Notes on Bound-State QED
Ingvar LindgrenDepartment of Physics, Chalmers University of Technology
and the Goteborg University
Based on notes 1987-91Latex version (C/Dok/QED/Bound8791)
March 10, 2003
Contents
1 Semiclassical theory of radiation 3
1.1 Classical electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Maxwell’s equations in covariant form . . . . . . . . . . . . . . . . . 3
1.1.2 Interaction between electron and e-m field . . . . . . . . . . . . . . . 5
1.1.3 The Coulomb gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 The quantized radiation field . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The transverse radiation field . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 The Breit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 The transverse photon propagator . . . . . . . . . . . . . . . . . . . 10
1.2.4 Comparison with the covariant treatment . . . . . . . . . . . . . . . 11
2 Covariant theory of Quantum-ElectroDynamics 13
2.1 The photon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The electron propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Single-photon exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 The bound-electron self energy . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 Renormalization in the nonrelativistic limit (Bethe’s treatment) . . . 21
2.4.3 Relativistic renormalization . . . . . . . . . . . . . . . . . . . . . . . 22
A Notations 27
A.1 Four-component vector notations . . . . . . . . . . . . . . . . . . . . . . . . 27
A.2 Alpha and gamma matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
B Representations of states and operators 29
1
B.1 The Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
B.2 Dirac notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
B.3 Momentum eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
B.4 Representation of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
B.5 Representation of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
C Dirac free particle 34
C.1 Solutions to the Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . 34
C.2 Projection operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
C.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
C.4 The free-electron propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 39
D The free-electron self energy 41
E Evaluation of the free-electron self-energy integral 42
F Some fourier transforms 43
F.1 Evaluation of the integral∫
d3k(2π)3
eik·r12
q2−k2+iη. . . . . . . . . . . . . . . . . . . 43
F.2 Evaluation of the integral∫
d3k(2π)3
(α1·k)(α2·k)eik·r12
q2−k2+iη. . . . . . . . . . . . . 43
G Feynman diagrams for bound states 45
H Dimensional analysis 47
2
Chapter 1
Semiclassical theory of radiation
ch:ClassElDynThRad
1.1 Classical electrodynamicssec:ClassElDyn
1.1.1 Maxwell’s equations in covariant form
The Maxwell’s equations in vector form are1
∇ · E = ρ/ε0 (1.1a) Maxw.a
∇×B =1c2∂E
∂t+ µ0 j (1.1b) Maxw.b
∇ · B = 0 (1.1c) Maxw.c
∇×E +∂B
∂t= 0 (1.1d) Maxw.d
ρ is the electric charge density and j the electric current density. (Maxw.c1.1c) gives
B = ∇×A, (1.2) VectPot
where A is the vector potential. (Maxw.d1.1d) gives
E = −∂A
∂t−∇φ, (1.3) ScalPot
where φ is the scalar potential. (Maxw.a1.1a) and (
Maxw.b1.1b) give together with (
ScalPot1.3) and (
VectPot1.2)
−∇2φ− ∂
∂t∇ · A = ρ/ε0 = µ0 j
0,(∇2A− 1
c2∂2A
∂2t
)−∇
(∇ · A +
1c2∂φ
∂t
)= −µ0 j, (1.4) FieldEq
using ∇ × (∇ × A) = ∇(∇ · A) −∇2A. j0 = c2ρ (with ε0µ0 = c−2) is the scalar or’time-like’ part of the four-dimensional current
j = jµ = (c2ρ, j),1The formulas are here given for use in a consistent unit system, like the SI system (see Appendix
ch:DimAnalH).
Note that equations for use in a mixed system, like the Gaussian (c.g.s) unit system, used in many books,like in
MS84[1, Sect.1.2], will look differently.
3
where the vector part is the three-dimensional current j. Similarly, the four-dimensionalvector potential
A = Aµ = (φ/c,A)
has the scalar part φ/c and the vector part A. With the d’Alambertian operator, theseequations can be expressed2
�φ− ∂
∂t
(∇A
)= µ0 j
0 (1.5)
�A + ∇(∇A
)= µ0 j, (1.6)
which leads to Maxwell’s equations in covariant form
�A−∇(∇A) = µ0 j (1.7) MaxwCov
or
Aµ − ∂µ(∂νAν) = µ0 j
µ.
Lorentz condition
∇A = ∂µAµ = ∇ · A +
1c2∂φ
∂t= 0 (1.8) Lorentz
gives
�A = µ0 j (1.9) MaxwLor
Continuity equation
Operate on Maxwells’ equations (MaxwCov1.7) with ∇:
∇ (�A)−∇∇(∇A) = µ0∇j.
Since � = ∇2 and ∇ commute, this leads to the continuity equation
∇j = ∂µjµ = 0 (1.10) ContEq
Gauge invariance
Gauge transformation: A⇒ A+∇Λ Λ scalar.
Insert in (MaxwCov1.7): � (∇Λ)−∇(∇∇Λ) = � (∇Λ)−∇(�Λ) = 0
Maxwells’s equations are gauge invariant.2Concerning covariant notations, see Appendix
ch:NotA.
4
1.1.2 Interaction between electron and e-m field
The relativistic interaction between an electron of charge −e and the electromagnetic fieldis
Hint = ecα · A− eφ (1.11) ClassInt0
or in covariant notations
Hint = −ec αA = −ec αµAµ. (1.12) ClassInt
1.1.3 The Coulomb gauge
Transverse and longitudinal field components
The vector part of the electromagnetic field can be separated into transverse (divergence-free) and longitudinal (rotation-free) components
A = A⊥ + A‖ ; ∇ · A⊥ = 0 ; ∇×A‖ = 0 . (1.13) ASep
The electric field can be similarly separated
E = E⊥ + E‖ ; E⊥ = −∂A⊥∂t
; E‖ = −∂A‖
∂t−∇φ ,
while the magnetic field has only transverse components due to (VectPot1.2). The separated field
equations (FieldEq1.4) then become
∇2φ+∂
∂t∇ · A‖ = −ρ/ε0 (1.14a)
(∇2A‖ −
1c2∂2A‖
∂2t
)−∇
(∇ · A‖ +
1c2∂φ
∂t
)= −µ0 j‖ (1.14b)
(∇2 − 1
c2∂2
∂2t
)A⊥ = −µ0jT . (1.14c)
The longitudinal and the scalar or ’time-like’ components (A‖, φ) represent the instanta-neous Coulomb interaction and the transverse components (A⊥) represent retardation ofthis interaction and all magnetic interactions, as well as the electromagnetic radiation field(see section
sec:PhotEx41.2.4).
The energy of the electromagnetic field is given by
Erad =12
∫d3x
[ 1µ0
∣∣B∣∣2 + ε0∣∣E∣∣2]
=12
∫d3x
[ 1µ0
∣∣B∣∣2 + ε0∣∣E⊥
∣∣2] +12
∫d3x ε0
∣∣E‖∣∣2 . (1.15) RadEn
The last term represents the energy of the instantaneous Coulomb field, which is normallyalready included in the hamiltonian of the system. The first term represents the radiationenergy.
5
Semiclassically, only the transverse part of the field is quantized, while the longitudinalpart is treated classically
Sak67[2, Ch. 2,3]. It should be noted that the separation into trans-
verse and longitudinal components is is not Lorentz invariant and therefore not physicallyjustified, when relativity is taken into account. The retardation of the Coulomb interactionis represented by the transverse component (sec.
sec:PhotEx41.2.4), and therefore only the combination
of transverse and longitudinal components has physical significance.
In a fully covariatn treatment also the longitudinal component is quantized. The fieldis then represented by virtual photons with four directions of polarizations. A real photoncan only have transverse polarizations.
In the Coulomb gauge we have
∇ · A = 0, (1.16) CoulGauge
also known as the transversality condition. In this gauge there is no longitudinal compo-nent of A, and Maxwell’s equations then reduce to
∇2φ = −ρ/ε0 . (1.17) MaxwCoul
This has the solution
φ(x) =1
4πε0
∫d3x′
ρ(x′)|x− x′|
, (1.18) InstCoul
which is the instantaneous Coulomb interaction.
In free space the scalar potential φ can be eliminated by a gauge transformation.Then the Lorentz condition (
Lorentz1.8) is automatically fulfilled in the Coulomb gauge. The
field equation (FieldEq1.4) then becomes
∇2A− 1c2∂2A
∂2t= 0 . (1.19)
The relativistic interaction with an atomic electron (ClassInt01.11) is in the Coulomb gage given
by
Hint = ecα · A⊥ (1.20) IntCoul
and in second quantization
Hint =∑ij
c†j 〈i|ecα · A⊥|j〉 cj , (1.21) IntCoulSQ
where c†, c represent creation/annihilation operators for electrons. In the interactionpicture this becomes
Hint,I(t) =∑ij
c†i 〈i|ecα · A⊥|j〉 cj ei(εi−εj)t/~ . (1.22) IntCoulIP
1.2 The quantized radiation fieldsec:QuantRad
1.2.1 The transverse radiation fieldTransField
Classically a radiation field can be represented by the vector potentialSak67[2, Eq. 2.14]
A(x, t) =∑k
2∑p=1
[ckp εp e
i(k·x−ωt) + c∗kp εp e−i(k·x−ωt)
], (1.23) RadField
6
where k is the wave vector, ω = c|k| the frequency, and ckp /c∗kp represent the amplitude
of the wave with the a certain k vector and a certain polarization εp. The energy of thisradiation can be shown to be equal to
Sak67[2, p.22]
Erad = 2ε0∑kp
ω2 c∗kp ckp = ε0∑kp
ω2(c∗kp ckp + ckp c
∗kp
). (1.24) RadEn2
By making the substitution
ckp →√
~2ε0 ωV
akp and c∗kp →√
~2ε0 ωV
a†kp ,
where a†kp, /akp are photon creation/annihilation operators, the radiation energy goes overinto the hamiltonian of a collection of harmonic oscillators
Hharm.osc = 12
∑kp
~ω (akp a†kp + a†kp akp).
Therefore, we can motivate that the quantized transverse radiation field can be representedby
Sak67[2, Eq. 2.60]
A⊥(x, t) =∑k
√~
2ε0ωV
2∑p=1
[akp εp e
i(k·x−ωt) + a†kp εp e−i(k·x−ωt)
](1.25) Acl
1.2.2 The Breit interactionsec:SingPhotCl
6r
6s
6a
6
s
s
1
2
6b
6r 6s
6a
B12s s2 1
6b
Figure 1.1: Diagrammatic representation of the exchange of a single, transverse photon between twoelectrons (left). This is equivalent to a potential (Breit) interaction (right). Fig:SingPhotCl
The exchange of a single transverse photon between two electrons is illustrated by thetime-ordered diagram (left) in Fig.
