Renormalized Electron Mass in Nonrelativistic
QED
Jürg Fröhlich∗
Institute of Theoretical Physics; ETH Zürich;
CH-8093 Zürich, Switzerland
(juerg@itp.phys.ethz.ch)
Alessandro Pizzo
Institute of Theoretical Physics; ETH Zürich;
CH-8093 Zürich, Switzerland
(pizzo@itp.phys.ethz.ch)
27-June-2008
Abstract
Within the framework of nonrelativistic QED, we prove that, for
small values of the coupling constant, the energy function, EP~ , of
a dressed electron is twice differentiable2 in the momentum P~ in a
∂ EP~
neighborhood of P~ = 0. Furthermore,
~ |)2 is bounded from below
(∂|P
by a constant larger than zero. Our results are proven with the help
of iterative analytic perturbation theory.
∗
also at IHES, Bures-sur-Yvette
FP-Ren. Elect. Mass., 27-June-2008
I
1
Description of the problem, definition of
the model, and outline of the proof
In this paper, we study problems connected with the renormalized electron
mass in a model of quantum electrodynamics (QED) with nonrelativistic
matter. We are interested in rigorously controlling radiative corrections to
the electron mass caused by the interaction of the electron with the soft
modes of the quantized electromagnetic field. The model describing interactions between nonrelativistic, quantum-mechanical charged matter and the
quantized radiation field at low energies (i.e., energies smaller than the rest
energy of an electron) is the “standard model”, see [6]. In this paper, we
consider a system consisting of a single spinless electron, described as a nonrelativistic particle that is minimally coupled to the quantized radiation field,
and photons. Electron spin can easily be included in our description without
substantial complications.
The physical system studied in this paper exhibits space translations
invariance. The Hamiltonian, H, generating the time evolution, commutes
with the vector operator, P~ , representing the total momentum of the system,
which generates space translations. If an infrared regularization, e.g., an
infrared cutoff σ on the photon frequency, is imposed on the interaction
Hamiltonian, there exist single-electron or dressed one-electron states, as
long as their momentum is smaller than the bare electron mass, m, of the
electron. This means that a notion of mass shell in the energy momentum
spectrum is meaningful for velocities |P~ |/m smaller than the speed of light c;
(with c ≡ m ≡ 1 in our units). Vectors {Ψσ } describing dressed one-electron
states are normalizable vectors in the Hilbert space H of pure states of the
system. They are characterized as solutions of the equations
H σ Ψσ = EPσ~ Ψσ , |P~ | < 1
(I.1)
where H σ is the Hamiltonian with an infrared cutoff σ in the interaction term
and EPσ~ , the energy of a dressed electron, is a function of the momentum
operator P~ . If in the joint spectrum of the components of P~ the support of
the vector Ψσ is contained in a ball centered at the origin and of radius less
than 1 ≡ mc then Eq. (I.1) has solutions; see [5]. Since [H, P~ ] = 0, Eq.
(I.1) can be studied for the fiber vectors, Ψσ~¯ , corresponding to a value, P~ , of
P
the total momentum; (both the total momentum operator and points in its
spectrum will henceforth be denoted by P~ – without danger of confusion).
Thus we consider the equation
HPσ~ ΨσP~ = EPσ~ ΨσP~ ,
(I.2)
FP-Ren. Elect. Mass., 27-June-2008
2
where HPσ~ is the fiber Hamiltonian at fixed total momentum P~ , and EPσ~ is the
value of the function E~zσ at the point ~z ≡ P~ . Physically, states {Ψσ } solving Eq. (I.1) describe a freely moving electron in the absence of asymptotic
photons.
It is an essential aspect of the “infrared catastrophe” in QED that Eq. (I.1)
does not have any normalizable solution in the limit where the infrared cutoff σ tends to zero, and the underlying dynamical picture of a freely moving
electron breaks down; see [4]. Nevertheless, the limiting behavior of the
function EPσ~ is of great interest for the following reasons.
As long as σ > 0, a natural definition of the renormalized electron mass, mr ,
is given by the formula
mr (σ) :=
h
∂ 2 E|σP~ |
(∂|P~ |)2
|P~ =0
i−1
.
(I.3)
(Note that EPσ~ ≡ E|σP~ | is invariant under rotations). Equation (I.3) is expected
to remain meaningful in the limit σ → 0. In particular, the quantity on
the R.H.S. of Eq. (I.3) is expected to be positive and bounded from above
uniformly in the infrared cutoff σ.
More importantly, one aims at mathematical control of the function
mr (σ, P~ ) :=
h
∂ 2 E|σP~ | i−1
(∂|P~ |)2
(I.4)
in a full neighborhood, S, of P~ = 0, corresponding to a slowly moving electron; (i.e., in the nonrelativistic regime). When combined with a number
of other spectral properties of the Hamiltonian of nonrelativistic QED the
condition
∂ 2 E|σP~ |
> 0 , P~ ∈ S ,
(I.5)
(∂|P~ |)2
uniformly in σ > 0, suffices to yield a consistent scattering picture in the
limit when σ → 0 in which the electron exhibits infraparticle behavior. In
fact, (I.5) is a crucial ingredient in the analysis of Compton scattering presented in [12], [4].
Main results
Assuming the coupling constant, α, small enough, the following results follow.
1) The function
∂ 2 E|σP~ |
Σ|P~ | := lim
(I.6)
σ→0 (∂|P
~ |)2
FP-Ren. Elect. Mass., 27-June-2008
3
is well defined for P~ ∈ S := {P~ | |P~ | < 31 }; furthermore, it is Höldercontinuous in P~ .
2) The function
(I.7)
EP~ := lim EPσ~
σ→0
is twice differentiable in P~ ∈ S and
∂ 2 E|P~ |
= Σ|P~ | .
(∂|P~ |)2
(I.8)
3)
2>
∂ 2 E|σP~ |
∂ 2 |P~ |
>0 ,
P~ ∈ S ,
(I.9)
uniformly in σ.
We wish to mention some related earlier results. Using operator-theoretic
renormalization group methods, results (I.6) and (I.9) have been proven in
[1] for the special value P~ = 0. The point P~ = 0 is exceptional, because
the Hamiltonian HP~ is infrared regular at P~ = 0; it has a normalizable
ground state. A highly non-trivial extension of the analysis of [1] to arbitrary
momenta P~ ∈ S has been described in [2].
With regard to ultraviolet corrections to the electron mass, we refer the reader
to [9], [10], [7], and [8].
I.1
Definition of the model
Hilbert space
The Hilbert space of pure state vectors of a system consisting of one nonrelativistic electron interacting with the quantized electromagnetic field is
given by
H := Hel ⊗ F ,
(I.10)
where Hel = L2 (R3 ) is the Hilbert space for a single Schrödinger electron;
for expository convenience, we neglect the spin of the electron. The Hilbert
space, F, used to describe the states of the transverse modes of the quantized
electromagnetic field (the photons) in the Coulomb gauge is given by the Fock
space
F :=
∞
M
N =0
F (N ) ,
F (0) = C Ω ,
(I.11)
FP-Ren. Elect. Mass., 27-June-2008
4
where Ω is the vacuum vector (the state of the electromagnetic field without
any excited modes), and
F (N ) := SN
N
O
h,
N ≥1,
(I.12)
j=1
where the Hilbert space h of state vectors of a single photon is
h := L2 (R3 × Z2 ) .
(I.13)
Here, R3 is momentum space, and Z2 accounts for the two independent
transverse polarizations (or helicities) of a photon. In (I.12), SN denotes
N
the orthogonal projection onto the subspace of N
j=1 h of totally symmetric
N -photon wave functions, which accounts for the fact that photons satisfy
Bose-Einstein statistics. Thus, F (N ) is the subspace of F of state vectors
corresponding to configurations of exactly N photons.
Units
In this paper, we employ units such that Planck’s constant ~, the speed of
light c, and the mass of the electron m are equal to 1.
Hamiltonian
The dynamics of the system is generated by the Hamiltonian
H :=
~ x)
~ ~x + α1/2 A(~
− i∇
2
2
+ Hf .
(I.14)
The (three-component) multiplication operator ~x ∈ R3 represents the posi~ ~x .
tion of the electron. The electron momentum operator is given by p~ = −i∇
Furthermore, α > 0 is the fine structure constant (which, in this paper,
~ x) denotes the vector potential
plays the rôle of a small parameter), and A(~
of the transverse modes of the quantized electromagnetic field in the Coulomb
gauge,
~ ~x · A(~
~ x) = 0 ,
∇
(I.15)
cutoff at high photon frequencies.
H f is the Hamiltonian of the quantized, free electromagnetic field. It is
given by
XZ
(I.16)
H f :=
d3 k |~k| a~∗k,λ a~k,λ ,
λ=±
FP-Ren. Elect. Mass., 27-June-2008
5
where a~∗k,λ and a~k,λ are the usual photon creation- and annihilation operators
satisfying the canonical commutation relations
[a~k,λ , a~∗k0 ,λ0 ] = δλλ0 δ(~k − ~k 0 ) ,
(I.17)
[a~#
k,λ
(I.18)
,
a~#
]
k0 ,λ0
= 0
for ~k, ~k 0 ∈ R3 and λ, λ0 ∈ Z2 ≡ {±}, where a# = a or a∗ . The vacuum vector
Ω ∈ F is characterized by the condition
a~k,λ Ω = 0 ,
(I.19)
for all ~k ∈ R3 and λ ∈ Z2 ≡ {±}.
