EFFECTIVE DYNAMICS OF AN ELECTRON COUPLED TO AN EXTERNAL
POTENTIAL IN NON-RELATIVISTIC QED
VOLKER BACH, THOMAS CHEN, JÉRÉMY FAUPIN, JÜRG FRÖHLICH, AND ISRAEL MICHAEL SIGAL
Abstract. In the framework of non-relativistic QED, we establish the relationship between the
renormalized mass of the electron due to its interaction with the quantized electromagnetic field,
and the kinematic mass appearing in its response to a slowly varying external force. Specifically,
we study the dynamics of an electron in a slowly varying external potential and with slowly varying
initial conditions and prove that, for a long time, it is accurately described by the effective dynamics
of a Schrödinger electron in the same external potential and for the same initial data, with a kinetic
energy operator determined by the renormalized dispersion law of the translation-invariant QED
model.
1. Introduction
In this paper we establish the relationship between the renormalized mass of the electron due to
its interaction with the quantized electromagnetic field, and the kinematic mass appearing in its
response to a slowly varying external force. We work in the standard framework of non-relativistic
quantum electrodynamics (QED). The renormalized electron mass is defined as the inverse curvature of the dispersion relation E = E(p), the energy of a dressed electron as a function of its
momentum p (no external potentials present), while the kinematic mass is defined in terms of the
effective dynamics of the electron under the influence of an external force.
Our starting point is the dynamics generated by the Hamiltonian H V describing a non-relativistic
electron interacting with the quantized electromagnetic field and under the influence of a slowly
varying potential V . We consider the time evolution of states parametrized by wave functions
u0 ∈ H 1 (R3 ), with ku0 kL2 = 1 and k∇u0 kL2 ≤ , and prove that this evolution is accurately
approximated, on a long time interval, by an effective Schrödinger dynamics generated by the
one-particle Schrödinger operator
Heff := E(−i∇x ) + V (x) ,
(1.1)
with the kinetic energy given by the dispersion relation E(p). This result is in line with the idea
that every physical dynamics is an effective one, derived from a more precise theory. While results
of a similar nature were proven for massive bosons, [26], ours is the first result involving massless
photons and the resulting renormalized mass of the electrons. (Similarly, one may analyze the
effects of interactions with phonons and other massless bosons.)
In the framework of non-relativistic QED, the Hilbert space of states of the system is given by
H := L2 (R3 ) ⊗ F ,
(1.2)
where L2 (R3 ) denotes the Hilbert space associated with the electron degrees of freedom, neglecting
spin (for notational convenience). The space F is the Fock space of photons in the Coulomb gauge,
1
2
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
L
F = n≥0 Fn . Here, Fn := Sym( L2 (R3 × {+, −} ) )⊗n denotes the physical Hilbert space of states
of n photons. The Hamiltonian is given on this space by the expression
H V = H + V ⊗ 1f ,
(1.3)
where H is the generator of the dynamics of a single, freely moving non-relativistic electron minimally coupled to the quantized electromagnetic field, i.e.,
H :=
√
1
( −i∇x ⊗ 1f + αA(x) )2 + 1el ⊗ Hf ,
2
(1.4)
and where V (x) := V (x) is a slowly varying potential, with > 0 small; its precise properties are
formulated in Theorem 1.1 below. Furthermore,
XZ
dk
A(x) :=
{ λ (k) e−ikx ⊗ aλ (k) + h.c. }
(1.5)
1/2
|k|
|k|≤1
λ
denotes the quantized electromagnetic vector potential with an ultraviolet cutoff imposed, |k| ≤ 1,
and
XZ
(1.6)
Hf =
dk |k| a∗λ (k) aλ (k)
λ
denotes the photon Hamiltonian. In (1.5) and (1.6), a∗λ (k), aλ (k) are the usual photon creationand annihilation operators; λ = ± indicates photon helicity, and λ (k) is a polarization vector
perpendicular to k corresponding to helicity λ.
To give a precise formulation of our result, we observe that the Hamiltonian H is translation
invariant. We thus represent the Hilbert space of the system as a direct integral,
Z ⊕
H =
dp Hp ,
(1.7)
over the spectrum of the total momentum operator
Ptot := −i∇x ⊗ 1f + 1el ⊗ Pf ,
(1.8)
where Pf = λ dk k a∗λ (k)aλ (k) is the momentum operator associated to the quantized radiation
field, and each fiber Hilbert space Hp is isomorphic to F. We let H(p) = H|Hp denote the fiber
Hamiltonian corresponding to total momentum p. We define E(p) = inf specH(p), with p ∈ S,
where
1
S := p ∈ R3 |p| ≤
(1.9)
3
P R
and continued suitably to p ∈ R3 \ S. For ρ > 0, we introduce the family of maps J0ρ : L2 (R3 ) 7→ H,
from the quantum mechanical one-electron state space L2 (R3 ) to a subset of dressed one-electron
states, as
Z
J0ρ ( u ) :=
dp u
b(p) e−ix(p−Pf ) χSµ (p) Φρ (p) ,
(1.10)
S
where x is the electron position, χSµ is a smooth approximate characteristic function of the set
Sµ := (1 − µ)S ⊂ S ⊂ R3 , (0 < µ < 1), and Φρ (p) is an approximate ground state of H(p) (dressed
by a cloud of soft photons with frequencies ≤ ρ).
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
3
In this paper we study the time evolution of one-electron states, J0ρ (u0 ), with electron wave
functions u0 , dressed by an infrared cloud of photons with frequencies ≤ ρ. More precisely, we
study solutions of the Schrödinger equation
i∂t Ψ(t) = H V Ψ(t) ,
with Ψ(0) = J0ρ (u0 ) .
(1.11)
V
The key idea is to relate the solution Ψ(t) = e−itH J0ρ (u0 ) of this Schrödinger equation to the
solution of the Schrödinger equation
i∂t u(t) = Heff u(t) ,
with
u(t = 0) = u0 ,
(1.12)
corresponding to the one-particle Schrödinger operator (1.1), where E(p) has been defined above.
We consider the comparison state
J0ρ (u(t)) ∈ H ,
(1.13)
where u(t) := e−itHeff u0 is the solution of (1.12), and show that Ψ(t) remains close to J0ρ (u(t)),
for a long time. The choice of initial data satisfying k∇u0 kL2 ≤ guarantees that u
b(t) remains
concentrated in S during the time scales relevant in this problem.
Theorem 1.1. Let 0 < µ < 1 and 0 < < µ/3, and define J0ρ by (1.10). Assume that u0 ∈ L2 (R3 )
is a normalized vector obeying k∇u0 kL2 (R3 ) ≤ . Furthermore, assume that V ∈ L∞ (R3 ; R) is such
that Vb ∈ L1 (R3 ) and that Vb is supported in the unit ball,
supp(Vb ) ⊂ k ∈ R3 | |k| ≤ 1 .
(1.14)
2
Let 0 < δ < 2/3 and choose ρ = 3 −δ =: ρ . There exists αδ > 0 such that, for all 0 ≤ α ≤ αδ , the
bound
V
k e−itH J0ρ ( u0 ) − J0ρ ( e−itHeff u0 ) kH ≤ Cδ 2/3−δ t + 4/3−δ/2 t2 ,
(1.15)
holds for all times t.
1.1. Outline of proof strategy. To prove Theorem 1.1, we introduce an infrared regularized
version of the model defined by (1.3), (1.4), obtained by restricting the integration domain in the
quantized electromagnetic vector potential (1.5) to the region {σ ≤ |k| ≤ 1}, for an arbitrary
infrared cutoff σ > 0. Thereby, we obtain infrared regularized Hamiltonians HσV and Hσ , as well
as an infrared regularized family of maps Jσρ corresponding to J0ρ .
