation
[φ(t, ~x), π(t, ~x′)] = i~δ3(~x− ~x′), (~ = 1)
[φ(t, ~x), φ(t, ~x′)] = [π(t, ~x), π(t, ~x′)] = 0,
Heisenberg equation (equation of motion)
φ(~x, t) = i[H,φ(~x, t)],
π(~x, t) = i[H,π(~x, t)].
For any physical quantity F ,
F = i[H,F ].
Wei Wang(SJTU) Lectures on QFT 2017.10.12 14 / 41
Quantization of real scalar field
Lagrangian density
L =1
2∂µφ∂µφ−
1
2m2φ2.
Euler-Lagrange equation (Klein-Gordon equation)
(∂µ∂µ +m2)φ(x) = 0
canonical momentum π(x) = ∂L∂φ(x)
= φ(x)
Hamiltonian density
H = π∂0φ− L =1
2
[(∂0φ)2 + (~∇φ)2
]+
1
2m2φ2.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 15 / 41
quantization of real scalar field
Introducing commutation relation for φ and π
[φ(~x, t), π(~x′, t)] = iδ3(~x− ~x′),[φ(~x, t), φ(~x′, t)] = [π(~x, t), π(~x′, t)] = 0
Heisenberg equation
φ(~x, t) = i[H,φ(~x, t)],
π(~x, t) = i[H,π(~x, t)].
Wei Wang(SJTU) Lectures on QFT 2017.10.12 16 / 41
Mode expansion
Plane-wave expansion
φ(x) =
∫d3k
(2π)32ωk[a(~k)e−ik·x + a†(~k)eik·x],
with ωk =√~k2 +m2.
For π(x, t), we have
π(x) = φ(x) =
∫d3k
(2π)32ωk(−iωk)[a(~k)e−ik·x − a†(~k)eik·x],
a and a† can be expressed by the field operator
a(~k) = i
∫d3xeik·x
←→∂0φ(x, t), a†(~k) = −i
∫d3xe−ik·x
←→∂ 0φ(x, t)
Wei Wang(SJTU) Lectures on QFT 2017.10.12 17 / 41
Commutation relation for a and a†
[a(~k), a†(~k′)] = (2π)32ωkδ3(~k − ~k′),
[a(~k), a(~k′)] = [a†(~k), a†(~k′)] = 0.
Hamiltonian
H =1
2
∫d3x[π(x, t)2 + |∇φ(x, t)|2 +m2φ(x, t)2]
=1
2
∫d3k
(2π)32ωkωk[a(~k)a†(~k) + a†(~k)a(~k)].
Momentum
P = −∫d3xπ(x, t)∇φ(x, t)
=1
2
∫d3k
(2π)32ωk~k[a(~k)a†(~k) + a†(~k)a(~k)]
Wei Wang(SJTU) Lectures on QFT 2017.10.12 18 / 41
vacuum zero point energy
H =1
2
∫d3k
(2π)32ωkωk[a(~k)a†(~k) + a†(~k)a(~k)]
=
∫d3k
(2π)32ωk[ωka
†(~k)a(~k) +ωk2δ3(0)]
=
∫d3k
(2π)32ωkωk
[a†(~k)a(~k) +
V
2(2π)3
].
〈0|H|0〉 =
∫d3k
(2π)32ωkωk
V
2(2π)3.
The vacuum is not empty! Infinity!The infinity can be dropped (no worry).
Wei Wang(SJTU) Lectures on QFT 2017.10.12 19 / 41
Normal ordering
Operator ordering in quantum theory: normal (Wick), anti-normal,Weyl-Wigner,...
Normal ordering : O(a, a†) : ,move all a†(k) to the left of a(k). e.g.,
: a(~k)a†(k) :=: a†(~k)a(k) := a†(~k)a(~k).
zero point energy is dropped
〈0| : O(a, a†) : |0〉 = 0.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 20 / 41
Hamiltonian and particle number operator
Define Hamiltonian by normal ordering
H =1
2
∫d3k
(2π)32ωkωk : [a(~k)a
†(~k) + a
†(~k)a(~k)] :
=
∫d3k
(2π)32ωkωka†(~k)a(~k)
Momentum
~P =
∫d3k
(2π)32ωk
~ka†(~k)a(~k).
four-momentum
Pµ
=
∫d3k
(2π)32ωkkµa†(~k)a(~k).