Fig:SingPhotCl1.1, where one photon is emitted at the time t1 and
absorbed at a later time t2. The second-order evolution operator for this process is givenby
U (2)γ (0,−∞) =
(−i~
)2∫ 0
−∞dt2Hint,I(t2)
∫ t2
−∞dt1Hint,I(t1) eγ(t1+t2) (t1 < t2) , (1.26) U2SP
where γ is the parameter for the adiabatic damping of the perturbation. The interactionhamiltonians are in the Coulomb gauge given by (
IntCoulIP1.22) with the vector potential (
Acl1.25)
Hint,I(t1) =∑k1
√~
2ε0ω1V
2∑p1=1
c†r⟨r∣∣(a†kp ecα · εkp e
−ik·x)1
∣∣a⟩ ca e−it1(εa−εr−~ω1)/~
Hint,I(t2) =∑k2
√~
2ε0ω2V
2∑p2=1
c†s⟨r∣∣(akp ecα · εkp e
ik·x)2
∣∣a⟩ cb e−it2(εb−εs+~ω2)/~, (1.27) Hint1
7
which leads to the evolution operator
U (2)γ (0,−∞) = −c†rcac†scb
∑k
e2c2
2~ ε0V√ω1ω2
×∑p1p2
⟨rs
∣∣(akp α · εp eik·x)
2
(a†kp α · εp e
−ik·x)1
∣∣ab⟩× I , (1.28) U2SP2
where I is the time integral. The contraction between the creation and annihilationoperators (
Comma2.2) yields (ω = ω1 = ω2)∑
p1p2
⟨rs
∣∣(akp α · εp eik·x)
2
(a†kp α · εp e
−ik·x)1
∣∣ab⟩=
2∑p=1
⟨rs
∣∣(α · εp)2 (α · εp)1 e−ik·r12 (r12 = x1 − x2). (1.29) PolSum2
The time integral in (U2SP21.28) is
I =∫ 0
−∞dt2 e−it2(εb−εs+~ω+iγ)/~
∫ t2
−∞dt1 e−it1(εa−εr−~ω+iγ)/~
= − ~(εa + εb − εr − εs + 2iγ)(εa − εr − ~ω + iγ)
. (1.30) TimeInt
The result of the opposite time ordering t1 > t2 is obtained by the exchange 1 ↔ 2(r12 ↔ −r12), a ↔ b, and r ↔ s, and the total evolution operator, including both time-orderings, can be expressed
U (2)γ (0,−∞) = c†rcac
†scb
e2c2
ε0V
∑k
2∑p=1
⟨rs
∣∣(α · εp)1(α · εp)2M∣∣ab⟩
2ω(1.31) S2SP12
withM =
e−ik·r12
(εa + εb − εr − εs + 2iγ)(εa − εr − ~ω + iγ)
+eik·r12
(εa + εb − εr − εs + 2iγ)(εb − εs − ~ω + iγ)
= cos (k · r12)εa + εb − εr − εs − 2~ω + 2iγ
(εa + εb − εr − εs + 2iγ)(εa − εr − ~ω + iγ)(εb − εs − ~ω + iγ)
+i sin (k · r12)εa − εr − εa + εs
(εa + εb − εr − εs + 2iγ)(εa − εr − ~ω + iγ)(εb − εs − ~ω + iγ). (1.32) M
This can be compared with the evolution operator corresponding to a potential interactionB12 between the electrons, as illustrated in the right diagram of Fig.
Fig:SingPhotCl1.1,
U (2)η (0,−∞) = c†rcac
†scb 〈rs|B12|ab〉
(−i~
) ∫ 0
−∞dt e−it(εa+εb−εr−εs+iη)/~
= c†rcac†scb
〈rs|B12|ab〉εa + εb − εr − εs + iη
. (1.33) PotScatt
Identification then leads to
B12 =e2c2
2ε0ωV
∑kp
(α · εp)1(α · εp)2
×[cos (k · r12)
εa + εb − εr − εs − 2~ω + 2iγ(εa − εr − ~ω + iγ)(εb − εs − ~ω + iγ)
+i sin (k · r12)εa − εr − εb + εs
(εa − εr − ~ω + iγ)(εb − εs − ~ω + iγ)
]. (1.34) B12
8
We assume that energy is conserved by the interaction, i.e.,
εa − εr = εs − εb = ~cq. (1.35) cq
Then (B121.34) becomes
B12 = − e2c2
2ε0ωV
∑kp
(α · εp)1(α · εp)2[cos (k · r12)
2ω(cq − ω + iγ)(−cq − ω + iγ)
−i sin (k · r12)2cq
(cq − ω + iγ)(−cq − ω + iγ)
]or
B12 =e2
ε0V
∑kp
(α · εp)1(α · εp)2cos (k · r12)− iq/k sin (k · r12)
q2 − k2 + iγ(1.36) B12b
with ω = ck. Since the contribution to the sum comes from k ≈ ±q, we can replace q/kby sgn(q). In addition, it is found that the sign of the imaginary part is immaterial (seeAppendix
sec:FT1F.1), and therefore we can replace the numerator by eik·r12 ,
B12 =e2
ε0V
∑kp
(α · εp)1(α · εp)2eik·r12
q2 − k2 + iγ. (1.37) B12c
The εp vectors are orthogonal unit vectors, which leads toSak67[2, Eq. 4.312]
3∑p=1
(α · εp)1(α · εp)2 = α1 · α2 . (1.38) SumEps3
This gives 2∑p=1
(α · εp)1(α · εp)2 = α1 · α2 − (α1 · k)(α2 · k), (1.39) SumEps2
assuming ε3 = k to be the unit vector in the k direction. The interaction (B12c1.37) then
becomes in the limit of continuous momenta (App.sec:ReprB.4)
B12 =e2
ε0
∫d3k
(2π)3[α1 · α2 − (α1 · k)(α2 · k)
] eik·r12
q2 − k2 + iγ. (1.40) B12d
With the fourier transforms in Appendixch:FTF this yields the retarded Breit interaction
BRet12 = − e2
4πε0
[α1 · α2
ei|q|r12
r12− (α1 · ∇1)(α2 · ∇2)
ei|q|r12 − 1q2 r12
](1.41) BreitRet
Setting q = 0, we obtain the instantaneous Breit interaction (real part)
BInst12 = − e2
4πε0
[α1 · α2 + 1
2 (α1 · ∇1)(α2 · ∇2) r12]
or using
(α1 · ∇1)(α2 · ∇2) r12 = −α1 · α2
r12+
(α1 · r12)(α1 · r12)r312
we arrive at
BInst12 = − e2
4πε0 r12
[12 α1 · α2 +
(α1 · r12)(α1 · r12)2r212
](1.42) BreitInst
which is the standard form of the instantaneous Breit interaction.
9
6r
6s
6a
6
s
s
1
2
6b
+
6r
6s
6a6
s
s1
26b
=
6r 6s
6a
-s s1 2
6b
Figure 1.2: The two time-orderings of a single-photon exchange can be represented by a single Feynmandiagram. Fig:SingPhotF
1.2.3 The transverse photon propagatorsec:PhotProp2
We shall now consider both time-orderings of the interaction represented in the figuresimultaneously. The evolution operator can then be expressed
U (2)γ (0,−∞) =
(−i~
)2∫ 0
−∞dt2
∫ 0
−∞dt1 T
[Hint,I(t2)Hint,I(t1)
]e−γ(|t1|+|t2|), (1.43) U2SP3
where
T[Hint,I(t2)Hint,I(t1)
]=
Hint,I(t2)Hint,I(t1) t2 > t1
Hint,I(t1)Hint,I(t2) t1 > t2 .(1.44) T
In the Coulomb gauge the interaction is given by (IntCoulIP1.22) and the vector potential is given
by (Acl1.25). The evolution operator for the combined interactions will then be
U (2)γ (0,−∞) = −c†rcac†scb
e2c2
~2×∫ 0
−∞dt2
∫ 0
−∞dt1 T
[(α · A⊥)1 (α · A⊥)2
]e−it1(εa−εr+iγ)/~ e−it2(εb−εs+iγ)/~. (1.45) U2SP4
HereT
[(α · A⊥)1 (α · A⊥)2
]=
∑kp
~2ωε0V
(α · εp)1 (α · εp)2
×
e−i(k1·x1−ωt1) ei(k2·x2−ωt2) t2 > t1
e−i(k2·x2−ωt2) ei(k1·x1−ωt1) t1 > t2
or with r12 = x1 − x2 and t12 = t1 − t2
T[(α · A⊥)1 (α · A⊥)2
]=
~ε0
2∑p=1
(α · εp)1 (α · εp)21V
∑k
e∓i(k·r12−ωt12)
2ω, (1.46) PhProp2
where the upper sign is valid for t2 > t1. This yields
U (2)γ (0,−∞) = −c†rcac†scb
e2c2
ε0~2
∫ 0
−∞dt2
∫ 0
−∞dt1
×2∑
p=1
(α · εp)1 (α · εp)21V
∑k
e∓i(k·r12−ωt12)
2ωe−icq t12 eγ(t1+t2), (1.47) U2SP4a
10
utilizing the energy conservation (cq1.35).
The boxed part of the equation above is essentially the photon propagator (PhotProp2.10)
DF(2, 1) =1V
∑k
e∓i(k·r12−ωt12)
2ω⇒
∫d3k
(2π)3e∓i(k·r12−ωt12)
2ω. (1.48) PhProp
This can be represented by a complex integral
DF(2, 1) = i∫
d3k(2π)3
∫dz2π
eizt12
z2 − ω2 + iηeik·r12 , (1.49) PhotPropInt
where η is a small, positive quantity. As before, the sign of the exponent ik · r12 isimmaterial. The integrand has poles at z = ±(ω − iη), assuming ω to be positive. Fort2 > t1 integration over the negative half plane yields 1
2ω eiω t12 eik·r12 and for t1 > t2
integration over the positive half plane yields 12ω e
−iω t12 eik·r12 , which is identical to (PhProp1.48).
The evolution operator (U2SP4a1.47) can then be expressed
U (2)γ (0,−∞) = −c†rcac†scb
e2c2
ε0~
∫ 0
−∞dt2
∫ 0
−∞dt1
×2∑
p=1
(α · εp)1 (α · εp)2 DF(1, 2) e−icq t12/~ eγ(t1+t2)~. (1.50) U2SP5
1.2.4 Comparison with the covariant treatmentsec:PhotEx4
It is illustrating to compare the quantization of the transverse photons with the fullycovariant treatment, to be discussed in the next chapter. Then we simply have to replacethe sum in (
U2SP21.28) by the corresponding covariant expression
2∑p1p2=1
(akp α · εp)2 (a†kp α · εp)1 ⇒3∑
p1p2=0
(akp αµεµp)2 (a†kp α
νενp)1 . (1.51) eq:PolSum4
The commutation relation (Comma2.2) yields
3∑p1p2=0
(akp αµεµp)2 (a†kp α
νενp)1 = α1 · α2 − 1 . (1.52) Replace
We then find that the equivalent potential interaction (B12c1.37) becomes
V12 = −e2
ε0
∫d3k
(2π)3(1−α1 · α2
) eik·r12
q2 − k2 + iγ(1.53) V12
and with the fourier transform given in Appendixsec:FT1F.1
V12 =e2
4πε0 r12
(1−α1 · α2
)ei|q|r12 (1.54) CoulBreit
We shall now compare this with the exchange of transverse photons, treated above.We then make the decomposition
1−α1 · α2 =
1− (α1 · k)(α2 · k)
−α1 · α2 + (α1 · k)(α2 · k) .(1.55) Decompose
11
The last part, which represents the exchange of transverse photons, is identical to (SumEps21.39),
which led to the Breit interaction. The first part, which represesents the exchange oflongitudinal and scalar photons, corresponds to the interaction
VC = −e2
ε0
∫d3k
(2π)3[1− (α1 · k)(α2 · k)
] eik·r12
q2 − k2 + iγ. (1.56) VC
This fourier transform is evaluated in Appendixsec:FT2F.2 and yields
VC = −e2
ε0
∫d3k
(2π)3(1− q2
k2
) eik·r12
q2 − k2 + iγ=e2
ε0
∫d3k
(2π)3eik·r12
k2 − iγ, (1.57) VC2
provided that the orbitals are generated in a local potential. Using the transform inAppendix
sec:FT1F.1, this becomes
VCoul =e2
4ıε0 r12. (1.58) CoulInst
Thus, we see that the exchange of longitudinal and scalar photons corresponds to theinstantaneous Coulomb interaction, while the exchange of the transverse photons corre-sponds to the Breit interaction. Note that this is true only if the orbitals are generated ina local potential.