We note that, unlike H(p), the infrared cut-off fiber Hamiltonian Hσ (p) has a ground state
Ψσ (p) ∈ Hp ∼
= F, for every p ∈ S and for σ > 0, but Ψσ (p) does not possess a limit in Hp ∼
= F, as
σ & 0. In particular, we expect that the number of photons in the state Ψσ (p) diverges, as σ & 0,
(thus the lack of convergence of Ψσ (p) in F). This is a well-known aspect of the infrared problem in
ρ
QED, [8, 9, 10, 11, 22]. It is remedied by applying a dressing transformation, W∇E
, defined in
σ (p)
ρ
ρ
(2.9), below, to Ψσ (p), where Eσ (p) = inf specHσ (p). The resulting vector, Φσ (p) := W∇Eσ (p) Ψσ (p),
describes an infraparticle (or dressed electron) state containing infrared photons with frequencies
in [σ, ρ]. As σ & 0, the limit
Φρ (p) = lim Φρσ (p)
σ→0
(1.16)
exists in F, see Proposition 2.2. This allows us to construct the map J0ρ as the limit of the maps
Jσρ , as σ & 0.
4
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
We note that Φρσ (p) is the ground state eigenvector of the fiber Hamiltonian
ρ
ρ
Kσρ (p) := W∇E
Hσ (p) (W∇E
)∗
σ (p)
σ (p)
(1.17)
which is obtained by applying to Hσ (p) the Bogoliubov transformation corresponding to the dressing
ρ
transformation W∇E
.
σ (p)
In Theorem 2.3, below, we prove that an estimate similar to (1.15) holds for the infrared regularized model,
V
(1.18)
k e−itHσ Jσρ ( u0 ) − Jσρ ( e−itHeff,σ u0 ) kH ≤ Cδ 2/3−δ t + 4/3−δ/2 t2 ,
2
uniformly in σ ≥ 0, for ρ = 3 −δ . This result crucially uses the regularity properties of the dressed
electron states Φρσ (p), which allow us to take advantage of the fact that V is slowly varying. An
1
1
additional key ingredient is the bound k(Hσ (p) − Kσρ (p))Φρσ (p)kF ≤ Cα 2 ρ 2 |p|, for p ∈ S, proven
in Appendix C.
In section 3, we control the limit σ & 0, thus concluding the proof of Theorem 1.1. This requires
control of the radiation emitted by the electron due to its acceleration in the external potential V ,
in the limit σ & 0.
Remark 1.2. Theorem 1.1 implies that, for all δ 0 such that δ < δ 0 < 2/3,
0
V
k e−itH J0ρ ( u0 ) − J0ρ ( e−itHeff u0 ) kH ≤ Cδ δ −δ
− 23 +δ 0
(1.19)
− 23 +δ 0
holds for all times t with 0 ≤ t ≤ . We note that the time scale of order O(
) is
not sharp; we leave further investigation of an optimal time scale to future work. Indeed, for
a regularized model based on massive photons (see [26]), or nonzero infrared cutoff σ > 0, the
dynamics can be controlled up to a time scale O(−1 ). The main obstacle against passing beyond a
2
0
time scale of order O(− 3 +δ ) within our current approach comes from the θ-Hölder continuity of
ρ
Φσ (p) in p, only established for θ < 2/3.
2. Infrared cut-off and construction of Φρ (p)
As noted in the introduction, we analyze the original dynamics by first imposing an infrared
cut-off, and controlling the dynamics generated by the resulting Hamiltonian. Thus, we define the
IR regularized Hamiltonian
HσV = Hσ + V (x) ⊗ 1f ,
(2.1)
where
√
1
( −i∇x ⊗ 1f + αAσ (x) )2 + 1el ⊗ Hf
(2.2)
2
is the generator of the dynamics of a single, freely moving non-relativistic electron minimally coupled
to the electromagnetic radiation field. In (2.2),
XZ
dk
Aσ (x) =
{ λ (k) e−ikx ⊗ aλ (k) + h.c. }
(2.3)
1/2
|k|
σ≤|k|≤1
λ
Hσ :=
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
5
denotes the quantized electromagnetic vector potential with an infrared and ultraviolet cutoff corresponding to σ ≤ |k| ≤ 1. Since V ∈ L∞ (R3 ) is a bounded operator, D(HσV ) = D(Hσ ) =
D(−∆x ⊗ 1f + 1el ⊗ Hf ).
The Hamiltonian Hσ is translation
invariant. Representing the Hilbert space of the system
R⊕
as a direct integral, H =
dp Hp , over the spectrum of the total momentum operator, Ptot
(see (1.7) and (1.8)), we let Hσ (p) = Hσ |Hp denote the fiber Hamiltonian corresponding to total
momentum p. While H(p) has a ground state only for p = 0, it is proven in [2, 6] that, for
p ∈ S := {p ∈ R3 ||p| ≤ 1/3} and σ > 0,
Eσ (p) = inf specHσ (p)
(2.4)
is a non-degenerate eigenvalue (the fiber ground state energy) of Hσ (p). This motivates the introduction of the cut-off. We let Ψσ (p) ∈ F, with kΨσ (p)kF = 1, denote the corresponding normalized
fiber ground state,
Hσ (p)Ψσ (p) = Eσ (p) Ψσ (p) ,
(2.5)
for p ∈ S. Properties of the fiber ground state energy Eσ (p), and of dressed electron states Φσ (p),
for p ∈ S, are given in the following proposition proven in [2, 6, 9, 10]:
Proposition 2.1. The infimum of the spectrum of the fiber Hamiltonian, Eσ (p) = inf spec(Hσ (p)),
satisfies:
(1) For any σ > 0, Eσ ∈ C 2 (S), and for all p ∈ S = p ∈ R3 | |p| ≤ 13 , Eσ (p) is a simple
eigenvalue.
(2) For α > 0 sufficiently small, there exists a constant c, with 0 < c < ∞, such that, for any
p ∈ S,
|∇p Eσ (p) − p| ≤ c α |p| ,
and
2
1 − c α ≤ ∂|p|
Eσ (p) ≤ 1 ,
(2.6)
hold uniformly in α (0 < α 1) and σ ≥ 0.
(3) The following limit exists in C 2 (S)
lim Eσ ( · ) = E( · ).
σ&0
(2.7)
Let b∗λ (k), bλ (k) denote creation- and annihilation operators on the fiber space, see Appendix A.
For 0 < σ < ρ ≤ 1 and p ∈ S, we introduce the Weyl operators
h 1 XZ
∇Eσ (p) · λ (k)bλ (k) − h.c. i
ρ
W∇Eσ (p) := exp α 2
dk
,
(2.8)
|k|1/2 (|k| − ∇Eσ (p) · k)
σ≤|k|≤ρ
λ
with ∇Eσ (p) ≡ ∇p Eσ (p), which are unitary on F, for σ > 0. Moreover, we define dressed electron
states
ρ
Φρσ (p) := W∇E
Ψσ (p) .
σ (p)
(2.9)
The properties of these states are described in the following proposition
Proposition 2.2. For any p ∈ S, 0 < ρ ≤ 1, and for sufficiently small values of the finestructure
ρ
ρ
constant 0 < α 1, the ground state eigenvector Φρσ (p) of Kσρ (p) := W∇E
Hσ (p) (W∇E
)∗
σ (p)
σ (p)
satisfies:
6
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
(1) The strong limit
Φρ (p) := lim Φρσ (p)
(2.10)
σ→0
exists in F.
(2) For θ < 23 , the vectors Φρσ (p) are θ-Hölder continuous in p,
sup
p,q∈S
k Φρσ (p) − Φρσ (q) k
≤ C(θ) ,
|p − q|θ
(2.11)
uniformly in σ and ρ, with 0 ≤ σ < ρ ≤ 1.
The proof of θ-Hölder continuity for θ < 23 is given in Appendix B; (see also [9, 10, 22] for earlier
results covering the range θ < 14 , in the case where ρ = 1).
For arbitrary u ∈ L2 (R3 ) (with Fourier transform denoted by u
b), we define the linear map
Z
Jσρ : u 7→
dp u
b(p) e−ix(p−Pf ) χSµ (p) Φρσ (p) ,
(2.12)
S
where x is the electron position, χSµ is a smooth approximate characteristic function of the set
Sµ := (1 − µ) S ⊂ S ⊂ R3 ,
Jσρ
(2.13)
L2 (R3 )
and 0 < µ < 1. Note that
:
→ M ⊂ H, where
nZ
o
M :=
dp u
b(p) e−ix(p−Pf ) χSµ (p) Φρσ (p) u ∈ L2 (R3 ) ,
(2.14)
R3
the space of vectors in H supported on the one-particle shell of the operator
R⊕
S
dp Kσρ (p).