The quantized electromagnetic vector potential is given by
~ x) :=
A(~
XZ
λ=± BΛ
o
d3 k n
~
∗ i~k·~
~ε~k,λ e−ik·~x a~∗k,λ + ~ε~k,λ
e x a~k,λ ,
|~k|
q
(I.20)
where ~ε~k,− , ~ε~k,+ are photon polarization vectors, i.e., two unit vectors in
R3 ⊗ C satisfying
∗
~ε~k,λ
· ~ε~k,µ = δλµ ,
~k · ~ε~ = 0 ,
k,λ
(I.21)
for λ, µ = ±. The equation ~k · ~ε~k,λ = 0 expresses the Coulomb gauge condition. Moreover, BΛ is a ball of radius Λ centered at the origin in momentum
space. Here, Λ represents an ultraviolet cutoff that will be kept fixed throughout our analysis. The vector potential defined in (I.20) is thus cut off in the
ultraviolet.
Throughout this paper, it will be assumed that Λ ≈ 1 (the rest energy of
an electron), and that α is sufficiently small. Under these assumptions, the
Hamitonian H is selfadjoint on D(H 0 ), i.e., on the domain of definition of
the operator
~ ~x )2
(−i∇
+ Hf .
(I.22)
H 0 :=
2
The perturbation H − H 0 is small in the sense of Kato.
The operator representing the total momentum of the system consisting
of the electron and the electromagnetic radiation field is given by
P~ := p~ + P~ f ,
(I.23)
~ ~x , and where
with p~ = −i∇
P~ f :=
XZ
λ=±
d3 k ~k a~∗k,λ a~k,λ
(I.24)
FP-Ren. Elect. Mass., 27-June-2008
6
is the momentum operator associated with the photon field.
The operators H and P~ are essentially selfadjoint on a common domain, and since the dynamics is invariant under translations, they commute,
[H, P~ ] = ~0. The Hilbert space H can be decomposed into a direct integral
over the joint spectrum, R3 , of the three components of the momentum operator P~ . Their spectral measure is absolutely continuous with respect to
Lebesgue measure, and hence we have that
H :=
Z ⊕
HP~ d3 P ,
(I.25)
where each fiber space HP~ is a copy of Fock space F.
Remark Throughout this paper, the symbol P~ stands for both a vector in R3
and the vector operator on H, representing the total momentum, depending
on context. Similarly, a double meaning is given to arbitrary functions, f (P~ ),
of the total momentum operator.
Given any P~ ∈ R3 , there is an isomorphism, IP~ ,
IP~ : HP~ −→ F b ,
(I.26)
from the fiber space HP~ to the Fock space F b , acted upon by the annihilation~
and creation operators b~k,λ , b~∗k,λ , where b~k,λ corresponds to eik·~x a~k,λ , and b~∗k,λ
~
~
to e−ik·~x a~∗k,λ , and with vacuum Ωf := IP~ (eiP ·~x ). To define IP~ more precisely, we consider a vector ψ(f (n) ;P~ ) ∈ HP~ with a definite total momentum
describing an electron and n photons. Its wave function in the variables
(~x; ~k1 , λ1 ; . . . , ~kn , λn ) is given by
~
~ ~
ei(P −k1 −···−kn )·~x f (n) (~k1 , λ1 ; . . . ; ~kn , λn )
(I.27)
where f (n) is totally symmetric in its n arguments. The isomorphism IP~ acts
by way of
~
~ ~
IP~ ei(P −k1 −···−kn )·~x f (n) (~k1 ,1 ; . . . ; ~kn , λn )
(I.28)
Z
X
1
= √
d3 k1 . . . d3 kn f (n) (~k1 , λ1 ; . . . ; ~kn , λn ) b~∗k1 ,λ1 · · · b~∗kn ,λn Ωf .
n! λ1 ,...,λn
Because the Hamiltonian H commutes with the total momentum, it preserves
the fibers HP~ for all P~ ∈ R3 , i.e., it can be written as
H =
Z ⊕
HP~ d3 P ,
(I.29)
FP-Ren. Elect. Mass., 27-June-2008
7
where
HP~ : HP~ −→ HP~ .
(I.30)
Written in terms of the operators b~k,λ , b~∗k,λ , and of the variable P~ , the fiber
Hamiltonian HP~ is given by
HP~ :=
~
P~ − P~ f + α1/2 A
2
+ Hf ,
2
(I.31)
where
P~ f =
X Z
d3 k ~k b~∗k,λ b~k,λ ,
(I.32)
d3 k |~k|b~∗k,λ b~k,λ ,
(I.33)
o
dk n
∗
~ε~k,λ b~∗k,λ + ~ε~k,λ
b~k,λ .
|~k|
(I.34)
λ
H
f
=
X Z
λ
and
~ :=
A
X Z
BΛ
λ
q
Let
1
}.
(I.35)
3
In order to give a mathematically precise meaning to the constructions presented in the following, we introduce an infrared cut-off at a photon frequency
σ > 0 in the vector potential. The calculation of the second derivative of
the energy of a dressed electron – in the following called the “ground state
energy” – as a function of P~ in the limit where σ → 0, and for P~ ∈ S, represent the main problem solved in this paper. Hence we will, in the sequel,
study the regularized fiber Hamiltonian
S := { P~ ∈ R3 : |P~ | <
HPσ~ :=
~σ
P~ − P~ f + α1/2 A
2
2
+ Hf ,
(I.36)
acting on the fiber space HP~ , for P~ ∈ S, where
~ σ :=
A
X Z
λ
BΛ \Bσ
o
dk n
∗
b~k,λ
~ε~k,λ b~∗k,λ + ~ε~k,λ
|~k|
q
(I.37)
and where Bσ is a ball of radius σ centered at the origin. In the following,
we will consider a sequence of infrared cutoffs
σj := Λj
with 0 < <
1
2
and j ∈ N.
(I.38)
FP-Ren. Elect. Mass., 27-June-2008
I.2
8
Outline of the proof
Next, we outline the key ideas used in the proofs of our main results in Eqs.
(I.6), (I.8), and (I.9).
For P~ ∈ S, α small enough, and σ > 0, EPσ~ is an isolated eigenvalue of HPσ~ |Fσ ;
see Section II and Eq. (II.4). Because of the analyticity of HPσ~ in the variable
P~ , it follows that
∂ 2 E|σP~ |
(∂|P~ |)2
= ∂i2 E|σP~ | |P~ =Pi î =
(I.39)
1
1 Z
1
= 1 − 2h
[∂i HPσ~ ] σ
dz ΨσP~ , [∂i HPσ~ ]ΨσP~ i|P~ =Pi î ,
σ
2πi γσ HP~ − z
HP~ − z
where ∂i = ∂/∂Pi , î is the unit vector in the direction i, ΨσP~ is the normalized
ground state eigenvector of HPσ~ constructed in [5]; γσ is a contour path in
the complex energy plane enclosing EPσ~ and no other point of the spectrum
of HPσ~ , and such that the distance of γσ from spec (HPσ~ ) is of order σ.
At first glance, the expression on the R.H. S. of (I.39) might become singular
as σ → 0, because the spectral gap above EPσ~ = inf spec (HPσ~ |Fσ ) is of order
σ. To prove that the limit σ → 0 is, in fact, well defined, we make use
~ σ ); (see Section II, Eq.
of a σ-dependent Bogoliubov transformation, Wσ (∇E
~
P
(II.3)). This transformation has already been employed in [5] to analyze mass
~ σ ) yields an infrared
shell properties. In fact, conjugation of HPσ~ by Wσ (∇E
~
P
regularized Hamiltonian
~ σ~ )H σ~ Wσ∗ (∇E
~ σ~ )
KPσ~ := Wσ (∇E
P
P
P
(I.40)
with the property that the corresponding ground state, ΦσP~ , has a non-zero
limit, as σ → 0. The Hamiltonian KPσ~ has a “canonical form” derived in [5]
(see also [11], where a similar operator has been used in the analysis of the
Nelson model):
KPσ~ =
(~ΓσP~ )2 X Z
+
|~k|δPσ~ (k̂)b~∗k,λ b~k,λ d3 k + EPσ~ ,
3
2
R
λ
(I.41)
where EPσ~ is a c-number, and ~ΓσP~ is a vector operator defined in Section II.1.,
Eq. (II.40). By construction,
hΦσP~ , ~ΓσP~ ΦσP~ i = 0 .
(I.42)
This is a crucial property in the inductive construction of the limit of proof
of {ΦσP~ } as σ → 0.
FP-Ren. Elect. Mass., 27-June-2008
9
Eq. (I.42) is also an important ingredient in the proof of (I.6), because, by
~ σ ) to each term of the scalar product
applying the unitary operator Wσ (∇E
~
P
on the R.H.S. of (I.39) and using (I.42), one finds that
ΦσP~
ΦσP~
1
1
1 Z
σ i
σ i
(Γ
)
dz
,
(Γ
)
i| ~ i
~
P
2πi γσ KPσ~ − z P~ KPσ~ − z kΦσP~ k
kΦσP~ k P =P î
(I.43)
remains uniformly bounded in σ.
To see this we use the inequality
(I.39) = 1 − 2h
D
(ΓσP~ )i ΦσP~
,
1 2 σ i σ E
1
),
(ΓP~ ) ΦP~ ≤ O( 1
σ
KP~ − z
α 2 σ 2δ
(I.44)
for an arbitrarily small δ > 0, with z ∈ γσ . This inequality will be proven
inductively and will be combined with an improved (as compared to the
result in [5]) estimate of the rate of convergence of {ΦσP~ } as σ → 0 .
By telescoping, one can plug these improved estimates into (I.43) to end up
with the desired uniform bound. The control of the rate of convergence of
the R.H.S. in (I.43), as σ → 0, combined with the smoothness in P~ , for
arbitrary infrared cutoff σ > 0, finally entails the Hölder-continuity in P~ of
the limiting quantity
ΦσP~
ΦσP~
1 Z
1
1
σ i
σ i
Σ|P~ | := 1−lim 2h
(Γ )
dz
, (ΓP~ )
i| ~ i .
