Spohn H. Dynamics of Charged Particles and their Radiation Field
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19.1 Infrared photons 303
average
ψ(t), N
f
ψ(t)
F
= e
2
λ=1,2
d
3
k|ϕ|
2
ω
−1
(e
λ
· v)
2
(ω − k · v)
−2
1 − cos((ω − v · k)t )
∼
=
v
2
log t (19.8)
for large t.Onthe other hand, ψ(t), H
f
ψ(t)
F
stays bounded because of the extra
factor of ω from the definition of the energy. Also, in every bounded region in posi-
tion space and in any region in momentum space avoiding the origin, the number of
photons is Poisson-distributed with a finite mean. The photons in (19.8) are bound
to the charge, i.e. virtual in the usual parlance. For the Pauli–Fierz Hamiltonian
virtual photons can be probed only indirectly, e.g. through the effective dynamics
discussed in chapter 16. As long as the energy remains finite no qualitative changes
are expected, as confirmed by the fact that the g-factor and the effective mass are
infrared convergent at least to order e
2
.
As a second example let us study the generation of photons through accelerated
motion. We prescribe the trajectory q
t
with velocity v
t
of the classical charge, and
thus the current j (x, t) = eϕ(x − q
t
)v
t
. The scattering process is captured most
conveniently through the S-matrix defined by
S = lim
t→∞
U (t, 0)
∗
e
2iH
f
t
U (0, −t). (19.9)
From (19.7) we conclude
S = exp
− i
∞
−∞
dt
λ=1,2
d
3
k(2ω)
−1/2
e
λ
· v
t
eϕ
∗
e
−i(ωt−k·q
t
)
a(k,λ)
+eϕe
i(ωt−k·q
t
)
a
∗
(k,λ)
−
1
2
iIm
∞
−∞
ds
∞
−∞
ds
(s − s
)
×
λ=1,2
d
3
ke
2
|ϕ|
2
(2ω)
−1
(e
λ
· v
s
)(e
λ
· v
s
)e
i(ωs−ωs
−k·q
s
+k·q
s
)
.
(19.10)
Note that for constant velocity, v
t
= v, |v| < 1, the time-integration yields the δ-
function δ(ω − k · v) and therefore the S-matrix is trivial, S = 1. For the sake of
an example let us assume that there are no incoming photons. Then the scattering
304 Behavior at very large and very small distances
state of interest is S, which is a coherent state with average number of photons
S, N
f
S
F
= e
2
λ=1,2
d
3
k|ϕ|
2
(2ω)
−1
|
dt (e
λ
· v
t
)e
i(ωt−k·q
t
)
|
2
. (19.11)
In standard scattering v
t
→ v
±
for t →±∞.Ifv
+
= v
−
, from the previous ar-
gument one concludes that S, N
f
S
F
< ∞.However if v
+
= v
−
, then from
the time-integration a factor |k|
−2
appears which together with the factor 1/ω
makes the integral in (19.11) logarithmically divergent at small k.Asbefore,
S, H
f
S
F
< ∞.Also the number of photons is finite in any region of the
form {k||k| >δ} with δ>0.
If an electron is scattered by, say, a short-range electrostatic potential then in the
collision process a large number of infrared photons is generated. Strictly speak-
ing, there is no channel with elastic scattering. Since the total energy of scattered
photons is bounded, the collision cross-section is slightly modified but remains fi-
nite. These infrared photons are however somewhat elusive objects. For example,
for the state S the photon density in position space decays as |x|
−3
for large |x|,
which means that there is a small probability for the photons to have been created
very far away from the source. A real detector necessarily makes a cutoff in the
energy range and in position, thus necessarily misses the infrared part.
19.2 Energy renormalization in Nelson’s scalar field model
On the classical level we consider a scalar wave field and couple it to a mechanical
particle in such a way that the interaction is linear in the field, local, and translation
invariant. This fixes the Hamiltonian function to be of the form
H =
1
2m
p
2
+
1
2
d
3
x
π(x)
2
+ (∇φ(x))
2
+ m
2
ph
φ(x)
2
+ eφ
ϕ
(q). (19.12)
Here q, p are the position and momentum of the particle with bare mass m and
π(x) is the momentum field canonically conjugate to the scalar wave field φ(x).
