Spohn H. Dynamics of Charged Particles and their Radiation Field
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20.2 Quasi-static limit 333
Theorem 20.5 (Coulomb Hamiltonian and correction). Let ψ ∈ L
2
with
ψ, H
coul
ψ
L
2
< ∞. Then
lim
c→∞
e
−iH (c)c
2
t
− e
−i(H
ϕdarw
+cH
f
)c
2
t
ψ ⊗=0 . (20.27)
Since the limit (20.27) is on the long-time scale c
2
, the Darwin correction is mean-
ingfully singled out.
Except for operator ordering, (20.27) is in accordance with the results in section
11.2. However in L
darw
of (11.27) the kinetic energy is modified and the Coulomb
potential is not smeared out, which reflects the fact that limits here and in section
11.2 differ somewhat.
For the limit m
j
→∞we can also rely on methods developed before. We start
with the classical symbol
H(q, p) =
N
j=1
1
2m
j
p
2
j
− ε
e
j
m
j
p
j
· A
ϕ
(q
j
)
− ε
2
e
j
2m
j
σ
j
· B
ϕ
(q
j
) + ε
2
e
2
j
2m
j
: A
ϕ
(q
j
)
2
:
+ V
ϕcoul
(q) + H
f
,
(20.28)
q = (q
1
,... ,q
N
), p = ( p
1
,... , p
N
). The Weyl quantization of H(q, p) is
H(c) of (20.18) with m
j
replaced by ε
−2
m
j
, where for convenience we returned
to c = 1. The leading symbol for H (q, p) is
H
0
( p, q) =
N
j=1
1
2m
j
p
2
j
+ V
ϕcoul
(q) + H
f
. (20.29)
Its ground state band has the projection P
0
(q, p) = 1 ⊗ P
,independent of q, p,
and the eigenvalue e
0
(q, p) =
N
j=1
( p
2
j
/2m
j
) + V
ϕcoul
(q). Thus, following sec-
tion 16.4, the Coulomb Hamiltonian can be understood as Peierls’ substitution for
(20.28). It approximates on the time scale ε
−1
t the true unitary evolution projected
to 1 ⊗ P
.
To obtain corrections we have to first compute h
1
. Since , H
int
=0 and
since P
0
does not depend on p, q, h
1
= 0, in accordance with the previous find-
ings that the Darwin correction is of order ε
−2
. Thus we need h
2
.Insection 16.4 no
explicit formula was given, since it is already somewhat lengthy. In our particular
334 Many charges, stability of matter
case, many simplifications occur and as a net result one finds that
h
2
(q, p) =−
N
i, j=1
e
i
e
j
m
i
m
j
d
3
k|ϕ(k)|
2
1
2ω
2
( p
i
· Q
⊥
(k) p
j
)e
ik·(q
i
−q
j
)
+
1
12
(σ
i
· σ
j
)
d
3
k|ϕ(k)|
2
k
2
ω
2
e
ik·(q
i
−q
j
)
. (20.30)
While, since in agreement with the previous result, (20.30) is very satisfactory on
a formal level, a complete proof has to deal with the fact that the ground state band
is not separated by a gap from the remainder of the spectrum. If one is willing to
impose a gap by hand through a massive dispersion ω,then a suitable version of
the results described in section 16.4 becomes available. The picture so derived is
somewhat different from the c →∞limit: the almost invariant subspace is tilted
by order ε relative to (1 ⊗ P
)H.Over the time scale ε
−2
t the motion in this
subspace is governed by h
0
+ ε
2
h
2
.
20.3 H-stability
For the (no-cutoff) Coulomb Hamiltonian
H
coul
=
N
j=1
1
2
p
2
j
+ V
coul
,
V
coul
=
e
2
4π
1≤i< j≤N
|x
i
− x
j
|
−1
− Z
N
i=1
K
j=1
|x
i
−r
j
|
−1
+ Z
2
1≤i< j≤K
|r
i
−r
j
|
−1
, (20.31)
the H-stability is a famous result by Dyson and Lenard. An independent proof
wasachieved by Lieb and Thirring, who succeeded in a fairly realistic estimate
of the stability constant. For stability to hold the electrons must satisfy the Pauli
exclusion principle, as they do in nature. For bosons the energy would decrease as
−N
5/3
.Ifthe nuclei have a finite mass, for a H-stable system at least one of the
two species must be fermions.
To extend H-stability to the realm of nonrelativistic quantum electrodynamics,
one has to establish a lower bound on H
v
(N ) = H, see (20.7), linear in K + N.
Note that for spinless electrons H ≥ H
coul
by the diamagnetic inequality (14.69)
and one is back to the H-stability in (20.31). Thus the difficult point is to deal with
electron spin and the associated magnetic energy. The Schr
¨
odinger representation,
as explained in Chapter 14, suggests that for the purpose of a lower bound, H
f
20.3 H-stability 335
could be substituted by the classical field energy stored in the A-field, i.e. by
E
magn
=
1
2
d
3
xB(x)
2
. (20.32)
If it could be established that H
f
− E
magn
≥ 0, then
H = H + E
magn
− E
magn
≥
N
j=1
1
2
σ
j
· ( p
j
− eA
ϕ
(x
j
))
2
+ V
coul
+ E
magn
.
