# Quantum Boltzmann dynamics and bosonized particle-hole interactions in
fermion gases

Esteban Cárdenas  Department of Mathematics, University of Texas at Austin,
2515 Speedway, Austin TX, 78712, USA<EMAIL_ADDRESS>and  Thomas Chen
Department of Mathematics, University of Texas at Austin, 2515 Speedway,
Austin TX, 78712, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study a cold gas of $N\gg 1$ weakly interacting fermions. We
describe the time evolution of the momentum distribution of states relative to
the Fermi ball by simultaneously analyzing the dynamical behavior of excited
particles and holes. Our main result states that, for small values of the
coupling constant, and for appropriate initial data, the effective dynamics of
the above system is driven by an energy-mollified quantum Boltzmann collision
operator, plus a an interaction term with virtual bosonized particle-hole
pairs around the Fermi surface.

###### Contents

  1. 1 Introduction
  2. 2 Main results
  3. 3 Preliminaries
  4. 4 Tool Box I: Analysis of $b$\- and $D$-operators
  5. 5 Tool Box II: Excitation Estimates
  6. 6 Leading Order Terms I: Emergence of $Q$
  7. 7 Leading Order Terms II: Emergence of $B$
  8. 8 Subleading Order Terms
  9. 9 Proof of Theorem 1
  10. 10 The Fixed Volume Case

##  1\. Introduction

The quantum Boltzmann equation was first proposed on phenomenological grounds
by Nordheim [40] and Uehling and Uhlenbeck [47] as a quantum-mechanical
correction to the classical Boltzmann equation, taking into account the
statistics of the particles. For a spatially homogeneous gas of fermions in
three dimensions, it reads

$\displaystyle\partial_{T}F=\int_{\mathbb{R}^{9}}B(p\,p_{*}p^{\prime}p_{*}^{\prime})\Big{(}F^{\prime}F^{\prime}_{*}(1-F)(1-F_{*})-FF_{*}(1-F^{\prime})(1-F_{*}^{\prime})\Big{)}\mathrm{d}p_{*}\mathrm{d}p^{\prime}\mathrm{d}p_{*}^{\prime}$
(1.1)

Here, the unknown $F=F_{T}(p)$ is a probability density function in momentum
space, and we use the short-hand notations $F=F(p)$,
$F^{\prime}=F(p^{\prime})$, $F_{*}=F(p_{*})$ and
$F_{*}^{\prime}=F(p_{*}^{\prime})$. The scattering amplitude is calculated
quantum-mechanically in Born approximation, namely

$B(p\,p_{*}p^{\prime},p_{*}^{\prime})=4\pi\delta(p+p_{*}-p^{\prime}-p^{\prime}_{*})\delta(E_{p}+E_{p_{*}}-E_{p^{\prime}}-E_{p_{*}^{\prime}})\big{|}\hat{v}(p-p^{\prime})-\hat{v}(p-p_{*}^{\prime})\big{|}^{2}\
,$ (1.2)

where $\hat{v}(k)$ is the Fourier transform of the interaction potential, and
$E_{p}$ is the kinetic energy of a particle with momentum
$p\in\mathbb{R}^{3}$.

Despite important mathematical efforts, the rigorous derivation of (1.1) from
first principles remains an important open problem in mathematical physics.
This amounts to analyzing the microscopic dynamics governed by the Schrödinger
equation with respect to microscopic variables
$(t,x)\in\mathbb{R}\times\mathbb{R}^{3}$ and coupling $\lambda>0$, and
studying the asymptotic dynamics in a suitable scaling regime. In this regard,
two main approaches have been previously investigated in the literature.

  1. (1)

In the weak-coupling limit one considers the macroscopic set of variables
$(X,T)$, defined in terms of the microscopic variables $(t,x)$ as follows

$X=\varepsilon x\ ,\qquad T=\varepsilon t\ ,\qquad\lambda=\varepsilon^{1/2}\
.$ (1.3)

The scaling reimge $t=\lambda^{2}T$ is also known as the kinetic scaling
limit. It is then conjectured that the equation (1.1) emerges as
$\lambda\downarrow 0$ as the time evolution for the Wigner transform of the
density matrix of the system. For partial results in this direction, see e.g.
[3, 5, 6, 21, 31, 37, 43]. Most notably, it has been proven [21, 22] that
under conditional regularity assumptions, the associated $N$-particle BBGKY
hierarchy converges to a hierarchy that does not yield the quantum Boltzmann
equation (1.1) as a fully tensorized solution. The limiting equation is
quadratic in $F$ and resembles the classical Boltzmann equation.

  2. (2)

On the other hand, in the low density limit one considers instead the scaling
regime

$X=\varepsilon x,\qquad T=\varepsilon t,\qquad\lambda=1\ .$ (1.4)

Similarly, in the limit $\varepsilon\downarrow 0$, it is expected that the
kinetic equation (1.1) still holds, but with a quadratic collision operator
emerging on the right hand side, with full scattering kernel (in contrast to
(1.1), where only the first Born approximation appears). See e.g. [4].

Certainly, deriving the quantum Boltzmann equation for the kinetic time scale
$t=\lambda^{-2}T$ is a challenging open problem. On the other hand, one may
ask an easier question: whether it is possible to prove that the operator
appearing on the right hand side of (1.1) is the leading order term of the
time evolution on smaller time scales, with complete error analysis. This
would entail finding an appropiate norm and prove that the error terms are
dominated by the collision operator, over the scaling regime under
consideration. To the authors best knowledge, it seems that there are no
available results in the literature that directly answer this question.

###  1.1. Description of main results

In this work, we analyze the emergence of quantum Boltzmann dynamics for
perturbations of the Fermi ball. To this end, we consider a spatially
homogeneous system of $N$ fermions moving in a periodic box
$\Lambda=(\mathbb{R}/L\mathbb{Z})^{d}$, and study the microscopic dynamics of
its momentum distribution $F_{t}(p)$ relative to the characteristic function
of the set

$\mathcal{B}:=\\{p\in(2\pi\mathbb{Z}/L)^{d}:|p|\leqslant p_{F}\\}\ ,$ (1.5)

where $p_{F}>0$ is the Fermi momentum. We consider the regime in which
$\lambda$ is small, and the density of particles $N/|\Lambda|$ is large. In
this situation, it is extremely convenient to change variables and work in
particle-hole space. That is, we study the the momentum distribution
$f_{t}(p)$ of excited particles and empty holes, which can be defined through
the relations

$F_{t}(p)=\begin{cases}f_{t}(p)\quad&|p|>p_{F}\\\ 1-f_{t}(p)\quad&|p|\leqslant
p_{F}\end{cases}\ .$ (1.6)

Our attention is on states that arise as external perturbations of the
system–in particular, not close to the ground state (see Condition 2 for more
details).

Our main goal is to determine the effective dynamics of $f_{t}(p)$. In Theorem
1, we consider the following asymptotic expansion in terms of the coupling
$\lambda>0$ and microscopic time $t\in\mathbb{R}$

$f_{t}(p)=f_{0}(p)+\lambda^{2}t\,B_{t}[f_{0}]+\lambda^{2}t\,Q_{t}[f_{0}]+\cdots\
$ (1.7)

and give an explicit estimate on the remainder, in terms of the unconstrained
parameters of the theory. In particular, our estimates focus on the
description of the dynamics of fermions whose momenta are away from the Fermi
surface. This is encoded in the norm $\|\cdot\|_{\ell_{m}^{1*}}$ that we use
to measure distances. Note that this consistent with the fact that particles
and holes near the Fermi surface couple together to form bosonized particle-
hole pairs, and experience different dynamical behaviour. This bosonization
phenomenon was first explained by Sawada [44] and Sawada, Brueckner, Fukuda
and Brout [45] in the 1950s, and has recently been proven rigorously in a
series of papers [8, 9, 10, 13, 34, 24, 25, 26].

The operator $Q_{t}$ is given in Definition 2, and corresponds to an energy-
mollified collisional operator of quantum Boltzmann form, of cubic order after
taking into account appropiate cancellations. It includes collisions of the
form particle-particle, particle-hole, and hole-hole. On the other hand, the
operator $B_{t}$ is quadratic and is given in Definition 3. It describes the
interaction between fermions that are mediated by virtual bosonized particle-
hole pairs near the Fermi surface. In particular, our assumptions on the
initial data and on the dynamics guarantee that the Fermi surface is depleted
throughout the time scale under consideration i.e. interactions with real
particle-hole pairs are negligible (see Proposition 5.2).

In Theorem 2 we consider the dynamics on the three-dimensional torus
$\Lambda=(2\pi\mathbb{T})^{3}$ for longer time scales. More precisely, we
consider the scaling regime

$\lambda=1/N^{\frac{3}{2}+\delta}\qquad\textnormal{and}\qquad
t=N^{\frac{1}{6}+\frac{\delta}{2}}T\ ,$ (1.8)

where $\delta\in(0,1)$ is small enough, and $0\leqslant T\leqslant 1$ is a
time scale that is long enough to observe completed collisions. Here, we
identify the leading order terms of the operators $B_{t}$ and $Q_{t}$ as
$t\rightarrow\infty$, with rigorous error control. These are given by the
formal limits

$\mathscr{Q}:=\lim_{t\rightarrow\infty}\lim_{\lambda\downarrow
0}\frac{1}{t}Q_{t}\qquad\textnormal{and}\qquad\mathscr{B}:=\lim_{t\rightarrow\infty}\lim_{\lambda\downarrow
0}\frac{1}{t}B_{t}\ .$ (1.9)

For small enough perturbations, we construct initial data such that
$\mathscr{B}[f]\sim N^{1/3}$ dominates the dynamics, and $\mathscr{Q}[f]$
becomes subleading.

From a conceptual level, the last observation is crucial. Namely, it states
that the emergence of the operator that drives the quantum Boltzmann equation
(1.1) for states $F$ near the Fermi ball, is associated to a quadratic
operator $\mathscr{B}$ in particle-hole space, describing interactions with a
boson-field, i.e. the bosonized particle-hole pairs. Our main result is a
rigorous version of this statement.

###  1.2. Comparison

Let us compare our work with previous results that are available in the
literature.

  1. (1)

Benedikter, Nam, Porta, Schlein and Seiringer [10] studied the dynamics of a
three-dimensional Fermi gas around the Fermi ball, in the semi-classical
$\hbar=N^{-1/3}$, mean-field regime $\lambda=1/N$. The main focus is on the
bosonization of particle-hole pairs inside of a suitable neighborhood of the
Fermi surface. The initial states that are considered arise naturally as
thermal fluctuations of the ground state problem. In fact, their methods have
been used to prove upper and lower bounds for the correlation energy in the
so-called Random Phase Approximation. In their terms, here instead we consider
the contribution to the dynamics associated to the “non-bosonizable terms”.
These interaction terms describe the dynamics of particles and holes away from
the Fermi surface. See also [8, 9, 10, 13, 34, 24, 25, 26] for related results
in this direction.

  2. (2)

Hott and the second author [18] have studied the emergence of quantum
Boltzmann dynamics for the fluctuations around a Bose-Einstein condensate. In
particular, our scaling regimes are similar to one another, in the sense that
both of them contain a large number of particles per unit volume. In other
words, the density of particles acts as an expansion parameter. Similarly,
they both employ quasi-free initial states. While of course the difference in
statistics plays a crucial role in the analysis, the approach presented here
is largely inspired by that of [18].

  3. (3)

Erdős [29] studied the weak-coupling limit of an electron interacting with a
thermal bath of phonons. In particular, a linear Boltzmann equation is shown
to emerge from the long-time dynamics. It turns out that this is not very
different than the situation under consideration. Namely, the dynamics of
particles and holes outside of the Fermi surface can also be described as
particles that interact with a boson field, i.e. the bosonized electron-hole
pairs around the Fermi surface, as described by [10]. The situation here is
more complicated, however. On the one hand, bosonization is only approximate.
On the other hand, several other interactions influence the dynamics of
particles and holes, and rigorous error control over these interactions is
already demanding. Finally, let us recall that the weak coupling limit of
electrons interacting with a random medium is intimately related to the model
studied in [29]; for results in this direction see e.g [17, 19, 20, 30, 46].

  4. (4)

The scaling regime (1.8) contains an interaction strength that is much weaker
than the microscopic mean-field scaling regime–of order $N^{-1/3}$ in three
dimensions for fermions. Here instead, the kinetic and potential energies are
not in balance over equal time scales, and it is necessary to consider longer
time scales in order to observe the aggregated effect of particle
interactions. The literature of mean-field theory for fermions is vast, see
e.g [1, 2, 33, 41, 11, 12, 28, 32, 42, 36, 39, 10, 8, 13, 24, 34] for a non-
exhaustive list of references.

###  1.3. Organization of this article

In Section 2 we state the main result of this article, and in Section 3 we
introduce the preliminaries that are needed to set up the proof. In Sections 4
and 5 we introduce and develop the machinery that we use in our analysis. In
Section 6 and 7 we show how the operators $Q$ and $B$, respectively, emerge
from the many-body dynamics, giving rise to the leading order terms. In
Section 8 we estimate subleading order terms and in Section 9 we prove our
main result, Theorem 1. Finally, in Section 10 we analyze the fixed volume
case.

###  1.4. Notation

The following notation is going to be used throughout this article.

  * -

$\Lambda^{*}\equiv(2\pi\mathbb{Z}/L)^{d}$ denotes the dual lattice of
$\Lambda$.

  * -

We write
$\int_{\Lambda^{*}}F(p)\mathrm{d}p\equiv|\Lambda|^{-1}\sum_{p\in\Lambda^{*}}F(p)$
for any function $F:\Lambda^{*}\rightarrow\mathbb{C}$.

  * -

$\delta_{p,q}$ denotes the standard Kronecker delta.