If instead of the separation (Decompose1.55) we would separate the photons into the scalar part
(p = 0) and the vector part (p = 1, 2, 3),
1−α1 · α2 =
1
−α1 · α2 ,(1.59) Decompose2
then the result would be
V RetCoul =
e2
4ıε0 r12ei|q|r12
V RetGaunt = − e2
4πε0 r12α1 · α2 e
i|q|r12 . (1.60) CoulGauntRet
which represents the retarded Coulomb and the retarded magnetic (Gaunt) interaction.This implies that the longitudinal photon represents the retardation of the Coulombinteraction, which is included in the Breit interaction (
BreitRet1.41).
If we would set q = 0, then we would from (CoulGauntRet1.60) retrieve the instantaneous Coulomb
interaction (CoulInst1.58) and
− e2
4πε0α1 · α2 , (1.61) GauntInst
which is known as the Gaunt interaction. The Breit interaction will then turn into theinstanteneous interaction (
BreitInst1.42). This will still have some effect of the retardation of the
Coulomb interaction, although it is instantaneous.
We shall see later that the interactions (CoulGauntRet1.60) correspond to the interactions in the
Feynman gauge (VF2.48), while the instantaneous Coulomb and Breit intinteractions corre-
spond to the Coulomb gauge.
12
Chapter 2
Covariant theory ofQuantum-ElectroDynamics
ch:CovQED
2.1 The photon propagatorsec:PhotProp
The covariant electromagnetic radiation field is in analogy with (Acl1.25) represented by the
four-component vector potential1MS84[1, Eq. 5.16]
Aµ(x) = A+µ (x) +A−µ (x) =
√1
2ε0ωV
∑kr
εµr
[akr e
−ikx + a†kr eikx
], (2.1) A
where
k = kµ = (k0,k) ; k0 = ω/c = |k| ; kx = ωt− k · x
are the covariant notations, defined in Appendixch:NotA, and r = (0, 1, 2, 3) represents the four
polarization states – two transverse (r = 1, 2), one longitudinal (r = 3) and one scalar(r = 0).
The creation and annihilation operators satisfy the following commutation relationMS84[1,
Eq.5.28] [akr, a
†k′r′
]= δk,k′δrr′ζr, (2.2) Comma
where ζr = (−1, 1, 1, 1) for r = (0, 1, 2, 3), which leads to the relationMS84[1, Eq.5.19]∑
rr′
[εµr akr, ενr′ a
†k′r′
]= −gµν δk,k′ . (2.3) AComm2
The contraction between the photon fields is defined as the difference between the timeand normal orderings
Aν(x2)Aµ(x1) = T[Aν(x2)Aµ(x1)
]−N
[Aν(x2)Aµ(x1)
]. (2.4) PhotContr0
Since the vacuum expectation value vanishes for every normal ordered product, it followsthat the contraction is equal to the vacuum expectation of the time-ordered product
Aν(x2)Aµ(x1) =⟨0∣∣T [
Aν(x2)Aµ(x1)]∣∣0⟩
. (2.5) PhotContr
1From now on we set ~ = 1.
13
The Feynman photon propagator is defined by means of the photon-field contrac-tion
Aν(x2)Aµ(x1) =⟨0∣∣T [
Aν(x2)Aµ(x1)]∣∣0⟩
=ic ε0
DFνµ(x2 − x1). (2.6) PhotPropDef
With the radiation field (A2.1) this becomes
Aν(x2)Aµ(x1) =⟨0∣∣T [
Aν(x2)Aµ(x1)]∣∣0⟩
=⟨0∣∣Θ(t2 − t1)Aν(x2)Aµ(x1) + Θ(t1 − t2)Aµ(x1)Aν(x2)
∣∣0⟩=
⟨0∣∣∣Θ(t2 − t1)
[A+
ν (x2), A−µ (x1)]+ Θ(t1 − t2)
[A+
µ (x1), A−ν (x2)]∣∣∣0⟩
=∑
k1k2r1r2
12ε0V
√ω1ω2
⟨0∣∣∣Θ(t2 − t1)
[εµr2 ak2r2 , ενr1 a
†k1r1
]e−i(k2x2−k1x1)
+ Θ(t1 − t2)[εµr1 ak1r1 , εµr2 a
†k2r2
]ei(k2x2−k1x1)
∣∣∣0⟩= −gνµ
∑k
12ε0ωV
[Θ(t2 − t1) e−ik(x2−x1) + Θ(t1 − t2) eik(x2−x1)
]= −gνµ
∑k
12ε0ωV
e−ik·r12[Θ(t2 − t1) e−iω(t2−t1) + Θ(t1 − t2) eiω(t2−t1)
](2.7) PhotContr1
with r12 = x1 − x2. (The sign of k is immaterial).
The expression in the square brackets of (PhotContr12.7) can as in (
PhotPropInt1.49) be written as a complex
integral
I = Θ(t2 − t1) e−iω(t2−t1) + Θ(t1 − t2) eiω(t2−t1) = 2iω∫ ∞
−∞
dz2π
e−iz(t2−t1)
z2 − ω2 + iη. (2.8) ComplInt
Thus, ⟨0∣∣T [
Aν(x2)Aµ(x1)]∣∣0⟩
= −gνµi
ε0V
∑k
eik·r12
∫ ∞
−∞
dz2π
e−iz(t2−t1)
z2 − ω2 + iη
→ −gνµiε0
∫d3k
(2π)3eik·r12
∫ ∞
−∞
dz2π
e−iz(t2−t1)
z2 − ω2 + iη, (2.9) PhotProp0
and the photon propagator (PhotPropDef2.6) becomes (c.f (
PhotPropInt1.49))
DFνµ(x2 − x1) = −gνµ c
∫d3k
(2π)3eik·r12
∫ ∞
−∞
dz2π
e−iz(t2−t1)
z2 − ω2 + iη. (2.10) PhotProp
With z = ck0 , ω2 = c2k2, k2 = k20 − k2 this becomes in four-component form
DFνµ(x2 − x1) = −gνµ
∫d4k
(2π)4e−ik(x2−x1)
k2 + iη(2.11) PhotProp4
The photon propagator (PhotProp2.10) can be expressed
DFνµ(x2 − x1) =∫
dz2π
e−iz(t2−t1)DFνµ(x2,x1; z), (2.12)
14
where DFνµ(x2,x1; z) is the fourier transform with respect to time
DFνµ(x2,x1; z) = −gνµ c
∫d3k
(2π)3eik·r12
z2 − c2k2 + iη(2.13) PhotPropFT
The photon propagator (PhotPropFT2.13) can also be expressed⟨
x2
∣∣DFνµ(z)∣∣x1
⟩= −gνµ c
〈x2|ks〉〈ks|x1〉z2 − c2k2 + iη
, (2.14) PhotPropCoord
where
〈x|ks〉 = (2π)−3/2us(k) eik·x
is the momentum eigenfunction (MomEigB.33) and summation over s and integration over k are
understood. (PhotPropCoord2.14) is the coordinate representation (see section
sec:ReprOpB.5) of the operator
DFνµ(z) = −gνµ c|ks〉〈ks|
z2 − c2k2 + iη. (2.15) PhotPropOp
The corresponding momentum representation is
〈p2r2 |DFνµ(z)|p1r1〉 = −gνµc〈p2r2|ks〉〈ks|p1r1〉z2 − c2k2 + iη
= −gνµ c∑r1r2
δs,r2δs,r1
∫d3k δ3(p2 − k)δ3(p1 − k)
1z2 − c2k2 + iη
, (2.16) DFMom1
which leads to
〈p2r2 |DFνµ(z)|p1r1〉 = −gνµ c δ3(p2 − p1) δr2,r1
1z2 − c2p2 + iη
, (2.17) DFMom
where p = p1 = p2. The corresponding fourier transform is according to (OpFTB.38)∑
r1r2
ur2(p2) 〈p2r2|DFνµ(z)|p1r1〉u†r1(p1)
= DFνµ(p2,p1; z) = −gµν c δ3(p2 − p1)
1z2 − c2p2 + iη
. (2.18) DFFour
2.2 The electron propagatorsec:ElProp
The electron field is in the bound-interaction picture given by
ψ(x) =∑
j
cj(t)φj(x) ψ†(x) =∑
j
c†j(t)φ†j(x) , (2.19) ElField
where {φj} is a complete set of single-electron orbitals – with positive as well as negativeenergy – and cj(t), c
†j(t) are the corresponding time-dependent annihilation and creation
operators. The orbitals are generated by the Dirac hamiltonian in the field of the nucleus(with charge Z) and with an additional potential U
hD φa(x) =(cα · p + β mec
2 − Ze2
4πε0 r+ U
)φa(x) = εa φa(x), (2.20) DiracEq
where εa is the orbital energy eigenvalue. The operators have in this picture the timedependence
cj(t) = cj e−iεjt ; c†j(t) = c†j e
iεjt, (2.21) ctime
where cj , c†j are the time-independent operators.
15
The Feynman electron propagator is defined by means of the contraction betweenthe electron field operators in analogy with (
PhotPropDef2.6)
ψ(x)ψ†(x′) =⟨0∣∣T [
ψ(x)ψ†(x′)]∣∣0⟩
= iSF(x2, x1). (2.22) ElPropDef
It should be noted that the propagator is a matrix with the elements defined by
iSF(x2, x1)α,β =⟨0∣∣T [
ψ(x)ψ†(x′)]∣∣0⟩
α,β=
⟨0∣∣T [
ψ(x)αψ†(x′)β
]∣∣0⟩. (2.23)
The time-independent operators satisfy the fermion anti-commutation ruleMS84[1,
Eq.4.39]
{ci, c†j} = ci c†j + c†j ci = δi,j . (2.24) Anticomm
We then get in analogy with (PhotContr12.7) for t1 6= t2 (t1 = t2 will be considered below)
ψ(x2)ψ†(x1) =⟨0∣∣T [
ψ(x2)ψ†(x1)]∣∣0⟩
=⟨0∣∣T [
Θ(t2 − t1)ψ(x2)ψ†(x1)−Θ(t1 − t2)ψ†(x1)ψ(x2)]∣∣0⟩
=⟨0∣∣T [
Θ(t2 − t1){ψ+(x2), ψ†+(x1)} −Θ(t1 − t2){ψ†−(x1), ψ−(x2)}
]∣∣0⟩=
⟨0∣∣∣T[
Θ(t2 − t1)∑pp′
{cp, c†p′} e−iεpt2 eiεp′ t1 φp(x2)φ
†p′(x1)
−Θ(t1 − t2)∑hh′
{ch′ , c†h} eiεh′ t1 e−iεht2 φh(x2)φ
†h′(x1)
]∣∣∣0⟩= Θ(t2 − t1)e−iεp(t2−t1)φp(x2)φ†p(x1)−Θ(t1 − t2)
∑h
e−iεh(t2−t1)φh(x2)φ†h(x1). (2.25)
Here, ψ±(x1) represent the electron field operators for particles and holes, respectively,and p, h represent particles and holes in the single-particle picture. The contraction abovecan now be expressed as a complex integral in analogy with (
PhotPropInt1.49)
⟨0∣∣T [
ψ(x1)ψ†(x1)]∣∣0⟩
= i∑
j
∫dz2π
φj(x2)φ†j(x1)
z − εj + iηje−iz(t2−t1), (2.26)
where ηj is a small real quantity with the same sign as εj . This gives the electron propagator(ElPropDef2.22)
SF(x2, x1) =∑
j
∫dz2π
φj(x2)φ†j(x1)
z − εj + iηje−iz(t2−t1) (2.27) ElProp
Here, j runs over positive and negative energy states, and η is a small positive quantity.