The wave speed c is set equal to one. e is the coupling strength, and m
ph
≥ 0isthe
mass of the bosons. The equations of motion read
∂
2
t
φ(x, t) = ( − m
2
ph
)φ(x, t) − eϕ(x −q
t
), (19.13)
m ¨q
t
=−e∇φ
ϕ
(q
t
). (19.14)
The solutions to (19.13) and (19.14) bear a fair qualitative similarity to the
Abraham model, in particular, our discussion of the energy–momentum relation,
the radiation reaction, and the center manifold could be repeated almost word for
word.
19.2 Energy renormalization in Nelson’s scalar field model 305
The quantization of (19.12) is straightforward. π(x) and φ(x) become a scalar
Bose field with commutation relations
[φ(x), π(x
)] = iδ(x − x
), (19.15)
setting
= 1. It is convenient to introduce the scalar creation and annihilation
operators a
∗
(k), a(k) in momentum space. Then
φ(x) =
d
3
k
1
√
2ω
(2π)
−3/2
e
ik·x
a(k) + e
−ik·x
a
∗
(k)
, (19.16)
π(x) =
d
3
k
ω/2(2π)
−3/2
− ie
ik·x
a(k) + ie
−ik·x
a
∗
(k)
(19.17)
with ω(k) = (k
2
+ m
2
ph
)
1/2
. The quantized Hamiltonian reads
H =
1
2m
p
2
+ H
f
+ eφ
ϕ
(x), (19.18)
where the momentum operator p =−i∇
x
is canonically conjugate to the position
x and
φ
ϕ
(x) =
d
3
kϕ(k)
1
√
2ω
e
ik·x
a(k) + e
−ik·x
a
∗
(k)
(19.19)
with ϕ assumed to be real. H acts on L
2
(R
3
) ⊗ F ;wecall it the Nelson
Hamiltonian.If
d
3
k|ϕ|
2
(ω
−2
+ 1)<∞, then the interaction eφ
ϕ
(x) is infinites-
imally bounded with respect to
1
2m
p
2
+ H
f
and, by the Kato–Rellich theorem, H
is self-adjoint with domain D(( p
2
/2m) + H
f
).
Since H is invariant under translations, the total momentum
P = p + P
f
, P
f
=
d
3
kka
∗
(k)a(k), (19.20)
is conserved. As in section 15.2, H can be unitarily transformed to fixed total
momentum with the result
H(P) =
1
2m
(P − P
f
)
2
+ H
f
+ eφ
ϕ
(19.21)
and φ
ϕ
= φ
ϕ
(0). The ground state energy of (19.21) defines the energy–
momentum relation E(P).Ifonesets
H(0) =
1
2m
P
2
f
+ H
f
+ eφ
ϕ
, (19.22)
then the effective mass is given by
m
m
eff
= 1 −
2
3m
ψ
g
, P
f
·
1
H(0) − E(0)
P
f
ψ
g
F
, (19.23)
where ψ
g
is the ground state of H (0), i.e. H(0)ψ
g
= E(0)ψ
g
.
306 Behavior at very large and very small distances
With this somewhat rapid introduction the problem under consideration is
whether the Nelson Hamiltonian (19.18) remains well-defined in the point-charge
limit ϕ(x) → δ(x).Following the usual convention to denote the ultraviolet cutoff
in momentum space by , the point-charge limit means scaling the form factor as
ϕ
(x) =
3
ϕ(x), respectively ϕ
(k) = ϕ(k/) (19.24)
with →∞.
The interaction φ
ϕ
(x) is bounded relative to H
f
only if
d
3
k|ϕ|
2
/ω
2
< ∞.At
=∞this condition is violated indicating that the limit →∞is singular. To
find out how singular we compute the ground state energy to second order in e
2
,
regarding in (19.22) eφ
ϕ
as a perturbation. Then
E(0) =−e
2
d
3
k|ϕ
(k)|
2
1
2ω
ω +
1
2m
k
2
−1
+ O(e
4
) (19.25)
which diverges as −log for →∞.Physically only energy differences count
and one may want to subtract E(0) from H (0). After all, in the definition of H
f
an infinite zero-point energy was already subtracted. There are two caveats to this.