  * -

$\ell^{p}(\Lambda^{*})$ denotes the space of functions with finite norm
$\|f\|_{\ell^{p}}\equiv(\int_{\Lambda^{*}}|f(p)|^{p}\mathrm{d}p)^{1/p}$.

  * -

$B(X)$ denotes the space of bounded linear operators acting on $X$.

  * -

We denote $\widetilde{f}\equiv 1-f$ for any function
$f:\Lambda^{*}\rightarrow\mathbb{C}$ .

  * -

$\widehat{G}(k)\equiv(2\pi)^{-d/2}\int_{\Lambda}e^{-ik\cdot x}G(x)\mathrm{d}x$
denotes the Fourier transform of $G:\Lambda\rightarrow\mathbb{C}.$

  * -

We say that a positive real number $C>0$ is a constant, if it is independent
of the physical parameters $N$, $|\Lambda|$, $\lambda$, $n$ and $t$.

  * -

Given two real-valued quantities $A$ and $B$, we say that $A\lesssim B$ if
there exists a constant $C>0$ such that

$\displaystyle A\leqslant C\,B\ .$ (1.10)

Additionally, we say that $A\simeq B$ if both $A\lesssim B$ and $B\lesssim A$
hold true.

  * -

We shall frequently omit subscripts from Hilbert spaces norms throughout
proofs.

  * -

We denote by $\left\langle t\right\rangle=(1+t^{2})^{1/2}$ the standard
bracket.

##  2\. Main results

The main result of this article is a rigorous interpretation of the expansion
(1.7) that arises from the many-body fermionic dynamics. First, we present the
model that we study. Secondly, we present our main result in Theorem 1. It
contains an estimate in a weighted $\ell^{\infty}$ norm for the difference
between the momentum distribution of the full dynamics, and the dynamics
associated to the leading order $Q_{t}$ and $B_{t}$ terms. Finally, in Theorem
2 we discuss the consequences of this estimate in the scaling regime (1.8).

###  2.1. The model

We consider an interacting gas of $N$ identical fermions, moving in the
$d\geqslant 1$ dimensional torus $\Lambda\equiv(\mathbb{R}/L\mathbb{Z})^{d}$,
where $L>0$ is its linear length. We work in the grand canonical ensemble, and
introduce the fermionic Fock space over the one-particle space
$L^{2}(\Lambda)$ as follows

$\mathscr{F}\equiv\mathbb{C}\oplus\bigoplus_{n\geqslant
1}\mathscr{F}_{n}\qquad{\textnormal{where}}\qquad\mathscr{F}_{n}\equiv
L^{2}_{a}(\Lambda^{n}),\ \forall n\geqslant 1\ ,$ (2.1)

where we neglect the spin of the particles, and we denote by $L_{a}^{2}$ the
subspace of antisymmetric $L^{2}$-functions. As usual, $\mathscr{F}$ is
endowed with creation and annihilation operators $a_{p}$ and $a_{q}^{*}$. In
momentum space, they correspond to bounded operators that satisfy the
Canonical Anticommutation Relations (CAR)

$\\{a_{p},a_{q}^{*}\\}=\delta(p-q)\equiv|\Lambda|\delta_{p,q}\qquad
p,q\in\Lambda^{*}\ ,$ (2.2)

and zero otherwise. Here, $\Lambda^{*}=(2\pi\mathbb{Z}/L)^{d}$ is the
corresponding dual lattice, $\delta_{p,q}$ stands for the Kronecker delta, and
$\\{\cdot,\cdot\\}$ denotes the anticommutator. The Fock vacuum vector will be
denoted by $\Omega\in\mathscr{F}$, and for notational convenience we denote
sums by
$\int_{\Lambda^{*}}\mathrm{d}p\equiv|\Lambda|^{-1}\sum_{p\in\Lambda^{*}}$.

The Hamiltonian on Fock space that we study corresponds to the following self-
adjoint operator, written in appropiate microscopic units as

$\mathcal{H}\equiv\frac{1}{2}\int_{\Lambda^{*}}|p|^{2}a_{p}^{*}a_{p}\mathrm{d}p+\frac{\lambda}{2}\int_{(\Lambda^{*})^{3}}\hat{V}(k)\
a^{*}_{p+k}a^{*}_{q-k}a_{q}a_{p}\ \mathrm{d}p\,\mathrm{d}q\,\mathrm{d}k\ ,$
(2.3)

where $\hat{V}(k)\equiv(2\pi)^{-d/2}\int_{\Lambda}V(x)\mathrm{d}x$ is the
Fourier transform of a two-body potential $V(x)$, and $\lambda$ denotes the
the coupling strength of the interaction. In particular, the time evolution
generated by $\mathcal{H}$ can be described as follows. Denote by
$B(\mathscr{F})$ the $C^{*}$-algebra of bounded operators on Fock space, and
consider an initial state $\rho$. Then, the dynamics of the system is given by

$\rho(t,\mathcal{O})=\rho(e^{it\mathcal{H}}\mathcal{O}e^{-it\mathcal{H}})\
,\qquad\mathcal{O}\in B(\mathscr{F})\ .$ (2.4)

Here and in the sequel, the time variable $t\in\mathbb{R}$ should be
understood as being measured in microscopic units. In particular, the momentum
distribution per unit volume of the system is defined as

$F_{t}(p)\equiv|\Lambda|^{-1}\rho(t,a_{p}^{*}a_{p})\ ,$ (2.5)

for all $t\in\mathbb{R}$ and $p\in\Lambda^{*}$.

The initial states that we study are external perturbation of the Fermi ball
That is, $\rho$ corresponds to a translation invariant state, which is a
suitable perturbation of the pure state

$\Psi_{F}\equiv\prod_{p\in\mathcal{B}}a^{*}_{p}\Omega\qquad\textnormal{where}\qquad\mathcal{B}\equiv\\{p\in\Lambda^{*}:|p|\leqslant
p_{F}\\}\ .$ (2.6)

The state $\Psi_{F}$ corresponds to the Slater determinant of plane waves
$e_{p}(x)\equiv|\Lambda|^{-1/2}e^{ip\cdot x}$ with momenta in $\mathcal{B}$,
minimizing the kinetic energy of the system in compliance with Pauli’s
exclusion principle.

We refer interchangeably to $\Psi_{F}$ and $\mathcal{B}$ as the Fermi ball
(or, Fermi sea), defined in terms of the Fermi momentum $p_{F}$. For
simplicity, we assume here that $p_{F}$ is given, and define the number of
particles to be $N\equiv|\mathcal{B}|$. The relationship between $p_{F}$ and
$N$ is then given by the formula

$p_{F}\,=\,C\big{(}N/|\Lambda|\big{)}^{1/d}\ ,$ (2.7)

where $C=C_{d}+o(1)$ is a constant depending only on the dimension, and lower
order terms on $p_{F}$. In particular, it is important to note that the high-
density regime that we study corresponds to large values of the Fermi momentum
$p_{F}\gg 1$.

###  2.2. Assumptions and definitions

Let us state the precise mathematical conditions under which our main theorems
are formulated. We first discuss the conditions on the interaction potentials,
and then the conditions on the initial data.

Potentials. Throughout this work, we shall consider real-valued functions
$V:\Lambda\rightarrow\mathbb{R}$ that satisfy Condition 1 below. In
particular, under these conditions, the many-body Hamiltonian $\mathcal{H}$
defined in (2.3) is self-adjoint, and its dynamics is well-defined.

######  Condition 1.

$V:\Lambda\rightarrow\mathbb{R}$ is a real-valued function whose Fourier
transform $\hat{V}(k)$ satisfies the following conditions.

  1. (1)

It has compact support in a ball of radius $r>0$ .

  2. (2)

$\hat{V}(-k)=\hat{V}(k)$ for all $k\in\Lambda^{*}$. Thus, $\hat{V}$ is real-
valued .

  3. (3)

$\hat{V}(0)=0$ .

  4. (4)

$V$ is chosen relative to the box $\Lambda$ so that
$\sup_{|\Lambda|>0}\|\hat{V}\|_{\ell^{1}(\Lambda^{*})}<\infty$.

######  Remark 2.1.

The radius $r>0$ will be fundamental in our analysis. From a physical point of
view, it determines a momentum scale that allows for particle interactions. In
particular, it will determine an $\mathcal{O}(1)$ neighborhood around the
Fermi surface which separates excited particles and holes, into either
bosonizable or non-bosonizable.

States. As we have already mentioned, the initial states that we consider are
regarded as external perturbations of the Fermi ball $\Psi_{F}$. It will be
extremely convenient to introduce the following notations describing the
momentum distribution of $\Psi_{F}$, as well as its complement

$\chi(p)\equiv\mathds{1}(|p|\leqslant p_{F})\qquad\textnormal{ and
}\qquad\chi^{\perp}\equiv 1-\chi\ ,$ (2.8)

where $\mathds{1}$ stands for a characteristic function. In particular, it is
now a standard calculation using the Canonical Anticommutation Relations to
show that one may write the two-point function of $\Psi_{F}$ as follows

$\left\langle\Psi_{F},a_{p}^{*}a_{q}\Psi_{F}\right\rangle=\delta(p-q)\chi(p)$
(2.9)

for all $p,q\in\Lambda^{*}$.

We analyze the dynamics of fermions relative to the Fermi ball. In this
regard, we think of excited fermions with momentum $|p|>p_{F}$ as particles,
and the anti-particles they leave behind inside of the Fermi ball as holes,
with momentum $|h|\leqslant p_{F}$ This change of variables is implemented by
a particle-hole transformation. It corresponds to the unitary transformation
on Fock space

$\mathcal{R}:\mathscr{F}\longrightarrow\mathscr{F}$ (2.10)

that can be explicitly defined through its action on creation- and
annihilation operators as follows

$\displaystyle\mathcal{R}^{*}a_{p}^{*}\mathcal{R}\ =\ \begin{cases}\
a_{p}^{*}\qquad|p|>p_{F}\\\ \ a_{p}\qquad|p|\leqslant
p_{F}\end{cases}\qquad\textnormal{and}\qquad\mathcal{R}\Omega\equiv\Psi_{F}\
.$ (2.11)

In particular, we are interested in describing the time evolution of the
momentum distribution of states relative to the Fermi ball. In particle-hole
space, the dynamics of these states is described by the particle-hole
Hamiltonian

$\mathfrak{h}\equiv\mathcal{R}^{*}\mathcal{H}\mathcal{R}\ .$ (2.12)

A more explicit representation of the Hamiltonian $\mathfrak{h}$ will be given
in the next section.

Thus, we study the evolution-in-time of the corresponding momentum
distribution, defined as follows. Recall that a state is a positive linear
functional on $B(\mathscr{F})$, with $\nu(\mathds{1})=1$.

######  Definition 1.

Given an initial state $\nu$, we define the momentum distribution per unit
volume of particles and holes as

$\displaystyle
f_{t}(p)\equiv|\Lambda|^{-1}\nu(e^{\mathrm{i}t\mathfrak{h}}a_{p}^{*}a_{p}e^{-\mathrm{i}t\mathfrak{h}})\
,$ (2.13)

for $(t,p)\in\mathbb{R}\times\Lambda^{*}$.

######  Remark 2.2.

Let $F_{t}(p)$ be the momentum distribution (2.5) of a state $\rho$, evolving
accrording to the Hamiltonian $\mathcal{H}$, in the original many-body
problem. Then, a straightforward calculation using the CAR shows that

$F_{t}(p)=\begin{cases}f_{t}(p)\quad&|p|>p_{F}\\\ 1-f_{t}(p)\quad&|p|\leqslant
p_{F}\ ,\end{cases}$ (2.14)

where $f_{t}(p)$ is given as in Definition 1, with respect to the unitarily
transformed initial state

$\nu(\mathcal{O})\equiv\rho(\mathcal{R}\mathcal{O}\mathcal{R}^{*})\
,\qquad\mathcal{O}\in B(\mathscr{F})\ .$ (2.15)

Note that, if $\rho$ and $\nu$ are determined by pure states $\Psi$ and
$\psi$, respectively, then (2.15) is equivalent to $\Psi=\mathcal{R}\psi$. In
particular, $\Psi=\Psi_{F}$ if and only if $\psi=\Omega$.

We find it convenient to introduce conditions on the initial data with respect
to the particle-hole variables, which we now describe. In order to motive
them, let us recall that two-body interactions around the Fermi ball induce a
collective bosonization of particle-hole pairs around its boundary; the modes
of excitation of these quasiparticles belong to the support of the interaction
potential $\hat{V}$, which we denote by $r>0$. Since such phenomena will arise
in our analysis, it is convenient to introduce the following neighborhood
around the surface of the Fermi ball

$\displaystyle\mathcal{S}\equiv\\{p\in\Lambda^{*}:p_{F}-3r\leqslant|p|\leqslant
p_{F}+3r\\}$ (2.16)

which (under a slight abuse of notation) we shall refer to as the Fermi
surface. The pre-factor $3$ is included for technical reasons.

The conditions for the initial data in the particle-hole representation are
given as follows.

######  Condition 2.

The initial state $\nu$ verifies the following conditions.

  1. (C1)

$\nu$ is a mixed state: there exists sequences $0\leqslant\nu_{j}\leqslant 1$
and $\Psi_{j}\in\mathscr{F}$ such that
$\nu(\mathcal{O})=\sum_{j=1}^{\infty}\nu_{j}\left\langle\Psi_{j},\mathcal{O}\Psi_{j}\right\rangle_{\mathscr{F}}$
with $\sum\nu_{j}=1$ and $\|\Psi_{j}\|_{\mathscr{F}}=1$ .

