quantization of scalar field - sjtuweiwang/sites/… · quantization of scalar field wei wang...

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


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


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.


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.


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.


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 ].


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.


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.


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〉.


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.


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).


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)].


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 ].


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.


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)].


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)


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)]


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).


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.


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.


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.


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.


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).


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).


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)].



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).


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∗.



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.

√


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.



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| .



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′)


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)


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.


Green’s function

C


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


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).


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


CF


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).


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