First, E(0) from (19.25) is only a second-order perturbation and a priori one does
not know which energy to subtract. More importantly, it must be ensured that phys-
ical properties are not distorted as →∞.Inthe classical model the effective
mass is the relevant indicator and we adopt the same criterion here. From (19.23)
we compute, compare with (15.36),
m
m
eff
= 1 −
2
3m
e
2
d
3
k|ϕ
(k)|
2
1
2ω
k
2
ω +
1
2m
k
2
−3
+ O(e
4
), (19.26)
which stays finite as →∞,atleast to second order, fostering our hope that
H(0) − E(0) is a well-defined Hamiltonian as →∞.
The Nelson model has the simplifying feature that the energy renormalization
can be made explicit through a unitary transformation originally introduced by
E. P. Gross. It is constructive to work out the case of N charges coupled to the
Bose field. The Hamiltonian (19.18) then generalizes to
H
N
=
N
j=1
1
2m
j
p
2
j
+ H
f
+
N
j=1
e
j
φ
ϕ
(x
j
). (19.27)
Here the j-th particle has position x
j
, momentum p
j
=−i∇
x
j
, mass m
j
, and
charge e
j
.Wedefine
T =−
N
j=1
e
j
d
3
kϕ
1
√
2ω
β
j
e
ik·x
j
a(k) − e
−ik·x
j
a
∗
(k)
(19.28)
19.2 Energy renormalization in Nelson’s scalar field model 307
with β
j
= (ω +
1
2m
j
k
2
)
−1
.e
−T
is the Gross transformation. Since
d
3
kβ
2
j
/ω <
∞ provided m
ph
> 0, e
−T
is unitary and well defined in H even at =∞. Let us
set
A
−
ϕ j
(x) =−
d
3
kϕ
1
√
2ω
kβ
j
e
ik·x
a(k),
A
−
ϕ j
(x)
∗
= A
+
ϕ j
(x), A
ϕ j
(x) = A
+
ϕ j
(x) + A
−
ϕ j
(x). (19.29)
3 We here use on purpose the same notation as for the transverse vector potential,
since through the Gross transformation A
ϕ j
(x) appears in the Hamiltonian in the
same way as the transverse vector potential does for the Pauli–Fierz model. How-
ever A
ϕ j
(x) is longitudinal and [ p, A
ϕ j
(x)] = 0. It is better behaved at small x
because the factor β
j
gains one extra power in decay at large k. Only for section
19.2, A
ϕ
is defined through (19.29). 3
e
−T
acts as
e
T
p
j
e
−T
= p
j
− e
j
A
−
ϕ j
(x
j
) − e
j
A
+
ϕ j
(x
j
), e
T
x
j
e
−T
= x
j
,
e
T
a(k)e
−T
= a(k) −
N
j=1
e
j
ϕ
1
√
2ω
β
j
e
−ik·x
j
,
e
T
a
∗
(k)e
−T
= a
∗
(k) −
N
j=1
e
j
ϕ
1
√
2ω
β
j
e
ik·x
j
. (19.30)
When normally ordered, the Gross-transformed Hamiltonian becomes
e
T
H
N
e
−T
=
N
j=1
1
2m
j
p
2
j
− 2e
j
p
j
· A
−
ϕ j
(x
j
) − 2e
j
A
+
ϕ j
(x
j
) · p
j
+e
2
j
A
−
ϕ j
(x
j
)
2
+ e
2
j
A
+
ϕ j
(x
j
)
2
+ 2e
2
j
A
+
ϕ j
(x
j
) · A
−
ϕ j
(x
j
)
+ H
f
−
N
i=j=1
e
i
e
j
d
3
k|ϕ|
2
1
2ω
(β
i
+ β
j
− ωβ
i
β
j
)e
ik·(x
i
−x
j
)
−
N
j=1
e
2
j
d
3
k|ϕ|
2
1
2ω
β
j
. (19.31)
Note that A
ϕ j
(x
j
) does not commute with p
j
. The last term in (19.31) is the en-
ergy renormalization, granted for a moment that the remainder is a well-defined
Hamiltonian with energy bounded from below. The energy renormalization co-
incides with E(0) as computed from second-order perturbation theory, compare
with (19.25), and diverges as −N log .