σ→0
2πi γσ KPσ~ − z P~ KPσ~ − z kΦσP~ k
kΦσP~ k P =P î
(I.45)
Our paper is organized as follows.
In Section II, we recall how to construct the ground states of the Hamiltonians
HPσ~ and KPσ~ by iterative analytic perturbation theory. This section contains
an explicit derivation of the formula of the transformed Hamiltonians and of
related algebraic identities that will be used later on.
In Section III, we first derive inequality (I.44) and the improved convergence
rate of {ΦσP~ } as σ → 0. Section III.1 is devoted to an analysis of (I.39) that
culminates in the following main results.
Theorem
∂ 2 E σ~
For α small enough, (∂|P~||)P2| converges as σ → 0. The limiting function Σ|P~ | :=
limσ→0
∂E σ~
|P |
~ |)2
(∂|P
is Hölder continuous in P~ ∈ S. The bounds
2 > Σ|P~ | > 0
(from above and below) hold true uniformly in P~ ∈ S.
Corollary
(I.46)
FP-Ren. Elect. Mass., 27-June-2008
10
For α small enough, the function EP~ := limσ→0 EPσ~ , P~ ∈ S, is twice differentiable, and
∂ 2 E|P~ |
= Σ|P~ | for all P~ ∈ S .
(I.47)
(∂|P~ |)2
II
Sequences of ground state vectors
In this section, we report on results contained in [5] concerning the ground
σ
states of the Hamiltonians HP~j , where P~ ∈ S and j ∈ N, and the existence of
a limiting vector for the sequence of ground state vectors of the transformed
σ
σ
Hamiltonians, KP~j , where the Bogoliubov transformation used to obtain KP~j
σ
from HP~j (derived in [3]) is determined by
∗
b~k,λ
→
~ σj )b~∗ Wσ∗ (∇E
~ σj )
Wσj (∇E
~
~
j
k,λ
P
P
=
∗
b~k,λ
−α
1
2
~ σj · ~ ∗
∇E
~k,λ
~
P
3 σ
|~k| 2 δP~j (k̂)
~ σj · ~~
1 ∇E
~ σj )b~ Wσ∗ (∇E
~ σj ) = b~ − α 2
b~k,λ → Wσj (∇E
~
~
k,λ
k,λ
j
P
P
~
P
|~k|
3
2
k,λ
σj
δP~ (k̂)
(II.1)
,
(II.2)
for ~k ∈ BΛ \ Bσj , with
~ σj ) := exp α 21
Wσj (∇E
~
P
XZ
λ
3
dk
BΛ \Bσj
~ σj
∇E
~
P
σ
|~k| δP~j (k̂)
3
2
· (~~k,λ b~∗k,λ − h.c.) . (II.3)
In [5], the first step consists in constructing the ground states of the reguσ
σ
larized fiber Hamiltonians HP~j . As shown in [5], HP~j has a unique ground
σ
state, ΨP~j , that can be constructed by iterative analytic perturbation theory,
as developed in [11]. To recall how this method works some preliminary
definitions and results are needed:
• We introduce the Fock spaces
Fσj := F b (L2 ((R3 \Bσj )×Z2 ))
,
j
Fσσj+1
:= F b (L2 ((Bσj \Bσj+1 )×Z2 )) .
(II.4)
Note that
j
Fσj+1 = Fσj ⊗ Fσσj+1
.
(II.5)
If not specified otherwise, Ωf denotes the vacuum state in anyone of
these Fock spaces. Any vector φ in Fσj can be identified with the
σ
corresponding vector, φ ⊗ Ωf , in F, where Ωf is the vacuum in F0 j .
FP-Ren. Elect. Mass., 27-June-2008
11
• Momentum-slice interaction Hamiltonians are defined by
1
~ ~ H σj · A|
~ σσj + α (A|σσj )2 ,
∆HP~ |σσjj+1 := α 2 ∇
~
P P
j+1
j+1
2
(II.6)
where
~ σσj :=
A|
j+1
X Z
λ
Bσj \Bσj+1
o
d3 k n
∗
~ε~k,λ b~∗k,λ + ~ε~k,λ
b~k,λ ;
|~k|
q
(II.7)
• Four real parameters, , ρ+ , ρ− , and µ, will appear in our analysis.
They have the properties
0 < ρ− < µ < ρ+ < 1 − Cα <
0<<
2
3
ρ−
ρ+
(II.8)
(II.9)
where Cα , with 13 < Cα < 1, for α small enough, is a constant such
that the inequality
EPσ~ −~k > EPσ~ − Cα |~k|
(II.10)
holds for all P~ ∈ S and any ~k 6= 0. Here EPσ~ −~k := inf specHPσ~ −~k . We
note that Cα → 13 , as α → 0; (see Statement (S4) of Theorem III.1. in
[5]).
By iterative analytic perturbation theory (see [5]), one derives the following
results, valid for sufficiently small α (depending on our choice of Λ, , ρ− , µ,
and ρ+ ):
σ
σ
1) EP~j is an isolated simple eigenvalue of HP~j |Fσj with spectral gap larger
σ
or equal to ρ− σj . Furthermore, EP~j is an isolated simple eigenvalue of
σ
HP~j |Fσj+1 with spectral gap at least ρ+ σj+1 .
σ
σ
σ
2) The ground-state energies EP~j and EP~j+1 of the Hamiltonians HP~j and
σ
HP~j+1 , respectively, (acting on the same space Fσj+1 ) satisfy the inequalities
σ
σ
0 ≤ EP~j ≤ EP~j+1 + c α σj2 ,
(II.11)
where c is a Λ-dependent, but j- and α-independent constant.
σ
σ
3) The ground state vectors, ΨP~j+1 , of HP~j+1 can be recursively constructed
starting from ΨσP~0 ≡ Ωf with the help of the spectral projection
1
1 I
dzj+1 σj+1
.
2πi γj+1
HP~ − zj+1
FP-Ren. Elect. Mass., 27-June-2008
12
More precisely,
σ
ΨP~j+1 :=
=
1 I
1
σ
dzj+1 σj+1
ΨP~j ⊗ Ωf
2πi γj+1
HP~ − zj+1
(II.12)
∞ I
1
1
1 X
σ
dzj+1 σj
[−∆HP~ |σσjj+1 σj
]n ΨP~j ⊗ Ωf , (II.13)
2πi j=0 γj+1
HP~ − zj+1
HP~ − zj+1
σ
where γj+1 := {zj+1 ∈ C | |zj+1 − EP~j | = µσj+1 }, with µ as in (II.8).
σ
σ
ΨP~j+1 is the ground state of HP~j+1 |Fσ for any 0 ≤ σ ≤ σj+1 .
II.1
Transformed Hamiltonians.
In this section, we consider the (Bogoliubov-transformed) Hamiltonians
σ
~ σj )
~ σj )H σj Wσ∗ (∇E
KP~j := Wσj (∇E
~
~
~
j
P
P
P
(II.14)
σ
with ground state vectors ΦP~j , j = 0, 1, 2, 3, . . . . Some algebraic manipulaσ
tions to express KP~j in a “canonical form” appear to represent a crucial step
before iterative perturbation theory can be applied to the sequence of these
transformed Hamiltonians. In addition, some intermediate Hamiltonians,
σ
denoted K̂P~j , must be introduced to arrive at the right kind of convergence
estimates.
The same algebraic relations that are used to obtain the “canonical form” of
σ
KP~j also play an important role in the proof of our main result concerning
the limiting behavior, as σ → 0, of the second derivative of the ground state
σ
energy EPσ~ . It is therefore useful to derive the ”canonical form” of KP~j and
the relevant algebraic identities in some detail.
σ
The Feynman-Hellman formula (which holds because (HP~j )P~ ∈S is an anσ
alytic family of type A, and EP~j is an isolated eigenvalue) yields the identity
~ σj = P~ − hP~ f − α 12 A
~ σj i σj ,
∇E
~
ψ
P
(II.15)
~
P
where, given an operator B and a vector ψ in the domain of B, we use the
notation
hψ , B ψi
hBiψ :=
.
(II.16)
hψ , ψi
FP-Ren. Elect. Mass., 27-June-2008
13
We define
1
~ σj
β~ σj := P~ f − α 2 A
(II.17)
σ
~ σj ,
δP~j (k̂) := 1 − k̂ · ∇E
~
P
∗
∗
c~k,λ
:= b~k,λ
+α
1
2
k̂ :=
(II.18)
~ σj · ~ ∗
∇E
~k,λ
~
P
3 σ
|~k| 2 δP~j (k̂)
~ σj · ~~
1 ∇E
c~k,λ := b~k,λ + α 2
~k
|~k|
~
P
k,λ
σ
|~k| δP~j (k̂)
3
2
(II.19)
.
(II.20)
σ
We rewrite HP~j as
σ
HP~j =
(P~ − β~ σj )2
+ Hf ,
2
(II.21)
and, using (II.15) and (II.17),
~ σj + hβ~ σj i σj .
P~ = ∇E
~
ψ
P
(II.22)
~
P
We then obtain
σ
~ σj 2
P~ 2
~ σj + hβ~ σj i σj ) · β~ σj + β
− (∇E
+ Hf
~
ψ~
P
2
2
P
P~ 2 β~ σj 2
=
+
− hβ~ σj iψσj · β~ σj
~
2 Z 2
P
X
σ
+
|~k|δ j (k̂)b~∗ b~ d3 k
HP~j =
R3 \(BΛ \Bσj )
λ
+
XZ
BΛ \Bσj
λ
−α
(II.24)
(II.25)
k,λ k,λ
σ
∗
|~k|δP~j (k̂)c~k,λ
c~k,λ d3 k
XZ
λ
~
P
(II.23)
BΛ \Bσj
σ
|~k|δP~j (k̂)
~ σj · ~ ∗
~ σj · ~ ∗ ∇E
∇E
~k,λ
~k,λ
~
~
P
P
3 σ
3 σ
|~k| 2 δP~j (k̂) |~k| 2 δP~j (k̂)
(II.26)
d3 k .