The contraction has so far been defined only for t1 6= t2. For the bound-state problemit is necessary to consider also equal-time contrations. We then define the time-orderingfor equal time as
T[ψ(x2)ψ†(x1)
]=
12
[ψ(x2)ψ†(x1)− ψ†(x1)ψ(x2)
](t1 = t2) (2.28)
16
ψ(x2)ψ†(x1) =⟨0
∣∣∣T [ψ(x2)ψ†(x1)
]∣∣∣ 0⟩
=12
⟨0
∣∣∣ψ(x2)ψ†(x1)− ψ†(x1)ψ(x2)∣∣∣ 0
⟩=
12
∑p
φp(x2)φ†p(x1)−12
∑h
φh(x2)φ†h(x1) =
12
∑j
sgn(εj)φj(x2)φ†j(x1), (2.29)
where j as before runs over particles and holes. This can still be expressed by the integralabove, as can be seen from the relation
1εj − z − iεj η
=εj − z
(εj − z)2 + (εj η)2+
iεj η(εj − z)2 + (εj η)2
= P1
εj − z+ iπ sgn(εj) δ(εj − z). (2.30)
P stands for the principal-value integration, which does not contribute here. Therefore,the electron propagator (
ElProp2.27) is valid also for equal times.
The fourier transform with respect to time of the electron propagator (ElProp2.27) is
SF(x2,x1; z) =∑
j
φj(x2)φ†j(x1)
z − εj + iηj(2.31) ElPropFT
This can be regarded as the coordinate representation
SF(x2,x1; z) = 〈x2|SF(z)|x1〉 =〈x2|j〉 〈j|x1〉z − εj + iηj
(2.32) SFCoord
of the operatorSF(z) =
|j〉 〈j|z − εj (1− iη)
. (2.33) ElPropOp
Using the relation (OpExpB.39), this can also be expressed
SF(z) =1
z − hD (1− iη), (2.34) ElPropOp2
where hD is the Dirac hamiltonian (DiracEq2.20), hD |j〉 = εj |j〉. The corresponding momentum
representation is
SF(p2r2,p1r1; z) = 〈p2r2|SF(z)|p1r1〉 =〈p2r2|j〉 〈j|p1r1〉z − εj + iηj
. (2.35) SFMom
2.3 Single-photon exchangesec:SingPhotEx
The exchange of a single photon between two electrons is represented by the Feynmandiagram in Fig.
Fig:SingPhot2.1, and the scattering amplitude is given by 〈cd|S(2)|ab〉, where S(2) is
the second-order S-matrixMS84[1, Eq.7.1]
S(2) = −12
∫ ∞
−∞dt1
∫ ∞
−∞dt2 T
[Hint,I(t2)Hint,I(t1)
]e−η1|t1|−η1|t2|
= −12
∫∫d4x1 d4x2 T
[Hint,I(x2)Hint,I(x1)
]e−η1|t1|−η1|t2|. (2.36) S2
17
6c 6d
6a
-s s1 µ 2 ν
6b
Figure 2.1: The Feynman representation of the exchange of a single, virtual photon between two electrons.The heavy lines represent electronic states in the bound-interaction picture. Fig:SingPhot
Here, Hint,I(t) is the interaction hamiltonian (c.f. Eq. (ClassInt1.12))
Hint,I(t) = c
∫d3xHint(x)
withHint,I(x) = −ec ψ†(x)αµAµ(x)ψ(x). (2.37) Hint
η1, η2 are adiabatic damping coefficients, which eventually go to zeroGML51,Su57[3, 4]. ψ(x) and
ψ†(x) are the electron-field operators in the bound-interaction picture (ElField2.19) (Note that the
interaction (Hint2.37) is defined without the normal-ordering in order to allow for equal-time
contractions discussed above). The exchange of a virtual photon represents a contractionbetween the photon fields (
PhotContr2.5)
S(2) = −e2c2
2
∫∫d4x1 d4x T
[(ψ†(x)ανAν(x)ψ(x)
)2
(ψ†(x)αµAµ(x)ψ(x)
)1
]e−η1|t1|−η2|t2|
= −e2c2
2
∑abcd
∫∫d4x1 d4x2
[c†dφ
†d(x)α
νcbφb(x)]2〈0|T [Aν(x2)Aµ(x1)]|0〉
×[c†cφ
†c(x)α
µcaφa(x)]1e−η1|t1|−η1|t2|. (2.38) S22
Identification with the second-quantized form
S(2) = 12
∑abcd
c†cc†d
⟨cd
∣∣S(2)∣∣ab⟩ cbca (2.39) Secquant
gives⟨cd
∣∣S(2)∣∣ab⟩ = −e2c2
∫∫d4x1 d4x2
[φ†d(x)α
νφb(x)]2〈0|T [Aν(x2)Aµ(x1)]|0〉
×[φ†c(x)α
µφa(x)]1e−η1|t1|−η1|t2|. (2.40) S23
Using the definition of the photon propagator (PhotPropDef2.6) this becomes⟨
cd∣∣S(2)
∣∣ab⟩ =
− ie2cε0
∫∫d4x1 d4x2
[φ†d(x)α
νφb(x)]2DFνµ(x2 − x1)
[φ†c(x)α
νφa(x)]1e−η1|t1|−η1|t2|. (2.41) S24
The time integral is ∫∫dt1 dt2 eit2(εd−εb) eit1(εc−εa) eiz(t1−t2) e−η1|t1|−η1|t2|
= 2π∆η1(εa − εc − z) ∆η2(εb − εd + z), (2.42) Timeint
18
where the ∆ function is defined in Appendixsec:DiracfcnB.1.
Integration over z gives z = εa− εc = εd− εb = cq and the scattering amplitude (S242.41)⟨
cd∣∣S(2)
∣∣ab⟩ = −2πi∆η1+η2(εc + εd − εa − εb)
× e2c
ε0
∫∫d3x1 d3x2
[φ†d(x)ανφb(x)
]2DFνµ(x2 − x1; cq)
[φ†c(x)αµφa(x)
]1. (2.43) S25
With the photon propagator in the Feynman gauge (PhotPropFT2.13) this becomes⟨
cd∣∣S(2)
∣∣ab⟩ = 2πi∆η1+η2(εa + εb − εc − εd)
× e2
ε0
∫∫d3x1 d3x2
[φ†d(x)αµφb(x)
]2
[φ†c(x)αµφa(x)
]1
∫d3k
(2π)3eik·r12
q2 − k2 + iη. (2.44) S26
The time integration leads – in the limit η → 0 – to energy conservation at eachvertex with z treated as the energy parameter of the propagator.
6c 6d
6a
V12s s1 2
6b
Figure 2.2: The single-photon exchange is compared with the scattering of a potential, V12. Fig:SingPot
The single-photon exchange can be compared with the scattering by an energy-dependentpotential interaction, V12(cq), (see Fig.
Fig:SingPot2.2) with the scattering amplitude⟨
cd∣∣S(2)
∣∣ab⟩ = −2πi∆η1+η2(εc + εd − εa − εb)⟨cd
∣∣V12(cq)∣∣ab⟩. (2.45) PotInt
Identification then gives
⟨cd
∣∣V12(cq)∣∣ab⟩ = −e
2
ε0
∫∫d3x1 d3x2
[φ†d(x2)αµφb(x2)
] [φ†c(x1)αµφa(x1)
]×
∫d3k
(2π)3eik·r12
q2 − k2 + iη= −e
2
ε0
⟨cd
∣∣∣αµ1α2µ
∫d3k
(2π)3eik·r12
q2 − k2 + iη
∣∣∣ab⟩ . (2.46) PotEl
Thus, with αµ1α2µ = 1−α1 · α2, the equivalent potential in the Feynman gauge
becomes
V F12(cq) = −e
2
ε0(1−α1 · α2)
∫d3k
(2π)3eik·r12
q2 − k2 + iη(2.47) EqPot
which is identical to the previous result (V121.53). The Fourier transform then yields
V F12 = (1−α1 · α2)
e2
4πε0 r12ei|q|r12 (2.48) VF
19
which is the retarded Coulomb and Gaunt interactions (CoulGauntRet1.60).
In calculating the first-order energy shift, corresponding to the exchange of a singlephoton, we employ the Gell-Mann-Low-Sucher
GML51,Su57[3, 4] prescription and set all ηi equal to η.
Then the energy shift in the (closed-shell) state Φ is given by
∆E = limη→0
iη2
∑n〈Φ|S(n)|Φ〉〈Φ|S|Φ〉
, (2.49) GMLS
which in lowest order becomes
∆E = limη→0
iη 〈Φ|S(2)|Φ〉. (2.50) GMLS1
Using (I.9), we get from (I.12)
limη1+η2→0
i(η1 + η2)2
〈cd|S(2)|ab〉 = δεc+εd, εa+εb〈cd|V12(cq)|ab〉
and the first-order energy shift becomes
∆E = 〈Φ|V12(cq)|Φ〉 = 〈ab|V12(cq)|ab〉 − 〈ba|V12(cq)|ab〉, (2.51) DeltaE1
where Φ is the antisymmetrized, unperturbed state |{ab}〉. This result is consistent withthe interpretation of V12 as an equivalent perturbing potential.
Due to the energy conservation of the scattering process, only diagonal (on-the-energy-shell) matrix elements of the potential are obtained from the analysis of the single-photonexchange. Off-diagonal elements are obtained from the analysis of the two-photon ex-change
Li89[5].
2.4 The bound-electron self energy
2.4.1 General
6a
6t 6
s
sz
6b
Figure 2.3: Diagram representing the first-order self-energy of a bound electron. Fig:SE
With the rules given in Appendixch:FeynmanG, the first-order electron self energy in the state
|a〉 is (see Fig.Fig:SE2.3)
∆SE(a) = i∫∫
d3x1 d3x2 φ†a(x2) iecαν
∫dx2π
iSF(x2,x1; εa − z)
× icε0
DFνµ(x2 − x1; z) iecαµ φa(x1) , (2.52) SE
20
where the orbitals are generated by the Dirac equation (DiracEq2.20). SF(x2,x1; εa − z) is the
electron propagator (ElPropFT2.31), (
SFCoord2.32) and DFνµ(x2 − x1; z) is the photon propagator (
PhotPropFT2.13).