We also note that in (2.14) we do not require that supp(b
u) ⊂ Sµ ; instead, we cutoff u
b outside
the region Sµ by multiplying it by χSµ .
Furthermore, we consider an initial wave function u0 (x) satisfying ku0 kL2 = 1 and k∇u0 kL2 ≤ .
Our main goal in this paper is to study the solution of the Schrödinger equation
i∂t Ψ(t) = HσV Ψ(t) ,
with Ψ(0) = Jσρ (u0 ) ,
(2.15)
which is given by
V
Ψ(t) = e−itHσ Jσρ (u0 ) ∈ H ,
(2.16)
and, in particular, to determine its properties for large t.
The key idea is to relate Ψ(t) to the solution of the Schrödinger equation
i∂t u(t, x) = (Heff,σ u)(t, x)
,
u(0, x) = u0 (x)
(2.17)
corresponding to the one-particle Schrödinger operator
Heff,σ := Eeff,σ (−i∇x ) + V (x) ,
(2.18)
where (t, x) ∈ R × R3 . The kinetic energy operator Eeff,σ (p) ∈ C 2 (R3 ) is defined by
Eeff,σ (p) := Eσ (p) , if p ∈ S ,
(2.19)
and suitably continued to R3 \ S. We consider the comparison state
Jσρ (u(t)) ∈ H ,
(2.20)
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
7
where u(t) := e−itHeff,σ u0 is the solution of (2.17), and show that Ψ(t) remains close to Jσρ (u(t)),
for a long time. The choice of initial data satisfying k∇u0 kL2 ≤ guarantees that u
b(t) remains
concentrated in S during the time scales relevant in this problem.
As a first step in proving Theorem 1.1, we will prove the following result.
Theorem 2.3. Under the conditions of Theorem 1.1, there exists αδ > 0 such that, for all 0 ≤
α ≤ αδ , the bound
2
V
1
1
k e−itHσ Jσρ ( u0 ) − Jσρ ( e−itHeff,σ u0 ) kH ≤ Cδ (1 + ln(ρ−1 )) 3 −δ t + C α 2 ρ 2 t (1 + t) , (2.21)
holds uniformly in the infrared cutoff σ.
Proof. Our proof makes crucial use of the properties of the fiber ground state energy Eσ (p) and of
the corresponding dressed electron states Φρσ (p), for p ∈ S, given in Propositions 2.1 and 2.2 above.
We define the Bogoliubov-transformed fiber Hamiltonians
ρ
ρ
Kσρ (p) := W∇E
Hσ (p) (W∇E
)∗
σ (p)
σ (p)
for p ∈ S, and the operator Kσρ acting on H,
Z ⊕
ρ
Kσ :=
Ip−1 Kσρ (p)Ip dp,
(2.22)
(2.23)
and the perturbed operator KσV := Kσρ + V . Note that the dressed electron states Φρσ (p), for p ∈ S,
are the ground states of Kσρ (p), i.e.,
Kσρ (p) Φρσ (p) = Eσ (p) Φρσ (p) ,
and the operator
Kσρ
(2.24)
has the property that
Kσρ Jσρ = Jσρ Eeff,σ (−i∇).
(2.25)
Next, we proceed to the proof of (2.21). We write the difference on the LHS of (2.21) as the
integral of a derivative, and separate it into
Z t
V
−itHσV
ρ
ρ −itHeff,σ −itHσV
e
Jσ ( u0 ) − Jσ ( e
u0 ) = −i e
ds eisHσ (HσV Jσρ (u(s)) − Jσρ (Heff u(s))
0
1
2
=: φ (t) + φ (t) ,
HσV
HσV
KσV
(2.26)
KσV
inside the integral and grouping terms suitably. The first
+
by substituting
→
−
term on the r.h.s. accounts for the radiation of infrared photons due to the motion of the dressed
electron in the external potential, while the second term accounts for the influence of the external
potential V on the full QED dynamics (2.16), as compared to the effective Schrödinger evolution
(2.20).
The first term on the r.h.s. of (2.26) has the form
Z t
V
1
−itHσV
φ (t) := −i e
ds eisHσ Hσ − Kσρ Jσρ ( u0 ) ,
(2.27)
0
where we have used the cancelation of V in HσV − KσV = Hσ − Kσρ . Using the fiber integral
decomposition, we obtain
n1
oZ t
1
ρ
ρ
kφ (t)kH ≤ sup
k(Hσ − Kσ )(p) Φσ (p)kF
k ∇u(s) kL2 (R3 ) ds .
(2.28)
p∈S |p|
0
8
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
In Appendix C we prove the following key result:
n1
o
1
1
sup
k(Hσ − Kσρ )(p) Φρσ (p)kF ≤ Cα 2 ρ 2 ,
p∈S |p|
uniformly in σ ≥ 0. Furthermore, we have the estimate
Z t
k ∇u(s) kL2 (R3 ) ds ≤ C t (1 + t) ,
(2.29)
(2.30)
0
as shown below by using the condition k∇u0 kL2 (R3 ) ≤ on u0 , and the fact that the potential V
satisfies (1.14). We obtain
1
1
kφ1 (t)kH ≤ C t (1 + t) α 2 ρ 2 ,
(2.31)
which yields the second contribution to the r.h.s. of (2.21).
For the second term on the r.h.s. of (2.26), we have that
Z t
V
V
ds eisHσ KσV Jσρ (u(s)) − Jσρ (Heff u(s))
φ2 (t) := −i e−itHσ
0
Z t
V
V
= −i e−itHσ
ds eisHσ V Jσρ (u(s)) − Jσρ (V u(s)) ,
(2.32)
0
using the fiber decomposition and the equation Kσρ (p) Φρσ (p) = Eσ (p) Φρσ (p). Let kΦkC θ (S) :=
supp,q∈S
k Φ(p)−Φ(q) k
.
|p−q|θ
Below, we prove an estimate of the form
\
θ V k 1 3 (1 + kΦρ k θ
kφ2 (t)kH ≤ t C k|∇|
L (R )
σ C (S) ) ,
(2.33)
for θ < 32 . The key point here is that the θ-Hölder continuity of the fiber ground state Φρσ (p) enables
\
θ V k 1 3 ≤ Cθ . To summarize, we have
us to gain a θ derivative of the potential, yielding k|∇|
L (R )
made use of the θ-Hölder continuity of Φρσ ( · ), which holds uniformly in σ, with 0 < σ < ρ, and we
have used that
\
θV k 1 3 ≤ γ ,
(2.34)
k|∇|
where γ := kVb (k)kL1 < ∞ ,
L (R )
(see (1.14)).
Moreover, in Proposition B.4, we prove that kΦρσ kC θ (S) ≤ Cδ (1 + ln(ρ−1 )). Collecting the
estimates above, we arrive at
kφ2 (t)kH ≤ Cδ t θ (1 + ln(ρ−1 )),
which yields the first term on the RHS of (2.21).
Proof of (2.30). To verify (2.30), a simple calculation shows that
Z s
∇u(s) = e−isHeff,σ ∇u0 − i
dv e−ivHeff,σ ∇V (x) e−i(s−v)Heff,σ u0 .
(2.35)
(2.36)
0
Using that k∇u0 kL2 ≤ , and that
d kL1 ≤ γ ,
k∇V kL∞ ≤ k∇V
(2.37)
where γ is defined in (2.34), we conclude that
k ∇u(s) kL2
≤ C (1 + s) ,
(2.38)
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
9
and thus,
t
Z
ds k ∇u(s) kL2 ≤ C t (1 + t) .
(2.39)
0
This proves (2.30).
Proof of (2.33). We define
ψs := V Jσρ (u(s)) − Jσρ (V u(s)).
Moreover, we define the generalized Fourier transform in the electron position, x, by
Z
1
b
ψ(p)
=
dx e−i(p−Pf )x ψ(x) .