(II.27)
FP-Ren. Elect. Mass., 27-June-2008
14
Adding and subtracting 1/2 hβ~ σj i2ψσj , one finds that
~
P
σ
HP~j
hβ~ σj i2ψσj
(β~ σj − hβ~ σj iψσj )2
P~ 2
~
~
P
P
=
−
+
2 Z 2
2
X
σ
+
|~k|δ j (k̂)b~∗ b~ d3 k
R3 \(BΛ \Bσj )
λ
+
XZ
BΛ \Bσj
λ
−α
(II.29)
k,λ k,λ
σ
∗
|~k|δP~j (k̂)c~k,λ
c~k,λ d3 k
XZ
λ
~
P
(II.28)
BΛ \Bσj
σ
|~k|δP~j (k̂)
(II.30)
~ σj · ~ ∗ ∇E
~ σj · ~ ∗
∇E
~k,λ
~k,λ
~
~
P
P
3 σ
3 σ
|~k| 2 δP~j (k̂) |~k| 2 δP~j (k̂)
d3 k .
(II.31)
Next, we implement the Bogoliubov transformation
∗
b~k,λ
~ σj · ~ ∗
∇E
~k,λ
~
P
~ σj )b~∗ Wσ∗ (∇E
~ σj ) = b~∗ − α 12
→ Wσj (∇E
~
~
j
k,λ
k,λ
P
P
σ
3 σ
|~k| 2 δP~j (k̂)
~ σj · ~ ∗
1 ∇E
σ
~k,λ
σj
δP~ (k̂)
~
P
~ j )b~ Wσ∗ (∇E
~ j ) = b~ − α 2
b~k,λ → Wσj (∇E
~
~
k,λ
k,λ
j
P
P
|~k|
3
2
, (II.32)
, (II.33)
~ σj ) is defined in (II.3). It is evident that Wσ
for ~k ∈ BΛ \ Bσj , where Wσj (∇E
j
~
P
acts as the identity on F b (L2 (Bσj × Z2 )) and on F b (L2 ((R3 \ BΛ ) × Z2 )).
We define the vector operators
~ σj )β~ σj Wσ∗ (∇E
~ σj )β~ σj Wσ∗ (∇E
~ σj := Wσ (∇E
~ σj ) − hWσ (∇E
~ σj )iΩ , (II.34)
Π
j
j
~
~
~
~
~
f
j
j
P
P
P
P
P
noting that, by (II.15), (II.17), and (II.34)
~ σj
hβ~ σj iψσj = P~ − ∇E
~
P
(II.35)
~
P
σ
~ σj Φ σj i
hΦP~j , Π
~
~
P
P
~ σj )β~ σj Wσ∗ (∇E
~ σj )iΩ , (II.36)
+ hWσj (∇E
=
σj
σj
~
~
f
j
P
P
hΦP~ , ΦP~ i
σ
where ΦP~j is the ground state of the Bogoliubov-transformed Hamiltonian
σ
~ σj ) .
~ σj )H σj Wσ∗ (∇E
KP~j := Wσj (∇E
~
~
~
j
P
P
P
(II.37)
It is easy to see that
σ
σ
σ
σ
~ j )β~ σj Wσ∗ (∇E
~ j ) − hβ~ σj i σj = Π
~ j − hΠ
~ j i σj .
Wσj (∇E
~
~
~
~ Φ
j
Ψ
P
P
P
P
~
P
~
P
(II.38)
FP-Ren. Elect. Mass., 27-June-2008
15
σ
The “canonical form” of KP~j is given by
σ
KP~j
σ
(~ΓP~j )2 X Z
σ
σ
=
+
|~k|δP~j (k̂)b~∗k,λ b~k,λ d3 k + EP~ j ,
3
2
R
λ
(II.39)
where
~Γσj := Π
~ σj − hΠ
~ σj i σj ,
~
~
~ Φ
P
P
P
(II.40)
σ
h~ΓP~j iΦσj = 0 ,
(II.41)
~
P
so that
~
P
and
σ
EP~ j
~ σj )2
P~ 2 (P~ − ∇E
~
P
−
:=
2
2
−α
XZ
BΛ \Bσj
λ
(II.42)
σ
|~k|δP~j (k̂)
~ σj · ~ ∗ ∇E
~ σj · ~ ∗
∇E
~k,λ
~k,λ
~
~
P
P
3 σ
3 σ
|~k| 2 δP~j (k̂) |~k| 2 δP~j (k̂)
d3 k . (II.43)
Eq. (II.38) follows by using that
~ σj )c~∗ Wσ (∇E
~ σj ) = b~∗ ,
Wσj (∇E
j
~
~
k,λ
k,λ
P
P
(II.44)
σ
σ
~ j ) = b~ ,
~ j )c~ Wσ (∇E
Wσj (∇E
j
~
~
k,λ
k,λ
P
P
(II.45)
for ~k ∈ BΛ \ Bσj .
σ
An intermediate Hamiltonian, K̂P~j+1 , is defined by
σ
σ
σ
σ
~ j )H j+1 Wσ∗ (∇E
~ j) ,
K̂P~j+1 := Wσj+1 (∇E
~
~
~
j+1
P
P
P
(II.46)
where
~ σj ) := exp α 12
Wσj+1 (∇E
~
P
We decompose
that
σ
K̂P~j+1
XZ
λ
3
dk
BΛ \Bσj+1
~ σj
∇E
~
P
|~k|
3
2
σ
δP~j (k̂)
into several different terms, similarly as
σ
HP~j+1 =
(P~ − β~ σj+1 )2
+ Hf ,
2
· (~~k,λ b~∗k,λ − h.c.) .
σ
KP~j .
(II.47)
We recall
(II.48)
and, by (II.35),
~ σj + hβ~ σj i σj .
P~ = ∇E
~
ψ
P
~
P
(II.49)
FP-Ren. Elect. Mass., 27-June-2008
16
It follows that (see also (II.28)-(II.31))
σ
~ σj+1 2
P~ 2
~ σj + hβ~ σj i σj ) · β~ σj+1 + β
− (∇E
+ Hf
~
ψ
P
~
2
2
P
σj+1 2
2
~
~
β
P
=
+
− hβ~ σj iψσj · β~ σj+1
~
2 Z 2
P
X
σ
+
|~k|δ j (k̂)b~∗ b~ d3 k
HP~j+1 =
+
XZ
BΛ \Bσj+1
λ
−α
~
P
R3 \(BΛ \Bσj+1 )
λ
(II.51)
(II.52)
k,λ k,λ
σ
∗
|~k|δP~j (k̂)c~k,λ
c~k,λ d3 k
XZ
BΛ \Bσj+1
λ
(II.50)
σ
|~k|δP~j (k̂)
(II.53)
~ σj · ~ ∗
~ σj · ~ ∗ ∇E
∇E
~k,λ
~k,λ
~
~
P
P
3 σ
3 σ
|~k| 2 δP~j (k̂) |~k| 2 δP~j (k̂)
d3 k .
(II.54)
We now add and subtract 1/2 hβ~ σj i2ψσj and conjugate the resulting operator
~
P
~ σj ). After inspecting straightforward
with the unitary operator Wσj+1 (∇E
~
P
operator domain questions (see [5]), we find that
σ
j
~ σσjj+1 + I
~ σσj+1
(~ΓP~j + L
)2
=
2
XZ
σ
σ
∗
+
|~k|δP~j (k̂)b~k,λ
b~k,λ d3 k + ÊP~ j+1 ,
σ
K̂P~j+1
λ
(II.55)
(II.56)
R3
where
~ σσj
L
:= −α
j+1
1
2
XZ
~k
~ σj · ~ ∗ b~ + h.c.
∇E
~k,λ k,λ
~
P
Bσj \Bσj+1
σj
3 σ
|~k| 2 δP~j (k̂)
d3 k ,
(II.57)
1
~ σσj
−α 2 A|
j+1
~ σσj
I
:= α
j+1
XZ
XZ
λ
σ
ÊP~ j+1
~k
Bσj \Bσj+1
λ
+α
(II.58)
Bσj \Bσj+1
σ
∗
∗ ~
∇EP~j · ~~k,λ
· ~~k,λ
d3 k ,
3 σj
3 σj
~
~
|k| 2 δP~ (k̂) |k| 2 δP~ (k̂)
~ σj · ~ ∗
∇E
d3 k
~k,λ
~
P
q
[~~k,λ
+
h.c.]
3 σ
|~k| 2 δP~j (k̂)
|~k|
~ σj
∇E
~
P
(II.59)
~ σj )2
P~ 2 (P~ − ∇E
~
P
:=
−
2
2
−α
XZ
λ
BΛ \Bσj+1
σ
|~k|δP~j (k̂)
(II.60)
~ σj · ~ ∗ ∇E
~ σj · ~ ∗
∇E
~k,λ
~k,λ
~
~
P
P
3 σ
3 σ
|~k| 2 δP~j (k̂) |~k| 2 δP~j (k̂)
d3 k .