  2. (C2)

$\nu$ is number-conserving and quasi-free: for all
$k,k^{\prime}\in\mathbb{N}$, $p_{1},\ldots,p_{k}\in\Lambda^{*}$ and
$q_{1},\ldots,q_{k^{\prime}}\in\Lambda^{*}$ there holds

$\nu\Big{(}a_{p_{1}}^{*}\cdots a_{p_{k}}^{*}a_{q_{k^{\prime}}}\cdots
a_{q_{1}}\Big{)}\,=\,\delta_{k,k^{\prime}}\det\big{[}\nu(a_{p_{i}}^{*}a_{q_{j}})\big{]}_{1\leqslant
i,j\leqslant k}\ .$ (2.17)

  3. (C3)

$\nu$ is translation invariant: for all $p,q\in\Lambda^{*}$ there holds
$\nu(a_{p}^{*}a_{q})=\delta(p-q)\nu(a_{p}^{*}a_{p})\ .$

  4. (C4)

$\nu$ has zero charge:
$\int_{\mathcal{B}}\nu(a_{p}^{*}a_{p})\mathrm{d}p=\int_{\mathcal{B}^{c}}\nu(a_{p}^{*}a_{p})\mathrm{d}p\
.$

  5. (C5)

There exists a constant $C\geqslant 0$ such that
$\int_{\mathcal{S}}\nu(a_{p}^{*}a_{p})\mathrm{d}p\leqslant
C(\lambda|\Lambda|p_{F}^{d-1})^{2}$.

######  Remark 2.3.

We observe that in the spirit of Erdős, Salmhofer and Yau [31], we could have
considered states that are only restricted quasi-free up to a certain finite
degree, and our main results would remain unchanged. Since all the concrete
examples that we consider are indeed quasi-free, we leave it as is. Further,
we note that translation invariance is a natural assumption that greatly
simplifies the analysis, while at the same time being physically relevant. The
same comments apply to the requirement of states having zero charge. Finally,
let us comment on Condition (C5). We refer to it as the depletion of the Fermi
surface, and it makes precise the idea that the initial data that we are
considering arises as an external perturbation to the physical system. In
other words, these states are not related to thermal fluctuations around the
Fermi surface.

Example. We may construct a pure state $\nu$ that verifies Condition 2 as a
Slater determinant. Namely, given $n\in\mathbb{N}$, let
$h_{1},\ldots,h_{n}\in\mathcal{B}\backslash\mathcal{S}$ and
$p_{1},\ldots,p_{n}\in\mathcal{B}^{c}\backslash\mathcal{S}$. Then, we set

$\nu(\mathcal{O})\equiv\left\langle\Psi_{0},\mathcal{O}\Psi_{0}\right\rangle_{\mathscr{F}}\qquad\textnormal{where}\qquad\Psi_{0}\equiv
a^{*}_{h_{1}}\cdots a^{*}_{h_{n}}a^{*}_{p_{1}}\cdots a^{*}_{p_{n}}\Omega\ .$
(2.18)

Since Slater determinants are always number-conserving and quasi-free, this
verifies (C2). One may verify that translation invariance in (C3) is satisfied
by direct computation of the two-point function

$\nu(a_{p}^{*}a_{q})=\delta(p-q)\Big{(}\delta(p-h_{1})+\ldots+\delta(p-h_{n})+\delta(p-p_{1})+\ldots+\delta(p-p_{n})\Big{)}\
.$ (2.19)

The state $\nu$ has zero charge in (C4) because we have chosen an equal number
of $h_{i}^{\prime}s$ and $p_{i}^{\prime}s$ in $\mathcal{B}$ and
$\mathcal{B}^{c}$, respectively. Finally, (C5) follows from
$\nu(a_{p}^{*}a_{p})=0$ for all $p\in\mathcal{S}.$

###  2.3. Statement of the main theorem

Our main result identifies the time evolution of the momentum distribution
$f_{t}(p)$ in terms of two non-linear operators that act on functions on
$\Lambda^{*}$. In order to define these, we introduce the following three
notations. First, we will denote by $E_{p}$ the dispersion relation of
particles and holes, given as follows for $p\in\Lambda^{*}$

$E_{p}\,\equiv\,-\chi(p)\Big{(}\
\frac{p^{2}}{2}\,+\,\frac{\lambda}{2}\,(\hat{V}*\chi^{\perp})(p)\
\Big{)}+\chi^{\perp}(p)\Big{(}\
\frac{p^{2}}{2}\,-\,\frac{\lambda}{2}\,(\hat{V}*\chi)(p)\ \Big{)}\ .$ (2.20)

Second, for $(t,E)\in\mathbb{R}^{2}$ we denote by $\delta_{t}(E)$ the
following mollified Delta function

$\displaystyle\delta_{t}(E)\equiv t\delta_{1}(tE)\qquad\textnormal{ where
}\qquad\delta_{1}(E)\equiv\frac{2}{\pi}\frac{\sin^{2}\big{(}\frac{E}{2}\big{)}}{E^{2}}\
.$ (2.21)

In this context, the presence of an approximate delta function is a
consequence of the Heisenberg uncertainty principle for the conjugate
variables $(t,E)$ of time and energy: it is a manifestation that energy cannot
be exactly conserved in microscopic interactions. Third, we introduce the
following convenient notation for products of $\chi$. Namely, for any
$k\in\mathbb{N}$ we will write

$\chi(p_{1},\ldots,p_{k})\equiv\chi(p_{1})\cdots\chi(p_{k})$ (2.22)

for all $p_{1},\cdots,p_{k}\in\Lambda^{*}$ and similarly for $\chi^{\perp}.$
We are now ready to give the relevant definitions.

The following operator describes describes Boltzmann-type interactions between
particles/particles, particles/holes and holes/holes. Here and in the sequel
we denote

$\widetilde{f}\equiv 1-f$

for any function $f:\Lambda^{*}\rightarrow\mathbb{R}$.

######  Definition 2.

For $f\in\ell^{1}(\Lambda^{*})$ and $t\in\mathbb{R}$ we define

$\displaystyle Q_{t}[f](p)$
$\displaystyle\equiv\pi\int_{\Lambda^{*4}}\mathrm{d}\vec{p}\,\sigma(\vec{p})\,\Big{[}\delta(p-p_{1})+\delta(p-p_{2})-\delta(p-p_{3})-\delta(p-p_{4})\Big{]}$
(2.23)
$\displaystyle\times\delta_{t}[E_{p_{1}}+E_{p_{2}}-E_{p_{3}}-E_{p_{4}}]\Big{(}f(p_{3})f(p_{4})\widetilde{f}(p_{1})\widetilde{f}(p_{2})-f(p_{1})f(p_{2})\widetilde{f}(p_{3})\widetilde{f}(p_{4})\Big{)}\
.$

The coefficient function $\sigma:(\Lambda^{*})^{4}\rightarrow\mathbb{R}$ is
decomposed as

$\sigma=\sigma_{HH}+\sigma_{PP}+\sigma_{HP}+\sigma_{PH}$ (2.24)

where the coefficient functions are defined for
$\vec{p}=(p_{1},p_{2},p_{3},p_{4})\in(\Lambda^{*})^{4}$ as follows

$\displaystyle\sigma_{HH}(\vec{p})$
$\displaystyle=\chi(p_{1},p_{2},p_{3},p_{4})\delta(p_{1}+p_{2}-p_{3}-p_{4})|\hat{V}(p_{1}-p_{4})-\hat{V}(p_{1}-p_{3})|^{2}$
(2.25) $\displaystyle\sigma_{PP}(\vec{p})$
$\displaystyle=\chi^{\perp}(p_{1},p_{2},p_{3},p_{4})\delta(p_{1}+p_{2}-p_{3}-p_{4})|\hat{V}(p_{1}-p_{4})-\hat{V}(p_{1}-p_{3})|^{2}$
(2.26) $\displaystyle\sigma_{HP}(\vec{p})$
$\displaystyle=2\chi(p_{1},p_{3})\chi^{\perp}(p_{2},p_{4})\delta(p_{1}-p_{2}-p_{3}+p_{4})|\hat{V}(p_{1}-p_{3})|^{2}$
(2.27) $\displaystyle\sigma_{PH}(\vec{p})$
$\displaystyle=2\chi^{\perp}(p_{1},p_{3})\chi(p_{2},p_{4})\delta(p_{1}-p_{2}-p_{3}+p_{4})|\hat{V}(p_{1}-p_{3})|^{2}\
.$ (2.28)

The following operator describes the effect of the bosonized excitations
around the Fermi surface, with holes and particles.

######  Definition 3.

For all $t\in\mathbb{R}$ we define in terms of particle and hole interactions

$\displaystyle B_{t}$ $\displaystyle\equiv
B_{t}^{(H)}+B_{t}^{(P)}:\ell^{1}(\Lambda^{*})\rightarrow\ell^{1}(\Lambda^{*})$
(2.29)

where $B_{t}^{(H)}:\ell^{1}(\Lambda^{*})\rightarrow\ell^{1}(\Lambda^{*})$ and
$B_{t}^{(P)}:\ell^{1}(\Lambda^{*})\rightarrow\ell^{1}(\Lambda^{*})$ are
defined as follows

$\displaystyle B_{t}^{(H)}[f](h)$ $\displaystyle\equiv
2\pi\int_{\Lambda^{*}}|\hat{V}(k)|^{2}\Big{(}\alpha^{H}_{t}(h-k,k)f(h-k)\widetilde{f}(h)-\alpha^{H}_{t}(h,k)f(h)\widetilde{f}(h+k)\Big{)}\mathrm{d}h\
,$ $\displaystyle B_{t}^{(P)}[f](p)$ $\displaystyle\equiv
2\pi\int_{\Lambda^{*}}|\hat{V}(k)|^{2}\Big{(}\alpha^{P}_{t}(p+k,k)f(p+k)\widetilde{f}(p)-\alpha^{P}_{t}(p,k)f(p)\widetilde{f}(p-k)\Big{)}\mathrm{d}p\
,$

for $f\in\ell^{1}$ and $p,h\in\Lambda^{*}$. Here, the coefficients
$\alpha^{H}_{t}$ and $\alpha^{P}_{t}$ are defined as

$\displaystyle\alpha^{H}_{t}(h,k)$
$\displaystyle\equiv\chi(h)\chi(h+k)\int_{\Lambda^{*}}\chi(r)\chi^{\perp}(r+k)\delta_{t}\big{[}E_{h}-E_{h+k}-E_{r}-E_{r+k}\big{]}\mathrm{d}r\
,$ (2.30) $\displaystyle\alpha^{P}_{t}(p,k)$
$\displaystyle\equiv\chi^{\perp}(p)\chi^{\perp}(p-k)\int_{\Lambda^{*}}\chi(r)\chi^{\perp}(r+k)\delta_{t}\big{[}E_{p}-E_{p-k}-E_{r}-E_{r+k}\big{]}\mathrm{d}r\
,$ (2.31)

for all $p,h,k\in\Lambda^{*}.$

We will analyze error terms with respect to a weighted norm which we now
introduce. Indeed, for $m>0$ we introduce the following weight

$w_{m}(p)\equiv\begin{cases}\left\langle p\right\rangle^{m}\
,&p\in\mathcal{S}\\\ 1\ ,&p\in\Lambda^{*}\backslash\mathcal{S}\end{cases}\ .$
(2.32)

where $\left\langle p\right\rangle\equiv(1+p^{2})^{1/2}$ denotes the standard
Japanese bracket. We define the Banach space
$\ell_{m}^{1}\equiv\ell^{1}_{m}(\Lambda^{*})$ of functions
$\varphi:\Lambda^{*}\rightarrow\mathbb{C}$ for which the norm

$\displaystyle\|\varphi\|_{\ell^{1}_{m}}\equiv\int_{\Lambda^{*}}|\varphi(p)|w_{m}(p)\mathrm{d}p$
(2.33)

is finite. We will measure distances in the norm associated to the dual space
of $\ell^{1}_{m}(\Lambda^{*})$. Namely, we regard
$\ell_{m}^{1*}\equiv[\ell^{1}_{m}(\Lambda^{*})]^{*}$ as the Banach space of
functions $f:\Lambda^{*}\rightarrow\mathbb{C}$ endowed with the norm

$\|f\|_{\ell^{1*}_{m}}\equiv\sup_{p\in\Lambda^{*}}w_{m}(p)^{-1}|f(p)|=\sup_{\varphi\in\ell^{1}_{m}}\frac{|\left\langle\varphi,f\right\rangle|}{\|\varphi\|_{\ell_{m}^{1}}}$
(2.34)

where we denote by
$\left\langle\varphi,f\right\rangle\equiv\int_{\Lambda^{*}}\overline{\varphi(p)}f(p)\mathrm{d}p$
the coupling between $\ell_{m}^{1}$ and $\ell^{1*}_{m}$.

######  Remark 2.4.

As vector spaces, $\ell_{m}^{1}(\Lambda^{*})=\ell^{1}(\Lambda^{*})$ and
$\ell_{m}^{1*}(\Lambda^{*})=\ell^{\infty}(\Lambda^{*})$ for all $m>0$.
However, we choose to equip these spaces with the norms
$\|\cdot\|_{\ell_{m}^{1}}$ and $\|\cdot\|_{\ell_{m}^{1*}}$ since the weight
$w_{m}(p)$ appropriately records the decay near the Fermi surface
$\mathcal{S}$–this point will be crucial in our analysis. For completeness, we
record here the following inequality

$\|f\|_{\ell^{\infty}(\Lambda^{*}\backslash\mathcal{S})}\leqslant\|f\|_{\ell_{m}^{1*}}\leqslant\|f\|_{\ell^{\infty}(\Lambda^{*})}\
,\qquad\forall f\in\ell^{\infty}(\Lambda^{*})$ (2.35)

which we shall make use of, when studying the fixed volume case in the next
subsection.

######  Remark 2.5.

If $f\in\ell_{m}^{1*}$ is real-valued, one may restrict the supremum over
$\varphi\in\ell_{m}^{1}$ on the right hand side of (2.34) to be real-valued as
well.