3
(2π) 2
(2.40)
(2.41)
By definitions (2.26) and (2.40) and the unitarity of the generalized Fourier transform proven in
Appendix A, we have that
Z t
Z t
2
cs kL2 ⊗F .
kφ (t)kH ≤
ds kψs kL2x ⊗F =
ds kψ
(2.42)
p
0
0
Jσρ
and computing the Fourier transform, we find that
Z
ψ̂s (p) =
dq Vb (p − q) u
b(s, q) χSµ (q)Φρσ (q) − χSµ (p)Φρσ (p) .
Using the definition of
(2.43)
R3
It is important to note that, for any function f ∈ L2 (R3 ) with supp(f ) ⊂ Sµ ,
supp(Vb ∗ f ) ⊂ S,
(2.44)
for ≤ µ/3, since we are assuming supp(Vb ) ⊂ {k||k| ≤ 1}, so that supp(Vb ) ⊂ {k||k| ≤ }. Since
the term in the integrand given by (b
u(s)χSµ Φρσ )(q) is supported in q ∈ Sµ , so that, by (2.44), its
convolution with Vb has support in S, we find
Z
ψ̂s (p) = 1S (p)
dq Vb (p − q) u
b(s, q) (χSµ (q) Φρσ (q) − χSµ (p)Φρσ (p)) ,
(2.45)
R3
for ≤ µ/3, where 1S is the characteristic function of the set S. Inserting |p − q|θ |p − q|−θ = 1 into
(2.45), using the definition of |∇|θ by its Fourier transform and using that, since χSµ is a smooth
function,
sup |p − q|−θ k χSµ (q)Φρσ (q) − χSµ (p)Φρσ (p) kF ≤ C(1 + kΦρσ kC θ (S) ) ,
(2.46)
p,q∈S
\
θV | k 2
we obtain the bound kψ̂s kL2x ⊗F ≤ C(1 + kΦρσ kC θ (S) )k |1S u
b(s)| ∗ ||∇|
L (S) . Next, using Young’s
inequality, kf ∗ gkLr ≤ kf kL1 kgkLr , we find that
\
θ V k 1 3 sup k1 u
kψ̂s kL2x ⊗F ≤ C(1 + kΦρσ kC θ (S) ) k|∇|
L (R )
S b(s)kL2 (R3 ) .
(2.47)
s∈[0,t]
Finally, observing that
k1S u
b(s)kL2 (R3 ) ≤ kb
u(s)kL2 (R3 ) = ku(s)kL2 (R3 ) = ku0 kL2 (R3 ) = 1 ,
by unitarity of e−itHeff,σ , and using (2.42), we arrive at (2.33).
(2.48)
10
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
3. The limit σ & 0
In this section we remove the infrared cut-off from the evolution.
Proposition 3.1. Under the conditions of Theorem 2.3, the strong limits
V
V
s − lim e−itHσ Jσρ ( u0 ) = e−itH J0ρ ( u0 )
(3.1)
s − lim Jσρ ( e−itHeff,σ u0 ) = J0ρ ( e−itHeff u0 )
(3.2)
σ&0
and
σ&0
exist, for arbitrary |t| < ∞.
Proof. We write
V
V
V
V
V
e−itHσ Jσρ (u0 ) − e−itH J0ρ (u0 ) = (e−itHσ − e−itH )J0ρ (u0 ) + e−itHσ (Jσρ − J0ρ )(u0 ).
(3.3)
Clearly,
V
e−itHσ (Jσρ − J0ρ )(u0 )
= k(Jσρ − J0ρ )(u0 )k ≤ ku0 kL2 sup kΦρσ (p) − Φρ (p)kF .
p∈Sµ
Thus,
V
lim e−itHσ (Jσρ − J0ρ )(u0 )
σ&0
= 0,
follows from Proposition B.1.
Next, we discuss the first term on the right side of (3.3). In order to prove that it converges to
0, as σ & 0, it suffices to show that HσV converges to H V in the norm resolvent sense; (see [24,
Theorem VIII.21]), i.e.,
lim (HσV + i)−1 − (H V + i)−1
σ&0
= 0.
From the second resolvent equation and the fact that k(HσV + i)−1 k ≤ 1, it follows that
(HσV + i)−1 − (H V + i)−1
= (H V + i)−1 Qσ (HσV + i)−1 ,
(3.4)
where
1
Qσ := H V − HσV = α 2 A<σ (x) · vσ +
α
(A<σ (x))2 ,
2
and
1
vσ := −i∇x + α 2 Aσ (x)
is the velocity operator. Here Aσ (x) is defined in (2.3), and
XZ
dk
A<σ (x) :=
{ λ (k) e−ikx ⊗ aλ (k) + h.c. } .
1/2
|k|
|k|≤σ
λ
In order to estimate the norm of Qσ (H V + i)−1 , we use the following well-known lemma.
(3.5)
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
11
Lemma 3.2. Let f, g ∈ L2 (R3 × {+, −}; B(Hel )) be operator-valued functions such that k(1 +
|k|−1 )1/2 f k, k(1 + |k|−1 )1/2 gk < ∞. Then
1
1
ka# (f )(Hf + 1)− 2 k ≤ k(1 + |k|−1 ) 2 f k,
(3.6)
1
2
1
2
ka# (f )a# (g)(Hf + 1)−1 k ≤ k(1 + |k|−1 ) f kk(1 + |k|−1 ) gk,
(3.7)
where a# stands for a or a∗ .
In particular, using the Kato-Rellich theorem, one easily shows that, for α small enough, D(H V ) =
D(−∆x ⊗ I + I ⊗ Hf ) ⊂ D(Hf ). Thus, we have that
(Hf + 1)(H V + i)−1
which when combined with Lemma 3.2 yields
α
(A<σ (x))2 (H V + i)−1
2
Likewise one verifies that
1
α 2 A<σ (x) · vσ (H V + i)−1
≤ C,
≤ C α σ.
1
(3.8)
1
≤ C α2 σ2 ,
(3.9)
since 0 ≤ vσ2 ≤ H V + kV kL∞ is bounded relative to H V . Estimates (3.8) and (3.9) yield
Qσ (H V + i)−1
1
1
≤ C α2 σ2 .
By (3.4), we have shown that HσV converges to H V , as σ & 0, in the norm resolvent sense.
4. Proof of Theorem 1.1
In this section, we prove the bound in Theorem 1.1, which compares the full dynamics to the
effective dynamics for the system without infrared cutoff. We have that
V
k e−itH J0ρ ( u0 ) − J0ρ ( e−itHeff u0 ) kH
V
≤ k e−itHσ Jσρ ( u0 ) − Jσρ ( e−itHeff,σ u0 ) kH
V
V
+ k e−itHσ Jσρ ( u0 ) − e−itH J0ρ ( u0 ) kH
+ k Jσρ ( e−itHeff,σ u0 ) − J0ρ ( e−itHeff u0 ) kH ,
(4.1)
for any t and 0 < σ < ρ ≤ 1. It follows from Theorem 2.3 that the first term on the r.s. of the
2
1
1
inequality sign is bounded by Cδ (1 + ln(ρ−1 )) 3 −δ t + C α 2 ρ 2 t (1 + t), uniformly in σ > 0.
From Proposition 3.1, it follows that the second and third term on the r.s. converge to zero, as
σ & 0.
We thus conclude that
V
2
1
1
k e−itH J0ρ ( u0 ) − J0ρ ( e−itHeff u0 ) kH ≤ Cδ (1 + ln(ρ−1 )) 3 −δ t + C α 2 ρ 2 t (1 + t) ,
by taking σ to zero. This concludes the proof of Theorem 1.1.
(4.2)
12
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
Appendix A. Generalized Fourier transform
For ψ ∈ H, we define the Fourier transform in the electron position, x, by
Z
1
b
ψ(p) =
dx e−i(p−Pf )x ψ(x) .
3
(2π) 2
(A.1)
Our claim is that
Z
1
∨
φ (x) =
3
dp ei(p−Pf )x φ(p)
(A.2)
(2π) 2
corresponds to the inverse Fourier transform, in the following sense.
Lemma A.1. The above linear operations define unitary maps H → H, and are mutual inverses.