FP-Ren. Elect. Mass., 27-June-2008
17
We also define the operators
~ˆ σj := Wσ (∇E
~ σj−1 )Wσ∗ (∇E
~ σj )Π
~ σj Wσ (∇E
~ σj )Wσ∗ (∇E
~ σj−1 ) ,
Π
j
j
~
~
~
~
~
~
j
j
P
P
P
P
P
P
and
ˆ σj
~Γ
~ˆ σj − hΠ
~ˆ σj i σj ,
~ := ΠP
~
~ Φ̂
P
P
(II.61)
(II.62)
~
P
which are used in the proofs of convergence of the ground state vectors. Here,
σ
Φ̂P~j denotes the ground state vector of the Hamiltonian
σ
~ σj−1 )Wσ∗ (∇E
~ σj )K σj Wσ (∇E
~ σj )Wσ∗ (∇E
~ σj−1 )
K̂P~j := Wσj (∇E
j
~
~
~
~
~
j
j
P
P
P
P
P
An important identity used in [5] and in the sequel of the present paper is
ˆ σj ~ σj−1
~Γ
~ σj − ∇E
~ σj−1 + L
~ σσj−1
= ∇E
~ − ΓP
~
~
~
j
P
P
P
+α
XZ
Bσj−1 \Bσj
λ
+α
XZ
Bσj−1 \Bσj
λ
(II.63)
~ σj−1 · ~ ∗
~ σj−1 · ~~ ∇E
∇E
~k,λ 3
~
~
k,λ
P
P
~k
dk
3 σj−1
3 σj−1
|~k| 2 δP~ (k̂) |~k| 2 δP~ (k̂)
~ σj−1 · ~ ∗
∇E
d3 k
~k,λ
~
P
q
[~~k,λ
+
h.c.]
.
3 σ
|~k| 2 δP~j−1 (k̂)
|~k|
Eq. (II.63) can be derived using (II.34), (II.36), (II.38), (II.40), (II.61), and
(II.62).
II.2
σ
Convergence of the sequence {ΦP~j }∞
j=0 .
To pass from momentum scale j to j + 1, we proceed in two steps: First, we
σ
construct an intermediate vector, Φ̂P~j+1 , defined by
σ
Φ̂P~j+1
∞
X
1
1
1 Z
σ
dzj+1 σj
[−∆KP~ |σσjj+1 σj
]n ΦP~j ,
:=
KP~ − zj+1
KP~ − zj+1
n=0 2πi γj+1
(II.64)
where
σ
σ
σ
σ
∆KP~ |σσjj+1 := K̂P~j+1 − ÊP~ j+1 + EP~ j − KP~j
(II.65)
1 ~ σj ~ σj
~ σj + I
~ σσj )2 . (II.66)
~ σσj ) + h.c. + 1 (L
ΓP~ · (Lσj+1 + I
=
j+1
j+1
2
2 σj+1
σ
Subsequently, we construct ΦP~j+1 by setting
σ
~ σj+1 )Wσ∗ (∇E
~ σj )Φ̂σj+1 .
ΦP~j+1 := Wσj+1 (∇E
~
~
~
j+1
P
P
P
(II.67)
FP-Ren. Elect. Mass., 27-June-2008
18
The series in (II.64) is termwise well-defined and converges strongly to a nonzero vector, provided α is small enough (independently of j). The proof of
this claim is based on operator-norm estimates of the type used in controlling
the Neumann expansion in (II.13), which requires an estimate of the spectral
~ σj ) and Result 1) (after Eq.
gap that follows from the unitarity of Wσj (∇E
~
P
(II.10)).
σ
A key point in our proof of convergence of the sequence {ΦP~j } is to show
that the term
~Γσj · (L
~ σσj + I
~ σσj ) + h.c.
(II.68)
~
j+1
j+1
P
appearing in (II.66), which is superficially ”marginal” in the infrared, by
power counting, is in fact ”irrelevant” (using the terminology of renormalization group theory). This is a consequence of the orthogonality condition
σ
σ
σ
hΦP~j , ~ΓP~j ΦP~j i = 0 ,
(II.69)
which, combined with an inductive argument, implies that
k(
σ
KP~j
1
1
1
σ
σj
~ σσj (+) + I
~ σσj )] ( σ 1
2 Φ
) 2 [~ΓP~j · (L
)
j
~ k
j+1
j+1
P
− zj+1
KP~ − zj+1
(II.70)
σ (+)
~ σjj+1 stands for the part which contains only photon creation oper(where L
ators) is of order O(ηj ), for some η > 0 specified in [5]. In particular, this
suffices to show that
σ
σ
kΦ̂P~j+1 − ΦP~j k ≤ O(
II.2.1
j+1
(1−δ
2
).
(II.71)
Key ingredients
σ
To prove convergence of the sequence of {ΨP~j } of ground state vectors of the
σ
Hamiltonians KP~j , some further conditions on α, , and µ are required, in
particular an upper bound on µ and an upper bound on strictly smaller
than the ones imposed by inequalities (II.80), (II.81); (see Lemma A.3 in [5]).
We note that the more restrictive conditions on µ and imply new bounds
1
on ρ− and ρ+ . Moreover, must satisfy a lower bound > Cα 2 , with a
multiplicative constant C > 0 sufficiently large.
Some key inequalities needed in our analysis of the convergence properties
σ
of {ΦP~j } are summarized below. They will be marked by the symbol (B).
In order to reach some important improvements in our estimates of the conσ
vergence rate of ΦP~j , as j → ∞ (discussed in the next section), a refined
estimate is needed that is stated in (B2), and a new inequality, see (B5),
(analogous to (B3) and (B4)) is required.
FP-Ren. Elect. Mass., 27-June-2008
19
• Estimates on the shift of the ground state energy and its gradient
There are constants C1 , C20 such that the following inequalities hold.
(B1)
σ
σ
|EP~j − EP~j+1 | ≤ C1 α j ;
(II.72)
see [5].
• (B2)
~ σj+1 − ∇E
~ σj | ≤ C20 kΦ̂σj+1 − Φσj kF + α 14 j+1
|∇E
~
~
~
~
P
P
P
P
(II.73)
This is an improvement over a corresponding estimate in [5]: It can be
proven after the results stated in Theorem III.1 in [5], in particular the
σ
σ
σ
σ
uniform bound from below on hΦP~j , ΦP~j i, hΦP~j , ΦP~j i > 23 , and following
the steps in the proof of Lemma A.2. in [5].
• Bounds relating expectations of operators to expectations of their absolute values
There are constants C3 , C4 , C5 > 1 such that the following inequalities
hold.
(B3) For zj+1 ∈ γj+1 ,
E
1
σj i σj
(Γ ~ ) Φ ~
P
− zj+1 P
D
E
1
σ
σ
σ
σ
(ΓP~j )i ΦP~j ,
≤ C3 (ΓP~j )i ΦP~j , σj
KP~ − zj+1
D
σ
σ
(ΓP~j )i ΦP~j , σ
KP~j
(II.74)
(II.75)
σ
σ
where (ΓP~j )i is the ith component of ~ΓP~j .
(B4) For zj+1 ∈ γj+1 ,
E
1
σj (+) l σj i σj
(Lσj+1 ) (Γ ~ ) Φ ~
(II.76)
P
P
− zj+1
D
E
1
σ
σ
σ
σ
(+) l
j (+) l
(Lσσj+1
) (ΓP~j )i ΦP~j (II.77)
,
≤ C4 (Lσσjj+1
) (ΓP~j )i ΦP~j , σj
KP~ − zj+1
D
σ
σ
(+) l
(Lσσjj+1
) (ΓP~j )i ΦP~j , σ
KP~j
σ (+)
j (+)
~ σσj+1
where (Lσjj+1 )l is the lth component of L
.
(B5) For zj+1 ∈ γj+1 ,
2
E
1
σj
σj
(Γ ~ )i Φ ~
P
P
− zj+1
D
E
1
σ
σ
σ
σ
≤ C5 (ΓP~j )i ΦP~j , ( σj
)2 (ΓP~j )i ΦP~j .
KP~ − zj+1
D
σ
σ
(ΓP~j )i ΦP~j , σ
KP~j
(II.78)
(II.79)
FP-Ren. Elect. Mass., 27-June-2008
20
To prove (B3) and (B4), it suffices to exploit the fact that the specσ
σ
σ
tral support (with respect to KP~j ) of the two vectors (ΓP~j )i ΦP~j and
σ (+)
σ
σ
σ
(Lσjj+1 )l (ΓP~j )i ΦP~j is strictly above the ground state energy of KP~j ,
σ
since they are both orthogonal to the ground state ΦP~j of this operator. In the proof of bound (B5), it is also required that ρ− > 3µ, as
will be assumed in the following.
Remark
The constants C1 , . . . , C5 are independent of α, , µ, and j ∈ N, provided that
α, , and µ are sufficiently small, and > Cα.
Remark
For the convenience of the reader, we recapitulate the relations between the
parameters entering the construction:
0 < ρ− < µ < ρ+ < 1 − Cα <
0<<
2
,
3
ρ−
,
ρ+
(II.80)
(II.81)
> Cα ,
ρ− > 3µ .
(II.82)
(II.83)
Moreover, we stress that the final result is a small coupling result, i.e., for
α small, and that, for technical reasons, small values of the parameters
, µ, ρ− , ρ+ are required within the constraints above.
σ
σ
The crucial estimate for the bound on ΨP~j+1 − ΨP~j obtained in [5] (see
(II.71)) is
D
E
R0
1
σj i σj σj
σj
(Γ
)
Φ
,
(II.84)
(Γ ~ )i Φ ~ ,
σj
~
~ ≤
P
P
P
P
αjδ
KP~ − zj+1
where δ, 0 < δ < 1, can be taken arbitrarily small, and R0 is independent of
j. This estimate will be improved in the next section. As a consequence, the
σ
estimate of the convergence rate of {ΦP~j } will be improved. As a corollary,
the second derivative of EPσ~ is proven to converge, as σ → 0.
FP-Ren. Elect. Mass., 27-June-2008
III
21
Improved estimate of the convergence rate
of {ΦσP~ }, as σ → 0, and uniform bound on
the second derivative of EPσ~ .
By repeating some steps of the proof of (II.84), we derive the inequality
D
σj
σj
(Γ ~ )i Φ ~
P
,
P
σ
KP~j
E
2
1
R0
σ
σ ,
(ΓP~j )i ΦP~j ≤ 1
− zj+1
α 2 2jδ
(III.1)
where 0 < δ < 1 and R0 is a constant independent of j ∈ N, for α and sufficiently small.