Particle number operator
N =
∫d3k
(2π)22ωka†(~k)a(~k). [N,P
µ] = 0 for free field.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 21 / 41
Fock space and particle interpretation
Basis: all the eigenstate of N .
|~k〉 = a†(~k)|0〉,|~k1,~k2〉 = a†(~k1)a†(~k2)|0〉,...
vacuum state
a(~k)|0〉 = 0, 〈0|0〉 = 1.
One-particle state: Pµ|~k〉 = Pµa†(~k)|0〉 = kµ|~k〉. With energy
momentum relation |~k|2 +m2 = ω2k.
normalization
〈~k′|~k〉 = (2π)32ωkδ3(~k − ~k′)
Wei Wang(SJTU) Lectures on QFT 2017.10.12 22 / 41
Quantization for many real scalar fields
n scalar fields φr(x, t), (r = 1, · · · , n)
π(x, t)r =∂L
∂φr(x, t)
Hamiltonian
H(πr, · · · , φr, · · · ) =
n∑r=1
πrφr − L,
H =
∫d3xH.
Commutation relation
[φr(x′, t), πs(x, t)] = iδrsδ
3(x− x′),[φr(x, t), φs(x
′, t)] = [πr(x, t), πs(x′, t)] = 0.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 23 / 41
Quantization for many real scalar fields
Heisenberg equation
φr(x, t) = i[H,φr(x, t)],
πr(x, t) = i[H,πr(x, t)].
or
∂
∂xµφr(x) = i[Pµ, φr(x)].
solution:
φr(x+ b) = eiP ·bφr(x)e−iP ·b.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 24 / 41
Complex scalar field
real scalar field: Hermitian, particle=anti-particle.
complex scalar field: particle6=anti-particle, e.g. π±, K±, etc.
Lagrangian for free complex scalar field
L = (∂µφ∗)∂µφ−m2φ∗φ
expressed with two real scalar field
φ =φ1 + iφ2√
2, φ∗ =
φ1 − iφ2√2
Lagrangian in terms of φ1 and φ2
L =1
2(∂µφ1∂
µφ1 + ∂µφ2∂µφ2)− 1
2m2(φ2
1 + φ22).
Wei Wang(SJTU) Lectures on QFT 2017.10.12 25 / 41
complex scalar field
Euler-Lagrange equation: Klein-Gordon equation
(� +m2)φ(x) = 0,
(� +m2)φ∗(x) = 0.
conjugate momentum
π =∂L∂φ
= φ∗,
π∗ =∂L∂φ∗
= φ.
Hamiltonian
H =
∫d3x(πφ+ π∗φ∗ − L) =
∫d3x(π∗π +∇φ∗ · ∇φ+m2φ∗φ).
Wei Wang(SJTU) Lectures on QFT 2017.10.12 26 / 41
Quantization of complex scalar field
commutation relation (and 0 for others)
[φ(x, t), π(x, t)] = [φ∗(x, t), π∗(x, t)] = iδ3(x− y),
Mode expansions
φ(x) =
∫d3k
(2π)32ωk[a(~k)e
−ik·x+ b†(~k)e
ik·x],
φ∗(x) =
∫d3k
(2π)32ωk[a†(~k)e
ik·x+ b(~k)e
−ik·x]
φ annihilates a and creates b.Plane wave expansion for the two real field
φi(x) =
∫d3k
(2π)32ωk[ai(~k)e
−ik·x+ a†i (~k)e
ik·x].
a,b can be expressed by ai as
a(~k) =1√
2[a1(~k) + ia2(~k)], a
†(~k) =
1√
2[a†1(~k)− ia†2(~k)],
b(~k) =1√
2[a1(~k)− ia2(~k)], b
†(~k) =
1√
2[a†1(~k) + ia
†2(~k)].
