Spohn H. Dynamics of Charged Particles and their Radiation Field
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14.3 Some applications 193
usual Feynman–Kac formula. The first two terms define a multidimensional Brow-
nian motion. It has a position-dependent diffusion matrix, which by inspection is
strictly positive. Thus the “free” measure is positivity improving, a property which
is preserved when adding the potential.
(ii) Diamagnetic inequality
In (14.44) the fluctuating magnetic field appears as a phase, which leads immedi-
ately to the diamagnetic inequality
|F, U e
−tH
U
−1
G
H
s
|≤|F|, U e
−t (H
p
+H
f
)
U
−1
|G|
H
s
. (14.69)
As one application we derive a bound on the electronic charge density in the
ground state. We assume the existence of a ground state, H ψ
g
= E
g
ψ
g
, with
ground state energy E
g
. Then the electronic charge density is
ρ
g
(x) =ψ
g
(x, ·)
2
F
=
∞
n=0
λ
d
3n
k|ψ
gn
(x, k
1
,λ
1
,...,k
n
,λ
n
)|
2
. (14.70)
We choose F = f (x)U ψ
g
, f ≥ 0and bounded, G = U ψ
g
. Since e
−tH
ψ
g
=
e
−tE
g
ψ
g
and since e
−tH
f
is a contraction, one concludes from the diamagnetic
inequality that
e
−(t+τ)E
g
d
3
xf(x)ρ
g
(x) ≤fρ
1/2
g
, e
−(t+τ)H
p
ρ
1/2
g
L
2
=e
−tH
p
fρ
1/2
g
, e
−τ H
p
ρ
1/2
g
L
2
. (14.71)
From the Feynman–Kac formula (14.5) it follows that (e
−τ H
p
ρ
1/2
g
)(x) ≤ c
1
. Using
this bound in (14.71) and letting f shrink to a δ-function at x we obtain
ρ
g
(x) ≤ [c
1
e
tE
g
(e
−tH
p
1)(x)]
2
(14.72)
with 1 the constant function. Inequality (14.72) is the desired bound on the elec-
tronic charge density.
To make this bound explicit we rewrite
(e
−tH
p
1)(x) = E
x
e
−
t
0
dsV(q
s
)
(14.73)
according to (14.5) for the particular choice ψ(x) = 1. If the potential has a lower
bound as V (x) ≥ c
0
+ c
1
|x|
γ
, c
1
> 0, γ>1, then for fixed t the weight in (14.73)
is dominated by the potential and one has ρ
g
(x) ≤ ce
−V (x)
.Onthe other hand if
V (x) → 0as|x|→∞, then the expression in (14.73) tends to 1 as x →∞. Thus
we should optimize in t for fixed x.Very crudely this means minimizing the action
t
0
ds
1
2
˙q
2
s
+ V (q
s
)
for a fixed initial condition q
0
= x and then to optimize in t.
In this variation one has to include the contribution from the exponentially growing
194 The statistical mechanics connection
factor e
E
g
t
.Tohaveabound state at all, V
min
= min
x
V (x)<0. For sufficiently
small e also V
min
− E
g
< 0 and the variational bound decays exponentially in x,
i.e. ρ
g
(x) ≤ ce
−γ |x|
.Forlarger e one can no longer balance E
g
and the bound
(14.72) becomes vacuous.
The diamagnetic inequality suggests that the decay of the electronic charge den-
sity in the ground state does not worsen by the coupling to the Maxwell field. If one
imagines, rather crudely, the effective mass of the electron to be increased through
the interaction with the field, then the electron density should become even better
localized for larger e and point-like as e →∞.