With the notations of Appendixch:CoordMomB, the self energy can be written
∆SE(a) = 〈a|Σ(εa)|a〉 = 〈a|x2〉〈x2|Σ(εa)|x1〉〈x1|a〉 , (2.53)
where〈x2|Σ(εa)|x1〉 =
i e2cαν
ε0
∫dz2π
SF(x2,x1; εa − z)DFνµ(x2 − x1; z)αµ
= −ie2c2
ε0
∫dz2π
∫d3k
(2π)3〈x2|αµe
ik·x|t〉〈t|αµe−ik·x|x1〉(εa − z − εt + iηt)
)(z2 − c2k2 + iη
) (2.54) SECoord
is the coordinate representation of the electron self-energy operator, Σ(εa). This is amass operator, which should be added to the Dirac hamiltonian in order to include theelectron self energy – together with a counter term to take care of the renormalization.Integration over z then yields
〈x2|Σ(εa)|x1〉 = − e2c
2ε0
∫d3k
(2π)31k
〈x2|αµeik·x|t〉〈t|αµe−ik·x|x1〉
εa − εt − ck sgn(εt), (2.55) SECoord2
where k = |k|. Integration over the angular part of k yields
〈x2|Σ(εa)|x1〉 = − e2c
4πε0 r12〈x2|αµ|t〉
∫ ∞
0
dk sin kr12
εa − εt − ck sgn(εt)〈t|αµ|x1〉, (2.56) SEInt
where r12 = |x1 − x2|. This yields the following expression for the self-energy operatoritself
Σ(εa) = − e2c
4πε0 r12αµ|t〉
∫ ∞
0
dk sin kr12
εa − εt − ck sgn(εt)〈t|αµ . (2.57) SEOp
Note that the self energy depends on the energy εa of the state |a〉.
2.4.2 Renormalization in the nonrelativistic limit (Bethe’s treatment)
For small k values and positive intermediate states, (SEOp2.57) reduces to
Σ(εa) = − e2c
4π2ε0αµ|t〉
∫ ∞
0
k dkεa − εt − ck
〈t|αµ . (2.58) SEOp1
The scalar part of αµαµ cancels in the renormalization, leaving only the vector part to be
considered,
Σ(εa) =e2c
4π2ε0α|t〉 ·
∫ ∞
0
k dkεa − εt − ck
〈t|α . (2.59) SEOp2
The corresponding operator for a free electron in the state p+ (see Fig. 2) is
ΣFree(p+) =e2c
4π2ε0α|q+〉 ·
∫ ∞
0
k dkεp+ − εq+ − ck
〈q+|α , (2.60) SEOpFree
restricting the intermediate states to positive energies. In the momentum representationthis becomes⟨
p′+∣∣ΣFree(p+)
∣∣p+
⟩=
e2c
4π2ε0〈p′+|α|q+〉 ·
∫ ∞
0
k dkεp+ − εq+ − ck
〈q+|α|p+〉 . (2.61) SEFreeMom0
21
But since α is diagonal with respect to the momentum, we must have q = p = p′. Thus,
⟨p′+
∣∣ΣFree(p+)∣∣p+
⟩= −δ3p′,p
e2
4π2ε0
∣∣〈p+|α|p+〉∣∣2 ∫ ∞
0dk . (2.62) SEFreeMom2
Obviously, this quantity is infinite. Inserting a set of complete states (IdentB.41), this becomes
⟨p′+
∣∣ΣFree(p+)∣∣p+
⟩= −δ3p′,p
e2
4π2ε0〈p+|α|t〉 · 〈t|α|p+〉
∫ ∞
0dk . (2.63) SEFreeMom3
The free-electron self-energy operator can then be expressed
ΣFree(p+) = −δ3p′,pe2
4π2ε0α|t〉 ·
∫ ∞
0dk 〈t|α , (2.64) SEFreeOp
which should be subtracted from the bound-electron self-energy operator (SEOp22.59). We can
assume the intermediate states {j} to be indentical to those in the bound case. This givesthe renormalized self-energy operator
ΣRenorm(εa) =e2
4π2ε0α|t〉 ·
∫ ∞
0dk
εa − εtεa − εt − ck
〈t|α . (2.65) SEOpRen
The expectation value of this operator in a bound state |a〉 yields the renormalized bound-electron self energy in this approximation, i.e., the corresponding contribution to thephysical Lamb shift,
⟨a
∣∣ΣRenorm(εa)∣∣ a⟩ =
e2
4π2ε0 r12〈a|α|t〉 · 〈t|α|a〉
∫ ∞
0dk
εa − εtεa − εt − ck
. (2.66) BetheSE1
This result is derived in a covariant Feynman gauge, where the quantized radiation hastransverse as well as longitudinal components. In the Coulomb gauge only the formerare quantized. Since all three vector components above yield the same contribution, wewill get the result in the Coulomb gauge by multiplying by 2/3. Furthermore, in thenon-relativistic limit we have α → p/c, which leads to
⟨a
∣∣ΣRenorm(εa)∣∣ a⟩ =
e2
6π2c2ε0 r12〈a|p|t〉 · 〈t|p|a〉
∫ ∞
0dk
εa − εtεa − εt − ck
, (2.67) BetheSE
which is essentially the result of Bethe.
2.4.3 Relativistic renormalization
For the renormalization in the general case we use the momentum representation of theelectron self energy
∆SE(a) = 〈a|pr〉⟨pr
∣∣Σ(εa)∣∣p′r′⟩〈p′r′|a〉 , (2.68)
where the self-energy operator in the momentum represention is obtained by transformingthe coordinate representation (
SECoord22.55)
〈pr|Σ(εa)|p′r′〉 = 〈pr|x〉〈x|Σ(εa)|x′〉〈x′|p′r′〉 (2.69) SEMom1
or
22
〈pr|Σ(εa)|p′r′〉 = − e2c
2ε01
(2π)3
∫∫d3x d3x′ u†r(p) e−ip·x
×∫
d3k(2π)3
1k
〈x|αµeik·x|qs〉〈qs|t〉〈t|q′s′〉〈q′s′|e−ik·x′αµ|x′〉
εa − εt − ck sgn(εt)ur′(p′) eip
′·x′
= − e2c
2ε0u†r(p) αµ
∑ss′
us(p− k)
×∫
d3k(2π)3
1k
〈p− k, s|t〉〈t|p′ − k, s′〉εa − εt − ck sgn(εt)
u†s′(p′ − k) αµ ur′(p′). (2.70) SEMom2
The corresponding expression for a free electron in the state p, r is obtained by replacingthe bound-electron energy εa by the free-electron energy εp,r and letting the intermediatestates run over free-electron states,⟨
pr∣∣ΣFree
∣∣p′r′⟩ = − e2c
2ε0δ3(p− p′) δrr′ αµ
∑s
∫d3k
(2π)31kus(p− k)
× 1εp,r − εp−k,s − ck sgn(s)
u†s(p− k) αµ . (2.71) SEFreeMom
The last two expressions contain some linearly divergent terms, which can be shown tocancel, making the expression only logarithmically divergent
MS84[1, p. 225].
6pr
6p− k, s 6
s
sk
1
2
6pr
Figure 2.4: Graphical illustration of the first-order free-electron self-energy (SEFreeMom2.71). Fig:SERen
The renormalized self-energy operator is the difference between the bound-stateand the free-electron operators⟨
pr∣∣ΣRenorm
∣∣p′r′⟩ =⟨pr
∣∣Σ(εa)− ΣFree∣∣p′r′⟩
= − e2c
2ε0u†r(p) αµ
∑ss′
∫d3k
(2π)31kus(p− k)
×[〈p− k, s|t〉〈t|p′ − k, s′〉
εa − εt − ck sgn(εt)− δ3(p− p′) δrr′
εp,r − εp+k,s − ck sgn(εs)
]u†s′(p
′ − k) αµ ur′(p′) . (2.72) SEFreeMom1
Using the relation (ClosProp4B.25) we can rewrite this as⟨
pr∣∣ΣRenorm
∣∣p′r′⟩ = − e2c
2ε0u†r(p) αµ
∑ss′
∫d3k
(2π)31kus(p− k) 〈p− k, s|t〉
×[ 1εa − εt − ck sgn(εt)
− 1εp,r − εp−k,s − ck sgn(s)
]×〈t|p′ − k, s′〉u†s′(p
′ − k)αµ ur′(p′) . (2.73) SERen
23
The renormalized self energy is the average of this operator in the state a,
∆RenormSE =
∑pp′rr′
〈a|pr〉〈pr|ΣRenorm|p′r′〉〈p′r′|a〉 . (2.74) SERenAv
6a
6εa − z t 6
s
sz
1
2
6a
- 〈a| |a〉
6pr
6p− k, s 6
s
sk
1
2
6pr
Figure 2.5: Graphical illustration of the renormalization of the first-order electron (SERenAv2.74). The second
diagram represents the average of the free-electron self-energy in the bound state |a〉. Fig:SERen1
So far, no approximations have been made. Let us now assume that the boundstates |t〉 contain only positive free-electron components when εt > 0 and vice versa, i.e.,sgn(εt) = sgn(s). Then
⟨pr
∣∣ΣRenorm∣∣p′r′⟩ = − e
2c
2ε0u†r(p) αµ
∑s
∫d3k
(2π)31kus(p− k) 〈p− k, s|t〉
×[ 1εa − εt − ck sgn(s)
− 1εp,r − εp−k,s − ck sgn(s)
]〈t|p′ − k, s〉u†s(p′ − k) αµ ur′(p′) . (2.75) SERen2
or using (OpExpB.39)
⟨pr
∣∣ΣRenorm∣∣p′r′⟩ = − e
2c
2ε0u†r(p) αµ
∑s
∫d3k
(2π)31kus(p− k)
×⟨p− k, s
∣∣∣ 1
εa − hD − ck sgn(s)− 1
εp,r − hFreeD − ck sgn(s)
∣∣∣p′ − k, s⟩u†s(p
′ − k) αµ ur′(p′) .(2.76) SERen3
As a test of the general treatment, we let k → 0 and neglect negative energy contri-butions. Then us(p)〈p, s|t〉 in (
SERen22.75) becomes the fourier component 〈p|t〉
⟨pr
∣∣ΣRenorm∣∣p′r′⟩ = − e
2c
2ε0u†r(p) αµ
∫d3k
(2π)31k〈p|t〉
×[ 1εa − εt − ck
+1ck
]〈t|p′〉αµ ur′(p′) (2.77)
and after integrating over the angular part of k (see Appendixch:FTF) and eliminating the
scalar part of αµαµ we retrieve the Bethe result (
BetheSE12.66)
⟨p∣∣ΣRenorm
∣∣p′⟩ = − e2
2ε0
∫d3k
(2π)31k2
αµ〈p|t〉[ εa − εtεa − εt − ck
]〈t|p′〉αµ
=e2
4π2ε0 r12
∫dk α · 〈p|t〉
[ εa − εtεa − εt − ck
]〈t|p′〉α, . (2.78)
24
6a
6εa − z t 6
s
sz
6a
-
6a
6εa − z qs 6
s
sz
6a
=
6a
6
6x s
6
s
sz
6a
=
=
6a
6
6x s
6
s
sz
6a
+
6a
6
x s6x s66
s
sz
6a
Figure 2.6: Diagram representing the first-order self-energy of a bound electron. Fig:SERen
In order to make contact with other treatment we will split the operator difference in(SERen32.76) into two parts. In the first part we replace εpr by εa, giving
1
εa − hD − ck sgn(s)− 1
εa − hFreeD − ck sgn(s)
=1
εa − hD − ck sgn(s)U
1
εa − hFreeD − ck sgn(s)
=1
εa − hFreeD − ck sgn(s)
U1
εa − hFreeD − ck sgn(s)
+1
εa − hFreeD − ck sgn(s)
U1
εa − hD − ck sgn(s)U
1
εa − hFreeD − ck sgn(s)
. (2.79)
The second part corrects for the difference,
1
εa − hFreeD − ck sgn(s)
− 1
εpr − hFreeD − ck sgn(s)
=1
εa − hFreeD − ck sgn(s)
[εpr − εa
] 1
εpr − hFreeD − ck sgn(s)
. (2.80)
25
The first part becomes after transforming to the coordinate representation
∆RenormSE;1 = − e
2c
2ε0〈a|αµ|x〉
∫d3k
(2π)31keik·(x−x′)
×⟨x∣∣∣ 1
εa − hD − ck sgn(s)U
1
εa − hFreeD − ck sgn(s)
∣∣∣x′⟩〈x′|αµ|a〉 (2.81) Se1
The second part represents the difference between the free-electron self energy inserted ona free-electron lind and a bound-electron line.