Proof. By density, we may assume that ψ is smooth in x and of rapid decay, i.e., belongs to the
space
(A.3)
SF := ψ ∈ H sup xα ∂xβ ψ F < ∞ , ∀α, β ∈ N30 .
x
Then,
b ∨ (x) =
(ψ)
Z
1
3
2
dp e−i(p−Pf )x
Z
1
3
2
0
dx0 ei(p−Pf )x ψ(x0 )
(2π)
(2π)
Z
Z
1
0
0
=
dx0
dp e−ip(x−x ) eiPf (x−x ) ψ(x0 )
(2π)3
Z
0
=
dx0 δ(x − x0 ) eiPf (x−x ) ψ(x0 )
= ψ(x) .
(A.4)
Likewise, for φ ∈ SF ,
Z
1
1
Z
dx e
dq e−i(q−Pf )x φ(q)
3
3
(2π) 2
(2π) 2
Z
Z
1
=
dq
dx ei(p−q)x φ(q)
(2π)3
Z
=
dq δ(p − q) φ(q)
∨ b
(φ ) (p) =
i(p−Pf )x
= φ(p) .
(A.5)
From the density of SF in H, we infer that (A.1) and (A.2) define linear maps H → H, and are
mutual inverses. Moreover, the identity
Z
Z
Z
D 1 Z
E
1
2
−i(p−Pf )x
0 −i(p−Pf )x0
0
b
dp ψ(p)
=
dp
dxe
ψ(x)
,
dx
e
ψ(x
)
3
3
F
F
(2π) 2
(2π) 2
Z
Z
D
E
Z
0
0
= (2π)−3
dx dx0
dp eip(x−x )
eiPf x ψ(x) , eiPf x ψ(x0 )
F
|
{z
}
= (2π)3 δ(x−x0 )
Z
=
dx ψ(x)
2
F
(A.6)
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
proves unitarity.
13
As an application, we observe that
P[
tot ψ(p) =
=
Z
1
dx ei(p−Pf )x (−i∇x + Pf )ψ(x)
3
(2π) 2
Z
1
dx (−i∇x + Pf )ei(p−Pf )x ψ(x)
3
2
(2π)
b .
= p ψ(p)
(A.7)
To each fiber space Hp there corresponds an isomorphism Ip : Hp → Fb where Fb is the Fock
space corresponding to creation- and annihilation operators b∗λ (k) (given by e−ikx a∗λ (k)), and bλ (k)
(given by eikx aλ (k)) commuting with Ptot . The vacuum Ωf ∈ Fb is given by Ip (eipx ). To define Ip
explicitly, we consider an improper state in Hp with a definite total momentum p describing the
electron at position x together with n photons. Its wave function has the form
ei(p−k1 −···−kn )x φ(n) (k1 , λ1 ; . . . ; kn , λn ) .
Then the equation
Ip ei(p−k1 −···−kn )x φ(n) (k1 , λ1 ; . . . ; kn , λn )
X Z
=
dk1 · · · dkn φ(n) (k1 , λ1 ; . . . ; kn , λn ) b∗λ1 (k1 ) · · · b∗λn (kn ) Ωf
(A.8)
(A.9)
λ1 ,...,λn
defines Ip .
Next, we discuss the Hamiltonian (2.2). It is easy to see that Hσ is translation-invariant, so that
[Hσ , Ptot ] = 0. Accordingly,
b
(Hσ ψ)b (p) = Hσ (p)ψ(p),
(A.10)
where Hσ (p) is the fiber Hamiltonian corresponding to total momentum p. Applying the isomorphism Ip : Hp → Fb , Hσ (p) is represented on Fb by
√
1
(A.11)
Hσb (p) := Ip Hσ (p)Ip−1 = ( p − Pfb + αAbσ )2 + Hfb
2
where
XZ
dk
b
Aσ =
dk 1/2 { λ (k) bλ (k) + h.c. }
(A.12)
|k|
σ≤|k|≤1
λ
and
Pfb =
XZ
k b∗λ (k) bλ (k) dk
(A.13)
|k| b∗λ (k) bλ (k) dk .
(A.14)
Ip−1 Hσb (p)Ip dp
(A.15)
λ
Hfb =
XZ
λ
We note that
Z
Hσ =
⊕
14
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
is the direct integral decomposition of Hσ over the spectrum of Ptot .
Remark A.2. Throughout this paper, we have usually dropped the superscripts in Hσb (p) and Fb ,
etc., while keeping the notation b∗λ (k), bλ (k) for the creation- and annihilation operators on Fb .
Appendix B. Hölder continuity of the ground state
We recall that Φρσ (p) denotes a normalized ground state of the Bogoliubov transformed fiber
ρ
ρ
Hamiltonian Kσρ (p) = W∇E
Hσ (p) (W∇E
)∗ , with infrared cutoff σ > 0 (see (2.22)). Our aim
σ (p)
σ (p)
in this appendix is to prove that, for a suitable choice of the vectors Φρσ (p), the map p 7→ Φρσ (p) is
θ-Hölder continuous, for θ < 2/3.
For ρ = 1, we set
Φσ (p) := Φ1σ (p),
Kσ (p) := Kσ1 (p).
(B.1)
We remark that
ρ,1
Kσρ (p) = W∇E
σ (p)
∗
ρ,1
Kσ (p) W∇E
,
σ (p)
ρ,1
Φρσ (p) = W∇E
σ (p)
∗
Φσ (p) ,
(B.2)
where
ρ,1
W∇E
σ (p)
h
:= exp α
1
2
XZ
λ
ρ≤|k|≤1
dk
∇Eσ (p) · λ (k) bλ (k) − h.c. i
.
|k|1/2 (|k| − ∇Eσ (p) · k)
Letting
Fσ :=
M
Sym( L2 ({k ∈ R3 , |k| ≥ σ} × {+, −} ) )⊗n
(B.3)
n≥0
denote the Fock space of photons of energies ≥ σ, and identifying Fσ with a subspace of F, we
observe that Kσ (p) leaves Fσ invariant. Let K̃σ (p) denote the restriction of Kσ (p) to Fσ . An
important property, proven in [3, 10, 13], is that there is an energy gap of size ησ, η > 0, in the
spectrum of K̃σ (p) above the ground state energy Eσ (p). Moreover, one can choose
Φσ (p) = Φ̃σ (p) ⊗ Ωf ,
(B.4)
in the representation F ' Fσ ⊗ F<σ , where
M
F<σ :=
Sym( L2 ({k ∈ R3 , |k| ≤ σ} × {+, −} ) )⊗n .
(B.5)
n≥0
Now, by [10, 13], Φ̃σ (p) can be chosen in the following way:
Φ̃σ (p) =
Π̃σ (p)Ωf
kΠ̃σ (p)Ωf k
,
(B.6)
where Ωf denotes the vacuum in Fock space and Π̃σ (p) is the rank-one projection onto the eigenspace
associated with Eσ (p) = inf spec(K̃σ (p)). We recall from [10, 13] that
1
kΠ̃σ (p)Ωf k ≥ ,
(B.7)
3
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
15
for arbitrary σ > 0 and |p| ≤ 1/3 provided that α is chosen sufficiently small.
Let N denote the number operator,
N =
XZ
dk b∗λ (k) bλ (k) .
(B.8)
λ
The following proposition has been proven in [8, 10, 13].
Proposition B.1. For α 1 and |p| ≤ 1/3, there exists a vector Φ(p) in the Fock space such that
Φσ (p) → Φ(p), strongly, as σ → 0. The following bound holds,
1
kN 2 Φσ (p)k ≤ C < ∞ ,
(B.9)
uniformly in σ ≥ 0. Moreover, For all δ > 0, there exists αδ > 0 and Cδ < ∞ such that, for all
0 ≤ α ≤ αδ , 0 ≤ σ 0 < σ ≤ 1 and |p| ≤ 1/3,
1
kΦσ (p) − Φσ0 (p)k ≤ Cδ α 4 σ 1−δ ,
(B.10)
1
4
|∇Eσ (p) − ∇Eσ0 (p)| ≤ Cδ α σ 1−δ .
(B.11)
As a consequence, we show the following corollary.