Wei Wang(SJTU) Lectures on QFT 2017.10.12 27 / 41
Quantization of complex scalar field
commutation relation for a and b (and 0 for others)
[a(~k), a†(~k′)] = [b(~k), b†(~k′)] = (2π)32ωkδ3(~k − ~k′).
commutation relation for φ and φ∗
[φ(x), φ∗(y)] = i∆(x− y),
[φi(x), φj(y)] = iδij∆(x− y).
Number operator for a and b
Na =
∫d3k
(2π)32ωka†(~k)a(~k),
Nb =
∫d3k
(2π)32ωkb†(~k)b(~k).
Wei Wang(SJTU) Lectures on QFT 2017.10.12 28 / 41
Quantization for complex scalar field
four-momentum
Pµ =
∫d3k
(2π)32ωkkµ[a†(~k)a(~k) + b†(~k)b(~k)]
vacuum stsate
a(~k)|0〉 = b(~k)|0〉 = 0.
U(1) symmetry of complex scalar field: invariant under U(1)transformation
φ→ eiαφ, φ∗ → e−iαφ∗
or
δφ = iδαφ, δφ∗ = −iδαφ∗.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 29 / 41
Quantization of complex scalar field
Noether current
jµ
= iφ∗←→∂µφ,
conserved charge
Q =
∫d3xj
0(x) = i
∫d3xφ∗←→∂
0φ
conserved charge as an operator
Q =
∫d3k
(2π)22ωk[a†(~k)a(~k)− b(~k)b
†(~k)]
=
∫d3k
(2π)32ωk[a†(~k)a(~k)− b†(~k)b(~k)] = Na −Nb.
In QM, φ is interpreted as wave function, jµ is probability density.Probability can be negative! ×In QFT, φ is interpreted as field operator, jµ is the charge current.Can be negative.
√
Wei Wang(SJTU) Lectures on QFT 2017.10.12 30 / 41
Propagator of Klein-Gordon field
propagator of real scalar field
[φ(x), φ(y)] =
∫d3k
(2π)32ωk
∫d3k′
(2π)32ω′k
×[[a(k), a†(k′)]e−ik·x+ik′·y + [a†(k), a(k′)]eik·x−ik
′·y]
=
∫d3k
(2π)32ωk
∫d3k′
(2π)32ω′k
×(2π)32ωkδ3(~k − ~k′)
[e−ik·x+ik·y − eik·x−ik·y
]=
∫d3k
(2π)32ωk(e−ik·(x−y) − eik·(x−y))
≡ i∆(x− y),
with ωk =
√|~k|2 +m2.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 31 / 41
Propagator of Klein-Gordon field
propagator of real scalar field
i∆(x− y) =
∫d3k
(2π)32ωk(e−ik·(x−y) − eik·(x−y))
=
∫d3k
(2π)32ωkei~k·(~x−~y)(e−iωk(x0−y0) − eiωk(x0−y0)),
with ωk =
√|~k|2 +m2. Introducing on-shell condition δ(k2 −m2),
then
∆(x) =1
i
∫d4k
(2π)42πδ(k2 −m2)ε(k0)e−ik·x,
where ε(k0) = k0
|k0| .