For R0 and α small enough, inequality (III.1) implies (see the paragraph
Proof by induction of the inequality (III.1) below):
σ
1
σ
kΦ̂P~j − ΦP~j−1 k ≤ α 4 j(1−δ) ,
(III.2)
for any (1 >)δ > 0.
From now on, we assume the lower bounds
σ
σ
σ
σ
hΦ̂P~j+1 , Φ̂P~j+1 i , hΦP~j , ΦP~j i >
2
3
(III.3)
uniformly in j ∈ N, which appear in the proof of Theorem III.1 of ref. [5],
(using the a-priori results stated as ingredients A1, . . . , A4).
Proof by induction of the inequality (III.1)
• Inductive hypothesis We assume that, at scale j − 1, the following estimate holds
D
σj−1
σj−1
(Γ ~ )i Φ ~
P
P
1
,
σ
KP~j−1
2
− zj
E
σ
σ
(ΓP~j−1 )i ΦP~j−1 ≤
R0
1
2
α 2(j−1)δ
. (III.4)
This estimate readily implies that, for R0 and α small enough, but
uniformly in j,
σ
σ
kΦ̂P~j − ΦP~j−1 k =
=
k
∞
X
1
n=1 2πi
Z
γj
dzj
(III.5)
1
σ
KP~j−1
− zj
[−∆KP~ |σσj−1
j
1
4
≤ α j(1−δ) .
1
σ
KP~j−1
σ
− zj
]n ΦP~j−1 k
(III.6)
σ
σ
An improved estimate on kΦ̂P~j+1 − ΦP~j k is based on the following
bounds:
FP-Ren. Elect. Mass., 27-June-2008
22
i)
k
1
σ
KP~j−1
1
σ
− zj
1
∆KP~ |σσj−1
ΦP~j−1 k ≤ O(R02 α 4 j(1−δ) ) ,
j
(III.7)
whose proof requires (III.4) and a slightly modified version of
Lemma A3 in [5];
ii)
k
1
σ
KP~j−1
1
− zj
∆KP~ |σσj−1
kFσj ≤ O(α 2 ) ,
j
(III.8)
where we use the notation kAkH = kA|H k for the norm of a
bounded operator A acting on Hilbert space H. This estimate
can be derived from standard bounds and using the pull-through
formula.
• Induction step from scale j − 1 to scale j
σ
σ
~ j−1 )Wσ∗ (∇E
~ j ), we have that
By unitarity of Wσj (∇E
~
~
j
P
P
2
E
1
σj i σj (Γ
)
Φ
~
~ =
P
P
P
P
− zj+1
D
E
2
1
σj i σj σ
σ
(
Γ̂
)
Φ̂
= (Γ̂P~j )i Φ̂P~j ,
~
~ .
σ
P
P
K̂P~j − zj+1
D
σj
σj
(Γ ~ )i Φ ~
,
σ
KP~j
(III.9)
1
As α is small enough and > C α 2 , where C > 0 is large enough, we
may use (B1) to re-expand the resolvent and find that
E
2
1
σj i σj (
Γ̂
)
Φ̂
~
~
P
P
P
P
− zj+1
D
2
E
1
σ
σ
σj
σj ≤ 2 (Γ̂P~j )i Φ̂P~j , σj−1
(Γ̂ ~ )i Φ̂ ~ .
P
P
KP~ − zj+1
D
σj
σj
(Γ̂ ~ )i Φ̂ ~
,
σ
K̂P~j
(III.10)
(III.11)
It follows that
D
σ
σ
2 (Γ̂P~j )i Φ̂P~j , 2
E
1
σj i σj (
Γ̂
)
Φ̂
σ
~
~ ≤
P
P
KP~j−1 − zj+1
(III.12)
1
σj
σj
σj−1
σj−1 2
((Γ̂ ~ )i Φ̂ ~ − (Γ ~ )i Φ ~ (III.13)
P
P
P
P
− zj+1
2
D
E
1
σj−1
σj−1 σ
σ
(Γ ~ )i Φ ~
.(III.14)
+4 (ΓP~j−1 )i ΦP~j−1 , σj−1
P
P
KP~ − zj+1
≤ 4
σ
KP~j−1
FP-Ren. Elect. Mass., 27-June-2008
23
Our recursion relates (III.14) to the initial expression in (III.1), with
j replaced by j − 1, while (III.13) is a remainder term that can be
controlled as follows:
1
σj i σj
σj−1 i σj 2
((
Γ̂
)
Φ̂
−
(Γ
)
Φ
)
σ
~
~
~
~ P
P
P
P
KP~j−1 − zj+1
1
σj
σj
σj−1
σj 2
((Γ̂ ~ )i Φ̂ ~ − (Γ ~ )i Φ̂ ~ )
≤ 8 σj−1
P
P
P
P
KP~ − zj+1
1
σj−1 2
σj−1 i σj
(Γ ~ ) (Φ̂ ~ − Φ ~ )
+8 σj−1
P
P
P
KP~ − zj+1
R2
R1
≤ 2jδ + 2jδ .
4
(III.15)
(III.16)
(III.17)
(III.18)
Here R1 ≤ O(−2 ) and R2 ≤ O(−2 ) are constants independent of α,
1
µ, and j ∈ N, provided that α, µ are sufficiently small, and > Cα 2 .
In detail:
– Property (B4) and the two norm-bounds
k
σ
KP~j−1
1
σ
(Γ j−1 )i kFσj ≤ O(−(j+1) ) ,
− zj+1 P~
σ
σ
1
kΦ̂P~j −ΦP~j−1 k ≤ α 4 j(1−δ)
(III.19)
(see (III.5)) justify the step from (III.17) to (III.18);
– concerning the step from (III.16) to (III.18), it is enough to consider Eq. (II.63) and the two bounds
1
σ
k(Lσσj−1
)i Φ̂P~j k ≤ O(α 4 j+1 ) ,
j
σ
σ
1
kΦ̂P~j − ΦP~j−1 k ≤ α 4 j(1−δ) .
(III.20)
To bound the term (III.14), we use (B5) and the key orthogonality
property (II.69). For zj ∈ γj and zj+1 ∈ γj+1 , we find that
2
E
1
σj−1
σj−1 (Γ ~ )i Φ ~
(III.21)
P
P
− zj+1
D
2
E
1
σ
σ
σ
σ
≤ 4C5 (ΓP~j−1 )i ΦP~j−1 ,
(ΓP~j−1 )i ΦP~j−1 (III.22)
σj−1
KP~ − zj+1
D
2
E
1
σj−1 i σj−1 σ
σ
(Γ
)
Φ
. (III.23)
≤ 8C52 (ΓP~j−1 )i ΦP~j−1 ,
σ
~
~
P
P
KP~j−1 − zj
D
σ
σ
4 (ΓP~j−1 )i ΦP~j−1 , σ
KP~j−1
In passing from (III.22) to (III.23), we again use the constraint on the
σ
σ
σ
spectral support (with respect to KP~j−1 ) of the vector (ΓP~j−1 )i ΦP~j−1 .
FP-Ren. Elect. Mass., 27-June-2008
24
Assuming that the parameters and α are so small that the previous
constraints are fulfilled and that
0 < R1 + R2 ≤ (1 − 8C52 2δ )
R0
1
α2
,
(III.24)
we then conclude that
D
σj
σj
(Γ ~ )i Φ ~
P
≤
P
,
σ
K̂P~j
E
2
1
σ
σ
(ΓP~j )i ΦP~j − zj+1
R1
R2
+
2jδ 2jδ
D
(III.26)
σ
σ
+8C52 (ΓP~j−1 )i ΦP~j−1 ,
≤
(III.25)
R0
1
2
α 2jδ
1
2
σ
KP~j−1 − zj
σ
σ
E
(ΓP~j−1 )i ΦP~j−1 (III.27)
.
(III.28)
• The zeroth step in the induction
Since
(ΓσP~0 )i ≡ (P~ f )i
ΦσP~0 ≡ Ωf ,
,
(III.29)
inequality (III.1) is trivially fulfilled for j = 0; thus (III.1) holds for
all j ∈ N ∪ 0 and for R0 arbitrarily small, provided α is small enough.
By standard arguments (see [11]), one obtains similar results for the ground
state vectors of the σ-dependent Hamiltonians KPσ~ , for arbitrary σ > 0. A
precise statement is as follows: For α small enough, the normalized ground
state vectors (that, with an abuse of notation, we call ΦσP~ )
ΦσP~
:=
1 H
2πi γσ
dz K σ1−z Ωf
~
P
1 H
1
k 2πi
γσ K σ −z Ωf k
~
,
(III.30)
P
where γσ := {z ∈ C | |z −EPσ~ | =
ΦP~ , as σ → 0, with
ρ−
2
σ}, converges strongly to a nonzero vector
1
kΦσP~ − ΦP~ k ≤ O(α 4
for any 0 < δ(< 1).
σ 1−δ
Λ
)
(III.31)
FP-Ren. Elect. Mass., 27-June-2008
III.1
25
Convergence of the second derivative of the ground
state energy EPσ~ .
Because of rotational symmetry we have that EPσ~ ≡ E|σP~ | . Moreover, (HPσ~ )P~ ∈S
is an analytic family of type A in P~ ∈ S, with an isolated eigenvalue E|σP~ | .
∂ 2 E σ~
|P |
Thus, the second derivative
∂ 2 E|σP~ |
(∂|P~ |)2
~ |)2
(∂|P
is well defined and
= ∂i2 E|σP~ | |P~ =Pi î ,
i = 1, 2, 3 ,
(III.32)
∂
where ∂i := ∂P
.
i
Without loss of generality, the following results are proven for the standard
sequence (σj )∞
j=0 of infrared cutoffs. By simple arguments (see [11]), limiting
behavior as σ → 0 is shown to be ”sequence-independent”.
