Download 1 Canonical Quantization of the Electromagnetic Field Peskin

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Canonical Quantization of the Electromagnetic Field
Peskin-Schroeder state the Feynman Rules for the photon field in §4.8:
and postpone the proof to Chapter 9 Functional Methods.
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Altland-Simons state the results of quantizing the photon field using non-covariant Coulomb gauge which
selects a preferred reference frame and breaks Lorentz-invariance
Z
1
∂
∇·A=0, L=
d3x ( A)2 − (∇ × A)2
2
∂t
in §1.5 and say on page 25 “Readers interested in learning more about EM field quantization are referred
to, e.g., L. H. Ryder, Quantum Field Theory (Cambridge University Press, 1996)”.
Outline of Canonical Quantization
The electromagnetic field is perhaps the most fundamental field in physics. Canonical quantization will be
outlined below, and path integral quantization in Topic 4.
The physical E and B fields of classical electrodynamics are not easy to quantize directly because: (1) If the
3 components of E are taken as canonical coordinates, then the components of B are canonical momenta.
(2) A free electromagnetic wave has only two independent transverse field components, and quantizing all
three would give three different quanta, whereas real photons have only two spin states ±~.
To solve problem (1) and preserve explicit Lorentz invariance, use Aµ(x)
E2 − B2
1
1
L=
= − Fµν F µν = − (∂µAν − ∂ν Aµ) ∂ µAν
2
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2
2
and derive the canonically conjugate momenta π µ
πµ =
∂L
,
∂tAµ
π0 =
∂L
=0,
∂ Ȧ0
π j = −Ȧj − ∂tA0 = E j ,
which shows that there is no time-like canonical momentum, and the electric field is the canonical momentum
conjugate to the vector potential! The classical Hamiltonian density in terms of canonical coordinates and
momenta is
3
X
E2 + B2
j
+ E · ∇A0 .
H=
π Aj =
2
j=1
If the field obeys Gauss’ Law the Hamiltonian is
Z
Z
1
H = d3x H =
d3x E 2 + B 2 ,
2
if ∇ · E = 0 .
It is not possible to maintain explicit Lorentz covariance!
Quantizing Transverse Modes
To solve problem (2), the standard Hamiltonian quantization procedure of imposing canonical equal time
commutation relations between canonically conjugate coordinates and momenta
j
j
0
0
π (t, x), A`(t, x ) = E (t, x), A`(t, x ) = −i~δ`j δ(x − x0)
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must be modified to remove one of these 3 conjugate pairs. The problem and solution involve gauge
invariance. The free electromagnetic field is transverse as a consequence of Gauss’ Law
∇·E=0,
k · Ek = 0 ,
which is a time-independent constraint on the electric field components. The time-like momentum π 0 = 0,
and there is a time-independent constraint on the 3 spatial components.
One way of implementing the Gauss Law constraint is to use a “transverse δ-function”
Z
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j `
d k ik·(x−x0) j` k k
j`
δj`δ(x − x0) −→ δ⊥
(x − x0) =
e
δ − 2
,
(2π)3
k
where the subscript ⊥ stands for “transverse”, and the canonical commutation relations
j
j
(x − x0) .
π (t, x), A`(t, x0) = −i~δ⊥`
Choice of Radiation Gauge
The transverse δ function implements the Gauss Law constraint but does not completely fix the gauge
degrees of freedom. This is because Gauss’ Law relates the time-like component to the longitudinal canonical
momentum
0 = ∇ · E = −∇2A0 − ∇ · Ȧ .
This residual gauge invariance allows A0 and a to change simultaneously.