26
Appendix A
Notations
ch:Not
A.1 Four-component vector notations
Contravariant vector : x = xµ = (x0,x) = (ct,x)
(µ=0,1,2,3 and x is the three-dimensional coordinate vector)
Covariant vector : xµ = gµνxν = (x0,−x) = (ct,−x)
(Summation over repeated indices.)
Four-dimensional differential:
d4x = cdt− d3x and d3x = dxdy dz
Metric tensor : gij =
1 0 0 00 −1 0 00 0 −1 00 0 0 −1
Scalar product : ab = aµbµ = a0b0 − a · b
Gradient operator (covariant) : ∂µ =∂
∂xµ=
(1c
∂
∂t,∇
)
Gradient operator (contravariant) : ∂µ =∂
∂xµ=
(1c
∂
∂t,−∇
)
Four − dimensional divergence : ∂µAµ =
1c
∂A0
∂t+ ∇ · A = ∇A
d′Alembertian operator : � = ∂µ∂µ =1c2∂2
∂t2−∇2 = ∇2
27
A.2 Alpha and gamma matrices
The alpha matrices are related to the standard gamma matrices by
γµ = γ0αµ ; αµ = (1,α) ; γ0 = β ; α = γ0γ = −γγ0
where α, β are the Dirac matrices.
6 a = aµγµ = a0γ
0 − a · γ
28
Appendix B
Representations of states andoperators
ch:CoordMom
B.1 The Dirac delta functionsec:Diracfcn
We consider the integral ∫ L/2
−L/2dx eikx . (B.1)
Assuming periodic boundary conditions, e−iLx/2 = eiLx/2, limits the possible k values tok = kn = 2πn/L. Then
1L
∫ L/2
−L/2dx eiknx = δkn,0 = δ(kn, 0) , (B.2) IntL
where δn,m is the Kronecker delta factor1
δn,m ={
1 if m = n0 if m 6= n .
(B.3) Kronecker
If we let L → ∞, then we have to add a ’damping factor ’ e−η|x|, where η is a smallpositive number, in order to make the integral meaningful,∫ ∞
−∞dx eikx e−η|x| =
2ηk2 + η2
= 2π∆η(k) . (B.4)
In the limit η → 0, we have
limη→0
∆η(k) =12π
limη→0
∫ ∞
−∞dx eikx e−η|x| = δ(k), (B.5)
which can be regarded as a definition of the Dirac delta function, δ(k). Formally, we writethis relation as
12π
∫ ∞
−∞dx eikx = δ(k) (B.6) Deltafcn
1This may for typographical reasons sometimes be written δ(n, m), not to be confused by the deltafunction δ(x).
29
The ∆ function also has the following properties
limη→0
πη∆η(x) = δx,0∫ ∞
∞dx∆η(x− a)∆κ(x− b) = ∆η+κ(a− b). (B.7) Deltaint
In three dimensions equation (IntLB.2) goes over into
1V
∫V
d3x eikn·x = δ3(kn, 0) = δ(knx, 0) δ(kny, 0) δ(knz, 0) . (B.8) IntV
In the limit where the integration is extended over the entire three-dimensional space, wehave in analogy with (
DeltafcnB.6) ∫
d3x
(2π)3eik·x = δ3(k) (B.9) Deltafcn3
B.2 Dirac notationssec:DiracNot
The ket vector |a〉 is a vector representation of the state a in a certain basis,
∣∣a⟩ =
a1
a2
··
. (B.10) Ket
The corresponding bra vector 〈a| is∣∣a⟩ = (a∗1, a∗2, · · · ) . (B.11) Bra
The basis states are represented by unit vectors |j〉
∣∣1⟩=
100··
∣∣2⟩=
010··
etc. (B.12) Basis
The basis set need not be numerable but can also form a continuum.
The scalar product of two vectors is denoted 〈a|b〉 and defined
〈a|b〉 =∑
t
a∗t bt . (B.13) ScalProd
It follows from the definitions that
at = 〈t|a〉 and a∗t = 〈a|t〉 . (B.14) aj
Then the scalar product (ScalProdB.13) can also be expressed
〈a|b〉 =∑
t
〈a|t〉〈t|b〉 = 〈a|t〉〈t|b〉 . (B.15) ScalProd2
30
With the Dirac notations, a summation over intermediate states will always beunderstood. Since the relation (
ScalProd2B.15) holds for arbitrary vectors, we have the formal
identity
|t〉〈t| ≡ 1 (B.16) Id
which is known as the resolution of the identity.
The coordinate representation of the ket vector |a〉 is denoted 〈x|a〉 and is identical tothe state (Schrodinger) function
〈x|a〉 = Ψa(x) ; 〈a|x〉 = Ψ∗a(x) (B.17) StateFcn
(assuming for simplicity that there is only a single particle in the system). In analogywith (
ScalProd2B.15) the scalar product becomes
〈a|b〉 =∫
d3x 〈a|x〉〈x|b〉 =∫
d3x Ψ∗a(x) Ψb(x) . (B.18)
Also here we shall assume that an integration is always understood, i.e.,
〈a|b〉 = 〈a|x〉〈x|b〉 , (B.19) ScalProd3
which leads to the formal identity
|x〉〈x| ≡ 1 (B.20) Id2
From the closure property we have∑t
φ∗t (x)φt(x′) = 〈x|t〉〈t|x′〉 = 〈x|I|x′〉 = δ3(x− x′) , (B.21) ClosProp
which with the identity (IdB.16) leads to the formal identity
〈x||x′〉 = δ3(x− x′) (B.22) ClosProp2
This is also the coordinate representation of the identity operator (IdB.16). Note that here
there is no integration over the space coordinates. Setting x = x′ and integrating, leadsto unity on both sides.
The results above hold also when the basis states form a continuum, as can be illus-trated by the momentum eigenstates (
PlaneWB.27),
〈x|pr〉〈pr|x′〉 =∑
r
∫d3pφ∗pr(x)φpr(x′) =
∑r
u†r(p)ur(p)∫
d3p(2π)3
eip(x−x′), (B.23)
which using (I2B.30) and (
Deltafcn3B.9) leads to
|pr〉〈pr| ≡ 1 (B.24) Id3
Here, a summation over r and an integration over p is understood.
Similarly, we find
〈pr|I|p′r′〉 = 〈pr|x〉〈x|I|x′〉〈x′|p′r′〉
= ur(p)u†r′(p′)
1(2π)3
∫∫d3x d3x′ δ3(x− x′) eix·(p−p′) = δrr′ δ
3(p− p′) , (B.25) ClosProp4
using (ClosPropB.21) and (
Deltafcn3B.9).
31
B.3 Momentum eigenfunctions
(See also Appendixsec:DiracC)
The Dirac eqn for a free particle(cα · p + βmec
2)φpr(x) = εpr φpr(x) (B.26) Dirac
has the normalized plane-wave solutions
φpr(x) =
√1Vur(p) eip·x → (2π)−3/2 eip·x ur(p) , (B.27) PlaneW
which are eigenfunctions of the momentum operator
p = −i∇ ; pφpr(x) = pφpr(x) .
r = + or − correspond to positive and negative energies, respectively. The momentumoperator p should be distinguished from the momentum vector p.
According to (uqC.10) we have
u+(p) = N
(p0 +mec
σ · p
); u−(p) = N
(−σ · pp0 +mec
); N =
1√2p0(p0 +mec)
(B.28) up
Epr = |εpr| = cp0 ; p0 =√
p2 +m2ec
2 (B.29)
With the normalization (NormIIC.24b) we have
u†r(p)ur′(p) = δr,r′ . (B.30) I2
The projection operators are with this normalization (uudagC.33a), (
vvdagC.33b)
Λ+(p) = u+(p)u†+(p) =p0 + α · p + βmec
2p0(B.31a)
Λ−(p) = u−(p)u†−(p) =p0 −α · p− βmec
2p0(B.31b)
∑r
ur(p)u†r(p) = u+(p)u†+(p) + u−(p)u†−(p) = I(4× 4). (B.32) uID
Note that Λ±(p) are functions of the momentum p and act as projection operators onlywhen operating on momentum eigenfunctions. The true projection operators, Λ±(p), areobtained by replacing p by the operator p.
B.4 Representation of statessec:Repr
Coordinate representation of a state |a〉 : φa(x) = 〈x|a〉Momentum representation of a state |a〉 : φa(pr) = 〈pr|a〉
32
The momentum representation is the expansion coefficient of the state in momentumeigenfunctions
φpr(x) = 〈x|pr〉 =
√1Vur(p) eip·x (B.33) MomEig
〈x|a〉 =∑pr
〈x|pr〉〈pr|a〉 (B.34)
〈pr|a〉 =∫
d3x 〈pr|x〉〈x|a〉 =
√1V
∫d3x e−ip·x u†r(p)φa(x). (B.35)
In the limit of continuous momenta,∑
pr is replaced by∑
r
∫d3p and
√1V by (2π)−3/2.
The momentum representation is distinct from the fourier transform. The latter is definedas
〈p|a〉 =∑
r
ur(p)〈pr|a〉 =
√1V
∫d3x e−ip·x φa(x)
→ (2π)3/2
∫d3x e−ip·x φa(x), (B.36) FT
using the identity (uIDB.32).