Wei Wang(SJTU) Lectures on QFT 2017.10.12 32 / 41
Propagator of Klein-Gordon field
The field φ is the sum of the positive and negative frequency parts:φ(x) = φ(+)(x) + φ(−)(x), with
φ(+)
(x) =
∫d3k
(2π)32ωka(~k)fk(x),
φ(−)
(x) =
∫d3k
(2π)32ωka†(~k)f
∗k (x),
where fk(x) = e−iωkt+i~k·~x
The Green’s function
∆(+)
(x− x′) =d3k
(2π)32ωkf∗k (x′)fk(x) =
∫d4k
(2π)3θ(k0)δ(k
2 −m2)e−ik·(x−x′)
,
∆(−)
(x− x′) =d3k
(2π)32ωkfk(x
′)f∗k (x) =
∫d4k
(2π)3θ(k0)δ(k
2 −m2)eik·(x−x′)
,
∆(x− x′) = −i(∆(+)(x− x′)−∆
(−)(x− x′)) = −
d3k
(2π)3
sinωk(t− t′)ωk
ei~k·(~x−~x′)
Wei Wang(SJTU) Lectures on QFT 2017.10.12 33 / 41
Properties of Green’s function
∆(+)(x) = ∆(−)(−x)
(� +m2)∆(x) = 0, (� +m2)∆(±)(x) = 0
∆(x)|t=0 = 0, and 0 for x2 < 0.[∂∂x0
∆(x)]x0=0
= −δ3(x)
[ ∂∂xi
∆(x)]~x=0 = 0
∆(~x, x0) = ∆(−~x, x0)
∆(~x, x0) = −∆(~x,−x0)
∆(~x, x0) = −∆(−~x,−x0)
Wei Wang(SJTU) Lectures on QFT 2017.10.12 34 / 41
Retarded and advanced Green’s functions
Define retarded and advanced Green’s functions
∆R(x) = −1
2(1 + ε(x0))∆(x) = −θ(x0)∆(x),
∆A(x) =1
2(1− ε(x0))∆(x) = θ(−x0)∆(x),
∆(x) = ∆A(x)−∆R(x).
Represented by contour integral:
∆(x) =
∫c
dk0
2π
∫d3k
(2π)3
e−ik·x
m2 − k2.
Wei Wang(SJTU) Lectures on QFT 2017.10.12 35 / 41
Green’s function
C
Wei Wang(SJTU) Lectures on QFT 2017.10.12 36 / 41
Green’s function
For retarded and advanced Green’s functions:
∆R/A(x) =dk0
2π
∫CR/A
d3k
(2π)3
e−ik·x
m2 − k2.
CR
CA
Wei Wang(SJTU) Lectures on QFT 2017.10.12 37 / 41
time-ordered product
Dyson’s time-ordered product
Tφ(x′)φ∗(x) = θ(t′ − t)φ(x′)φ∗(x) + θ(t− t′)φ∗(x)φ(x′)
satisfying
(�x′ +m2)iTφ(x′)φ∗(x) = δ4(x′ − x).
Feynman’s propagation function
i∆F (x′ − x) = 〈0|Tφ(x′)φ∗(x)|0〉,(�x′ +m2)∆F (x′ − x) = −δ4(x′ − x).
Wei Wang(SJTU) Lectures on QFT 2017.10.12 38 / 41
Feynman’s propagator
From the definition,
i∆F (x′ − x)
= θ(t′ − t)〈0|φ(x′)φ∗(x)|0〉+ θ(t− t′)〈0|φ∗(x)φ(x′)|0〉
=
∫d3k
(2π)32ωk[θ(t′ − t)e−ik·(x′−x) + θ(t− t′)eik·(x′−x)]
=
∫CF
d4k
(2π)4e−ik·(x
′−x) i
k2 −m2
=
∫d4k
(2π)4e−ik·(x
′−x) i
k2 −m2 + iε
Wei Wang(SJTU) Lectures on QFT 2017.10.12 39 / 41
CF
Wei Wang(SJTU) Lectures on QFT 2017.10.12 40 / 41
Homework
Derive the propagator for a scalar field:
〈0|Tφ(x)φ(y)|0〉 =
∫d4p
(2π)4
i
p2 −m2 + iεe−ip·(x−y).
Peskin and Schroeder’s book: Exercise 2.2
Peskin and Schroeder’s book: Exercise 2.3
Prove the following identity:
[Pµ, a(k)] = −kµa(k), [Pµ, a†(k)] = kµa
†(k)
with
Pµ =
∫d3k
(2π)32ωkkµa†(~k)a(~k).
Wei Wang(SJTU) Lectures on QFT 2017.10.12 41 / 41
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