(iii) Photon expectations
We discovered in section 14.2 that, through integrating over the Maxwell field,
one obtains a path integral (functional measure) for the electron paths which has
the structure of an equilibrium measure relative to an a priori weight given by the
Wiener measure. Here we expand on this observation by computing averages for
the photon field in the ground state. Let ψ
0
be the ground state of H
p
and let us
introduce the approximate ground state
ψ
T
= e
−TH
ψ
0
⊗ /e
−TH
ψ
0
⊗ (14.74)
of H , which is normalized to one and, if the limit does not vanish, converges as
T →∞to the unique ground state ψ
g
of H.For observables of the form f (x) the
same argument as for (14.48) leads to
ψ
T
, f (x)ψ
T
H
(14.75)
=ψ
0
⊗ , e
−2TH
ψ
0
⊗
−1
H
ψ
0
⊗ , e
−TH
f (x)e
−TH
ψ
0
⊗
H
= E
G
[−T ,T ]
f (q
0
)
.
The “volume” [−T, T ]isarranged symmetrically relative to the origin.
E
G
[−T ,T ]
refers to the normalized expectation
E
G
[−T ,T ]
◦
= Z (2T )
−1
E
dW
ψ
0
(q
−T
)ψ
0
(q
T
)e
−
T
−T
dtV(q
t
)
e
−S
[−T ,T ]
◦
(14.76)
with the normalizing partition function
Z(2T ) =
E
dW
ψ
0
(q
−T
)ψ
0
(q
T
) exp
−
T
−T
dtV(q
t
) − S
[−T ,T ]
(14.77)
and with the effective action
S
[−T ,T ]
=
1
2
e
2
T
−T
T
−T
dq
t
· W (q
t
− q
s
, t − s)dq
s
. (14.78)
14.3 Some applications 195
Clearly, in the infinite-volume limit
lim
T →∞
ψ
T
, f (x)ψ
T
H
=ψ
g
, f (x)ψ
g
H
=
d
3
xf(x)ρ
g
(x) =f (q
0
).
(14.79)
Thus the electronic charge density is the distribution of q
0
, the position of the
path at time t = 0, under the infinite-volume Gibbs measure ·, i.e. under the
probability measure obtained in the limit T →∞in (14.76) which we denote by
·.
With (14.79) we have opened the first page in the dictionary for the translation
from Fock space expectations to Gibbs averages. We plan to expand the dictionary
by considering a bounded operator 1 ⊗ B referring only to the photons and want
to compute the expectation
ψ
T
, 1 ⊗ Bψ
T
H
, (14.80)
which for large T goes over to the ground state expectation ψ
g
, 1 ⊗ Bψ
g
H
.
Using the basic identity (14.44) one can write
(Ue
−TH
ψ
0
⊗ )(q, A)
=
E
q
E
A
ψ
0
(q
t
) exp
−
T
0
dtV(q
t
) − ie
T
0
dq
t
· A
tϕ
(q
t
)
, (14.81)
where
E
A
refers to the Ornstein–Uhlenbeck process A
t
(x) with fixed initial field
A
0
= A. The Gaussian expectation E
A
can be carried out with the result
E
A
e
−ie
T
0
dq
t
·A
tϕ
(q
t
)
= e
−ieA( f
+
)
exp
−
1
2
e
2
T
0
T
0
d
3
k|ϕ(k)|
2
×dq
t
· Q
⊥
(k)dq
s
e
ik·(q
t
−q
s
)
1
2ω
e
−ω|t−s|
− e
−ωt
e
−ωs
, (14.82)
where
f
+
(k) =
T
0
dq
t
ϕe
−ik·q
t
e
−ωt
, (14.83)
which depends on the path q
t
,0≤ t ≤ T .