Finally, let us now introduce two important new parameters. Namely, these are
the numbers $n$ and $R$ defined as

$n\equiv|\Lambda|\int_{\Lambda^{*}}f_{0}(p)\mathrm{d}p\qquad\textnormal{and}\qquad
R\equiv|\Lambda|p_{F}^{d-1}\simeq|\mathcal{S}|\ .$ (2.36)

Here, $n$ corresponds to the the initial number of particles and holes in the
system, and it measures the size of the perturbation of the Fermi ball. On the
other hand $R$ corresponds to the maximal number of bosonized particle-hole
pairs that can populate the Fermi surface $\mathcal{S}$, defined in (2.16).
Our main result is now stated as follows.

######  Theorem 1.

Let $f_{t}(p)$ be the momentum distribution of particles and holes, as given
in Definition 1. We assume that Condition 1 and 2 are satisfied, as well as
the bounds $1\leqslant n\leqslant CR^{1/2}$. Then, for all $m>0$ there exists
$C=C(m,d)>0$ such that for all $t\geqslant 0$ there holds

$\displaystyle
f_{t}\,=\,f_{0}\,+\,\lambda^{2}t\,B_{t}[f_{0}]\,+\,\lambda^{2}t\,Q_{t}[f_{0}]\,+\,\lambda^{2}t\,\textnormal{Rem}_{1}(t)\
,$ (2.37)

where $\textnormal{Rem}_{1}(t)$ is a reminder term that satisfies

$\|\textnormal{Rem}_{1}(t)\|_{\ell_{m}^{1*}}\ \leqslant\ C\,t\,e^{C\lambda
R\left\langle t\right\rangle}\,\bigg{(}\lambda
R^{2}(R^{\frac{1}{2}}+n^{2})\left\langle
t\right\rangle+\frac{R^{3}}{p_{F}^{m}}\bigg{)}\ .$ (2.38)

######  Remark 2.6.

At time zero, there holds $Q_{0}[f]=B_{0}[f]=0$. Hence, in order to prove that
the collision operators dominate the remainders terms, we consider longer time
scales in the next subsection for dynamics in the box
$\Lambda=(2\pi\mathbb{T})^{3}$. Here, we explain that for appropiately chosen
initial data

$\|B_{t}[f_{0}]\|_{\ell_{m}^{1*}}\simeq
tN^{1/3}\qquad\textnormal{and}\qquad\|Q_{t}[f_{0}]\|_{\ell_{m}^{1*}}\leqslant
tn\ .$ (2.39)

We will compare these sizes with the right hand side of (2.38), for $n\ll
N^{1/6}$ and $t\gg 1$

######  Remark 2.7.

In view of Remark 2.4, the above result is describing the dynamics of
particles and holes away from the Fermi surface, i.e. on
$\Lambda^{*}/\mathcal{S}$. This is consistent with the fact [10] that
bosonization occurs inside of the Fermi surface, whose dynamics is different
than the one predicted by the right hand side of (2.37).

######  Remark 2.8.

The physical situation in which $|\Lambda|\rightarrow\infty$ and
$\Lambda^{*}\rightarrow\mathbb{R}^{3}$ (that is, the continuum approximation)
will not be addressed in this article. However, let us mention that such limit
is within the scope of the estimates given by Theorem 1, provided one
restricts the size of $\lambda$. In this regard, it would be possible to
follow the general lines of [18] and study the limits of the operators $B_{t}$
and $Q_{t}$.

###  2.4. Fixed volume

Let us discuss in this section a scaling regime for which Theorem 1 turns into
an effective approximation.

Namely, we consider the three-dimensional torus $\Lambda=(2\pi\mathbb{T})^{3}$
with dual lattice $\Lambda^{*}=\mathbb{Z}^{3}$. and specialize to the
following scaling regime

$\lambda=1/N^{3/2+\delta_{1}},\qquad t=N^{1/6+\delta_{2}}T\qquad n\leqslant
N^{\frac{1}{6}}$ (2.40)

for positive parameters $0<\delta_{1}$, $\delta_{2}<1$ specified below.

The time scales that we consider are short enough to maintain control over the
remainders terms of Theorem 1, but long enough to observe one complete
collision. More precisely, the mollified delta function $\delta_{t}(\Delta E)$
incorporated in the definition of $Q_{t}$ and $B_{t}$, now becomes a Kronecker
delta function with respect to the free energy. In other words, we prove in
Lemma 10.1 in a suitable sense that

$\delta_{t}(\Delta E)=\frac{2t}{\pi}\delta_{\Delta
e,0}\Big{(}1+\mathcal{O}(1/t^{2})+\mathcal{O}(\lambda^{2}t^{2})\Big{)}$ (2.41)

where

$\Delta e\equiv e(p_{1})+e(p_{2})-e(p_{3})-e(p_{4})\
,\qquad\textnormal{where}\qquad
e(p)\equiv\big{[}\chi^{\perp}(p)-\chi(p)\big{]}\frac{p^{2}}{2}\ .$ (2.42)

As a consequence, in Lemma 10.2 and 10.3 we are able to identify the leading
order terms of the operators $Q_{t}$ and $B_{t}$ as follows

$\displaystyle Q_{t}[f]$
$\displaystyle=t\mathscr{Q}[f]+\mathcal{O}_{\ell^{\infty}}(1/t)+\mathcal{O}_{\ell^{\infty}}(t^{3}\lambda^{2}\|\hat{V}\|_{\ell^{1}}^{2})\
,$ $\displaystyle B_{t}[f]$
$\displaystyle=t\mathscr{B}[f]+\mathcal{O}_{\ell^{\infty}}(1/t)+\mathcal{O}_{\ell^{\infty}}(t^{3}\lambda^{2}\|\hat{V}\|_{\ell^{1}}^{2})\
.$ (2.43)

where $f\in\ell^{1}(\mathbb{Z}^{3}).$ Here, the operator $\mathscr{Q}[f]$ is
defined as in Def. 2 but with $\delta_{t}(\Delta E)$ being replaced by the
discrete Delta function on the lattice $(2/\pi)\delta_{\Delta e,0}$. The
definition of $\mathscr{B}$ is anologous.

In this context, the following result now follows as a corollary of Theorem 1,
Lemma 10.2 and 10.3, and the inequalities found in Eq. (2.35).

######  Theorem 2 (Fixed volume. First collision time).

Consider the same assumptions of Theorem 1, with the scaling regime (2.40) in
three dimensions. Assume that $\delta_{2}\leqslant\delta_{1}/2\equiv\delta$.
Then, for all $m>5$ there exists $C>0$ such that for all
$T\in[N^{-\delta/2},1]$ there holds

$f_{\epsilon^{-1}T}=f_{0}+(\lambda/\epsilon)^{2}T^{2}\Big{(}\mathscr{B}[f_{0}]+\mathscr{Q}[f_{0}]+\textnormal{Rem}_{2}(T)\Big{)}$
(2.44)

where $\textnormal{Rem}_{2}$ is a remainder term that satisfies

$\|\textnormal{Rem}_{2}(T)\|_{\ell^{\infty}(\mathbb{Z}^{3}\backslash\mathcal{S})}\leqslant
CN^{\frac{1}{3}}\bigg{(}\frac{1}{N^{\delta}}+\frac{1}{N^{(m-5)/3}}\bigg{)}=o(N^{\frac{1}{3}})\
.$ (2.45)

######  Remark 2.9.

Theorem 2 contains information about the evolution of $f_{t}(p)$ for
$p\in\mathbb{Z}^{3}\backslash\mathcal{S}$. That is, away from the Fermi
surface. For $p\in\mathcal{S}$ and $T\in[0,1]$, one actually has an
$\ell^{1}$-bound

$\|f_{\epsilon^{-1}T}\|_{\ell^{1}(\mathcal{S})}\,\leqslant\,C/N^{5/3+2\delta}$
(2.46)

which follows as a propagation-in-time of the depletion of the Fermi surface,
as stated in Condition 2 for the initial data $f_{0}$. See Proposition 5.2. In
words, the scaling is chosen so that the Fermi surface remains almost entirely
depleted over the scale $T.$

######  Remark 2.10 (Sizes of $\mathscr{Q}$ and $\mathscr{B}$).

The inequality contained in Theorem 2 shows that $\mathscr{B}$ and
$\mathscr{Q}$ dominate the remainder terms if

$\|\mathscr{Q}[f_{0}]+\mathscr{B}[f_{0}]\|_{\ell^{\infty}(\mathbb{Z}^{3}\backslash\mathcal{S})}\,\gg\,\|\textnormal{Rem}(N,n,T)\|_{\ell^{\infty}(\mathbb{Z}^{3}\backslash\mathcal{S})}\
.$ (2.47)

In Section 10 we prove that this holds for well-chosen initial data. Here, we
make an additional assumption on $\hat{V}(k)$ in Condition 3. More precisely,
for $N\geqslant 1$ large enough:

  1. (1)

We take $f(p)$ as a linear combination of Kronecker deltas in $\mathbb{Z}^{3}$
(see Def. 9). Further, we assume that they are supported away from each other
by a distance $r>0$, and that at least one of their cartesian components
satisfies $|p_{i}|\sim p_{F}$ (see Condition 4). We show that

$\|\mathscr{B}[f]\|_{\ell^{\infty}(\mathbb{Z}^{3}/\mathcal{S})}\simeq N^{1/3}\
.$ (2.48)

The heuristics for the $N^{1/3}$ dependence are as follows: given
$k\in\textnormal{supp}\hat{V}/\\{0\\}$ a fermion can interact with any of the
particle-hole pairs in the lune set

$L(k)\equiv\\{q\in\mathbb{Z}^{3}:|q|\leqslant p_{F},\ |q+k|\geqslant p_{F}\\}\
,$ (2.49)

which is of order $|L(k)|\sim N^{2/3}$. On the other hand, energy conservation
$\delta_{\Delta e,0}=0$ introduces a geometric constraint that reduces this
number by a factor $N^{1/3}$.

  2. (2)

A rather standard bound for $f\in\ell^{1}(\Lambda^{*})$ with $0\leqslant
f\leqslant 1$ shows that

$\|\mathscr{Q}[f]\|_{\ell^{\infty}(\mathbb{Z}^{3})}\leqslant
C\|f\|_{\ell^{1}(\mathbb{Z}^{3})}=Cn\ ,$ (2.50)

##  3\. Preliminaries

In this section, we introduce preliminaries that are needed to prove our main
result. First, we give an explicit representation of the particle-hole
Hamiltonian $\mathfrak{h}$, introduced in (2.12). Second, based on this
representation, we introduce the interaction picture framework that we shall
use to study the dynamics of the momentum distribution $f_{t}(p)$, defined in
(2.13). Third, we perform a double commutator expansion and identify nine
terms, from which we shall extract leading order and subleading order terms.
Finally, we introduce number estimates that we use to analyze the nine terms
found in the double commutator expansion.

###  3.1. Calculation of $\mathfrak{h}$

Let us introduce two fundamental collection of operators. We shall refer to
them informally as $D$\- and $b$-operators, respectively.

######  Definition 4.

Let $k\in\Lambda^{*}$.

(1) We define the $D$-operators as

$D_{k}\equiv\int_{\Lambda^{*}}\chi^{\perp}(p)\chi^{\perp}(p-k)a_{p-k}^{*}a_{p}\,\mathrm{d}p-\int_{\Lambda^{*}}\chi(h)\chi(h+k)a_{h+k}^{*}a_{h}\,\mathrm{d}h\
.$ (3.1)

(2) We define the $b$-operators as

$b_{k}\equiv\int_{\Lambda^{*}}\chi^{\perp}(p)\chi(p-k)a_{p-k}a_{p}\,\mathrm{d}p\
.$ (3.2)

######  Remark 3.1.

For the rest of the article, we denote the corresponding adjoint operators by
$D_{k}^{*}\equiv[D_{k}]^{*}$ and $b^{*}_{k}\equiv[b_{k}]^{*}$, respectively.
Additionally, we shall extensively use the basic relation

$D^{*}_{k}=D_{-k}\qquad\forall k\in\Lambda^{*}\ .$ (3.3)

######  Remark 3.2 (Heuristics).

One should understand the operator $D$ as a combination fermionic operators;
they intertwine only holes and holes, together with particles and particles,
that are away from the Fermi surface. One the other hand, the operators $b$
should be understood as an approximate bosonic operators; they
create/annihilate bosonized particle-hole pairs near the Fermi surface. In
fact, the following commutation relation holds

$[b_{k},D^{*}_{k}]=0\qquad\forall k\in\Lambda^{*}\ .$ (3.4)

However, we shall not need any estimates on the commutation relations
satisfied by $b$’s, and this interpretation will remain at a heuristic level.

The following lemma contains the explicit representation for the particle-hole
Hamiltonian, in terms of a “solvable Hamiltonian”, plus interaction terms
depending on $D$ and $b$ operators.