Corollary B.2. Let 0 < ρ < 1. For all δ > 0, there exists αδ > 0 such that, for all 0 ≤ α ≤ αδ
and |p| ≤ 1/3, there exists a vector Φρ (p) in the Fock space such that Φρσ (p) → Φρ (p), strongly, as
σ → 0. Moreover, there exists a constant Cδ < ∞ such that, for all 0 ≤ α ≤ αδ , 0 ≤ σ 0 < σ ≤ 1
and |p| ≤ 1/3,
1
kΦρσ (p) − Φρσ0 (p)k ≤ Cδ α 4 σ 1−δ (1 + ln(ρ−1 )) .
Proof. Using (B.2), we split
∗
ρ,1
ρ,1
W∇E
− W∇E
Φρσ (p) − Φρσ0 (p) =
σ (p)
σ 0 (p)
∗ ρ,1
Φσ (p) + W∇E
(B.12)
∗
σ 0 (p)
Φσ (p) − Φσ0 (p) . (B.13)
ρ,1
By Proposition B.1 and unitarity of W∇E
, the second term is estimated as
σ (p)
∗
ρ,1
W∇E
σ 0 (p)
Φσ (p) − Φσ0 (p)
1
≤ Cδ α 4 σ 1−δ .
(B.14)
The first term in the right side of (B.13) is estimated as
∗
∗ ρ,1
ρ,1
ρ,1
ρ,1
W∇E
−
W
Φ
(p)
=
1 − W∇E
W∇E
σ
∇E 0 (p)
σ (p)
σ (p)
σ 0 (p)
σ
∗ Φσ (p)
≤ B(ρ)Φσ (p) ,
(B.15)
ρ,1
by unitarity of W∇E
and the spectral theorem, where
σ (p)
Z
∇E (p) · (k) b (k) − h.c. ∇E 0 (p) · (k) b (k) − h.c. 1 X
σ
λ
λ
λ
λ
σ
B(ρ) := α 2
dk
−
. (B.16)
1/2 (|k| − ∇E (p) · k)
1/2 (|k| − ∇E 0 (p) · k)
|k|
|k|
σ
σ
ρ≤|k|≤1
λ
To estimate kB(ρ)Φσ (p)k, we use the well known fact that, for any f ∈ L2 (R3 × {+, −}),
√
1
ka# (f )(N + 1)− 2 k ≤ 2kf kL2 .
(B.17)
16
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
Clearly,
∇Eσ (p) · λ (k)
∇Eσ0 (p) · λ (k)
−
|k|1/2 (|k| − ∇Eσ (p) · k) |k|1/2 (|k| − ∇Eσ0 (p) · k)
(∇Eσ (p) − ∇Eσ0 (p)) · λ (k)
∇Eσ0 (p) · λ (k)
(∇Eσ (p) − ∇Eσ0 (p)) · k
=
.
+ 1/2
1/2
|k| (|k| − ∇Eσ (p) · k)
|k| (|k| − ∇Eσ (p) · k) (|k| − ∇Eσ0 (p) · k)
(B.18)
Hence, by (B.11) and the facts that |∇Eσ (p)|, |∇Eσ0 (p)| ≤ 1/2 for α small enough (see Proposition
2.1 (2)), we obtain
1
∇Eσ0 (p) · λ (k)
Cδ α 4 σ 1−δ
∇Eσ (p) · λ (k)
−
≤
.
3
|k|1/2 (|k| − ∇Eσ (p) · k) |k|1/2 (|k| − ∇Eσ0 (p) · k)
|k| 2
(B.19)
Thus, (B.16) and (B.17) yield that
3
B(ρ)Φσ (p) ≤ Cδ α 4 σ 1−δ
1ρ≤|k|≤1 (|k|)
|k|
3
2
1
L2k
(N + 1) 2 Φσ (p)
3
4
≤ Cδ α σ 1−δ ln(ρ−1 ) .
(B.20)
where we used (B.9) in the last inequality. Together with (B.13) – (B.15), this concludes the proof
of the corollary.
The following result follows from [10, 13] (it is also a consequence of (2.6) in Proposition 2.1
(2)).
Proposition B.3. There exist αc > 0 and C > 0 such that, for all 0 ≤ α ≤ αc and p, p0 satisfying
|p| ≤ 1/3, |p0 | ≤ 1/3,
∇Eσ (p) − ∇Eσ (p0 ) ≤ C |p − p0 |,
(B.21)
uniformly in σ > 0.
We now prove the following proposition.
Proposition B.4. Let 0 < ρ < 1. For all δ > 0, there exist αδ > 0 and Cδ < ∞ such that, for all
0 ≤ α ≤ αδ , σ > 0 and p, k ∈ R3 satisfying |p| ≤ 1/3, |p + k| ≤ 1/3,
2
kΦρσ (p + k) − Φρσ (p)k ≤ Cδ (1 + ln(ρ−1 )) |k| 3 −δ .
(B.22)
We recall the following easy lemma (see e.g. [3]).
Lemma B.5. Let H be a Hilbert space and let Π1 and Π2 be two rank-one projections in H. Let
Φ1 ∈ Ran Π1 , kΦ1 k = 1 and Φ2 ∈ Ran Π2 , kΦ2 k = 1. We have that
1
1
kΠ1 − Π2 k = hΦ1 , (Π1 − Π2 )Φ1 i| 2 = hΦ2 , (Π2 − Π1 )Φ2 i| 2 .
(B.23)
Proof of Proposition B.4
Step 1. We first prove that, for all 0 < σ < ρ ≤ 1,
1
kΦρσ (p + k) − Φρσ (p)k ≤ C |k| σ − 2 .
(B.24)
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
17
We decompose
∗
∗
ρ,1
ρ,1
Φρσ (p + k) − Φρσ (p) = W∇E
Φσ (p + k) − W∇E
Φσ (p)
σ (p+k)
σ (p)
∗
∗
ρ,1
ρ,1
= W∇E
− W∇E
Φσ (p)
σ (p+k)
σ (p)
∗
ρ,1
+ W∇E
Φ
(p
+
k)
−
Φ
(p)
.
σ
σ
(p+k)
σ
(B.25)
To estimate the first term in the right side of (B.25), we proceed as in the proof of Corollary B.2.
Namely, we have that
∗
∗ ∗ ρ,1
ρ,1
ρ,1
ρ,1
W∇E
−
W
Φ
(p)
=
1
−
W
W
Φσ (p)
σ
∇Eσ (p)
∇Eσ (p+k)
∇Eσ (p)
σ (p+k)
≤ C(ρ)Φσ (p) ,
(B.26)
by the spectral theorem, where
Z
∇E (p + k) · (k̃) b (k̃) − h.c. ∇E (p) · (k̃) b (k̃) − h.c. 1 X
σ
σ
λ
λ
λ
λ
dk̃
−
.
C(ρ) := α 2
1/2 (|k̃| − ∇E (p + k) · k̃)
1/2 (|k̃| − ∇E (p) · k̃)
|
k̃|
|
k̃|
ρ≤|k̃|≤1
σ
σ
λ
Using Proposition B.3, one verifies that
∇Eσ (p + k) · λ (k̃)
1/2
|k̃| (|k̃| − ∇Eσ (p + k)
· k̃)
∇Eσ (p) · λ (k̃)
1/2
|k̃| (|k̃| − ∇Eσ (p)
−
· k̃)
≤
C |k|
3
|k̃| 2
.
(B.27)
Hence (B.17) implies that
C(ρ)Φσ (p) ≤ C |k|
1ρ≤|k̃|≤1 (k̃)
|k̃|
−1
≤ C |k| ln(ρ
3
2
1
(N + 1) 2 Φσ (p)
L2
k̃
),
(B.28)
where we used (B.9) in the last inequality. Equations (B.26) and (B.28) yield
∗ ∗
∗ ρ,1
ρ,1
ρ,1
ρ,1
1
−
W
W
Φσ (p)
W∇E
−
W
Φ
(p)
=
σ
∇Eσ (p+k)
∇Eσ (p)
∇Eσ (p)
σ (p+k)
1
1
≤ C |k| σ − 2 σ 2 ln(ρ−1 )
1
≤ C |k| σ − 2 ,
(B.29)
since 0 < σ < ρ ≤ 1.