We have to take the expectation of 1 ⊗ B with respect to the wave function
(14.81), for which it is convenient to regard the adjoint wave function as coming
from an integration relative to a Brownian motion running from 0 to −T .For
this purpose one time-reverses the Brownian motion, which starts then at q
−T
and
ends at q. Upon integrating over dq
0
one obtains the Wiener measure for Brownian
196 The statistical mechanics connection
paths t → q
t
, |t|≤T . The expectation E
A
for the adjoint wave function yields an
expression as in (14.82) where f
+
is replaced by f
−
and
f
−
(k) =−
0
−T
dq
t
ϕe
−ik·q
t
e
−ω|t|
(14.84)
with a minus sign, since dq
t
is odd under time-reversal. The expectation for B is
most easily written in Fock space. Then
ψ
0
⊗ , e
−TH
(1 ⊗ B)e
−TH
ψ
0
⊗
H
= E
dW
ψ
0
(q
−T
)ψ
0
(q
T
)e
−
T
−T
V (q
t
)dt
, e
ieA( f
−
)
Be
−ieA( f
+
)
F
× exp
−
1
2
e
2
0
−T
0
−T
+
T
0
T
0
d
3
k|ϕ|
2
dq
t
· Q
⊥
dq
s
e
ik·(q
t
−q
s
)
×
1
2ω
e
−ω|t−s|
− e
−ω|t|
e
−ω|s|
. (14.85)
To make further progress we have to choose particular observables. One exam-
ple is the generating function for the photon number density in momentum space,
i.e.
B = exp
−
λ=1,2
d
3
kµ(k)a
∗
(k,λ)a(k,λ)
(14.86)
with µ ≥ 0. Then
, e
ieA( f
−
)
Be
−ieA( f
+
)
F
= exp
−
1
4
f
−
,ω
−1/2
Q
⊥
ω
−1/2
f
−
h
−
1
4
f
+
,ω
−1/2
Q
⊥
ω
−1/2
f
+
h
−
1
2
f
−
,ω
−1/2
Q
⊥
e
−µ
ω
−1/2
f
+
h
.
(14.87)
Collecting all terms yields
ψ
T
, exp
−
λ=1,2
d
3
kµ(k)a
∗
(k,λ)a(k,λ)
ψ
T
H
= E
G
[−T ,T ]
exp
− e
2
0
−T
T
0
d
3
k|ϕ|
2
dq
t
· Q
⊥
dq
s
e
ik·(q
t
−q
s
)
×
1
2ω
e
−ω|t|
e
−ω|s|
e
−µ
− 1
. (14.88)
14.3 Some applications 197
We differentiate with respect to µ(k) and obtain the ground state photon number
density in momentum space,
λ=1,2
ψ
g
, a
∗
(k,λ)a(k, λ)ψ
g
H
=−e
2
1
2ω
|ϕ|
2
0
−∞
∞
0
dq
t
· Q
⊥
dq
s
e
−ω|t|
e
−ω|s|
e
ik·(q
t
−q
s
)
, (14.89)
the average with respect to the infinite-volume Gibbs measure. In particular one
has the remarkable identity that ψ
g
, N
f
ψ
g
H
equals the average interaction energy
between the right and left half-line in the statistical mechanics system. By the same
technique, the photon density in physical space is given by
λ=1,2
ψ
g
, a
∗
(x,λ)a(x, λ)ψ
g
H
=−e
2
λ=1,2
∞
0
0
−∞
dq
t
· f
λ
(x − q
t
, t)dq
s
· f
λ
(x − q
s
, s), (14.90)
where
f
λ
(k, t) = ϕ
1
√
2ω
e
−ω|t|
e
λ
(k). (14.91)
Equations (14.89) and (14.90) are only partially useful, since there is too lit-
tle information on the dq
t
· Q
⊥
dq
s
correlations, except for the soft photon bound
(15.9), (15.14) from which one concludes that
−c
0
(1 +|k|
3
) ≤
0
−∞
∞
0
dq
t
· Q
⊥
dq
s
e
−ω|t|
e
−ω|s|
e
ik·(q
t
−q
s
)
≤0 (14.92)
with some positive constant c
0
. The interaction energy between right and left is
bounded and negative on the average. One would expect also its exponential mo-
ments to be bounded. If so, ψ
g
, e
−λN
f
ψ
g
H
< ∞ for all λ by (14.88), which im-
plies that in the ground state the number of photons has a super-exponential decay.