######  Lemma 3.1.

Let $\mathfrak{h}$ be the operator defined in (2.12). Then, the following
identity holds

$\mathfrak{h}-\mu_{1}\mathds{1}-\mu_{2}\mathcal{Q}=\mathfrak{h}_{0}+\lambda\mathcal{V}$
(3.5)

for some real-valued constants $\mu_{1},\mu_{2}\in\mathbb{R}$. Here
$\mathcal{Q}$ corresponds to the charge operator

$\mathcal{Q}\equiv\int_{\Lambda^{*}}\chi^{\perp}(p)a_{p}^{*}a_{p}\mathrm{d}p-\int_{\Lambda^{*}}\chi(p)a_{p}^{*}a_{p}\mathrm{d}p;$
(3.6)

$\mathfrak{h}_{0}$ corresponds to the quadratic, diagonal operator

$\mathfrak{h}_{0}=\int_{\Lambda^{*}}E_{p}a_{p}^{*}a_{p}\mathrm{d}p$ (3.7)

with $E_{p}$ the dispersion relation defined in (2.20); and
$\mathcal{V}=V_{F}+V_{FB}+V_{B}$ contains the following three interaction
terms

$\displaystyle V_{F}$
$\displaystyle\equiv\frac{1}{2}\int_{\Lambda^{*}}\hat{V}(k)D^{*}_{k}D_{k}\
\mathrm{d}k$ (3.8) $\displaystyle V_{FB}$
$\displaystyle\equiv\int_{\Lambda^{*}}\hat{V}(k)D^{*}_{k}\big{[}b_{k}+b^{*}_{-k}\big{]}\mathrm{d}k$
(3.9) $\displaystyle V_{B}$
$\displaystyle\equiv\int_{\Lambda^{*}}\hat{V}(k)\big{[}b^{*}_{k}b_{k}+\frac{1}{2}\,b^{*}_{k}b^{*}_{-k}+\frac{1}{2}\,b_{-k}b_{k}\big{]}\mathrm{d}k\
.$ (3.10)

######  Remark 3.3.

The labeling of $V_{F}$, $V_{FB}$ and $V_{B}$ is of course related to Remark
3.2. Namely, $V_{F}$ contains fermion/fermion interactions, $V_{FB}$ contains
fermion/boson interactions and $V_{B}$ contains boson/boson interactions.

######  Remark 3.4.

The charge operator $\mathcal{Q}$ is irrelevant for the dynamics in the
system. Indeed, one may easily check that
$[\mathfrak{h}_{0},\mathcal{Q}]=[D,\mathcal{Q}]=[b,\mathcal{Q}]=0$ and,
therefore, $[\mathfrak{h},\mathcal{Q}]=0$. In other words, the charge is a
constant of motion and only the right hand side of (3.5) is relevant regarding
the time evolution of the momenum distribution of the system. We make this
argument precise in the next subsubsection.

The proof of the above Lemma will not be given here, for it has already been
considered in the literature in a very similary form. The reader is refered
for instance to [9, pps 897-899].

###  3.2. The interaction picture

Let us now exploit the identity found in (3.5). First, recalling that the
Hamiltonian $\mathfrak{h}_{0}$ is quadratic and diagonal with respect to
creation and annihilation operators, we may easily calculate the associated
Heisenberg evolution to be given by

$\displaystyle a_{p}(t)$ $\displaystyle\equiv
e^{it\mathfrak{h}_{0}}a_{p}e^{-it\mathfrak{h}_{0}}=e^{-itE_{p}}a_{p}\ ,$
(3.11) $\displaystyle a_{p}^{*}(t)$ $\displaystyle\equiv
e^{it\mathfrak{h}_{0}}a_{p}^{*}e^{-it\mathfrak{h}_{0}}=e^{+itE_{p}}a_{p}^{*}\
,$ (3.12)

for all $p\in\Lambda^{*}$ and $t\in\mathbb{R}$ ; the dispersion relation
$E_{p}$ was defined in (2.20). Secondly, we introduce the interaction
Hamiltonian

$\mathfrak{h}_{I}(t)\equiv\lambda\,e^{it\mathfrak{h}_{0}}\mathcal{V}e^{-it\mathfrak{h}_{0}}\qquad\forall
t\in\mathbb{R}\ ,$ (3.13)

where $\mathfrak{h}_{0}$ and $\mathcal{V}$ are defined in Lemma 3.5.

We now introduce the dynamics associated to the interaction picture.

######  Definition 5.

Given an initial state $\nu:B(\mathscr{F})\rightarrow\mathbb{C}$, we denote by
$(\nu_{t})_{t\in\mathbb{R}}$ the solution of the intial value problem

$\displaystyle\begin{cases}&i\partial_{t}\nu_{t}(\mathcal{O})=\nu_{t}\big{(}[\mathfrak{h}_{I}(t),\mathcal{O}]\big{)}\quad\forall\mathcal{O}\in
B(\mathscr{F})\\\ &\nu_{0}=\nu\end{cases}$ (3.14)

which we shall refer to as the interaction dynamics.

The momentum distribution of the system $f_{t}(p)$, introduced in Def. 1, is
now linked to the interaction dynamics. Indeed, a standard calculation shows
that for all $t\in\mathbb{R}$ and $p\in\Lambda^{*}$, there holds

$f_{t}(p)=|\Lambda|^{-1}{\nu_{t}(a_{p}^{*}a_{p})}\ .$ (3.15)

In the next subsection, we shall use Eq. (3.15) to expand $f_{t}(p).$

###  3.3. Double commutator expansion

Let $f_{t}(p)$ be as in Eq. (3.15), and let us recall that $\nu$ is an initial
state satisfying Condition 2. In particular, quasi-freeness and translation
invariance imply that

$\nu([a_{p}^{*}a_{p},a_{k_{1}}^{\hash}a_{k_{2}}^{\hash}a_{k_{3}}^{\hash}a_{k_{4}}^{\hash}])=0\
,\qquad\forall\,k_{1},k_{2},k_{3},k_{4}\in\Lambda^{*}\ .$ (3.16)

Thus, upon expressing the Hamiltonian $\mathfrak{h}_{I}(t)$ in terms of
creation- and annihilation operators, one finds that
$\partial_{t}|_{t=0}f_{t}(p)=i|\Lambda|^{-1}\nu([a_{p}^{*}a_{p},\mathfrak{h}_{I}(0)])=0$.
Hence, the following third-order expansion holds true

$f_{t}(p)=f_{0}(p)-|\Lambda|^{-1}\int_{0}^{t}\int_{0}^{t_{1}}\nu_{t_{2}}\Big{(}[[a_{p}^{*}a_{p},\mathfrak{h}_{I}(t_{1})],\mathfrak{h}_{I}(t_{2})]\Big{)}\mathrm{d}t_{1}\mathrm{d}t_{2}$
(3.17)

for any $t\in\mathbb{R}$ and $p\in\Lambda^{*}.$ We dedicate the rest of this
article to the study of the right-hand side of the above equation.

Let us identify all of the terms in the double commutator expansion found
above. A straightforward expansion of the interaction Hamiltonian yields the
decomposition

$\mathfrak{h}_{I}(t)=\lambda\big{(}V_{F}(t)+V_{FB}(t)+V_{B}(t)\big{)}\qquad\forall
t\in\mathbb{R}$ (3.18)

where the interaction terms above according to the Heisenberg picture. Namely,
we set

$V_{\alpha}(t)\equiv\,e^{it\mathfrak{h}_{0}}V_{\alpha}\,e^{-it\mathfrak{h}_{0}}\qquad\forall
t\in\mathbb{R}\ ,\alpha\in\\{F,FB,B\\}\ .$ (3.19)

Upon expanding the right hand side of (3.17), one finds the following nine
terms

$\displaystyle f_{t}-f_{0}=$
$\displaystyle-\lambda^{2}|\Lambda|^{-1}\Big{(}T_{F,F}(t)+T_{F,FB}(t)+T_{F,B}(t)\Big{)}$
$\displaystyle-\lambda^{2}|\Lambda|^{-1}\Big{(}T_{FB,F}(t)+T_{FB,FB}(t)+T_{FB,B}(t)\Big{)}$
$\displaystyle-\lambda^{2}|\Lambda|^{-1}\Big{(}T_{B,F}(t)+T_{B,FB}(t)+T_{B,B}(t)\Big{)}$
(3.20)

where we set, for $t\in\mathbb{R}$ and $p\in\Lambda^{*}$

$\displaystyle
T_{\alpha,\beta}(t,p)\equiv\int_{0}^{t}\int_{0}^{t_{1}}\nu_{t_{2}}\Big{(}[[a_{p}^{*}a_{p},V_{\alpha}(t_{1})],V_{\beta}(t_{2})]\Big{)}\mathrm{d}t_{1}\mathrm{d}t_{2}\qquad\alpha,\beta\in\\{F,FB,B\\}\
.$ (3.21)

We shall analyze in detail the quantities
$T_{\alpha,\beta}:\mathbb{R}\times\Lambda^{*}\rightarrow\mathbb{R}$ when
tested against a smooth function. To this end, let us introduce some notation
we shall be using for the rest of this work. For
$\varphi:\Lambda^{*}\rightarrow\mathbb{C}$ we let

$N(\varphi)\equiv\int_{\Lambda^{*}}\overline{\varphi(p)}a_{p}^{*}a_{p}\mathrm{d}p$
(3.22)

together with

$\displaystyle
T_{\alpha,\beta}(t,\varphi)\equiv\left\langle\varphi,T_{\alpha,\beta}(t)\right\rangle=\int_{0}^{t}\int_{0}^{t_{1}}\nu_{t_{2}}\Big{(}[[N(\varphi),V_{\alpha}(t_{1})],V_{\beta}(t_{2})]\Big{)}\,\mathrm{d}t_{1}\mathrm{d}t_{2}\
.$ (3.23)

###  3.4. Excitation operators

The following two operators will play a major role in our analysis. They
correspond to the number operator (per unit volume) that counts the total
number of particles and holes in the system, together with the number operator
that only counts the number of particles and hole in the Fermi surface
$\mathcal{S}$. More precisely, we consider

######  Definition 6.

We define the two following operators in $\mathscr{F}$.

(1) The number operator as

$\mathcal{N}\equiv\int_{\Lambda^{*}}a_{p}^{*}a_{p}\mathrm{d}p\ .$ (3.24)

(2) The surface-localized number operator as

$\mathcal{N}_{\mathcal{S}}\equiv\int_{\mathcal{S}}a_{p}^{*}a_{p}\mathrm{d}p$
(3.25)

where $\mathcal{S}$ is the Fermi surface, defined in (2.16) .

######  Remark 3.5.

Let us recall that in Section 2 we have introduced the parameter
$R=|\Lambda|\int_{\mathcal{S}}\mathrm{d}p$. In particular, it follows from the
boundedness of creation- and annihilation- operators that
$\mathcal{N}_{\mathcal{S}}$ is a bouned operator and
$\|\mathcal{N}_{\mathcal{S}}\|_{B(\mathscr{F})}\leqslant R$.

######  Remark 3.6 (Domains).

$\mathcal{N}$ is an unbounded self-adjoint operator in $\mathscr{F}$ with
domain $\mathcal{D}(\mathcal{N})=\\{\Psi=(\psi_{n})_{n\geqslant
0}\in\mathscr{F}:\sum_{n\geqslant
0}n^{2}\|\psi_{n}\|^{2}_{L^{2}(\Lambda^{n})}<\infty\\}\ .$ As initial data,
the mixed states that we work with satisfy

$\nu(\mathcal{N})\equiv\int_{\Lambda^{*}}\nu(a_{p}^{*}a_{p})\mathrm{d}p=\int_{\Lambda^{*}}f_{0}(p)\mathrm{d}p<\infty\
,$ (3.26)

and similarly for higher powers $\mathcal{N}^{k}$. It is standard to show that
the time evolution generated by the particle-hole Hamiltonian $\mathfrak{h}$,
as defined in (2.12), preserves $\mathcal{D}(\mathcal{N})$, in the sense that
$\nu_{t}(\mathcal{N}^{k})<\infty$ for $t\in\mathbb{R}$ and $k\in\mathbb{N}$.
In order to simplify the exposition, we shall purposefuly not refer to the
unbounded nature of the operator $\mathcal{N}$ in the rest of the article.

The proof of Theorem 1 relies on the fact that the subleading order terms that
arise from the double commutator expansion –written in terms of $b$\- and
$D$-operators– can be bounded above by expectations of the operators
$\mathcal{N}$ and $\mathcal{N}_{\mathcal{S}}$, with respect to the evolution
of the state $\nu$ driven by the interaction Hamiltonian
$\mathfrak{h}_{I}(t)$. This analysis is carried out in Section 4. Further, in
Section 5 we prove bounds for the growth-in-time of the expectations
$\nu_{t}(\mathcal{N})$ and $\nu_{t}(\mathcal{N}_{\mathcal{S}})$. This two-step
analysis is combined in Section 9 to prove Theorem 1.

##  4\. Tool Box I: Analysis of $b$\- and $D$-operators

In the last section, we introduced the time evolution of certain observables
in the Heisenberg picture, with respect to the solvable Hamiltonian
$\mathfrak{h}_{0}$, introduced in (3.7). In particular, the evolution of the
creation- and annihilation- operators $a$ and $a^{*}$ take the simple form

$\displaystyle a_{p}(t)=e^{-itE_{p}}a_{p}\qquad\textnormal{and}\qquad
a_{p}^{*}(t)=e^{+itE_{p}}a_{p}^{*}\ ,$ (4.1)

for all $p\in\Lambda^{*}$ and $t\in\mathbb{R}$ ; the dispersion relation
$E_{p}$ was defined in (2.20). Let us now introduce the Heisenberg evolution
of the $b$\- and $D$-operators as follow.

######  Definition 7.

Let $k\in\Lambda^{*}$ and $t\in\mathbb{R}$.
(1) The Heisenberg evolution of the $D$-operators is given by

$D_{k}(t)\equiv
e^{it\mathfrak{h}_{0}}D_{k}e^{-it\mathfrak{h}_{0}}=\int_{\Lambda^{*}}\chi^{\perp}(p,p-k)a_{p-k}^{*}(t)a_{p}(t)\,\mathrm{d}p-\int_{\Lambda^{*}}\chi(h,h+k)a_{h+k}^{*}(t)a_{h}(t)\,\mathrm{d}h$

and $D_{k}^{*}(t)\equiv[D_{k}(t)]^{*}=D_{-k}(t)$.

(2) The Heisenberg evolution of the $b$-operators is given by

$b_{k}(t)\equiv
e^{it\mathfrak{h}_{0}}b_{k}e^{-it\mathfrak{h}_{0}}=\int_{\Lambda^{*}}\chi^{\perp}(p)\chi(p-k)a_{p-k}(t)a_{p}(t)\,\mathrm{d}p$

and $b_{k}^{*}(t)\equiv[b_{k}(t)]^{*}$.