ρ,1
It remains to estimate the second term in the right side of (B.25). By unitarity of W∇E
, it
σ (p+k)
suffices to estimate kΦσ (p + k) − Φσ (p)k, and, by (B.4),
kΦσ (p + k) − Φσ (p)k = kΦ̃σ (p + k) − Φ̃σ (p)k .
(B.30)
It follows from (B.7) and Lemma B.5 that
1
1
kΦ̃σ (p + k) − Φ̃σ (p)k ≤
Π̃σ (p)Ωf − Π̃σ (p + k)Ωf
+
kΠ̃σ (p)Ωf k kΠ̃σ (p + k)Ωf k
≤ 6 Π̃σ (p) − Π̃σ (p + k)
= 6 Φ̃σ (p), Π̄σ (p + k)Φ̃σ (p)
1
2
,
(B.31)
18
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
where Π̄σ (p + k) := I − Π̃σ (p + k). Using that there is an energy gap of size ησ above Eσ (p + k) in
the spectrum of the operator K̃σ (p + k), we can estimate
Φ̃σ (p), Π̄σ (p + k)Φ̃σ (p)
1
2
≤ (K̃σ (p + k) − Eσ (p + k))−1 Π̄σ (p + k)
1
1
1
2
1
(K̃σ (p + k) − Eσ (p + k)) 2 Φ̃σ (p)
1
≤ η − 2 σ − 2 (K̃σ (p + k) − Eσ (p + k)) 2 Φ̃σ (p) .
(B.32)
Let
∗
,
∇p Kσ (p) := W∇Eσ (p) ∇p Hσ (p)W∇E
σ (p)
(B.33)
where
1
∇p Hσ (p) = p − Pf + α 2 Aσ .
(B.34)
Using the Feynman-Hellman formula,
hΦ̃σ (p), ∇p Kσ (p)Φ̃σ (p)i = ∇Eσ (p) ,
(B.35)
together with the mean-value theorem and Proposition B.3, we have that (see also [7, Lemma 3.6])
1
(K̃σ (p + k) − Eσ (p + k)) 2 Φ̃σ (p)
2
= Φ̃σ (p), (K̃σ (p + k) − Eσ (p + k))Φ̃σ (p)
= Φ̃σ (p), (K̃σ (p) + k · (∇p Kσ (p)) + k 2 /2 − Eσ (p + k))Φ̃σ (p)
= Eσ (p) − Eσ (p + k) + k · (∇p Eσ (p)) + k 2 /2
≤ C k2 .
(B.36)
Hence,
1
K̃σ (p + k) − Eσ (p + k) 2 Φ̃σ (p) ≤ C |k|.
(B.37)
Combining (B.31), (B.32) and (B.37), we obtain that
1
kΦ̃σ (p + k) − Φ̃σ (p)k ≤ C |k| σ − 2 ,
(B.38)
and hence (B.24) follows.
Step 2. We now prove that kΦρσ (p + k) − Φρσ (p)k ≤ Cδ |k|2/3−δ (with Cδ < ∞ for δ > 0).
Suppose first that σ ≥ |k|2/3 . Then by Step 1, we have that
1
2
kΦρσ (p + k) − Φρσ (p)k ≤ C |k| |k|− 3 = C |k| 3 .
(B.39)
Newt, we assume that σ ≤ |k|2/3 . We write
kΦρσ (p + k) − Φρσ (p)k ≤ kΦρσ (p + k) − Φρ (p + k)k + kΦρ (p + k) − Φρ|k|2/3 (p + k)k
+ kΦρσ (p) − Φρ (p)k + kΦρ (p) − Φρ|k|2/3 (p)k
+ kΦρ|k|2/3 (p + k) − Φρ|k|2/3 (p)k.
(B.40)
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
19
By Corollary B.2, the first two lines are bounded by
kΦρσ (p + k) − Φρ (p + k)k + kΦρ (p + k) − Φρ|k|2/3 (p + k)k
+ kΦρσ (p) − Φρ (p)k + kΦρ (p) − Φρ|k|2/3 (p)k
1
2
≤ Cδ α 4 (1 + ln(ρ−1 )) |k| 3 (1−δ) ,
(B.41)
whereas by Step 1, the last term is bounded by C |k|2/3 . Setting δ 0 = 2δ/3 and changing notations
concludes the proof of the proposition.
Appendix C. Proof of Estimate (2.29)
In this Appendix, we prove (2.29). It asserts that
1
1
k(Kσρ (p) − Hσ (p))Φρσ (p)kF ≤ C α 2 ρ 2 |p| ,
(C.1)
for all p ∈ S, for a constant independent of α and σ, ρ such that 0 < σ < ρ ≤ 1.
To begin with, we note that
k(Kσρ (p) − Hσ (p))Φρσ (p)kF = k(Kσρ (p) − Hσ (p))Ψσ (p)kF ,
(C.2)
which follows from
(Hσ − Kσρ )(p)Φσ (p) =
ρ
ρ
Hσ (p) − W∇E
Hσ (p)(W∇E
)∗ Φρσ (p)
σ (p)
σ (p)
ρ
= W∇E
( Kσρ (p) − Hσ (p) )Ψσ (p) ,
σ (p)
(C.3)
ρ
ρ
and unitarity of W∇E
. Here we recall that Ψρσ (p) = (W∇E
)∗ Φρσ (p).
σ (p)
σ (p)
Next, we let
1
vλ] (k) := α 2 1σ≤|k|≤ρ (|k|)
∇Eσ (p) · ]λ (k)
,
|k|1/2 (|k| − ∇Eσ (p) · k)
(C.4)
]λ (k)
|k|1/2
(C.5)
(scalar-valued) and
1
wλ] (k) := α 2 1σ≤|k|≤1 (|k|)
(vector-valued). We note that
1
|vλ (k)| ≤ C α 2 |p|
1σ≤|k|≤ρ (|k|)
3
|k| 2
(C.6)
and
1
|wλ (k)| ≤ C α 2
1σ≤|k|≤1 (|k|)
1
|k| 2
(C.7)
where we have used that |∇Eσ (p)| ≤ C |p|, uniformly in the infrared cutoff 0 ≤ σ ≤ 1.
Using that
ρ
ρ
W∇E
a] (k) (W∇E
)∗ = a]λ (k) + vλ] (k) ,
σ (p) λ
σ (p)
(C.8)
20
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
a straightforward calculation yields
Kσρ (p) − Hσ (p)
ρ
ρ
= W∇E
Hσ (p) (W∇E
)∗ − Hσ (p)
σ (p)
σ (p)
= (∇p Hσ (p)) · V (p) + V (p) · (∇p Hσ (p)) + V 2 (p) + Y (p) ,
where ∇p Hσ (p) is defined in (B.34), and
i
Xh
V (p) :=
aλ (kvλ ) + a∗λ (kvλ ) + 2Re(wλ , vλ ) + (vλ , kvλ ) ,
(C.9)
(C.10)
λ
(vector-valued operator) and
Y (p) :=
Xh
i
aλ (|k|vλ ) + a∗λ (|k|vλ ) + (vλ , |k|vλ ) ,
(C.11)
λ
(scalar-valued operator). We observe that both V (p) and Y (p) are proportional to |∇Eσ (p)| since
all terms are of first or higher order in vλ (which is proportional to |∇Eσ (p)| ≤ C|p|).
Let us first consider
kY (p) Ψσ (p)kF ≤
X
aλ (|k|vλ )Ψσ (p)
λ
F
X
+
a∗λ (|k|vλ )Ψσ (p)
λ
F
X
+
(vλ , |k|vλ )kΨσ (p)kF
λ
1
2
1
2
≤ 2 k|k| vλ kL2 (R3 ×{+,−}) k(Hf + 1) Ψσ (p)kF +
X
(vλ , |k|vλ ),
(C.12)
λ
where we used Lemma 3.2 in the second inequality. Since Hf ≤ Hσ (p), we have that
1
k(Hf + 1) 2 Ψσ (p)kF
≤
1
k(Hσ (p) + 1) 2 Ψσ (p)kF
p2
1
1
2
2
= (Eσ (p) + 1) =
≤ C,
+ O(α) + 1
2
where we used the Rayleigh-Ritz principle in the last equality. Since
1
1
1
1
1
k|k| 2 vλ kL2 (R3 ×{+,−}) ≤ C α 2 ρ 2 |p| ,
(C.13)
(C.14)
(see (C.6)) we conclude that
kY (p) Ψσ (p)kF ≤ C α 2 ρ 2 |p| .
