Quantization of Electromagnetic Field

Wei Wang

October 30, 2017

Wei Wang Lectures on QFT October 30, 2017 1 / 19

Contents

1 Review: Quantization of scalar field

2 Review: Quantization of spinor field

3 Electromagnetic field

4 Homework

Wei Wang Lectures on QFT October 30, 2017 2 / 19

Review: Quantization of real scalar field

1.Lagrangian density

L = 12∂µφ∂µφ−

1

2m2φ2.

2. canonical momentum π(x) = ∂L∂φ̇(x)

= φ̇(x)

3. 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

4. Plane-wave expansion ωk =√~k2 +m2

φ(x) =

∫d3k

(2π)32ωk[a(~k)e−ik·x + a†(~k)eik·x],

Wei Wang Lectures on QFT October 30, 2017 3 / 19

Review: Quantization of real scalar field

The commutation relation for a and a†

[a(~k), a†(~k′)] = (2π)32ωkδ3(~k − ~k′),

[a(~k), a(~k′)] = [a†(~k), a†(~k′)] = 0.

The time-ordered propagator is defined as:

T [φ(x)φ(x′)] = θ(t− t′)φ(x)φ(x′) + θ(t′ − t)φ(x′)φ(x),(�x +m)T [φ(x)φ(x

′)] = −iδ4(x− x′).

Then the vacuum expectation value is given as

〈0|T [φ(x)φ(x′)]|0〉 =∫

d4p

(2π)4e−ip·(x−x

′) i

p2 −m2 + i�.

Wei Wang Lectures on QFT October 30, 2017 4 / 19

Review: Spinor field Quantization

1. Writing down the Lagrangian density:

L = ψ̄(i∂/−m)ψ.

2. Define the canonical momentum

πψ =∂L∂ψ̇

= iψ†.

3. Use the anti-commutation relation as the quantization condition:

{ψρ(~x, t), iψ†σ(~y, t)} = iδ3(~x− ~y)δρσ,

{ψρ(~x, t), ψρ(~y, t)} = {iψ†σ(~x, t), iψ†σ(~y, t)} = 0,

4. Plane wave expansion:

ψ(x) =

∫d3p

(2π)32Ek

∑λ=±1

[bλ(~p)uλ(~p)e−ip·x

+ d†λ

(~p)vλ(~p)eip·x

],

ψ̄(x) =

∫d3p

(2π)32Ek

∑λ=±1

[b†λ

(~p)ūλ(~p)eip·x

+ dλ(~p)v̄λ(~p)e−ip·x

],

Wei Wang Lectures on QFT October 30, 2017 5 / 19

Review: Spinor field quantization

Nonzero commutation relation for creation and annihilation operators:

{bλ(~p), b†λ′(~k)} = {dλ(~p), d†λ′(~k)} = (2π)32k0δ3(~k − ~p)δλλ′ ,

and others are zero.

The time-ordered propagator 1 is given as

Tψ(x)ψ̄(x′) = θ(t− t′)ψ(x)ψ̄(x′)− θ(t′ − t)ψ̄(x′)ψ(x).

The vacuum expectation value is defined as

〈0|Tψ(x)ψ̄(x′)|0〉 =∫

d4p

(2π)4e−ip·(x−x

′) i(p/+m)

p2 −m2 + i�.

1This is defined at the spinor component levelWei Wang Lectures on QFT October 30, 2017 6 / 19

Electromagnetic field

The Lagrangian density is given as

L = −14FµνF

µν , Fµν = ∂µAν − ∂νAµ,

with the field strength tensor given as:

Fµν =

0 −E1 −E2 −E3E1 0 −B3 B2E2 B3 0 −B1E3 −B2 B1 0

.The field strength tensor is directly related to the physical electric andmagnetic field:

Ei = −F 0i, Bi = 12�ijkF jk.

Wei Wang Lectures on QFT October 30, 2017 7 / 19

Electromagnetic field

The equation of motion is given as:

∂µFµν = 0.

which is equivalent to the Maxwell equation

~5 · ~E = 0, ~5× ~B = ∂∂t~E.

We can define the dual of the field strength tensor:

F̃µν = −12�µνρσFρσ =

0 −B1 −B2 −B3B1 0 E3 −E2B2 −E3 0 −E1B3 E2 E1 0

.Since due to the antisymmetric property, we have

∂µF̃µν = 0.

This is equivalent to the other two components of Maxwell equation:

~5 · ~B = 0, ~5× ~E = − ∂∂t~B.

Wei Wang Lectures on QFT October 30, 2017 8 / 19

Electromagnetic field: Gauge Symmetry

There is extra degrees of freedom in Aµ

Aµ(x)→ Aµ(x) + ∂µα(x),

with α(x) being any complex function. One can find that this doesnot change the Fµν , and thus corresponds to the same sets ofphysical fields.

One can identify the above transformations as the extra degrees offreedom, or one may view them as the gauge symmetry.

To remove the additional degrees of freedom, one usually uses thegauge fixing condition, for instance the Coulomb gauge:

~5 · ~A = 0, A0 = 0.

Or the covariant Lorenz gauge (NOT Lorentz):

∂µAµ(x) = 0.

Wei Wang Lectures on QFT October 30, 2017 9 / 19

Electromagnetic Field Quantization

The canonical momentum is given as

πµ =∂L∂Ȧµ

= −F0µ,

Apparently, π0 = 0, and thus one can not quantize the A0.

One can choose the gauge fixing condition to eliminate the unphysicaldegrees of freedom in Aµ, such as the Coulomb gauge:

A0 = 0, ~5 · ~A = 0.

This gauge is not covariant.