The main goal of this section is to introduce a systematic calculus that lets
us deal with combination of the operators $b_{k}(t)$ and $D_{k}(t)$–together
with multiple combination of their commutators– as they show up in the
analysis of the double commutator expansion found in (3.20). First, we
introduce many useful identities required for the upcoming analysis. Secondly,
we state estimates for several combinations of $b$\- and $D$-operators.

###  4.1. Identities

In this subsection, we record useful identities between operators in
$\mathscr{F}$ that we shall use extensively in the rest of this article. Most
importantly, in the next subsection we shall use these identities to obtain
estimates of importantes commutator observables.

Preliminary identities . First, we write general time-independent relations.

1) For all $p,q,r\in\Lambda^{*}$ the CAR imply that

$\displaystyle[a_{r}^{*}a_{r},a_{p}^{*}a_{q}]=\big{(}\delta(r-q)-\delta(r-p)\big{)}a_{p}^{*}a_{q}$
(4.2)

2) For all $p,q\in\Lambda^{*}$ and $\varphi\in\ell^{1}(\Lambda^{*})$ there
holds

$[N(\varphi),a_{p}^{*}a_{q}]=\Big{(}\overline{\varphi(p)}-\overline{\varphi(q)}\Big{)}a_{p}^{*}a_{q}$
(4.3)

where we recall
$N(\varphi)=\int_{\Lambda^{*}}\overline{\varphi(p)}a_{p}^{*}a_{p}\mathrm{d}p.$

Commutator identities. The following lemma contains useful operator
identities, to be used in the next section. Since they only rely on the CAR
and straightforward commutator calculations, we leave their proof to the
reader.

######  Lemma 4.1.

Let $k,\ell\in\Lambda^{*}$ and $t,s\in\mathbb{R}$ .
(1) For $p\in\mathcal{B}^{c}$ and $h\in\mathcal{B}$ there holds

$\displaystyle[b_{k}(s),a_{p}^{*}(t)]$ $\displaystyle\ =\
\chi(p-k)\,e^{i(t-s)E_{p}}a_{p-k}(s)\ ,$ (4.4)
$\displaystyle[b_{k}(s),a_{h}^{*}(t)]$ $\displaystyle\ =\
-\chi^{\perp}(h+k)\,e^{i(t-s)E_{h}}a_{h+k}(s)\ .$ (4.5)

(2) There holds

$\displaystyle[b_{\ell}(s),D_{k}^{*}(t)]$
$\displaystyle=\int_{\Lambda^{*}}\chi^{\perp}(p)\chi^{\perp}(p-k)\chi(p-\ell)e^{i(t-s)E_{p}}a_{p-\ell}(s)a_{p-k}(t)\mathrm{d}p$
$\displaystyle+\int_{\Lambda^{*}}\chi(h)\chi(h+k)\chi^{\perp}(h+\ell)e^{i(t-s)E_{h}}a_{h+\ell}(s)a_{h+k}(t)\mathrm{d}h\
.$ (4.6)

In particular, $[b_{k}(t),D_{k}^{*}(s)]=0$.

(3) There holds

$\displaystyle[b_{k}(t),b_{\ell}^{*}(s)]$ $\displaystyle=\
\delta(k-\ell)\int_{\Lambda^{*}}\chi^{\perp}(p)\chi(p-k)e^{-i(t-s)(E_{p}+E_{p-k})}\mathrm{d}p$
$\displaystyle-\int_{\Lambda^{*}}\chi^{\perp}(p)\chi^{\perp}(p+\ell-k)\chi(p-k)e^{-i(t-s)E_{p-k}}a_{p}^{*}(t)a_{p+\ell-k}(s)\,\mathrm{d}p$
$\displaystyle-\int_{\Lambda^{*}}\chi(h)\chi(h+\ell-k)\chi^{\perp}(h+\ell)e^{-i(t-s)E_{h+k}}a_{h}^{*}(t)a_{h+\ell-k}(s)\,\mathrm{d}h\
.$ (4.7)

Besides the $b$\- and $D$\- operators, we shall work extensively with their
contracted versions. These are defined as follows in terms of
$N(\varphi)=\int_{\Lambda^{*}}\overline{\varphi(p)}a_{p}^{*}a_{p}\mathrm{d}p.$

######  Definition 8 (Contractions).

Let $\varphi:\Lambda^{*}\rightarrow\mathbb{C}$. Then, we define the
$D(\varphi)$-operators as the collection of operators for $t\in\mathbb{R}$ and
$k\in\Lambda^{*}$

$\displaystyle
D_{k}(t,\varphi)\equiv[N(\varphi),D_{k}(t)]\qquad\textnormal{and}\qquad
D_{k}^{*}(t,\varphi)\equiv[N(\varphi),D_{k}^{*}(t)]\ .$ (4.8)

Similarly, we define the $b(\varphi)$-operators as the collection of operators
for $t\in\mathbb{R}$ and $k\in\Lambda^{*}$

$\displaystyle
b_{k}(t,\varphi)\equiv[N(\varphi),b_{k}(t)]\qquad\textnormal{and}\qquad
b_{k}^{*}(t,\varphi)\equiv[N(\varphi),b_{k}^{*}(t)]\ .$ (4.9)

We call them the contractions of $b$ and $D$ with the external function
$\varphi$.

######  Remark 4.1.

We immediately note that

$\displaystyle[D_{k}(t,\varphi)]^{*}=-D^{*}_{k}(t,\bar{\varphi})\qquad[b_{k}(t,\varphi)]^{*}=-b^{*}_{k}(t,\bar{\varphi})$
(4.10)

for all $t\in\mathbb{R}$ and $k\in\Lambda^{*}$. Thus, the contractions are not
adjoints of each other. However, the following relations hold true for all
$\Psi\in\mathscr{F}$

$\|[D_{k}(t,\varphi)]^{*}\Psi\|_{\mathscr{F}}=\|D_{k}^{*}(t,\bar{\varphi})\Psi\|_{\mathscr{F}}\qquad\textnormal{and}\qquad\|[b_{k}(t,\varphi)]^{*}\Psi\|_{\mathscr{F}}=\|b_{k}^{*}(t,\bar{\varphi})\Psi\|_{\mathscr{F}}\
.$ (4.11)

Since final estimates are given in terms of $\ell^{p}$ norms of $\varphi$, the
complex conjugation does not affect the end result. Thus, when proving
estimates, one may regard them as adjoints of each other.

######  Remark 4.2.

The contractions can of course be calculated explicitly using the CAR. Let us
record here the two following calculations

$\displaystyle D_{k}^{*}(t,\varphi)$
$\displaystyle=\int_{\Lambda^{*}}\chi^{\perp}(p)\chi^{\perp}(p-k)\big{[}\varphi(p)-\varphi(p-k)\big{]}a_{p}^{*}(t)a_{p-k}(t)\mathrm{d}p$
$\displaystyle\quad-\int_{\Lambda^{*}}\chi(h)\chi(h+k)\big{[}\varphi(h)-\varphi(h+k)\big{]}a_{h}^{*}(t)a_{h+k}(t)\mathrm{d}h\
,$ (4.12) $\displaystyle b_{k}(t,\varphi)$
$\displaystyle=\int_{\Lambda^{*}}\chi^{\perp}(q)\chi(q-k)\Big{[}\varphi(q-k)+\varphi(q)\Big{]}a_{q-k}(t)a_{q}(t)\
\mathrm{d}q\ .$ (4.13)

Let us now state in the following Lemmas some useful commutation relations.
Since they all follow from straightforward manipulation of the CAR, we leave
them as an exercise for the reader.

######  Lemma 4.2.

Let $k,\ell\in\Lambda^{*}$, $t,s\in\mathbb{R}$ and $\varphi\in\ell^{1}$.
(1) There holds

$\displaystyle[b_{\ell}(s),$ $\displaystyle D_{k}^{*}(t,\varphi)]$ (4.14)
$\displaystyle=\int_{\Lambda^{*}}\chi^{\perp}(p)\chi^{\perp}(p-k)\chi(p-\ell)\big{[}\varphi(p)-\varphi(p-k)\big{]}e^{i(t-s)E_{p}}a_{p-\ell}(s)a_{p-k}(t)\mathrm{d}p$
$\displaystyle+\int_{\Lambda^{*}}\chi(h)\chi(h+k)\chi^{\perp}(h+\ell)\big{[}\varphi(h)-\varphi(h+k)\big{]}e^{i(t-s)E_{h}}a_{h+\ell}(s)a_{h+k}(t)\mathrm{d}h\
.$

######  Lemma 4.3 ($\mathcal{N}$ commutators).

For all $k\in\Lambda^{*}$ and $t\in\mathbb{R}$ the following holds true.

(1) For the $D$-operators

$[D_{k}(t),\mathcal{N}]=[D_{k}^{*}(t),\mathcal{N}]=0$ (4.15)

and similarly for the contracted operators $D_{k}(t,\varphi)$ and
$D_{k}^{*}(t,\varphi)$.

(2) For the $b$-operators, for any measurable function
$f:\mathbb{R}\rightarrow\mathbb{C}$ the pull-through formulae holds true

$\displaystyle
f(\mathcal{N})b_{k}(t)=b_{k}(t)f(\mathcal{N}-2)\quad\textnormal{and}\quad
f(\mathcal{N})b_{k}^{*}(t)=b_{k}^{*}(t)f(\mathcal{N}+2)$ (4.16)

and similarly for the contracted operators $b_{k}(t,\varphi)$ and
$b_{k}^{*}(t,\varphi)$.

######  Lemma 4.4 ($\mathcal{N}_{\mathcal{S}}$ commutators).

For all $k\in 3\textnormal{supp}\hat{V}$ and $t\in\mathbb{R}$ the following
commutation relations hold true

$\displaystyle[\mathcal{N}_{\mathcal{S}},b_{k}(t)]=-2b_{k}(t)\quad\textnormal{and}\quad[\mathcal{N}_{\mathcal{S}},b_{k}^{*}(t)]=+2b_{k}^{*}(t)\
.$ (4.17)

###  4.2. Estimates

In this subsection we state estimates that shall be used extensively for the
rest of this article. Most of these are operator estimates for observables in
$\mathscr{F}$ containing the fermionic creation- and annihilation- operators
$a_{p}$ and $a_{p}^{*}$. We remind the reader that these are bounded operators
with norm $\|a_{p}\|_{B(\mathscr{F})}\,=\,\|a_{p}^{*}\|_{B(\mathscr{F})}\
\leqslant\ |\Lambda|^{1/2}$ for all $p\in\Lambda^{*}\ .$

Preliminary estimates. Let us state without proof elementary estimates that we
shall make use of.

1) For any function $f:\Lambda^{*}\rightarrow\mathbb{C}$, $k\in\Lambda^{*}$
and $\Psi\in\mathscr{F}$ there holds

$\Big{\|}\int_{\Lambda^{*}}f(p)a_{p+k}^{*}a_{p}\mathrm{d}p\Psi\,\Big{\|}_{\mathscr{F}}\leqslant\
\|f\|_{\ell^{\infty}}\|\mathcal{N}\Psi\|_{\mathscr{F}}\ .$ (4.18)

2) The Heisenberg evolution of the creation- and annihilation- operators
$a_{p}(t)$ and $a_{p}^{*}(t)$ are bounded operators in $\mathscr{F}$, with
norms

$\|a_{p}(t)\|_{B(\mathscr{F})}\,=\,\|a_{p}^{*}(t)\|_{B(\mathscr{F})}\
\leqslant\ |\Lambda|^{1/2},\qquad\forall t\in\mathbb{R},\ p\in\Lambda^{*}\ .$
(4.19)

3) The Heisenberg evolution of the $b$-operators are bounded operators in
$\mathscr{F}$ with norms

$\|b_{k}(t)\|_{B(\mathscr{F})}\,=\,\|b^{*}_{k}(t)\|_{B(\mathscr{F})}\
\leqslant\ |\Lambda|\int_{\Lambda^{*}}\chi^{\perp}(p)\chi(p-k)\mathrm{d}p\
\lesssim\ R$ (4.20)

for all $k\in\textnormal{supp}\hat{V}$ and $t\in\mathbb{R}$. Let us recall
that $R=|\Lambda|p_{F}^{d-1}$ .

Commutator estimates. Let us now describe the most important estimates
concerning $b$\- and $D$-operators. Essentially, commutators between $b$\- and
$D$-operators–together with their contracted versions $b(\varphi)$ and
$D(\varphi)$–can be classified into four types, depending on the estimate they
verify. It turns out that these four type of estimate exhaust all
possibilities that show up in the double commutator expansion for $f_{t}(p)$.
In other words, these estimates are enough to analyze the nine terms
$\\{T_{\alpha,\beta}(t,p)\\}_{\alpha,\beta\in\\{F,FB,B\\}}$.

We remind the reader of the relation $D_{k}^{*}(t)=D_{-k}(t)$, valid for all
$k\in\Lambda^{*}$ and $t\in\mathbb{R}$. In particular, all of the upcoming
inequalities are valid if we replace $D$ by $D^{*}$. On the other hand, we
warn the reader that this property does not hold for $b$-operators in general.

The first type of estimate concerns the combination of operators that are
relatively bounded with respect to the number operator
$\mathcal{N}=\int_{\Lambda^{*}}a_{p}^{*}a_{p}\mathrm{d}p$, or any of its
powers. We call these Type-I estimates. They are contained in the following
lemma.