By analytic perturbation theory we have that
σ
∂i2 E|P~j| |P~ =Pi î =
= 1 − 2h
1
2πi
(III.33)
Z
σj
γj H ~
P
1
1
σ
σ
[P i − (β σj )i ] σj
dzj ΨP~j , [P i − (β σj )i ]ΨP~j i|P~ =Pi î ,
− zj
HP~ − zj
σ
σ
where ΨP~j is the normalized ground state eigenvector of HP~j .
~ σj )
Next, we make use of the Bogoliubov transformation implemented by Wσj (∇E
~
P
to show that
1 Z
1
1
σ
σ
(III.34)
[P i − (β σj )i ] σj
dzj ΨP~j , [P i − (β σj )i ]ΨP~j i
h
σj
2πi γj HP~ − zj
HP~ − zj
1
1 Z
1
σ
~ σj )(β σj )i Wσ∗ (∇E
~ σj )] σ 1
=
h
[P i − Wσj (∇E
dzj ΦP~j ,
σj 2
σj
j
~
~
j
P
P
2πi
kΦP~ k
KP~ − zj
γj K ~ − zj
P
σ
σ
σ
j
j
j
i
σ
i
∗
~
~
)(β j ) Wσ (∇E
, [P − Wσ (∇E
)]Φ i ,
(III.35)
~
P
j
σ
ΦP~j
~
P
j
~
P
σ
KP~j
(iteratively constructed in
is the ground state eigenvector of
where
Section II).
Recalling the definitions
~ σj := Wσ (∇E
~ σj )β~ σj Wσ∗ (∇E
~ σj ) − hWσ (∇E
~ σj )β~ σj Wσ∗ (∇E
~ σj )iΩ ,
Π
j
j
~
~
~
~
~
f
j
j
P
P
P
P
P
(III.36)
~Γσj := Π
~ σj − hΠ
~ σj i σj ,
(III.37)
~
~
~ Φ
P
P
P
~
P
and because of the identity (Feynman-Hellman, see (II.36))
~ σj
hβ~ σj iψσj = P~ − ∇E
~
P
(III.38)
~
P
σ
σ
σ
~ j i σj + hWσ (∇E
~ j )β~ σj Wσ∗ (∇E
~ j )iΩ ,
= hΠ
j
~ Φ
~
~
f
j
P
P
P
~
P
(III.39)
FP-Ren. Elect. Mass., 27-June-2008
26
we find that
~ σj )(β σj )i Wσ∗ (∇E
~ σj ) = −(Γσj )i + ∂i E σj ;
P i − Wσj (∇E
~
~
~
~
j
P
P
P
P
(III.40)
hence,
σ
∂i2 E|P~j| |P~ =P i î =
= 1−2
(III.41)
1
1
σj 2 h
kΦP~ k 2πi
Z
σj
γj K ~
P
1
1
σ
σ
σ
[∂i EP~j − (ΓP~j )i ] σj
dzj ΦP~j ,
− zj
KP~ − zj
σ
σ
σ
, [∂i EP~j − (ΓP~j )i ]ΦP~j i|P~ =P i î .
(III.42)
Using the eigenvalue equation
σ
σ
σ
σ
KP~j ΦP~j = EP~j ΦP~j ,
σ
the terms proportional to (∂i EP~j )2 and to the mixed terms – i.e., proportional
σ
σ
to the product of ∂i EP~j and (ΓP~j )i – are seen to be identically 0, because the
contour integral vanishes for each i = 1, 2, 3; e.g.,
Z
h
γj
σ
KP~j
1
1
σ
σ
σ
σ
[∂i EP~j ] σj
ΦP~j , [∂i EP~j ]ΦP~j idz̄j = (III.43)
− zj
KP~ − zj
σ
=
Z
γj
σ
hΦP~j
,
σ
ΦP~j i
∂i EP~j 2
dz̄j = 0 .
σ
EP~j − z¯j
It follows that
σ
∂i2 E|P~j| |P~ =P i î =
= 1+
(III.44)
1 Z
1
1
σ
dz̄j h σ
(ΓP~j )i σ
πi γj
KP~ − zj
KP~ −
σ
ΦP~j
σ ,
zj kΦP~j k
1 Z
1
1
σ
dz̄j σj
h (ΓP~j )i σj
= 1+
πi γj
EP~ − z¯j
KP~ −
σj
σj i ΦP
~
(ΓP~ )
σ i| ~
kΦP~j k P =Pi î
σ
σj
ΦP~j
σj i Φ P
~
(ΓP~ )
,
σ
σ i| ~
zj
kΦP~j k kΦP~j k P =Pi î
(III.45)
. (III.46)
We are now ready for the key estimate.
Lemma III.1. For α and small enough, the estimate below holds true:
σ
Z
γj−1
D
σ
(ΓP~j−1 )i
σ
j−1
E
ΦP~j−1
1
1
σj−1 i ΦP
~
,
(Γ
)
dz̄j−1
σj−1
σj−1
σ
σ
j−1
j−1
~
kΦP~ k KP~ − z̄j−1 P
kΦP~ k EP~ − z̄j−1
σ
−
Z
γj
D
σ
(ΓP~j )i
σ
j
E
ΦP~j
1
1
σj i Φ P
~
(ΓP~ )
dz̄j ≤ j(1−2δ) .(III.47)
σj ,
σj
σj
σj
kΦP~ k KP~ − z̄j
kΦP~ k EP~ − z¯j
FP-Ren. Elect. Mass., 27-June-2008
27
Proof.
~ σj−1 )Wσ∗ (∇E
~ σj ),
By unitarity of Wσj (∇E
~
~
j
P
P
σ
Z
D
γj
σ
(ΓP~j )i
σ
j
E
ΦP~j
1
1
σj i Φ P
~
(ΓP~ )
dz̄j =
σj ,
σj
σj
σj
kΦP~ k KP~ − z̄j
kΦP~ k EP~ − z̄j
σ
=
Z
γj
D
σ
(Γ̂P~j )i
(III.48)
σ
Φ̂P~j
j
E
1
1
σj i Φ̂P
~
,
(
Γ̂
)
dz̄j . (III.49)
σ
j
~
σj
σj
σ
j
kΦ̂P~ k K̂P~ − z̄j P kΦ̂P~ k EP~ − z̄j
By assumption, α is so small that the Neumann series expansions of the
resolvents below converge:
σ
K̂P~j
1
1
σj−1
= σj−1
+ Σ∞
, z̄j ) ,
1 (KP
~
KP~ − z̄j
− z̄j
(III.50)
1
1
σ
= σj−1
+ ∆(EP~j−1 , z¯j ) ,
− z̄j
EP~ − z¯j
(III.51)
σ
EP~j
where:
σ
j−1
Σ∞
, z̄j ) :=
1 (KP
~
=
∞
X
1
σj−1
~
l=1 KP
(III.52)
σ
− z̄j
1
σ
[−(∆KP~ |σσj−1
+ ÊP~ j − EP~ j−1 )
j
σ
KP~j−1
− z̄j
]l ,
σ
is defined in Eq. (II.66);
and ∆KP~ |σj−1
j
σ
∆(EP~j−1 , z̄j ) :=
σ
EP~j
1
1
σ
σ
(EP~j−1 − EP~j ) σj−1
.
− z̄j
EP~ − z̄j
(III.53)
We proceed by using the obvious identity:
σ
Z
D
σ
(Γ̂P~j )i
D
σ
(Γ̂P~j )i
γj
j
E
1
1
σj i Φ̂P
~
,
(
Γ̂
)
dz̄j
σ
j
~
σj
σj
σ
j
kΦ̂P~ k K̂P~ − z̄j P kΦ̂P~ k EP~ − z̄j
σ
=
Z
σ
Φ̂P~j
Φ̂P~j
1
σ
(Γ̂P~j )i
σ
Φ̂P~j E
1
(III.54)
, σj−1
dz̄j
(III.55)
σj−1
σ
σ
γj
kΦ̂P~j k KP~ − z̄j
kΦ̂P~j k EP~ − z̄j
σj
σj
Z D
E
1
σj−1
σj i Φ̂P
σj i Φ̂P
~
~
∞
,
Σ
(K
,
z̄
)(
Γ̂
)
+
(Γ̂P~ )
dz̄j (III.56)
j
σj−1
1
~
~
σj
σj
P
P
γj
kΦ̂P~ k
kΦ̂P~ k EP~ − z̄j
σj
σj
Z D
E
1
σj i Φ̂P
σj i Φ̂P
σ
~
~
+
(Γ̂P~ )
,
(
Γ̂
)
∆(EP~j−1 , z̄j )dz̄j (III.57)
~
σj
σj
σj
P
γj
kΦ̂P~ k K̂P~ − z̄j
kΦ̂P~ k
FP-Ren. Elect. Mass., 27-June-2008
28
Each of the expressions (III.55) and (III.56) can be rewritten by adding and
σ
subtracting
Φ j−1
σ
(ΓP~j−1 )i σP~j−1 .
k
kΦ
For (III.55) we get
~
P
(III.55) =
=
σj−1
σj−1
E
1
1
σj−1 i ΦP
σj−1 i ΦP
~
~
(ΓP~ )
(ΓP~ )
dz̄j (III.58)
σj−1 ,
σj−1
σj−1
σj−1
kΦP~ k KP~ − z̄j
kΦP~ k EP~ − z̄j
γj
σj−1
σj
Z D
σj−1 i ΦP
σj i Φ̂P
~
~
)
(III.59)
+
(Γ̂P~ )
σj−1 ,
~
σj − (ΓP
kΦ
k
γj
kΦ̂P~ k
~
P
σj
σj−1
E
1
1
σj i Φ̂P
σj−1 i ΦP
~
~
, σj−1
[(Γ̂P~ )
−
(Γ
]
dz̄j
)
σj−1
σj−1
~
σj
P
KP~ − z̄j
kΦP~ k EP~ − z̄j
kΦ̂P~ k
σj−1
σj
Z D
σj−1 i ΦP
σj i Φ̂P
~
~
)
(III.60)
+
(Γ̂P~ )
σj−1 ,
~
σj − (ΓP
kΦ
k
γj
kΦ̂P~ k
~
P
σj−1
E
Φ
1
1
σ
~
, σj−1
(ΓP~j−1 )i σPj−1
dz̄j
σj−1
KP~ − z̄j
kΦP~ k EP~ − z̄j
σj−1
Z D
σj−1 i ΦP
~
(III.61)
+
(ΓP~ )
σj−1 ,
kΦP~ k
γj
σj
σj−1
E
1
1
σj i Φ̂P
σj−1 i ΦP
~
~
, σj−1
[(Γ̂P~ )
)
dz̄j .