Our method may be applied to other observables of interest. For example the
ground state expectation and variance of the vector potential is given by
ψ
g
, A(x)ψ
g
H
= 0 , (14.93)
ψ
g
, A(x)
2
ψ
g
H
=, A(x)
2
F
−
λ=1,2
∞
−∞
dq
t
· f
Aλ
(x − q
t
, t)
2
,
(14.94)
198 The statistical mechanics connection
where
f
Aλ
= e
λ
ϕ
1
2ω
e
−ω|t|
. (14.95)
Similarly for the transverse electric field one has
ψ
g
, E
⊥
(x)ψ
g
H
= 0 , (14.96)
ψ
g
, E
⊥
(x)
2
ψ
g
H
=, E
⊥
(x)
2
F
+
λ=1,2
∞
−∞
dq
t
· f
Eλ
(x − q
t
, t)
2
(14.97)
with
f
Eλ
= ∂
t
f
Aλ
.Infact, the vacuum variances are infinite but become finite
when the fields are slightly smeared out. Through the presence of a bound electron
the electric field fluctuations are increased whereas the vector field fluctuations are
suppressed. Their product remains constant, as required by the uncertainty rela-
tion.
We recall that E=E
+E
⊥
, the second term being zero by (14.96). From
the equations of motion, ∇·ψ
g
, E(x)ψ
g
H
= eϕ(x − q
0
)=eϕ ∗ ρ
g
(x), ρ
g
be-
ing the electron ground state density of (14.70). Thus, at large distances the average
electric field generated by a charge bound in the ground state is the Coulomb field
with a strength determined through the bare charge e, from which we conclude
that in the Pauli–Fierz model there is no charge renormalization.
Notes and references
Section 14.1
Gentle introductions to path integrals are Schulman (1981) and Kleinert (1995)
emphasizing statistical mechanics aspects. Roepstorff (1994) treats in detail the
quantized Maxwell field. Path integrals with a focus on relativistic quantum field
theory are explained in the advanced textbook of Huang (1998). Simon (1979) is
abeautiful discussion on the connection between functional integration and the
Schr
¨
odinger equation. In particular, he explains the Feynman–Kac–Ito formula
used in (14.8). Gaussian processes, Wick ordering, and the Schr
¨
odinger represen-
tation are exhaustively covered in Simon (1974) and Glimm and Jaffe (1987). The
functional measure for the Pauli–Fierz Hamiltonian is discussed by Hiroshima
(1997b). A standard reference on infinite-dimensional Ornstein–Uhlenbeck pro-
cesses is Holley and Stroock (1978). In Giacomin et al. (2001) martingale-type
estimates are explained.
Functional integration has two historical roots which developed apparently
completely independently. Feynman (1948), cf. also the textbook by Feynman and
Notes and references 199
Hibbs (1965), uses space-time histories to visualize quantum processes. This led
to quantum propagators as a “sum over histories”. On the other hand, Wiener,
Levy, and many other probabilists developed the theory of probability measures
on function space (= the space of trajectories) to have a mathematical framework
for Brownian motion and diffusion processes. Kac (1950) realized that the two
approaches are related through the Wick rotation. The extension to models of
quantum fields is achieved by Nelson (1966, 1973). With his insights functional
integration became the “secret weapon” and is at the heart of the technical devel-
opment in constructive quantum field theory through the hands of Glimm, Jaffe,
Spencer, Simon, and many, many others. I refer to Glimm and Jaffe (1987).