######  Lemma 4.5 (Type-I estimates).

There exists a constant $C>0$ such that for any $\Psi\in\mathscr{F}$,
$k,\ell\in\Lambda^{*}$, and $t,s,r\in\mathbb{R}$ the following inequalities
hold true

$\displaystyle\|D_{k}(t)\Psi\|_{\mathscr{F}}$ $\displaystyle\ \leqslant\
C\|\mathcal{N}\Psi\|_{\mathscr{F}}$ (4.21)
$\displaystyle\|[D_{k}(t),D_{\ell}(s)]\Psi\|_{\mathscr{F}}$ $\displaystyle\
\leqslant\ C\|\mathcal{N}\Psi\|_{\mathscr{F}}$ (4.22)
$\displaystyle\|[D_{k}(t),D_{\ell}(s)D_{\ell}(r)]\Psi\|_{\mathscr{F}}$
$\displaystyle\ \leqslant\ C\|\mathcal{N}^{2}\Psi\|_{\mathscr{F}}\ .$ (4.23)

The second type of estimates concerns combination of operators that can be
bounded above by the surface-localized number operator
$\mathcal{N}_{\mathcal{S}}=\int_{\mathcal{S}}a_{p}^{*}a_{p}\mathrm{d}p$, up to
pre-factors that can grow with the recurring parameter
$R=|\Lambda|p_{F}^{d-1}$. We call these Type-II estimates, and they are
contained in the following lemma

######  Lemma 4.6 (Type-II estimates).

There exists a constant $C>0$ such that for any $\Psi\in\mathscr{F}$,
$k,\ell,q\in\textnormal{supp}\hat{V}$, and $t,s,r\in\mathbb{R}$ the following
inequalities hold true

$\displaystyle\|b_{k}(t)\Psi\|_{\mathscr{F}}$ $\displaystyle\ \leqslant\
CR^{\frac{1}{2}}\,\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|_{\mathscr{F}}$
(4.24) $\displaystyle\|[b_{\ell}(t),D_{k}(s)]\Psi\|_{\mathscr{F}}$
$\displaystyle\ \leqslant\
CR^{\frac{1}{2}}\,\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|_{\mathscr{F}}$
(4.25) $\displaystyle\|[[b_{\ell}(t),D_{k}(s)],D_{q}(r)]\Psi\|_{\mathscr{F}}$
$\displaystyle\ \leqslant\
CR^{\frac{1}{2}}\,\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|_{\mathscr{F}}\ .$
(4.26)

######  Remark 4.3.

In certain proofs, it will be convenient to use the upper bound

$\mathcal{N}_{S}\leqslant\mathcal{N}.$

The reader should then have in mind that the (weaker) version of the estimates
contained in Lemma 4.6, in which $\mathcal{N}_{\mathcal{S}}$ is replaced by
$\mathcal{N}$, also holds true.

The third type of estimate corresponds to combination of operators that have
been contracted with a test function $\varphi\in\ell_{m}^{1}$, and their
operator norm can be bounded above in terms of the integral

$\int_{\mathcal{S}}|\varphi(p)|\mathrm{d}p\ \lesssim\
p_{F}^{-m}\|\varphi\|_{\ell_{m}^{1}}\ .$ (4.27)

We call these Type-III estimates, and they are contained in the following
lemma.

######  Lemma 4.7 (Type-III estimates).

Let $m>0$. There exists a constant $C>0$ such that for all
$k,\ell,q\in\textnormal{supp}\hat{V},\,$ $t,s,r\in\mathbb{R}$ and
$\varphi\in\ell^{1}_{m}(\Lambda^{*})$ the following inequalities true

$\displaystyle\|b_{k}(t,\varphi)\|_{B(\mathscr{F})}$ $\displaystyle\
\leqslant\ C\,|\Lambda|\,p_{F}^{-m}\,\|\varphi\|_{\ell_{m}^{1}}$ (4.28)
$\displaystyle\|[b_{\ell}(t),D_{k}(s,\varphi)]\|_{B(\mathscr{F})}$
$\displaystyle\ \leqslant\
C\,|\Lambda|\,p_{F}^{-m}\,\|\varphi\|_{\ell_{m}^{1}}$ (4.29)
$\displaystyle\|[[b_{k}(t),D_{\ell}(s)],D_{q}(r,\varphi)]\|_{B(\mathscr{F})}$
$\displaystyle\ \leqslant\
C\,|\Lambda|\,p_{F}^{-m}\,\|\varphi\|_{\ell_{m}^{1}}$ (4.30)

######  Remark 4.4.

Type-III estimates are symmetric with respect to the exchange of $b$ and
$b^{*}$. This property follows from the relation
$\|\mathcal{O}\|_{B(\mathscr{F})}=\|\mathcal{O}^{*}\|_{B(\mathscr{F})}$ and
the symmetry $D_{k}^{*}(t)=D_{-k}(t)$.

The fourth and final type of estimate corresponds to combination of operators
that have been contracted with a test function $\varphi\in\ell_{m}^{1}$, and
their operator norm can be bounded above in terms of the integral

$\int_{\Lambda^{*}}|\varphi(p)|\mathrm{d}p\ =\ \|\varphi\|_{\ell^{1}}\
\lesssim\ \|\varphi\|_{\ell_{m}^{1}}\ ,$ (4.31)

and a pre-factor, depending on the volume of the box $|\Lambda|.$ We call
these Type-IV estimates, and they are contained in the following lemma.

######  Lemma 4.8 (Type-IV estimates).

There exists a constant $C>0$ such that for all $k,\ell,q\in\Lambda^{*},\,$
$t,s,r\in\mathbb{R}$ and $\varphi\in\ell^{1}(\Lambda^{*})$ the following
inequalities true the following holds true

$\displaystyle\|D_{k}(t,\varphi)\|_{B(\mathscr{F})}$ $\displaystyle\
\leqslant\ C\,|\Lambda|\|\varphi\|_{\ell^{1}}$ (4.32)
$\displaystyle\|[D_{k}(t,\varphi),D_{\ell}(s)]\|_{{B}(\mathscr{F})}$
$\displaystyle\ \leqslant\ C\,|\Lambda|\|\varphi\|_{\ell^{1}}\ .$ (4.33)

####  4.2.1. Proof of Lemmata

In this subsection, we provide sketches for the proofs of Lemmas 4.5, 4.6, 4.7
and 4.8.

###### Sketch of Proof of Lemma 4.5.

Let us fix $\Psi\in\mathscr{F}$, $k,\ell\in\Lambda^{*}$, and
$t,s,r\in\mathbb{R}$.

Proof of (1). We shall make use of the elementary estimate found in (4.18). To
this end, starting from (7) we decompose

$D_{k}(t)=\int_{\Lambda^{*}}f^{(1)}(t,k,p)a_{p-k}^{*}a_{p}\mathrm{d}p+\int_{\Lambda^{*}}f^{(2)}(t,k,h)a_{h+k}^{*}a_{h}\mathrm{d}h$
(4.34)

where $f^{(1)}(t,k,p)=\chi^{\perp}(p,p-k)e^{it(E_{p-k}-E_{p})}$ and
$f^{(2)}(t,k,h)=\chi(h,h+k)e^{it(E_{h+k}-E_{h})}$. Clearly,
$\|f^{(1)}(t,k)\|_{\ell^{\infty}}=\|f^{(2)}(t,k)\|_{\ell^{\infty}}=1$. Hence,
it follows that $\|D_{k}(t)\Psi\|_{\mathscr{F}}\leqslant
2\|\mathcal{N}\Psi\|_{\mathscr{F}}$.

Proof of (2) The proof is extremely similar–it suffices to note that the
commutator can be calculated explicitly to be

$\displaystyle[D_{k}(t),D_{\ell}(s)]=$
$\displaystyle\int_{\Lambda^{*}}\chi^{\perp}(p,p-\ell,p-k-\ell)e^{i(s-t)E_{p-\ell}}a^{*}_{p-k-\ell}(t)a_{p}(s)\mathrm{d}p$
$\displaystyle\quad-\int_{\Lambda^{*}}\chi^{\perp}(p,p-k,p-k-\ell)e^{i(t-s)E_{p-k}}a^{*}_{p-k-\ell}(s)a_{p}(t)\mathrm{d}p$
$\displaystyle\quad+\int_{\Lambda^{*}}\chi(h,h+\ell,h+k+\ell)e^{i(s-t)E_{h+\ell}}a^{*}_{h+k+\ell}(t)a_{h}(s)\mathrm{d}h$
$\displaystyle\quad-\int_{\Lambda^{*}}\chi(h,h+k,h+k+\ell)e^{i(t-s)E_{h+k}}a^{*}_{h+k\ell}(s)a_{h}(t)\mathrm{d}h\
.$ (4.35)

Hence, the same argument shows that
$\|[D_{k}(t),D_{\ell}(s)]\Psi\|_{\mathscr{F}}\leqslant
4\|\mathcal{N}\Psi\|_{\mathscr{F}}$.

Proof of (3). For simplicity, let us supress the time labels, and the momentum
variables. In what follows $C>0$ is a contant whose value may change from line
to line. We calculate using the previous results, and the commutation
relations $[\mathcal{N},D]=0$

$\displaystyle\|[D,DD]\Psi\|_{\mathscr{F}}$
$\displaystyle\leqslant\|D[D,D]\Psi\|_{\mathscr{F}}+\|[D,D]D\Psi\|_{\mathscr{F}}$
$\displaystyle\leqslant
C\|\mathcal{N}[D,D]\Psi\|_{\mathscr{F}}+C\|\mathcal{N}D\Psi\|_{\mathscr{F}}$
$\displaystyle=C\|[D,D]\mathcal{N}\Psi\|_{\mathscr{F}}+\|CD\mathcal{N}\Psi\|_{\mathscr{F}}$
$\displaystyle\leqslant C\|\mathcal{N}^{2}\Psi\|_{\mathscr{F}}\ .$ (4.36)

This finishes the proof. ∎

###### Sketch of Proof of Lemma 4.6.

Let us fix $\Psi\in\mathscr{F}$, $k,\ell,q\in\textnormal{supp}\hat{V}$, and
$t,s,r\in\mathbb{R}$.

Let us give the main ideas behind the proof. Let us recall that
$\textnormal{supp}\hat{V}$ is contained in a ball of radius $r>0$. For
$n\in\mathbb{N}$, define the Fermi surfaces

$\mathcal{S}(n)\equiv\\{p\in\Lambda^{*}\ :\ p_{F}-nr\leqslant|p|\leqslant
p_{F}+nr\\},$ (4.37)

and the number operators
$\mathcal{N}_{\mathcal{S}(n)}\equiv\int_{\mathcal{S}(n)}a_{p}^{*}a_{p}\mathrm{d}p$.
In particular, we are denoting $\mathcal{S}=\mathcal{S}(3)$ in (2.6). Given
$k,\ell\in\textnormal{supp}\hat{V}$, consider operators of the form

$\displaystyle\beta_{k}\equiv\int_{\Lambda^{*}}\,\mathds{1}_{\mathcal{S}(1)}(p)\,a_{p+k}a_{p}\,\mathrm{d}p\
,\qquad\textnormal{and}\qquad\mathcal{D}_{\ell}\equiv\int_{\Lambda^{*}}a_{p+\ell}^{*}a_{p}\mathrm{d}p\
.$ (4.38)

One should think generically of $\beta_{k}$ as $b_{k}(t)$ and
$\mathcal{D}_{\ell}$ as $D_{\ell}(s)$. We make the following two observations.
First, $\beta_{k}$ can be controlled by $\mathcal{N}_{\mathcal{S}(1)}$ in the
following sense

$\displaystyle\|\beta_{k}\Psi\|_{\mathscr{F}}$
$\displaystyle\leqslant|\Lambda|^{\frac{1}{2}}\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}(1)}(p)\|a_{p}\Psi\|_{\mathscr{F}}$
$\displaystyle\leqslant|\Lambda|^{\frac{1}{2}}\Big{(}\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}(1)}(p)\mathrm{d}p\Big{)}^{\frac{1}{2}}\Big{(}\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}(1)}(p)\|a_{p}\Psi\|_{\mathscr{F}}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim|\Lambda|^{\frac{1}{2}}p_{F}^{\frac{d-1}{2}}\|\mathcal{N}_{\mathcal{S}(1)}^{\frac{1}{2}}\Psi\|_{\mathscr{F}}=R^{\frac{1}{2}}\|\mathcal{N}_{\mathcal{S}(1)}^{\frac{1}{2}}\Psi\|_{\mathscr{F}}\
,$ (4.39)

where we used a basic geometric estimate to find that
$\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}(1)}(p)\mathrm{d}p\lesssim
p_{F}^{d-1}.$ Secondly, the commutator between $\beta_{k}$ and
$\mathcal{D}_{\ell}$ can be calculated to be

$\displaystyle[\beta_{k},\mathcal{D}_{\ell}]=\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}(1)}(p-\ell)a_{p+k-\ell}a_{p}\mathrm{d}p+\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}(1)}(p)a_{p+k-\ell}a_{p}\mathrm{d}p\
.$ (4.40)

Since both $k,\ell\in\textnormal{supp}\hat{V}$, it holds that
$\mathds{1}_{\mathcal{S}(1)}(p-\ell)\leqslant\mathds{1}_{\mathcal{S}(2)}(p)$,
and of course
$\mathds{1}_{\mathcal{S}(1)}(p)\leqslant\mathds{1}_{\mathcal{S}(2)}(p)$.
Consequently, the same argument that we used to obtain (4.39) can now be
repeated on each term of the above equation to obtain

$\|[\beta_{k},\mathcal{D}_{\ell}]\Psi\|_{\mathscr{F}}\lesssim
R^{\frac{1}{2}}\|\mathcal{N}_{\mathcal{S}(2)}^{\frac{1}{2}}\Psi\|_{\mathscr{F}}\
.$ (4.41)

The same argument can be repeated for the next commutator with
$\mathcal{D}_{q}$, provided one enlarges the Fermi surface from
$\mathcal{S}(2)$ to $\mathcal{S}(3)$. In other words, it holds that