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To completely fix this unphysical variation in the constrained variables we can first make the transformation
Z t
∂
Aµ → A0µ = Aµ − µ
dt0 A0(t0, x)
∂x 0
and eliminate A0 completely as a dynamical variable (recall π 0 = 0 already) by solving
0 = ∇ · A00 = ∇ · A0 + ∇2Λ(t, x)
for a gauge transformation function
1
Λ(x) =
4π
This “radiation gauge” choice ensures that
A0 = 0 ,
π0 = 0 ,
Z
3 0∇
dx
0
· A(t, x0)
.
|x − x0|
∇·A=0,
∂t(∇ · A) = 0; .
Covariance of Radiation Gauge Fixing
The radiation gauge conditions will not be preserved under boosts if Aµ transforms like a 4-vector. If the
boost velocity is not transverse to the wavevector k of every mode, the time-like component will be non-zero
in the boosted reference frame.
Covariance can be ensured by supplementing boost with a gauge transformation to restore the radiation
gauge condition
∂
Aµ(x) → A0µ(x0) = Λ(ω, ζ)µν Aν (x) + 0µ Λ(x0, ω, ζ) ,
∂x
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where ω, ζ are the rotation and boost parameters and Λ(x0, ω, ζ) is a gauge function.
The gauge function Λ(x0) can always be chosen so that
A00(x0) = 0 ,
∇0 · A0 = 0 .
Generalized Covariant Gauge Fixing
Radiation gauge is very convenient in low energy condensed matter, atomic physics and quantum optics applications involving emission and absorption of photons. Experiments are performed in a terrestrial laboratory
with non-relativistic charges in a fixed reference frame.
In high energy physics and cosmology, it is frequently useful to make transformations between laboratory
frames, center of mass frames of highly relativistic colliding particles, and comoving frames of fluid elements
in astrophysics and cosmology. It is much more convenient to use a manifestly covariant gauge fixing
procedure, such as the generalized covariant gauge for the photon propagator given in Peskin-Schroeder
Eq. (9.58)
(
µ ν
ξ = 0 Landau gauge ,
−i µν
k k
µν
e
DF (k) = 2 g − (1 − ξ) 2
,
k
k
ξ = 1 Feynman gauge .
where ξ is a gauge-fixing parameter. The residual gauge invariance must be compensated by introducing
fictitious particles called “ghosts”. Most QED calculations can be done correctly by ignoring the ghosts,
but they must be taken into account in non-abelian gauge theories and quantum gravity.
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Ladder Operators and Fock Space Representation
When the electromagnetic field is quantized using radiation gauge, the Fock space represents arbitrary
numbers of transversely polarized photons. Time-like and longitudinal modes are completely absent.
The vector potential can be expanded in transversely polarized plane waves
2 Z
X
d3k
A(t, x) =
(k, λ) A(t, k, λ) eik·x ,
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(2π)
λ=1
where k = (k 0, k) = (|k|, k) is the 4-momentum of the mode with wavevector k, and (k, λ) are two
transverse polarization vectors defined to form a right-handed triad with the propagation direction
(k, λ) · k = 0 ,
(k, λ) · (k, λ0) = δλλ0 ,
(k, 1) × (k, 2) = k̂ .
The Photon Field Operator
To define Feynman rules it is necessary to consider photons with energy k 0 = ±|k|. The relative phases of
polarization vectors with energy ±|k 0| is not determined by orthonormality. A convenient choice is
(−k, λ) = (−1)λ(k, λ) ,
(k, λ) · (−k, λ0) = (−1)λδλλ0 .
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The photon field operator can then defined
2 Z
X
d3k
−ik·x
†
+ik·x
√
â(k, λ) e
+ â (k, λ) e
(k, λ) ,
Â(x) =
3 2ω
(2π)
λ=1
where ω = k 0 = |k|. Commutation relations of the ladder operators are determined by the canonical
commutation relations of the field operators in radiation gauge
†
† 0 0
3 3
0
0 0
† 0 0
â(k, λ), â (k , λ ) = δλλ0 (2π) δ (k − k ) , [â(k, λ), â(k , λ )] = â (k, λ), â (k , λ ) .