B.5 Representation of operatorssec:ReprOp
Coordinate representation of an operator O: O(x2,x1) = 〈x2|O|x1〉Momentum representation of an operator O: O(p2r2,p1r1) = 〈p2r2|O|p1r1〉. Transfor-mation between the representations
〈p2r2|O|p1r1〉 =∫∫
d3x2 d3x1 〈p2r2|x2〉〈x2|O|x1〉〈x1|p1r1〉. (B.37) MomCoordTrans
The corresponding fourier transform is according to (FTB.36)∑
r1r2
ur2(p2)〈p2r2|O|p1r1〉u†r1(p1). (B.38) OpFT
Any operator with a complete set of eigenstates can be expanded as
O = |j〉 εt 〈j| where O|j〉 = εt |j〉 , (B.39) OpExp
using the summation convention (IdB.16). This gives the coordinate and momentum repre-
sentations
〈x2|O|x1〉 = 〈x2|j〉 εj 〈j|x1〉 (B.40a)
〈p2r2|O|p1r1〉 = 〈p2, r2|j〉 εj 〈j|p1r1〉. (B.40b)
The identity operator (IdB.16)
I = |j〉〈j| (B.41) Ident
has the coordinate and momentum representations
〈x2|I|x1〉 = 〈x2|j〉 〈j|x1〉 = δ3(x2 − x1) (B.42a)
〈p2r2|I|p1r1〉 = 〈p2r2|j〉 〈j|p1r1〉 = δ3(p′ − p) δr′,r. (B.42b)
33
Appendix C
Dirac free particle
sec:Dirac
C.1 Solutions to the Dirac equation
The time-dependent Dirac eqn for a free particle is
i∂ψq(x)∂t
=(cα · p + βmec
2)ψq(x) (C.1) DiracFree
or ( iβc
∂
∂t− βα · p−mec
)ψq(x) = 0(
γ0p0 − γ · p−mec)
=(6 p−mec
)ψq(x) = 0, (C.2) DiracFree2
where (see Chapterch:NotA)
ˆ6 p = γµpµ = γ0p0 − γ · p ; p0 =ic
∂
∂t; p = −i∇
α =(
0 σσ 0
); β = γ0 =
(1 00 −1
); γ = γ0α =
(0 σ−σ 0
)
α = γ0γ = −γγ0
Separation of the wave function into space and time parts
ψq(x) = φq(x) e−iεqt (C.3) DiracSp
leads to the time-independent eqn
hfreeD (p)φq(x) =
(cα · p + βmec
2)φq(x) = εq φq(x) (C.4) DiracP
or (γ0εq/c− γ · p−mec
)φq(x) = 0 . (C.5) Dirac3
φq(x) is a four-component wave function, which can be represented by
φq(x) ∝ 1√Vur(q) eiq· x ; pφq(x) = qφq(x) . (C.6) MomFcn
34
q is a momentum vector (to be distinguished from the momentum operator p) and ur(q)is a four-component vector function of the momentum q. For each q there are fourindependent solutions (r = 1, 2, 3, 4) to the Dirac eqn. The parameter q in φq and εqrepresents q and r or, more explicitly,
φq(x) = φq,r(x) ; εq = εq,r
With the wave function (MomFcnC.6), Eq. (
Dirac3C.5) leads to(
γ0εq/c− γ · q−mec)ur(q) = 0
or (εq/c−mec −σ · q
σ · q −εq/c−mec
)ur(q) = 0 , (C.7) Dirac4
where each element is a 2× 2 matrix. This eqn has two solutions for each q:
u+(q) = N+
(εq/c+mec
σ · q
); u−(q) = N−
(−σ · q
−εq/c+mec
)(C.8) DiracSol
corresponding to positive (r = 1, 2) and negative (r = 3, 4) eigenvalues, respectively.Defining q0 by
|εq| = Eq = cq0 ; q0 =√
q2 +m2ec
2 (C.9) q0
givesu+(q) = N+
(q0 +mec
σ · q
); u−(q) = N−
(−σ · qq0 +mec
)(C.10) uq
The corresponding eigenfunctions (MomFcnC.6) are
φq+(x) ∝ 1√Vu+(q) eiq· x ; φq−(x) ∝ 1√
Vu−(q) eiq· x . (C.11) MomFcn2
Note that the equation
(6 p−mec)φq+(x) = (γ0p0 − γ · p−mec)φq+(x)= (γ0q0 − γ · q−mec)φq+(x) = 0 (C.12a) MomPos
with p0 =√
p2 +m2ec
2 is satisfied by the positive energy solutions only, although thecorresponding time-dependent eqn (
DiracFree2C.2) is valid for all solutions, when is defined as a time
derivative. The corresponding negative energy solutions satisfy the
(−γ0p0 − γ · p−mec)φq−(x) = (−γ0q0 − γ · q−mec)φq−(x) = 0 (C.12b) MomNeg
or, after changing q to −q,
(−γ0p0 − γ · p−mec)φq−(x) = (−γ0q0 + γ · q−mec)φq−(x) = 0,
whereφq−(x) ∝ 1√
Vu−(−q) e−iq· x =
1√Vv(q) e−iq· x (C.13) MomNeg2
The vectors
u(q) = u+(q) and v(q) = u−(−q) = N−
(σ · q
q0 +mc
)(C.14) uv
then satisfy the eqns
(6 q −mc)u(q) = 0 and ( 6 q +mc) v(q) = 0, (C.15) uveqns
where q0 is given by (q0C.9). Note that the negative energy solution corresponds here to
the momentum −q for the electron (or +q for the hole/positron).
35
C.2 Projection operators
It follows from (uveqnsC.15) that
(6 q +mec)u(q) = 2mec u(q),
which implies that
Λ+(q) = ( 6 q +mec)/2mec (C.16a) ProjPos
has the property of a projection operator in momentum space, projecting out the twopositive energy solutions of the four solutions corresponding to a given q. Similarly,
Λ−(q) = (− 6 q +mec)/2mec (C.16b) ProjNeg
projects out the negative solutions,
Λ+(q)u(q) = u(q) and Λ+(q) v(q) = 0 (C.17a) ProjPos2
Λ−(q)u(q) = 0 and Λ−(q) v(q) = v(q) (C.17b) ProjNeg2
andΛ+(q) + Λ−(q) = 1. (C.18) ProjSum
It should be noted that the corresponding momentum operators, operating in coordi-nate space,
Λ±(p) = (±6 p+mec)/2mec (C.19) ProjOp
do not have the corresponding properties when operating on the eigenfunctions φq± . For
instance, with p0 =√
p2 +m2ec
2 and q0 =√
q2 +m2ec
2
Λ+ φq−(x) ∝ (γ0p0 − γ · p +mec)φq (x) = (γ0q0 − γ · q +mec)φq (x) 6= 0
and
Λ+ φ−q−(x) ∝ (γ0p0 − γ · p +mec)φ−q (x) = (γ0q0 + γ · q +mec)φ−q (x) 6= 0
Instead, we can get the correct projection operators in coordinate space from (Dirac3C.5),
(MomFcnC.6)
(±γ0p0 − γ · p−mec)φq±(x) = 0. (C.20) ProjOp2
It then follows that the operators
Λc±(p) =
(p0 ± γ0(γ · p +mec)
)/2p0 =
(p0 ± (α · p + γ0mec)
)/2p0 (C.21) ProjOp3
have the following projection properties in coordinate space
Λc+(p)φq+(x) = φq+(x) and Λc
+(p)φq−(x) = 0 (C.22) ProjOp4
etc.
36
C.3 Normalization
The norm of the vectors (uqC.10) is
u†+(q)u+(q) =∣∣N+
∣∣2(q0 +mec, σ · q)(q0 +mec
σ · q
)=
∣∣N+
∣∣2(q20 +m2ec
2 + (σ · q)2 + 2q0mec)I(2× 2)
=∣∣N+
∣∣22q0(q0 +mec) I(2× 2) = u†−(q)u−(q) (C.23)
using (σ · q)2 = q2 = q20 − m2ec
2. Two different normalization schemes are used in theliterature
I. u†r′(q)ur(q) = δr,r′q0mec
(C.24a) NormI
II. u†r′(q)ur(q) = δr,r′ . (C.24b) NormII
The first scheme is used, for instance, in ref.MS84[1], but we shall apply the second scheme.
Normalization I
This scheme gives
N+ = N− =1√
2mec (q0 +mec)(C.25) NI
and the normalized eigenfunctions (MomFcn2C.11)
φq±(x) =√mec
q0Vu±(q) eiq· x. (C.26) NormFcn
The adjoint vectors are defined
u(q) = u+(q) = u†+(q)γ0 = N+(q0 +mec, −σ · q) (C.27a) ubar
v(q) = u−(−q) = u†−(−q)γ0 = N−(−σ · q, −q0 −mec) (C.27b) vbar
u(q)u(q) =∣∣N+
∣∣2(q20 +m2ec
2 − (σ · q)2 + 2q0mec)
=∣∣N+
∣∣2 2mec (q0 +mec) I(2× 2) = −v(q) v(q). (C.28) baruuI
It also follows that
u(q) u(q) =∣∣N+
∣∣2 (q0 +mec
σ · q
)(q0 +mec, −σ · q)
=1
2mec (q0 +mec)
((q0 +mec)2 −(q0 +mec) σ · q
(q0 +mec) σ · q −q2
)1
2mec
((q0 +mec) −σ · q
σ · q −q0 +mec
)=6 q +mec
2mec= Λ+(q) (C.29a) ubaru
and similarly
v(q) v(q) =6 q −mec
2mec= −Λ−(q). (C.29b) vbarv
This gives
u(q) u(q)− v(q) v(q) = I(4× 4). (C.30) uvIdI
37
Normalization II
The normalization (NormIIC.24b) gives
N+ = N− =1√
2q0 (q0 +mec)(C.31) NII
and the normalized eigenfunctions (MomFcn2C.11)
φq±(x) =1√Vu±(q) eiq· x. (C.32) NormFcnII
In analogy with (baruuIC.28) we then get
u+(q)u†+(q) =∣∣N+
∣∣2 (q0 +mec
σ · q
)(q0 +mec, σ · q)
=1
2q0
(q0 +mec σ · q
σ · q q0 −mec
)=q0 + α · q + βmec
2q0= Λ+(q) (C.33a) uudag
and similarly
u−(q)u†−(q) =∣∣N+
∣∣2 (−σ · qq0 +mec
)(−σ · q, q0 +mec)
=1
2q0
(q0 −mec −σ · q−σ · q q0 +mec
)=q0 −α · q− βmec
2q0= Λ−(q) (C.33b) vvdag
This gives the analogue of (uvIdIC.30)
u+(q)u†+(q) + u−(q)u†−(q) = I(4× 4). (C.34) uudagID
We also observe that the expressions (ubaruC.29a), (
vbarvC.29b)) have the same form as the projec-
tion operators in coordinate space (ProjOp3C.21), i.e.,
Λc±(p) = u±(p)u†±(p). (C.35) ProjOpCoord
In the following we shall use the normalization II, where the basic formulas are
φq±(x) =
√1Vu±(q) eiq· x
u†r′(q)ur(q) = δr,r′
∑r
ur(q)u†r(q) = I(4× 4)
38
C.4 The free-electron propagator
The coordinate representation of the free-electron propagator is, using the wave functions(NormFcnIIC.32) in the normalization II, given by
SFreeF (x,x1) =
∑q,r
φq,r(x)φ†q,r(x′)z − εq,r(1− iη)
=1V
∑q,r
ur(q)u†r(q)eiq·(x−x′)
z − εq,r(1− iη)
→∫
d3q(2π)3
∑r
ur(q)u†r(q)eiq·(x−x′)
z − εq,r(1− iη)
=∫
d3q(2π)3
[u+(q)u†+(q)
1z − Eq(1− iη)
+ u−(q)u†−(q)1
z + Eq(1− iη)
]. (C.36) SFfree
The expression in the square brackets is the fourier transform of SFreeF (x,x′),
SFreeF (q, z) = u+(q)u†+(q)
1z − Eq(1− iη)
+ u−(q)u†−(q)1
z + Eq(1− iη)
=12
[1
z − Eq(1− iη)+
1z + Eq(1− iη)
]=
α · q + βmec
2q0
[1
z − Eq(1− iη)− 1z + Eq(1− iη)
], (C.37) SFfreeMom
using (uudagC.33a). With q0 = Eq/c =
√q2 +m2
ec2 this becomes
SFreeF (q, z) =
z + cα · q + βmec2
z2 − E2q + iη
=z + cα · q + βmec
2
z2 − c2q2 −m2ec
4 + iη
=1
z − (cα · q + βmec2)(1− iη)(C.38) SFfreeMom2
orSFree
F (q, z) =1
z − hFreeD (q)(1− iη)
, (C.39) SFfreeMom3
where hFreeD is the momentum representation of the Dirac hamiltonian (
DiracPC.4).