(20.33)
H-stability of the Coulomb Hamiltonian with magnetic field energy added would
have to be shown for an arbitrary external transverse vector potential.
To progress towards our goal we note that, for an arbitrary operator A,
|ψ, A
2
ψ| ≤ A
∗
ψAψ≤
1
2
ψ, (AA
∗
+ A
∗
A)ψ and therefore
(A + A
∗
)
2
≤ 2(AA
∗
+ A
∗
A). (20.34)
We split the magnetic field as B
ϕ
(x) = B
+
ϕ
(x) + B
−
ϕ
(x) and apply (20.34),
B
ϕ
(x)
2
≤ 4B
+
ϕ
(x)B
−
ϕ
(x) + 2[B
+
ϕ
(x), B
−
ϕ
(x)] , (20.35)
which remains true when multiplied by f (x) ≥ 0. Then
1
2
d
3
xf(x)B
ϕ
(x)
2
≤f
∞
λ=1,2
d
3
k(2π)
3
|ϕ(k)|
2
|k|a
∗
(k,λ)a(k,λ)
+ f
1
d
3
k|ϕ(k)|
2
|k|. (20.36)
Let us assume that |ϕ(k)|≤(2π)
−3/2
and
d
3
k|ϕ(k)||k|=C
< ∞.For f we
choose f (x) = 1if|x −r
j
|≤1 for some j and f (x) = 0 otherwise. Then
H ≥
N
j=1
1
2
σ
j
· ( p
j
− eA
ϕ
(x
j
))
2
+ V
coul
+
1
2
d
3
xf(x)B(x)
2
− KC
.
(20.37)
The energy stability with an arbitrary external B-field is difficult, but has been
done. Unfortunately the field energy balances the Coulomb attraction only for |e|
sufficiently small. To have H-stability for all e one also has to include the B-field
gradients. In addition, the choice of f should be optimized. As one result we state
Theorem 20.6 (H-stability of nonrelativistic QED). Let ϕ be the form factor
with sharp cutoff at . Then there exists a positive constant C(e, Z) such that
H ≥−C(e, Z)( + 1)K (20.38)
independently of N.
336 Many charges, stability of matter
The proof of Theorem 20.6 relies on the H-stability of the Hamiltonian on the right
hand side of (20.37) and thus requires the electrons to be fermions.
In the Pauli–Fierz Hamiltonian (20.7) the self-energy of the electrons is not
subtracted. Thus, in principle, the stability bound (20.38) could exclusively be due
to the positive contribution from the self-energy. To rule out such an unphysical
mechanism we employ a technique, briefly touched upon already in section 19.3.
The results available are sharper in the case of spinless electrons with Hamiltonian
H
N
=
N
j=1
1
2
p
j
− eA
ϕ
(x
j
)
2
+ H
f
+ V
coul
= T
N
+ H
f
+ V
coul
. (20.39)
Since N is the important parameter, it is displayed explicitly. The no-cutoff
Coulomb potential carries the information on the K nuclei located at r
1
,... ,r
K
.
Let E(N ) be the bottom of the spectrum of H
N
and E
0
(N ) that of T
N
+ H
f
. The
binding energy for H
N
is defined as in (20.10) with the nucleon repulsion as an
additive constant. Then
E
bin
(N ) ≤ E
0
(N ) − E(N ). (20.40)
Similar to (19.95) the Coulomb energy is bounded from below as
V
coul
≥−κe
2
N
j=1
|p
j
− eA
ϕ
(x
j
)| (20.41)
with κ =
(π/2)Z + (2.22)Z
2/3
+ 1.03
/4π. Therefore, using Schwarz’s in-
equality,
H
N
≥ T
N
+ H
f
− κe
2
√
2N
T
N
+ H
f
. (20.42)
The function f (x) = x − κe
2
√
2N
√
x takes its minimum at x
min
=
1
2
(κe
2
)
2
N ,
f (x
min
) =−
1
2
(κe
2
)
2
N . Thus, if
E
0
(N ) ≤
1
2
N (κe
2
)
2
(case I) , (20.43)
then H
N
≥−
1
2
(κe
2
)
2
N and E(N ) − E
0
(N ) ≥−(κe
2
)
2
N .Onthe other hand, if
E
0
(N ) ≥
1
2
N (κe
2
)
2
(case II) , (20.44)
we can use the fact that f is monotonically increasing to conclude that H
N
≥
f (E
0
(N )) and E(N ) ≥ E
0
(N ) − (e
2
κ)
√
2N
√
E
0
(N ).Wesummarize as
Theorem 20.7 (Upper bound for N -particle binding energy).For the Hamilto-
nian H
N
of (20.39), in case I
E
bin
(N ) ≤ (κe
2
)
2
N (20.45)
Notes and references 337
and in case II
E
bin
(N ) ≤ (κe
2
)
√
2N
E
0
(N ). (20.46)
Note that energies are in units of mc
2
.