308 Behavior at very large and very small distances
The next to last term in (19.31) is the instantaneous interaction between the
particles which dominates their dynamics at small velocities; see section 20.2.
Let us set m
ph
= 0 and =∞. Then the interaction potential for particles i and
j is
V
ij
(x) =−e
i
e
j
d
3
k
1
ω
(β
i
+ β
j
− ωβ
i
β
j
)e
ik·x
(19.32)
as a function of their relative distance. V
ij
(x)
∼
=
−e
i
e
j
/4π|x| for large |x|, and
V
ij
(x)
∼
=
e
i
e
j
log |x| for small |x |.Eveninthe point-charge limit the interaction
deviates from a strict Coulomb law at distances on the scale of the Compton wave-
length for particles i, j . This confirms our previous findings that it is natural to
regard the Compton wavelength as an effective size of the charged particles in the
quantized theory. Even more importantly, the sign of the interaction is −e
i
e
j
.In
the scalar theory particles of equal charge attract, those of opposite charge repel
each other. Thus particles of opposite charge tend to segregate and a big cluster of
one sign would be separated from a big cluster of the opposite sign. There could
not be the delicate balance between nuclei (ions) and electrons which is respon-
sible for the formation of atoms and molecules. If the photons were spinless, the
world would have no similarity to the one we know.
We are left with the first piece of (19.31). Since it is additive in the particles, for
notational simplicity we return to N = 1 and rewrite it as
lim
→∞
e
T
H + e
2
d
3
k|ϕ
|
2
1
2ω
β
e
−T
=
1
2m
p
2
−
e
m
( p · A
−
(x) + A
+
(x) · p)
+
e
2
2m
A
−
(x)
2
+ A
+
(x)
2
+ 2A
+
(x) · A
−
(x)
+ H
f
=
H
ren
. (19.33)
Here, using ϕ
(0) = (2π)
−3/2
,wehave
A
−
(x) =−
d
3
k
1
√
2ω
kβ(2π)
−3/2
e
ik·x
a(k),
A
+
(x) = A
−
(x)
∗
, A(x) = A
+
(x) + A
−
(x). (19.34)
H
ren
is the physical Hamiltonian in the point-charge limit. The splitting into A
−
and A
+
results from normal ordering. The A-field is longitudinal but otherwise
plays a role very similar to the vector potential in the Pauli–Fierz model.
19.2 Energy renormalization in Nelson’s scalar field model 309
In the following it will be convenient to rewrite
H
ren
in dimensionless form.
Through the canonical transformation (13.88) one obtains
H
ren
= mH
ren
= m
1
2
p
2
− e( p · A
−
(x) + A
+
(x) · p)
+
1
2
e
2
A
−
(x)
2
+ A
+
(x)
2
+ 2A
+
(x) · A
−
(x)
+ H
f
= m(H
0
+ H
int
) (19.35)
with ω = (k
2
+ (m
ph
/m)
2
)
1/2
, β = (ω +
1
2
k
2
)
−1
, and A
−
(x) as in (19.34). We
repeat the relative form bound estimates from section 13.3 with the result
|ψ, H
int
ψ| ≤
3e
2
(2π)
−3
d
3
kk
2
β
2
ω
−2
ψ, H
0
ψ. (19.36)
If
3e
2
(2π)
−3
d
3
kk
2
β
2
ω
−2
< 1, (19.37)
then H
int
is H
0
-form bounded with a bound less than 1, which implies that H
ren
is
a self-adjoint operator bounded from below.
The total momentum transforms as e
T
( p + P
f
)e
−T
= p + P
f
= P and
[H
ren
, P] = 0, (19.38)
as can also be checked directly. For fixed total momentum H
ren
becomes
H
ren
(P) =
1
2
(P − P
f
)
2
− e
(P − P
f
) · A
−
+ A
+
· (P − P
f
)
+
1
2
e
2
A
+
· A
+
+ A
−
· A
−
+ 2A
+
· A
−
+ H
f
(19.39)
as acting on
F with the shorthand A = A(0).