(C.15)
To bound kV 2 (p) Ψσ (p)k and kV (p) · (∇p Hσ (p))Ψσ (p)kF , we use the following lemma.
Lemma C.1. For all p ∈ S, 0 ≤ σ ≤ 1 and α small enough,
kH(p)Ψσ (p)k ≤ C,
uniformly with respect to p, σ and α.
The proof of Lemma C.1 is given below.
Let us then consider
kV 2 (p) Ψσ (p)k ≤ kV 2 (p)(Hf + 1)−1 k k(Hf + 1)(H(p) + 1)−1 k k(H(p) + 1)Ψσ (p)k.
(C.16)
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
21
The first term is bounded as above, using (3.7) in Lemma 3.2, which yields
kV 2 (p)(Hf + 1)−1 k ≤ C α ρ p2 .
(C.17)
The second term is bounded by using the fact that
D(H(p)) = D(Pf2 + Hf ) ⊂ D(Hf ),
(C.18)
and the last term is uniformly bounded by Lemma C.1. Combining these estimates we obtain
kV 2 (p) Ψσ (p)k ≤ C α ρ p2 .
(C.19)
Finally, to bound kV (p) · (∇p Hσ (p))Ψσ (p)kF , we recall that
1
∇p Hσ (p) = p − Pf + α 2 Aσ ,
(C.20)
where
Aσ =
X
aλ (wλ ) + a∗λ (wλ ) .
(C.21)
λ
We decompose
1
kV (p) · (∇p Hσ (p))Ψσ (p)kF ≤ kV (p) · pΨσ (p)kF + α 2 kV (p) · Aσ Ψσ (p)kF
+kV (p) · Pf Ψσ (p)kF .
(C.22)
The first two terms are estimated in the same way as in (C.15) and (C.19), which gives
1
1
kV (p) · pΨσ (p)kF ≤ C α 2 ρ 2 p2
1
1
α 2 kV (p) · Aσ Ψσ (p)kF ≤ C α ρ 2 |p|.
(C.23)
It remains to estimate kV (p) · Pf Ψσ (p)kF . Let
X
φjl := −i
aλ (ikj kl vλ ) + a∗λ (ikj kl vλ ) .
(C.24)
λ
Using that [Pf,j , V (p)l ] = φjl , we find that
X
kV (p) · Pf Ψσ (p)k2F =
hΨσ (p), Pf,j V (p)j V (p)l Pf,l Ψσ (p)i
j,l
X
X
=
hΨσ (p), φjj V (p)l Pf,l Ψσ (p)i +
hΨσ (p), V (p)j Pf,j V (p)l Pf,l Ψσ (p)i
j,l
j,l
X
X
=
hΨσ (p), φjj V (p)l Pf,l Ψσ (p)i +
hΨσ (p), V (p)j φjl Pf,l Ψσ (p)i
j,l
j,l
X
+
hΨσ (p), V (p)j V (p)l Pf,j Pf,l Ψσ (p)i .
j,l
Proceeding in the same way as in (C.19), we obtain that
kV (p)l φjj Ψσ (p)k ≤ C α ρ2 p2 ,
kφjl V (p)l Ψσ (p)k ≤ C α ρ2 p2 ,
kV (p)l V (p)j Ψσ (p)k ≤ C α ρ p2 .
(C.25)
22
V. BACH, T. CHEN, J. FAUPIN, J. FRÖHLICH, AND I.M. SIGAL
Furthermore,
kPf,l Ψσ (p)k ≤ kHf Ψσ (p)k
≤ kHf (H(p) + 1)−1 kk(H(p) + 1)Ψσ (p)k ≤ C ,
(C.26)
by Lemma C.1 and the fact that D(H(p)) ⊂ D(Hf ), and, likewise,
kPf,j Pf,l Ψσ (p)k ≤ kPf2 Ψσ (p)k
≤ kPf2 (H(p) + 1)−1 kk(H(p) + 1)Ψσ (p)k ≤ C ,
(C.27)
since D(H(p)) ⊂ D(Pf2 ). Combining (C.25) with the Cauchy-Schwarz inequality and the previous
estimates, we obtain
1
1
kV (p) · (∇p Hσ (p))Ψσ (p)kF ≤ C α 2 ρ 2 |p|.
(C.28)
The term k(∇p Hσ (p)) · V (p)Ψσ (p)kF can be estimated exactly in the same way, which, together
with (C.15) and (C.19), concludes the proof of Lemma 4.2.
We conclude with proving Lemma C.1.
Proof of Lemma C.1. Since Hσ (p)Ψσ (p) = Eσ (p)Ψσ (p) and kΨσ (p)k = 1, we can write
kH(p)Ψσ (p)k ≤ Eσ (p) + kQσ (p)Ψσ (p)k
p2
≤
+ O(α) + kQσ (p)Ψσ (p)k ,
2
(C.29)
where
1
1
Qσ (p) := H(p) − Hσ (p) = α 2 A<σ · (p − Pf + α 2 Aσ ) +
Here Aσ is defined in (A.12), and
XZ
A<σ :=
λ
|k|≤σ
dk
α 2
A .
2 σ
dk
{ λ (k) bλ (k) + h.c. } .
|k|1/2
(C.30)
From Lemma 3.2, it follows, using that D(H(p)) ⊂ D(Hf ), that
1
1
1
kα 2 (A<σ · p)Ψσ (p)k ≤ C α 2 σ 2 |p| k(H(p) + 1)Ψσ (p)k ,
1
kα(A<σ · Aσ )Ψσ (p)k ≤ C α σ 2 k(H(p) + 1)Ψσ (p)k ,
kα(Aσ )2 Ψσ (p)k ≤ C α k(H(p) + 1)Ψσ (p)k .
(C.31)
Moreover, since A<σ and Pf commute, we can estimate
1
kα 2 (A<σ · Pf )Ψσ (p)k2 ≤ α kA2<σ Ψσ (p)kkPf2 Ψσ (p)k
≤ C α σ k(H(p) + 1)Ψσ (p)k2 ,
(C.32)
where in the last inequality we used again Lemma 3.2 together with D(H(p)) ⊂ D(Hf ) ∩ D(Pf2 ).
From (C.29)–(C.32), we obtain that
1
kH(p)Ψσ (p)k ≤ C + C α 2 kH(p)Ψσ (p)k ,
(C.33)
for any p ∈ S. Assuming that α is sufficiently small, the result follows.
EFFECTIVE DYNAMICS IN NON-RELATIVISTIC QED
23
Acknowledgements. J.Fa., I.M.S., and T.C. are grateful to J.Fr. for hospitality at ETH Zürich.
T.C. thanks I.M.S. for hospitality at the University of Toronto. The research of I.M.S. has been
supported by NSERC under Grant NA 7901. The work of T.C. was supported by the NSF through
the grants DMS-0940145 and DMS-1009448.
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(V. Bach) Institut fuer Analysis und Algebra Carl-Friedrich-Gauss-Fakultaet, Technische Universitaet Braunschweig, 38106 Braunschweig, Germany
E-mail address: v.bach@tu-bs.de
(T. Chen) Department of Mathematics, University of Texas at Austin, Austin TX 78712, USA
E-mail address: tc@math.utexas.edu
(J. Faupin) Institut de Mathématiques de Bordeaux, UMR-CNRS 5251, Université de Bordeaux 1, 351
cours de la libération, 33405 Talence Cedex, France
E-mail address: jeremy.faupin@math.u-bordeaux1.fr
(J. Fröhlich) Institut für Theoretische Physik, ETH Hönggerberg, CH-8093 Zürich, Switzerland
E-mail address: juerg@phys.ethz.ch
(I.M. Sigal) Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
E-mail address: im.sigal@utoronto.ca