Wei Wang Lectures on QFT October 30, 2017 10 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

We introduce the Lagrangian density

L = −14FµνF

µν − 12α

(∂µ ·Aµ)2,

α is an arbitrary constant (different with α(x)!). Setting(∂µ ·Aµ) = 0 (Lorenz gauge) will recover the original Lagrangian.The canonical momentum is given as

πµ =∂L∂Ȧµ

= −F0µ −1

αg0µ(∂

ν ·Aν).

Then π0 is not zero!

For simplicity, one can use α = 1, and this is called Feynman gauge(notice the two meanings of gauge!)

Wei Wang Lectures on QFT October 30, 2017 11 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

The quantization condition is chosen as:

[Aµ(x, t), πν(y, t)] = igµνδ3(~x− ~y),

[Aµ(x, t), Aν(y, t)] = [πµ(x, t), πν(y, t)] = 0.

This is equivalent to:

[Aµ(x, t), Aν(y, t)] = [Ȧµ(x, t), Ȧν(y, t)] = 0,

[Aµ(x, t), Ȧν(y, t)] = −igµνδ3(~x− ~y).

Wei Wang Lectures on QFT October 30, 2017 12 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

Plane wave expansion

Aµ(x) =

∫d3p

(2π)32Ep

3∑λ=0

[a(λ)(~p)�(λ)µ (~p)e−ip·x + a(λ)†(~p)�(λ)∗µ (~p)e

ip·x],

Here the �µ are the polarization vectors. For massless particles, onemay choose

�(0) =

1000

, �(1) =

0100

, �(2) =

0010

, �(3) =

0001

.The left-handed and right-handed polarization is given as

�(±) = (�(1) ± �(2))/√

2.

Wei Wang Lectures on QFT October 30, 2017 13 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

The polarization vectors satisfy the orthonormality:∑λλ′

gλλ′�(λ)µ (

~k)�(λ′)

µ (~p) = gµν ,

�(λ)µ(~p)�(λ′)

µ (~p) = gλλ′ .

The creation and annihilation operators are expressed as:

a(λ)(~p) = i

∫d3xeip·x

←→∂ 0�

(λ)µ A

µ(~x, t),

a(λ)†(~p) = −i∫d3xe−ip·x

←→∂ 0�

(λ)µ A

µ(~x, t)

Wei Wang Lectures on QFT October 30, 2017 14 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

The commutation relations for creation and annihilation operators

[a(λ)(~p), a(λ′)†(~p′)] = −gλλ′2p0(2π)3δ3(~p− ~p′),

[a(λ)(~p), a(λ′)(~p′)] = [a(λ

′)†(~p), a(λ′)†(~p′)] = 0.

The above relations have a problem for the scalar polarization λ = 0:

〈0|a(0)(~p)a(0)†(~p)|0〉 < 0,

on the other hand, this is a normalization of the state a(0)†(~p)|0〉which must be larger than or equal to 0. This problem happens as wehave included non-physical polarizations at the beginning.

Wei Wang Lectures on QFT October 30, 2017 15 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

To remove the non-physical polarizations, we put the constraints:

〈ψ|∂µAµ|ψ〉 = 0.This is a weakened Lorenz gauge fixing condition. Actually, the aboveconstraint is equivalent to the following constraint:

∂µA(+)µ |ψ〉 = 0,

where A(+)µ is the positive energy component of Aµ.

For single particle states, apparently the transverse polarizations arephysical:

i∂µA(+)µ |ψT 〉 =∫

d3p

(2π)32Epe−ip·x

3∑λ=0

a(λ)(~p)�(λ)µ (~p)pµ|ψT 〉

=

∫d3p

(2π)32Epe−ip·x

∑λ=1,2

a(λ)(~p)�(λ)µ (~p)pµ|ψT 〉 = 0.

Wei Wang Lectures on QFT October 30, 2017 16 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

For scalar and longitudinal polarizations, we have∑λ=0,3

p · �(λ)(~p)a(λ)(~p)|φ〉 = 0

namely the scalar polarization and longitudinal polarizations mustexist simultaneously!The presence of scalar polarization and longitudinal polarizations willnot affect the physical observables. Taking the expectation value ofHamiltonian as the example, we have

〈ψ|H|ψ〉〈ψ|ψ〉

=

〈φ|〈ψT |∫ d3p

(2π)32Ep[∑3λ=1 a

(λ)†(~p)a(λ)(~p)− a(0)†(~p)a(0)(~p)]|ψT 〉|φ〉

〈φ|φ〉〈ψT |ψT 〉

=

〈ψT |∫ d3p

(2π)32Ep[∑2λ=1 a

(λ)†(~p)a(λ)(~p)]|ψT 〉

〈ψT |ψT 〉

Wei Wang Lectures on QFT October 30, 2017 17 / 19

Electromagnetic Field Quantization: Gupta-Bleulerformalism

The propagator of the electromagnetic field is given as

〈0|TAµ(x)Aν(y)|0〉 =∫

d4p

(2π)4e−ip·(x−y)

−ip2 + i�

gµν

In a general gauge, we have

〈0|TAµ(x)Aν(y)|0〉 =∫

d4p

(2π)4eip·(x−y)

−ip2i�

[gµν + (1− α)pµpνp2 + i�

],

with the α being a constant.

Wei Wang Lectures on QFT October 30, 2017 18 / 19

Homework

Please derive the propagator for the electromagnetic field

〈0|TAµ(x)Aν(y)|0〉 =∫

d4p

(2π)4eip·(x−y)

−ip2

[gµν + (1− α)pµpνp2

],

one may use the Feynman gauge with α = 1.

Wei Wang Lectures on QFT October 30, 2017 19 / 19

Review: Quantization of scalar fieldReview: Quantization of spinor fieldElectromagnetic fieldHomework