The expression in (19.37) is finite also for m
ph
= 0. Thus H
ren
and H
ren
(P) are
well-defined Hamiltonians even for massless bosons with infrared and ultraviolet
cutoffs removed. However, e
−T
is unitarily implemented only for m
ph
> 0. At
m
ph
= 0, H and H
ren
are not unitarily equivalent. As can be seen from (19.30) the
Gross-transformed φ-field has a vacuum expectation which decays as −e/4π|x|
for large x and thus singles out the P = 0 representation; compare with our dis-
cussion in section 19.1. H
ren
(0) has a ground state in Fock space, whereas H
ren
(P),
P = 0, has no ground state in Fock space, just as is the case for the Pauli–Fierz
model.
H
ren
is the result of a mathematical limit procedure and it is not automatically
guaranteed that the limit Hamiltonian inherits the physically desired properties.
310 Behavior at very large and very small distances
Foramodest check we compute the self-energy, the effective mass, and the bind-
ing energy for hydrogen-like atoms in low-order perturbation theory. While these
quantities are well defined, it is not known whether they can be expanded around
e = 0. Only if the bosons had the strictly positive mass m
ph
> 0, H
ren
(P) for small
|P| and H
ren
− Ze
2
V
coul
have a gap between their ground state and the continuous
spectrum, which implies a convergent Taylor expansion at e = 0.
(i) Self-energy
We expand E
ren
(0) in powers of e
2
. H
ren
(0) is split as
H
ren
(0) = H
0
+ eH
1
+
1
2
e
2
H
2
(19.40)
with H
0
=
1
2
P
2
f
+ H
f
, H
1
= P
f
· A
−
+ A
+
· P
f
, H
2
= A
−
· A
−
+ A
+
· A
+
+
2A
+
· A
−
. The unperturbed ground state is with energy 0. The expansion is
written as
E
ren
(0) =
1
2
e
2
E
(2)
+
1
4!
e
4
E
(4)
+
1
6!
e
6
E
(6)
+ O(e
8
). (19.41)
E
(2)
= 0,since H
1
= 0. The next order is
1
4!
e
4
E
(4)
=−e
4
1
2
, A
−
· A
−
1
H
0
A
+
· A
+
F
=−e
4
(2π)
−6
d
3
k
1
d
3
k
2
1
2ω
1
β
2
1
1
2ω
2
β
2
2
(k
1
· k
2
)
2
1
E
12
(19.42)
with ω
i
= ω(k
i
), β
i
=
ω(k
i
) +
1
2
k
2
i
−1
, i = 1, 2, and E
12
= ω
1
+ ω
2
+
1
2
(k
1
+
k
2
)
2
.
For the discussion below we still need the sixth order, which is given by
e
6
1
6!
E
(6)
= e
6
1
4
−, A
−
· A
−
1
H
0
P
f
· A
−
1
H
0
A
+
· P
f
1
H
0
A
+
· A
+
F
−, A
−
· A
−
1
H
0
A
+
· P
f
1
H
0
P
f
· A
−
1
H
0
A
+
· A
+
F
+, A
−
· A
−
1
H
0
A
+
· A
−
1
H
0
A
+
· A
+
F
. (19.43)
The integrals appearing in the expressions for E
(4)
and E
(6)
are convergent.
(ii) Effective mass
From the definition (15.23) and (19.39) one concludes
m
m
eff
= 1 −
2
3
ψ
g
,(P
f
+ eA) · (H
ren
(0) − E
ren
(0))
−1
(P
f
+ eA)ψ
g
F
(19.44)
19.2 Energy renormalization in Nelson’s scalar field model 311
with H
ren
(0)ψ
g
= E
ren
(0)ψ
g
. m/m
eff
is even in e and, provided m
ph
> 0, analytic
for small e. Expanding in (19.44) to order e
2
by the scheme already explained, one
finds
m
eff
m
= 1 +
2
3
e
2
(2π)
−3
d
3
k(k
2
β
3
/2ω) + O(e
4
) (19.45)
which agrees with (19.26) in the limit →∞.Form
ph
= 0,
m
eff
= m
1 +
1
6π
2
e
2
+ O(e
4
)
(19.46)
is obtained. Since the mass renormalization is finite, the relation (19.46) allows us
in principle to obtain the bare mass m from an acceleration experiment at small
velocities which measures m
eff
according to our discussion in section 16.6.