σj−1 ]
σj−1
~
σj − (ΓP
KP~ − z̄j
kΦP~ k EP~ − z̄j
kΦ̂P~ k
Z
D
The difference in Eq. (III.47) corresponds to the sum of the terms (III.56)(III.57) and of the terms (III.59)-(III.61). In fact, (III.58) corresponds to the
first term in (III.47) after a contour deformation from γj−1 to γj .
The sum of the remainder terms (III.56), (III.57), and (III.59)-(III.61) can
be bounded by j(1−2δ) , for R0 and α small enough but independent of j, for
any P~ ∈ S. (We recall that R0 can be taken arbitrarily small, provided α is
small enough). The details are as follows.
FP-Ren. Elect. Mass., 27-June-2008
29
• For (III.59)-(III.61) use the following inequalities
1
1
σ
KP~j−1 − z̄j
σj
[(Γ̂ ~ )i
P
1
σj−1
KP~
σ
σ
i
≤ O(
R02
1
α 4 (j−1)δ
),
(III.62)
1
(III.63)
(III.64)
1
σ
KP~j−1 − z̄j
P
σ
ΦP~j−1 − (ΓP~j−1 )i Φ̂P~j ≤ O(α 4 j(1−δ) ) ,
1
≤ O( j+1 ) ,
− z̄j Fσj
1
σj
Φ̂ ~
σ
(ΓP~j−1 )i
σ
(ΓP~j−1 )i
σ
σ
(Φ̂P~j
σ
ΦP~j−1 )
−
α 4 j(1−δ)
) (,III.65)
≤ O(
j+1
1
− ΦP~j−1 ≤ α 4 j(1−δ) .
(III.66)
In order to derive the inequality in Eq. (III.63), one uses Eqs. (II.63),
(II.73), and (II.57).
σ
σ
• For (III.56), after adding and subtracting (ΓP~j−1 )i
to use that
Φ ~j−1
P
σ
kΦ ~j−1 k
P
, one also has
1
σ
σ
k[−(∆KP~ |σσj−1
+ÊP~ j −EP~ j−1 )]
j
1
σ
KP~j−1 − zj
σ
σ
(ΓP~j−1 )i ΦP~j−1 k
1
2
≤ O(α j−1
R04
1
α 4 (j−1)δ
(III.67)
);
• To bound (III.57), note that
σ
(III.57) = −2πi
D
σ
(Γ̂P~j )i
σ
Φ̂P~j
, σ
σ
kΦ̂P~j k K̂P~j
+2πi
σ
(Γ̂P~j )i
Φ̂P~j
, σ
σ
kΦ̂P~j k K̂P~j
σ
= −2πi
σ
D
σ
(Γ̂P~j )i
Φ̂P~j
σ
kΦ̂P~j k
(III.68)
σ
σ
D
j
E
1
σj i Φ̂P
~
(
Γ̂
)
~
σ
σ
− EP~j P kΦ̂P~j k
j
E
1
σj i Φ̂P
~
~ )
σj−1 (Γ̂P
σj
− EP~
kΦ̂P~ k
σ
,
σ
σ
(EP~j−1 − EP~j )
σ
σ
K̂P~j − EP~j
(III.69)
σ
K̂P~j
j
E
1
σj i Φ̂P
~
~ )
σj−1 (Γ̂P
σj
− EP~
kΦ̂P~ k
σ
where |EP~j−1 − EP~j | ≤ O(α j−1 ). Then use the following inequality
analogous to (III.62)
1
σ
K̂P~j
1
R02
σj i σj (
Γ̂
Φ̂
)
≤
O(
).
1
~
~
σ
P
P
− EP~j
α 4 jδ
(III.71)
(III.70)
FP-Ren. Elect. Mass., 27-June-2008
30
Theorem III.2. For α small enough,
∂ 2 E σ~
|P |
~ |)2
(∂|P
converges, as σ → 0. The
(∂ 2 E σ~
|P |
~ |)2
(∂|P
, is Hölder-continuous in P~ ∈ S (for
limiting function, Σ|P~ | := limσ→0
an exponent η > 0). The bounds from above and below
2 > Σ|P~ | > 0
(III.72)
hold true uniformly in P~ ∈ S.
Proof
It is enough to prove the result for a fixed choice of a sequence {σj }∞
j=0 . The
σ
estimate in Lemma III.1 (with δ < 1/2) implies the existence of limj→∞ ∂i2 E|P~j| |P~ =Pi î .
We now observe that ∂i2 E|σP~0| |P~ =Pi î = 1 (see Eq. (III.46)), because
(ΓσP~0 )i ≡ (P~ f )i
,
ΦσP~0 ≡ Ωf .
(III.73)
For α small enough, we can take so small that Lemma III.1, combined with
(III.73), yields
σ
σ
j
ΦP~j
1 Z
1
1
σj i
σj i Φ P
~
dz̄j σj
h σj , (ΓP~ ) σj
(ΓP~ )
~ =Pi î < 1 ,
σj i|P
πi γj
EP~ − z̄j kΦP~ k
KP~ − z̄j
kΦP~ k
(III.74)
uniformly in j ∈ N. Hence the bound (III.72) follows.
The Hölder-continuity in P~ of Σ|P~ | is a trivial consequence of the analyticity
in P~ ∈ S of E σ , for any σ > 0, and of Lemma III.1; see [11] for similar
~|
|P
results.
Corollary III.3. For α small enough, the function EP~ := limσ→0 EPσ~ , P~ ∈ S,
is twice differentiable, and
∂ 2 E|P~ |
= Σ|P~ | .
(∂|P~ |)2
(III.75)
Proof
∂E σ
The result follows from the Hölder-continuity of Σ|P~ | , of limσ→0 ∂|P~P~| , and
from the fundamental theorem of calculus applied to the functions EP~ and
limσ→0
•
∂E σ~
P
~| ,
∂|P
because
∂EPσ~
∂|P~ |
and
∂ 2 EPσ~
(∂|P~ |)2
converge pointwise, for P~ ∈ S, as σ → 0,
(III.76)
FP-Ren. Elect. Mass., 27-June-2008
•
∂E σ ~
P
∂|P~ |
31
and
∂ 2 EPσ~ (∂|P~ |)2
(III.77)
are uniformly bounded in σ, for all P~ ∈ S.
Acknowledgements
The authors thank Thomas Chen for very useful discussions.
References
[1] V. Bach,T. Chen, J. Fröhlich, and I. M. Sigal. The Renormalized Electron Mass in Non-Relativistic Quantum Electrodynamics.
http://arxiv.org/abs/math-ph/0507043
[2] T. Chen. Infrared Renormalization in Nonrelativistic QED and
Scaling Criticality. http://arxiv.org/abs/math-ph/0601010
[3] T. Chen, J. Fröhlich. Coherent Infrared Representations in Nonrelativistic QED, To appear in Proc. Symp. Pure Math. (B. Simon
60-th Birthday Volume). http://arxiv.org/abs/math-ph/0601009
[4] T. Chen, J. Fröhlich, and A. Pizzo. Infraparticle Scattering
States in Non-Relativistic QED: I. The Bloch-Nordsieck paradigm
http://arxiv.org/abs/math-ph/07092493
[5] T. Chen, J. Fröhlich, and A. Pizzo. Infraparticle Scattering States in Non-Relativistic QED: II. Mass Shell Properties.
http://arxiv.org/abs/math-ph/07092812
[6] M. Fierz, W. Pauli. Nuovo Cimento, 15 167 (1938)
[7] F. Hiroshima, H. Spohn. Mass renormalization in non-relativistic
quantum electrodynamics. J. Math. Phys.. 46 (2005) (4)
[8] C. Hainzl, R. Seiringer. Mass renormalization and energy level
shift in non-relativistic QED. Adv. Theor. Math. Phys. (6) (2003)
(5), 847-871.
[9] E.T. Lieb, M. Loss. Self energy of electrons in non-perturbative
QED. Conference Moshe Flato 1999, vol I, Dijon, Math. Phys.
Stud. vol. 21, Kluwer Acad. Publ., Dordrecht (2000), pp. 327-344.
FP-Ren. Elect. Mass., 27-June-2008
32
[10] E.T. Lieb, M. Loss. A bound on binding energies and mass renormalization in model of quantum electrodynamics. J. Statist. Phys.
108 (2002)
[11] A. Pizzo. One-particle (improper) states in Nelson’s massless
model. Ann. H. Poincaré, 4 (3), 439–486, 2003.
[12] A. Pizzo. Scattering of an Infraparticle: The One-particle (improper) Sector in Nelson’s massless model. Ann. H. Poincaré, 4
(3), 439–486, 2003.
[13] Reed Simon. Methods of modern mathematical physics. Vol. I-IIIII-IV Academic Press.