Formally, (SFfreeMom2C.38) can be written in four-component form with z = q0, where q0 is now
disconnected from Eq =√c2q2 +m2
ec4 , and using the relations (
DiracFree2C.2)
SFreeF (q, z) =
1c
q0 + α · q + βmec
q20 − q2 −m2ec
2 + iη=
1c
(6 q + βmec)γ0
q2 −m2ec
2 + iη
=1c
16 q − βmec+ iη
γ0 =1cSFree
F (q) (C.40) SFfreeMom4
withSFree
F (q) =1
6 q − βmec+ iηγ0 . (C.41) SFfreeMom5
The coordinate representation of the free-electron propagator (SFfreeC.36) can also be ex-
pressed
SFreeF (x,x′; z) =
〈x|q, r〉〈q, r|x′〉z − εq,r(1− iη)
=⟨x
∣∣∣SFreeF (z)
∣∣∣ x′⟩
39
where SFreeF (z) =
|q, r〉〈q, r|z − εq,r(1− iη)
=1
z − hFreeD (1− iη)
(C.42) SFfreeOp
where is the Dirac operator (DiracPC.4).
The momentum representation of the operator (SFfreeOpC.42), which is obviously diagonal, is
given by the matrix representation with momentum eigenfunctions as basis functions⟨q, r
∣∣SFreeF (z)
∣∣q, r⟩ =⟨q, r
∣∣∣ 1
z − hFreeD (1− iη)
∣∣∣q, r⟩ =1
z − εq,r(1− iη), (C.43) SFmatrix
which is distinct from (SFfreeMom2C.38). The corresponding fourier transform is according to (
OpFTB.38)
and (uudagIDC.34) ⟨
q∣∣SFree
F (z)∣∣q⟩
=∑
r
ur(q)⟨q∣∣∣ 1
z − hFreeD (1− iη)
∣∣∣q⟩u†r(q) , (C.44) SEFT
which is identical to (SFfreeMomC.37) and (
SFfreeMom2C.38).
40
Appendix D
The free-electron self energy
The wave function for the free electron is in normalization II given by (A5.32) fq+(x)=u+(q) ei q.xu+(q) ei q.x The Feynman amplitude for the first-order free-electron selfenergy is M = e2c(x2) an SF(x2, x1, Eq - z) am fq+(x1) DFnm(x2 - x1, z) (A6.1) wherethe free-electron propagator is from (A5.36) SF(x2, x1, Eq - z) = = ei q.(x2-x1) (A6.2)and the photon propagator in the Feynman gauge (A3.13) DFnm(x2 - x1, z) = - gnm(A6.3) Integration over d3x1 and d3x2 gives M = - am am u+(q) = u+(q) (A6.4) whereis the free-electron self energy in the momentum representation. Using the relations amam = - 2 ; amam = - am am= 2 ; am b am = - 2 am am b = 4 b, we then get = (A6.5)With Eq = cqo and z = cko this can be expressed = = go (A6.6) which agrees with (MS9.20).
Alternatively, we can start from (A6.1) with the electron propagator in operator form(A5.41), M = e2c(x2) anam fq+(x1) DFnm(x2 - x1, z) (A6.7) which gives = - am am(A6.8) and this leads to the same result as before. By evaluating the integral in (A6.6),which is logarithmically divergent (A7.10), the expression (A6.4) becomes M = u+(q) =igo A u+(q) = iA u+(q) (A6.9) where A is a number which depends logarithmically on thecut-off momentum L (see Appendix; Mandl- Shaw p 191; Jauch - Rohrlich p 181) A - =- atomic units (A6.10) With the expressions (A5.11) and the normalization II (A5.32,24),we get M = - i u+(q) = Ago u+(q) = A (qo + mc, ) = A = A (1 - ) = A (1 - + . .)(A6.11) This is the expectation value of go A in the free-electron state q+, q+ = = =(A6.12)
41
Appendix E
Evaluation of the free-electronself-energy integral
(See Mandl- Shaw sect 10.2) The free-electron self energy is according to (A6.6) = go(A7.1) In (A6.4) this operates on the u+(q) vector and in view of the eqn (A5.16), (-mc) u+(q) = 0, we can then replace by mc, u+(q) = - go u+(q) = igo A u+(q) (A7.2)Regularizing the photon propagator by the substitution - = - (A7.3) and using the identity= 2 (A7.4) with a = k2 - t and b = k2 - 2qk gives A = - i (A7.5) With the identities = -= - (A7.6) this gives A = - = - (2 - x) (A7.7) This expression is logaritmically divergentwhen L (r) but convergent when l (r) 0. The leading term then becomes A - (2 - x) (A7.8)and (2 - x) = - 3x ln x = (A7.9) gives A - (A7.10)
42
Appendix F
Some fourier transforms
ch:FT
F.1 Evaluation of the integral∫ d3k
(2π)3eik·r12
q2−k2+iηsec:FT1
Using spherical coordinates k = (k, θ, φ), (k = |k|), we have with d3k = k2dk sinΘ dΘ dΦand r12 = |x1 − x2|∫
d3k(2π)3
eik·(x1−x2)
q2 − k2 + iη= (2π)−2
∫ ∞
0
k2 dkq2 − k2 + iη
∫ π
0dΘ sinΘ eikr12 cos Θ
=i
4π2 r12
∫ ∞
0
k dk(eikr12 − e−ikr12
)k2 − q2 − iη
=i
8π2 r12
∫ ∞
−∞
k dk(eikr12 − e−ikr12
)k2 − q2 − iη
, (F.1) IntH1
where we have in the last step utilized the fact that the integrand is an even function of k.The poles appear at k = ±q(1 + iη/2q). eikr12 is integrated over the positive and e−ikr12
over the negative half-plane, which yields −e±iqr12/(4π r12) with the upper sign for q > 0.The same result is obtained if we change the sign of the exponent in the numerator of theoriginal integrand. Thus, we have the result∫
d3k(2π)3
e±ik·(x1−x2)
q2 − k2 + iη= −e
i|q|r12
4π r12(F.2) FT2
The imaginary part of the integrand, which is an odd function, does not contribute to theintegral.
F.2 Evaluation of the integral∫ d3k
(2π)3 (α1·k)(α2·k) eik·r12
q2−k2+iηsec:FT2
The integral appearing in the derivation of the Breit interaction (B12d1.40)is
I2 =∫
d3k(2π)3
(α1 · k)(α2 · k)eik·r12
q2 − k2 + iη
= (α1 · ∇1)(α2 · ∇2)∫
d3k(2π)3
1k2
eik·r12
q2 − k2 + iη. (F.3) IntH2
Using (IntH1F.1), we then have
I2 =i
8π2 r12(α1 · ∇1)(α2 · ∇2)
∫ ∞
−∞
dk(eikr12 − e−ikr12
)k(k2 − q2 − iη)
. (F.4) IntH22
43
The poles appear at k = 0 and k = ±(q + iη/2q). The pole at k = 0 can be treated withhafl the pole value in each half plane. For q > 0 the result becomes
− 14π r12
eiqr12−1
q2
and for q > 0 the same result with −q in the exponent. The final result then becomes∫d3k
(2π)3(α1 · k)(α2 · k)
eik·r12
q2 − k2 + iη= − 1
4π r12(α1 · ∇1)(α2 · ∇2)
ei|q|r12−1
q2(F.5) IntH23
Assuming that our basis functions are eigenfunctions of the Dirac hamiltonian hD
(DiracEq2.20), we can process this integral further. Then the commutator with an arbitrary
function of the space coordinates is[hD, f(x)
]= cα · p f(x) +
[U, f(x)
]. (F.6) hComm
The last term vanishes, if the potential U is a local function, yielding[hD, f(x)
]= cα · p f(x) = −icα · ∇ f(x) . (F.7) hComm2
In particular [hD, e
ik·x]= −icα · ∇ eik·x = cα · k eik·x . (F.8) hComm3
We then find that
(α · ∇)1(α · ∇)2 eik·x =1c2
[hD, e
ik·x]1
[hD, e
ik·x]2
(F.9)
with the matrix element⟨rs
∣∣∣(α · ∇)1(α · ∇)2 eik·x∣∣∣ ab⟩ = q2 eik·x , (F.10)
using the notation in (cq1.35). The integral (
IntH2F.3) then becomes
I2 =∫
d3k(2π)3
(α1 · k)(α2 · k)eik·r12
q2 − k2 + iη=
∫d3k
(2π)3q2
k2
eik·r12
q2 − k2 + iη(F.11) VCA
provided that the orbitals are generated by a hamiltonian with a local potential.
44
Appendix G
Feynman diagrams for boundstates
ch:Feynman
The scattering matrix can in the bound-interaction picture be expressed in terms of aFeynman amplitude, M, as
S = 2π δ( ∑
(εin − εout))M
related to the matrix element of the corresponding effective potential by
iM = 〈 |Veff | 〉 .
The Feynman amplitude, M, is given by the following rules:
6a
s1. Incoming electron line: φa(x)
6a
s2. Outgoing electron line: φ†a(x)
-s sµ s 3. Vertex: iecαµ
-zs s1,µ 2,ν 4. Internal photon line – Photon propagator:
icε0
DFνµ(x2 − x1; z)
45
6a
s
s4. Internal electron line – Electron propagator:
iSF(x2,x1; z)
6. Closed electron loop: Factor of (-1) and trace symbol
7. Energy conservation at each vertex (z energy parameter)
8. Integration over all x’s and z’s. Factor of (2π)−1 for each non-trivial z-integration.
46
Appendix H
Dimensional analysis
ch:DimAnal
The formulas given here are valid in any consistent unit system, like the SI system. Theformulas can be checked by an dimensional analysis. The dimensions in the SI systemsfor some of the quantities are
[force] = N =kgm
s2
[energy] = J = Nm =kgm2
e
s2
[action, ~] = Js =kgm2
e
s
[electric potential] = V =J
As=kgm2
e
As3
[electric field, E] = V/m =kgm
As3
[magnetic field, B] = V s/m2e =
kg
As2
[vector potential, A] = V s/m =kgm
As2
[charge density, ρ] =As
m3
47
[current density, j] =A
m2e
[µ0] = N/A2 =kgm
A2s2
[ε0] = [1/µ0c2] =
A2s4
kgm3
[ρ/ε0] =kg
As3= [∇ · E] Eq. (
Maxw.a1.1a)
[µ0j] =kg
Ams2= [∇×B] Eq. (
Maxw.b1.1b)
[ ~ε0ωV
]=kg2m2
e
A2s4= [A2] Eq. (
A2.1)
[DFνµ(x2 − x1)] = m−2 [DFνµ(x− x′)] = s/m2
[~DFνµ(x2 − x1)ε0c
]=kg2m2
e
A2s4= [A2] Eq. (
PhotPropDef2.6)
[ecA] =kgm2
e
s2= [energy] Eq. (
Hint2.37)
[e2c2~2
]=
A2s2
kg2m2e
= [A−2s−2] Eq. (S222.38)
48
Bibliography
MS84 [1] F. Mandl and G. Shaw, Quantum Field Theory (John Wiley and Sons, New York,1986).
Sak67 [2] J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley Publ. Co., Reading,Mass., 1967).
GML51 [3] M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951).
Su57 [4] J. Sucher, Phys. Rev. 107, 1448 (1957).
Li89 [5] I. Lindgren, AIP Conference Series 189, 371 (1989).
49