(iii) Binding energy
We consider two charges, a nucleus of charge Ze of infinite mass nailed down
at the origin and a “meson” of charge e. According to (19.31) the renormalized
Hamiltonian for →∞reads then
H = H
ren
−
Ze
2
4π|x|
(19.47)
in units of m.For sufficiently small e, e = 0, H has a ground state. Denoting its
ground state energy by E,bydefinition the (positive) binding energy is
E
bin
= m
E
ren
(0) − E
, (19.48)
since mE
ren
(0) is the energy of the meson far away from the nucleus. E
bin
is even
in e and proportional to the bare mass m.Physically the natural units for E
bin
are
m
eff
c
2
and we write
E
bin
= m
eff
h
bin
(e
2
), (19.49)
which is regarded as a definition of h
bin
.
We expand E in powers of e
2
.Tobetter follow the subtraction of the self-energy
we first transform to the total momentum representation. Then the split-up for H
is
H = H
at
+ H
f
+
1
2
P
2
f
− p · P
f
+ e
(P
f
− p) · A
−
+ A
+
· (P
f
− p)
+
1
2
e
2
(A
−
· A
−
+ A
+
· A
+
+ 2A
+
· A
−
)
= H
0
+ eH
1
+
1
2
e
2
H
2
. (19.50)
312 Behavior at very large and very small distances
The atomic Hamiltonian is H
at
=
1
2
p
2
− Ze
2
/4π|x| with ground state en-
ergy E
at
=−
1
2
(Ze
2
/4π)
2
and ground state ψ
at
(x) = (πr
3
B
)
−1/2
e
−|x |/r
B
, r
B
=
4π/Ze
2
. The unperturbed ground state is ψ
at
⊗ with energy E
at
. The perturba-
tion expansion up to order e
6
is given as in (19.42), (19.43) with the corresponding
substitutions for H
0
, H
1
.Let us first consider those terms not containing either
p · A
−
or A
+
· p. The inverse operator (H
at
+ H
f
+
1
2
P
2
f
− p · P
f
)
−1
is expanded
in p · P
f
.Since ψ
at
, p
2
ψ
at
L
2
=−2E
at
, only the leading term contributes and all
the self-energy terms cancel including order e
6
. The only remaining contribution is
E
bin
=−E
at
+ e
2
ψ
at
⊗ , p · A
−
(H
0
)
−1
A
+
· pψ
at
⊗
H
+ O(e
8
)
=−E
at
+ e
2
(2π)
−3
d
3
k
1
2ω
β
2
ψ
at
, p · k
1
H
at
− E
at
+ E
1
− p · k
p · kψ
at
L
2
+O(e
8
) (19.51)
with E
1
= β
−1
. Expanding in p · k and in H
at
− E
at
yields
E
bin
=−E
at
+
1
3
e
2
(2π)
−3
d
3
k
1
2ω
β
2
k
2
ψ
at
,(E
1
)
−1
p
2
ψ
at
L
2
+ O(e
8
)
=−E
at
1 +
2
3
e
2
(2π)
−3
d
3
k(k
2
β
3
/2ω)
+ O(e
8
). (19.52)
Note that in (19.51) the Taylor coefficient of order e
10
is infrared divergent, which
implies that E
bin
cannot be analytic at e = 0.
As a final step, we carry out the mass renormalization to order e
2
as required
according to (19.49). The corrections
O(e
6
) cancel and
E
bin
=−m
eff
E
at
+ O(e
8
). (19.53)
h
bin
acquires a radiative correction at least as small as O(e
8
), which confirms
the conventional picture. For small coupling the predictions of the one-particle
theory are reliable. The coupling to the field generates to leading order the attrac-
tive Coulomb potential. Further effects of the interaction with the scalar field are
small. Having no compelling incentive, the strong coupling regime of H
ren
is ap-
parently little explored. It is conceivable that for large e the kinetic energy of the
meson cannot balance the singular Coulomb attraction. If so, H of (19.50) would
no longer be bounded from below.
19.3 Ultraviolet limit, energy and mass renormalization
The ultraviolet limit of the Pauli–Fierz model is a poorly understood subject.
All we can do is to explain the few hints available, which in their optimistic