Section 14.2
The integration over field degrees of freedom is discussed in Feynman and Hibbs
(1965) and in Feynman (1948). He tackled a variety of physical problems with
this technique. The most widely known is the ground state energy of the polaron
(Feynman 1955) for which the analog of (14.48) is estimated through a variational
method with a result which covered both the intermediate and strong coupling
regime for the first time. To view the effective mass as the stiffness of a polymer
is proposed in Spohn (1987). If the Maxwell field is replaced by a scalar field, cf.
section 19.2, the double stochastic integral becomes a double Riemann integral,
which is much easier to handle. In particular, one obtains reasonable bounds on the
effective stiffness with a technique borrowed from Brascamp, Lieb and Lebowitz
(1976). To view the path measure (14.51) as a Gibbs measure relative to Brownian
motion is stressed in L
˝
orinczi and Minlos (2001), Betz et al. (2002), and L
˝
orinczi
et al. (2002a, 2002b).
Section 14.3
Positivity-improving semigroups are treated in Reed and Simon (1978), Chapter
XIII.12. For the existence of the ground state we refer to section 15.1. Whenever
magnetic fields are involved, the diamagnetic inequality is very helpful; compare
for example with Cycon, Froese, Kirsch and Simon (1987). Carmona (1978) uses
Brownian motion to estimate ground state properties of − + V .His techniques
extend to a charge coupled to a scalar field as discussed in Betz et al. (2002). There
is also a functional analytic proof of exponential localization, which is patterned
after Agmon (1982) in the case of the Schr
¨
odinger equation, see Theorem 20.1.
15
States of lowest energy: statics
Quantizing the Abraham model results in the Pauli–Fierz Hamiltonian which is a
self-adjoint operator under rather general conditions. Thus the dynamics is well
defined and we can start to investigate some of its properties. The most basic item
is the states of lowest energy. They really come in two varieties: (i) If the electron
is bound by a strong external electrostatic potential, like the Coulomb potential of
a nailed-down nucleus, then the lowest energy state is the ground state, where the
electron is at rest modulo quantum fluctuations. (ii) If there are no external poten-
tials, then the total momentum is conserved and the state of lowest energy must be
determined for every fixed total momentum, which then describes the electron to-
gether with its surrounding photon cloud traveling at constant velocity. Physically
the most important information is the energy–momentum relation which gives the
lowest energy E at given total momentum P. Both item (i) and item (ii) are dis-
cussed in this chapter. In case (i) one expects to have always a ground state pro-
vided the external potential is binding. In case (ii) the infrared divergence of the
Pauli–Fierz model becomes visible. As will be explained in more detail in section
19.1, for total momentum P = 0 the state of lowest energy is not in Fock space.
An electron traveling at nonzero velocity binds an infinite number of photons. To
avoid such a subtlety, for item (ii) we proceed as if the photon had a tiny mass.
The external fields manufactured with macroscopic devices under laboratory
conditions are weak and have a slow variation when measured in units of the effec-
tive size of the charge, roughly given through the inverse size of the form factor ϕ.
Such external fields thus constitute a small perturbation in item (ii) and, as for the
Abraham model, an important dynamical issue is to understand the motion of the
charge in terms of an effective one-particle Hamiltonian. The energy–momentum
relation must play an important role, but there will be additional pieces accounting
for the spin precession. Our discussion of this topic is postponed to section 16, to
keep the lengths of the chapters in reasonable proportion.
200
15.1 Bound charge 201
15.1 Bound charge
The hydrogen atom has a stable ground state and thus makes the size of atoms of
the order of a few
˚
angstr
¨
oms. The problem under discussion is whether this ground
state persists as the quantized transverse modes of the Maxwell field are taken
into consideration. Since the electron now has the opportunity to bind photons,
one would expect it to have effectively a larger mass. This intuition is confirmed
through the path integral of chapter 14, which suggests that the fluctuations in the
stochastic trajectories are reduced due to the additional interaction energy W from
the integration over the Maxwell field. Thus the coupling to the photons should
enhance binding.