These are exactly the commutation relations of two independent massless Klein-Gordon scalar fields for the
two polarization modes λ = 1, 2. The Fock space is the can be constructed from a vacuum state |0i by
using the creation operators
Y â†(k, λ) nk,λ
p
|0i .
â(k, λ)|0i = 0 , |{k, λ}i =
nk,λ!
k,λ
This normalized state represents a set of quantum numbers k, λ with nk,λ identical bosons for each member
of the set.
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Energy, Momentum, and Spin
The Fock-space representation of Â(x) can be used to show that the Hamiltonian operator is represented
by
Z
2
2
X Z d3k
E
+
B
1
b = d3x
=
H
ω â†(k, λ)â(k, λ) +
(2π)3δ 3(k = 0) .
3
2
(2π)
2
λ
The expectation value of the Hamiltonian in the vacuum state
Z
2 Z
3
X
ω
1
d
k
3 3
4
3
b
×
(2π)
δ
(k
=
0)
=
lim
|k|
d
x,
h0|H|0i
=
max
(2π)3 2
8π |k|max→∞
λ=1
has a quartic divergence in the ultraviolet |k|max → ∞ limit and is proportional to the volume of space.
The infinite volume limit is an infrared divergence due to long wavelength modes. In a cube of side L with
periodic boundary conditions the modes are spaced in wavenumber by dkj = 2π/L and
3 Z
X
L
→
d3k .
2π
k
The momentum operator is represented by
Z
XZ
b = d3x E × B =
P
λ
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d3k
k â†(k, λ)â(k, λ) .
3
(2π)
The spin of a relativistic particle can be represented by the Pauli-Lubański pseudovector
1
J νρP σ
Wµ = − µνρσ
,
2
Pµ P µ
where J νρ is the generator of proper Lorentz transformations (4×4 antisymmetric angular momentum
tensor) and P σ is the generator of translations (4-momentum vector). In the rest frame of a massive
particle PµP µ = M 2 this reduces to the non-relativistic spin angular momentum.
The photon is massless with PµP µ = 0. Recall that Wigner’s little group of Lorentz transformations that
does not change the 4-momentum pµ of the photon is not the 3-dimensional rotation group SU(2), but
the subgroup SO(2) of rotations about the k direction. The Helicity of a massless particle is defined to be
component of the angular momentum in the direction of propagation
h = J · k̂ .
To find the helicity of the Fock-states, consider a photon propagating along the 3-axis (z axis of a righthanded triad). It is straightforward to show that
h
i
3 †
b
J , â (k, λ) = iε1(k, λ)â†(k, 2) − iε2(k, λ)â†(k, 1) ,
where λ = 1, 2 represent linear polarizations along x, y, respectively. We can define creation operators
eigenstates with h = ±1 by
h
i h
i
â†(k, 1) ± iâ†(k, 2)
†
†
†
3
b , â±(k) = ĥ, â±(k) = ±â†±(k) .
√
â±(k) =
,
J
2
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The Feynman Propagator in Radiation Gauge
It is straightforward to show that
Z
D E
µν
0 T µ(x)Âν (y) 0 = DF (x − y) =
2
d4k e−ik·(x−y) X µ
ε (k, λ)εν (k, λ) .
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2
(2π) k + i
λ=1
The polarization sum is non-covariant. It involves the two transverse space-like units vectors εµ. It can be
evaluated explicitly using the completeness property of any 4 linearly independent vectors in 4-dimensional
spacetime, for example
2
X
λ=1
µ
ν
ε (k, λ)ε (k, λ) = −g
µν
(k µ − (k · η) η µ) (k ν − (k · η) η ν )
,
+η η −
(k · η)2 − k 2
µ ν
η µ = (1, 0, 0, 0) .
The terms involving k µ and the unit time-like vector η µ are “ghostly” artifacts of gauge invariance. In
practical QED calculations they drop out of the final answers, which can usually be obtained by using just
the Feynman gauge term −g µν .
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