The bound (20.46) is unexpected, since the binding energy is estimated in terms
of the self-energy of a system of N electrons without Coulomb repulsion. The
Pauli exclusion principle has not yet been invoked.
To make further progress one needs a good estimate on E
0
(N ). Fermions like to
stay alone and the state of lowest energy should be achieved once they are infinitely
separated.
Conjecture 20.8 For fermions
E
0
(N ) = NE
0
(1). (20.47)
If Conjecture 20.8 is assumed to hold, then the condition for the two cases reads
(case I) : E
0
(1) ≤
1
2
(κe
2
)
2
, (case II) : E
0
(1) ≥
1
2
(κe
2
)
2
. (20.48)
As explained in section 19.3, E
0
(1) ≤ c
2
e
4/7
(λ
c
)
12/7
. Consequently
E
bin
≤ (κe
2
)
2
N (case I) ,
E
bin
≤ (κe
2
)
√
2c
2
e
2/7
(λ
c
)
6/7
N (case II) . (20.49)
The binding energy is extensive. However, our estimate on the stability bound di-
verges with the cutoff . Since energies are calibrated in units of mc
2
, the folklore
tells us that multiplying the true stability constant by m/m
eff
should result in a
-independent prefactor.
Notes and references
Stability for the Coulomb Hamiltonian is covered extensively and excellently in
survey articles. Particularly recommended are Lieb (1976, 1990), which have
become classics. Some of the original articles are reprinted in the Lieb Selecta
(2001), where the reader should in addition consult the introduction by Thirring,
see also Thirring (2002). The first proof of stability is Dyson and Lenard (1968).
The use of the Thomas–Fermi theory as a comparison standard is introduced in
Lieb and Thirring (1975). Extensions to Coulomb systems with relativistic kinetic
energy are investigated by Conlon (1984), Feffermann and de la Llave (1986), and
were finally settled in Lieb and Yau (1988a, b). The basic discovery is that stabil-
ity holds only under a smallness condition on Zα and α.Ifelectrons were bosons,
they would cluster with a density increasing with N .Inthe ground state energy
this can be seen in a faster than linear decrease with N .For bosons and fixed
338 Many charges, stability of matter
nuclei it is known that E
N
−N
5/3
(Lieb 1979 and references therein), while
with bosonic nuclei of finite mass, E
N
∼
=
−N
7/5
(Dyson 1967; Conlon, Lieb and
Yau 1988).
Thermodynamic stability for Coulomb systems is proved by Lieb and Lebowitz
(1972).
If photons were scalar, then the Coulomb potential has the “wrong” sign; see
section 19.2. This leads to instability, some partial aspects of which are studied in
Gallavotti, Ginibre and Velo (1970).
Section 20.1
For Schr
¨
odinger operators,
F = C,the exponential localization of Proposition
20.2 goes back to Agmon (1982). Griesemer (2002) observes that it remains valid
for general
F . Theorem 20.4 is proved by Lieb and Loss (2003). For the helium
atom, N = 2, the strict positivity of the binding energy is established by Barbaroux
et al. (2003).
Section 20.2
For the Nelson model, i.e. a scalar Bose field, the limit c →∞is studied by Davies
(1979), see also Hiroshima (1997a), and the limit m →∞by Teufel (2002). They
prove that the dynamics is well-approximated through the Coulomb Hamiltonian.
Our observation seems to be new, but could have been made already by Davies, if
he had chosen the Gross-transformed Nelson Hamiltonian as a starting point.
Section 20.3
The argument leading to Theorem 20.6 is taken from Fefferman, Fr
¨
ohlich and
Graf (1997), see also Bugliaro, Fr
¨
ohlich and Graf (1996). The harder part is to
establish stability for the Hamiltonian on the right hand side of (20.37), which is
achieved by Feffermann (1996) with a “sufficiently small” constant and is sub-
sequently improved and simplified by Lieb, Loss and Solovej (1995) to include
the physical case. Theorem 20.7 is a result by Lieb and Loss (2002). They also
establish that the self-energy for N electrons is bounded as c
1
α
1/2
(λ
c
)
3/2
N ≤
E
0
(N ) ≤ c
2
α
2/7
(λ
c
)
12/7
N with suitable constants c
1
, c
2
, which is somewhat
weaker than our Conjecture 20.8. The discussion does not change; only the
prefactors are less sharp. For bosons the bounds c
3
α
1/2
(λ
c
)
3/2
N
1/2
≤ E
0
(N ) ≤
c
4
α
2/7
(λ
c
)
12/7
N
5/7
are available, which together with Theorem 20.7 strongly
indicate that, as to be expected, bosons remain unstable when the quantized radi-
ation field is added. The basic inequality (20.41) holds also in the case where the
electron spin is included, see Lieb and Loss (2002).
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