To put such reasoning on more solid grounds, we recall that for a Schr
¨
odinger
operator H
S
=−(1/2m) + V with a Coulomb-like potential, i.e. a potential V
such that lim
|x |→∞
V (x) = 0, it is rather straightforward to ensure a stable ground
state. Let us assume that V is infinitesimally bounded with respect to −. Then
the bottom of the continuous spectrum, denoted by E
c
, satisfies E
c
= 0 and one
only has to make sure that the energy is lowered when the electron is moved from
infinity to the potential region. This means that one has to find a trial wave function
such that ψ,H
S
ψ < 0. By the Kato–Rellich theorem H
S
is bounded from below.
Thus H
S
must have an eigenvalue at the bottom of its spectrum. The ground state
wave function ψ
g
is nodeless, since e
−tH
S
is positivity improving; compare with
section 14.3(i). Hence the ground state is unique. To adapt such reasoning to the
Pauli–Fierz Hamiltonian
H =
1
2m
( p − eA
ϕ
(x))
2
+ H
f
+ V (x) = H
0
+ V, (15.1)
one faces the difficulty that there are photon excitations of arbitrarily small ener-
gies. Thus H has no spectral gap and a variational bound will not do. The
convential approach is to first assume an infrared cutoff in the form factor ϕ by
setting ϕ(k) = 0 for |k|≤σ and to adopt the construction explained in property
(vi) of section 15.2.1. This yields the existence of a ground state ψ
g,σ
for the cutoff
Hamiltonian H
σ
. One is then left to show that as σ → 0 the sequence of ground
states ψ
g,σ
has a limit ψ
g
which is the desired ground state for H. The difficulty is
that as σ → 0 the number of bound photons could increase without limit resulting
in the physical ground state lying outside of Fock space. This is one aspect of the
infrared problem to be discussed in more detail in section 19.1. Thus one has to es-
tablish a bound on the number of low-energy (soft) photons in the ground state. We
explain some parts of the argument which allow us to illustrate the pull-through
formula that will also be handy later on.
202 States of lowest energy: statics
Theorem 15.1 (Soft photon bound). Let ψ
g
be a ground state of the Pauli–Fierz
Hamiltonian H of (15.1), H ψ
g
= Eψ
g
. Then the average number of photons is
bounded as
ψ
g
, N
f
ψ
g
≤c
0
ψ
g
, x
2
ψ
g
. (15.2)
Proof:Clearly
ψ
g
, N
f
ψ
g
=
λ=1,2
d
3
ka(k, λ)ψ
g
2
. (15.3)
Through a virial-type argument we plan to make use of the fact that ψ
g
is an
eigenfunction, and start with the pull-through formula
[H, a(k,λ)] =−ω(k)a(k,λ)+ eϕ
1
√
2ω
e
−ik·x
1
m
e
λ
(k) · ( p − eA
ϕ
(x)). (15.4)
Note that
1
m
( p − eA
ϕ
(x)) = i[H, x]. (15.5)
Therefore
(H + ω)a(k,λ)− a(k,λ)H = eϕ
1
√
2ω
i[H, e
−ik·x
e
λ
(k) · x]
−i[H, e
−ik·x
]e
λ
(k) · x
. (15.6)
The commutator with e
−ik·x
is
[H, e
−ik·x
] =−
1
m
k · ( p − eA
ϕ
(x))e
−ik·x
−
1
2m
k
2
e
−ik·x
(15.7)
and applied to ψ
g
,
a(k, λ)ψ
g
= ieϕ
1
√
2ω
(H − E + ω)
−1
(H − E) +
1
2m
k
2
+
1
m
k · ( p − eA
ϕ
(x))
e
−ik·x
e
λ
(k) · xψ
g
= ieϕ
1
√
2ω
(φ
1
+ φ
2
+ φ
3
). (15.8)
Thus, choosing e > 0for notational convenience,
a(k, λ)ψ
g
≤e|ϕ|
1
√
2ω
(φ
1
+φ
2
+φ
3
) (15.9)