$\|[[\beta_{k},\mathcal{D}_{\ell}],\mathcal{D}_{q}]\Psi\|_{\mathscr{F}}\lesssim
R^{\frac{1}{2}}\|\mathcal{N}_{\mathcal{S}(3)}^{\frac{1}{2}}\Psi\|_{\mathscr{F}}\
.$ (4.42)

The above motivation contains the main ideas for the proof of the lemma. One
merely has to include additional bounded coefficients in the definition of
$\beta_{k}$ and $\mathcal{D}_{\ell}$ to account for the dependence on
$t\in\mathbb{R}$ and $k\in\Lambda^{*}$, that comes from $b_{k}(t)$ and
$D_{\ell}(s)$. We leave the details to the reader. ∎

###### Sketch of Proof of Lemma 4.7.

Let us fix $m>0$, $k,\ell,q\in\textnormal{supp}\hat{V},\,$
$t,s,r\in\mathbb{R}$ and $\varphi\in\ell^{1}_{m}(\Lambda^{*})$. Starting from
Eq. (4.13) we easily estimate that

$\displaystyle\|b_{k}(t,\varphi)\|_{B(\mathscr{F})}\leqslant
2|\Lambda|\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(p)|\varphi(p)|\mathrm{d}p\
.$ (4.43)

It suffices then to note that
$\int_{\mathcal{S}}|\varphi(p)|\mathrm{d}p\lesssim
p_{F}^{-m}\|\varphi\|_{\ell_{m}^{1}}$. For the next estimate, the same
analysis can be carried out, starting from the commutator identity found in
Eq. (4.14). For the last estimate, one has to calculate the upcoming
commutators and bound each term in the same way. ∎

###### Sketch of Proof of Lemma 4.8.

Let us fix $k\in\Lambda^{*}$ and $\varphi\in\ell^{1}$. Starting from Eq. (4.2)
we use $0\leqslant\chi,\chi^{\perp}\leqslant 1$ and
$\|a_{p}(t)\|_{B(\mathscr{F})}=\|a_{p}^{*}(t)\|_{B(\mathscr{F})}\leqslant|\Lambda|^{\frac{1}{2}}$
to find

$\displaystyle\|D_{k}(t,\varphi)\|_{B(\mathscr{F})}\leqslant
4|\Lambda|\int_{\Lambda^{*}}|\varphi(p)|\mathrm{d}p\ .$ (4.44)

A similar inequality can be found upon calculation of the commutator
$[D_{k}(t),D_{\ell}(s,\varphi)].$ This finishes the proof. ∎

##  5\. Tool Box II: Excitation Estimates

In Section 3 we introduced the two following observables:

$\mathcal{N}=\int_{\Lambda^{*}}a_{p}^{*}a_{p}\mathrm{d}p\qquad\textnormal{ and
}\qquad\mathcal{N}_{\mathcal{S}}=\int_{\mathcal{S}}a_{p}^{*}a_{p}\mathrm{d}p$
(5.1)

corresponding to the the Number Operator and Surface-Localized Number
Operator, respectively. The main purpose of this section is to prove estimates
that control the growth-in-time of the expectation of $\mathcal{N}$ and
$\mathcal{N}_{\mathcal{S}}$ with respect to the interaction dynamics
$(\nu_{t})_{t\in\mathbb{R}}$, defined in (3.14). These estimates are precisely
stated in the following two propositions, which we prove in the reminder of
this section.

######  Proposition 5.1.

Let $(\nu_{t})_{t\in\mathbb{R}}$ solve the interaction dynamics defined in
(3.14), with initial data $\nu_{0}\equiv\nu$ satisfying Condition 2. Assume
that $n=\nu(\mathcal{N})\geqslant 1$. Then, for all $\ell\in\mathbb{N}$ there
exists a constant $C>0$ such that

$\nu_{t}(\mathcal{N}^{\ell})\leqslant Cn^{\ell}\exp(C\lambda Rt)\
,\qquad\forall t\geqslant 0\ .$ (5.2)

######  Proposition 5.2.

Let $(\nu_{t})_{t\in\mathbb{R}}$ solve the interaction dynamics defined in
(3.14), with initial data $\nu_{0}\equiv\nu$ satisfying Condition 2. Further,
assume that $n=\nu(\mathcal{N})\lesssim R^{1/2}$. Then, there exists a
constant $C>0$ such that

$\nu_{t}(\mathcal{N}_{\mathcal{S}})\leqslant C(\lambda R\left\langle
t\right\rangle)^{2}\exp(C\lambda Rt)\ ,\qquad\forall t\geqslant 0\ ,$ (5.3)

where $\left\langle t\right\rangle=(1+t^{2})^{\frac{1}{2}}.$

The idea behind the proof of our estimates relies on a standard Grönwall
argument, in which we bound expectations of commutators
$[\mathcal{N},\mathfrak{h}_{I}(t)]$ and
$[\mathcal{N}_{\mathcal{S}},\mathfrak{h}_{I}(t)]$ in terms of combinations of
expectations of $\mathcal{N}$ and $\mathcal{N}_{\mathcal{S}}$. This proof
relies heavily in the fact that the interaction Hamiltonian decomposes into
three parts, corresponding to fermion-fermion, fermion-boson and boson-boson
interactions. Namely, there holds

$\displaystyle\mathfrak{h}_{I}(t)\ =\
\lambda\,\big{(}V_{F}(t)+V_{F,B}(t)+V_{B}(t)\big{)}\ ,\qquad\forall t\geqslant
0\ .$ (5.4)

Here, time-dependence corresponds to the Heisenberg evolution associated to
the solvable Hamiltonian $\mathfrak{h}_{0}$–see Eq. (3.19). In particular,
using the formulae (3.8), (3.9) and (3.10) for $V_{F}$, $V_{F,B}$ and $V_{B}$,
respectively, we may write that for all $t\in\mathbb{R}$

$\displaystyle V_{F}(t)$
$\displaystyle=\frac{1}{2}\int_{\Lambda^{*}}\hat{V}(k)D_{k}^{*}(t)D_{k}(t)\
\mathrm{d}k$ (5.5) $\displaystyle V_{FB}(t)$
$\displaystyle=\int_{\Lambda^{*}}\hat{V}(k)D_{k}^{*}(t)\big{[}b_{k}(t)+b_{-k}^{*}(t)\big{]}\mathrm{d}k$
(5.6) $\displaystyle V_{B}(t)$
$\displaystyle=\int_{\Lambda^{*}}\hat{V}(k)\big{[}b_{k}^{*}(t)b_{k}(t)+\frac{1}{2}\,b^{*}_{k}(t)b_{-k}(t)+\frac{1}{2}\,b_{-k}(t)b_{k}(t)\big{]}\mathrm{d}k$
(5.7)

where $b_{k}(t)$ and $D_{k}(t)$ correspond to the Heisenberg evolution of the
$b$\- and $D$-operators, respectively, as given in Definition 7.

###  5.1. Number Operator Estimates

The main purpose of this section is to prove the Proposition 5.1. The first
step in this direction is to prove appropiate commutator estimates between
$\mathcal{N}$ and the generator of the interaction dynamics,
$\mathfrak{h}_{I}(t)$. The commutator estimates that we prove are contained in
the upcoming Lemma. We recall that $R=|\Lambda|p_{F}^{d-1}$.

######  Lemma 5.1 (Commutator Estimates for $\mathcal{N}$).

For all $\ell\geqslant 1$ there exists a constant $C=C(\ell)>0$ such that:

  1. (1)

For all $\Psi\in\mathscr{F}$ and $t\geqslant 0$ there holds

$\displaystyle\left\langle\Psi,[\mathcal{N}^{\ell},V_{F}(t)]\Psi\right\rangle_{\mathscr{F}}=0$

  2. (2)

For all $\Psi\in\mathscr{F}$ and $t\geqslant 0$ there holds

$\displaystyle|\left\langle\Psi,[\mathcal{N}^{\ell},V_{F,B}(t)]\Psi\right\rangle_{\mathscr{F}}|\leqslant
CR\left\langle\Psi,(\mathcal{N}^{\ell}+\mathds{1})\Psi\right\rangle_{\mathscr{F}}$

  3. (3)

For all $\Psi\in\mathscr{F}$ and $t\geqslant 0$ there holds

$|\left\langle\Psi,[\mathcal{N}^{\ell},V_{B}(t)]\Psi\right\rangle_{\mathscr{F}}|\leqslant
CR\left\langle\Psi,(\mathcal{N}^{\ell}+\mathds{1})\Psi\right\rangle_{\mathscr{F}}$

######  Remark 5.1.

Let us recall that we assume that $\nu:B(\mathscr{F})\rightarrow\mathbb{C}$ is
a mixed state. Namely, there exists sequences
$(\lambda_{n})_{n=0}^{\infty}\subset(0,\infty)$ and
$(\Psi_{n})_{n=0}^{\infty}\subset\mathscr{F}$ satisfying the normalization
condition $\sum_{n=0}^{\infty}\lambda_{n}=1$ and
$\|\Psi_{n}\|_{\mathscr{F}}=1$, respectively, such that the following
decomposition holds true

$\displaystyle\nu(\mathcal{O})=\sum_{n=0}^{\infty}\lambda_{n}\left\langle\Psi_{n},\mathcal{O}\Psi_{n}\right\rangle_{\mathscr{F}}\
,\qquad\forall\mathcal{O}\in B(\mathscr{F})\ .$ (5.8)

In particular, the estimates contained in Lemma 5.1 can be easily converted
into estimates for mixed states. For instance, if
$\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}$ are operators such that

$|\left\langle\Psi,\mathcal{O}_{1}\Psi\right\rangle_{\mathscr{F}}|\leqslant
C\|\mathcal{O}_{2}\Psi\|_{\mathscr{F}}\|\mathcal{O}_{3}\Psi\|_{\mathscr{F}}\
,\qquad\forall\Psi\in\mathscr{F}$ (5.9)

for a constant $C>0$, then it follows from the above decomposition of $\nu$
and the Cauchy-Schwarz inequality that

$|\nu(\mathcal{O}_{1})|\leqslant
C\nu(\mathcal{O}_{2}^{*}\mathcal{O}_{2})^{\frac{1}{2}}\,\nu(\mathcal{O}_{3}^{*}\mathcal{O}_{3})^{\frac{1}{2}}\
.$ (5.10)

In most applications, $\mathcal{O}_{2}$ and $\mathcal{O}_{3}$ shall corespond
to either $\mathcal{N}$ or $\mathcal{N}_{\mathcal{S}}$.

Let us briefly postpone the proof of the above Lemma to the next
subsubsection. First, we turn to the proof of the important Proposition 5.1.

###### Proof of Proposition 5.1.

The decomposition for $\mathfrak{h}_{I}(t)$ from (5.4) combined with the
commutator estimates from Lemma 5.1 imply that for all $\ell\geqslant 1$ there
exists $C=C(\ell)>0$ such that

$\partial_{t}\nu_{t}(\mathcal{N}^{\ell}+\mathds{1})=\nu_{t}\big{(}i[\mathfrak{h}_{I}(t),\mathcal{N}^{\ell}]\big{)}\leqslant
C\lambda R\nu_{t}(\mathcal{N}^{\ell}+\mathds{1}),\qquad\forall t\geqslant 0\
.$ (5.11)

Gronwall’s inequality now easily implies that there exists a constant $C>0$
such that

$\displaystyle\nu_{t}(\mathcal{N}^{\ell})\,\leqslant\,C\lambda
R\,\nu_{t}(\mathcal{N}^{\ell}+\mathds{1})e^{C\lambda Rt}\ ,\qquad\forall
t\geqslant 0\ .$ (5.12)

To finalize the proof, we use the fact that for quasi-free states it holds
true that $\nu(\mathcal{N}^{\ell})\lesssim\nu(\mathcal{N})^{\ell}$, together
with the assumption $\nu(\mathcal{N})=n\geqslant 1$ . ∎

####  5.1.1. Commutator Estimates for $\mathcal{N}$

###### Proof of Lemma 5.1.

Throughout this proof, $\Psi\in\mathscr{F}$ denotes an element in
$\cap_{k=1}^{\infty}D(\mathcal{N}^{k})$, which will justify all of the
upcoming calculations. Let us now fix $\ell\in\mathbb{N}.$

Proof of (1). This is an immediate consequence of the fact that
$[D_{k}(t),\mathcal{N}]=0$ for all $k\in\Lambda^{*}$ and $t\in\mathbb{R}$. See
Lemma 4.3.

Proof of (2). Using the fact that $D_{k}^{*}(t)=D_{-k}(t)$ and
$[D_{k}^{*}(t),b_{k}(t)]=0$ we may re-write the fermion-boson interaction term
as

$\displaystyle
V_{FB}(t)=\int_{\Lambda^{*}}\hat{V}(k)D^{*}_{k}(t)b_{k}(t)\mathrm{d}k\ +\
\mathrm{h.c}\ .$ (5.13)

Thus, we find that for all $t\in\mathbb{R}$

$\left\langle\Psi,[\mathcal{N}^{\ell},V_{FB}(t)]\Psi\right\rangle=2\mathrm{Im}\int_{\Lambda^{*}}\hat{V}(k)\langle\Psi,[\mathcal{N}^{\ell},D^{*}_{k}(t)b_{k}(t)]\Psi\rangle\
.$ (5.14)

In view of Lemma 4.3, we see that $[D_{k}^{*}(t),\mathcal{N}^{\ell}]=0$.
Further, using the pull-through formulae for $b$-operators with
$f(x)=x^{\ell}$ we find the following useful identity

$\displaystyle[\mathcal{N}^{\ell},b_{k}(t)]=\sum_{n=0}^{\ell-1}\binom{\ell}{n}(-2)^{\ell-n}\mathcal{N}^{n}b_{k}(t)\
,\qquad\forall k\in\Lambda^{*},\ t\in\mathbb{R}\ .$ (5.15)

Consequently, we can estimate that

$\displaystyle|\left\langle\Psi,[\mathcal{N}^{\ell},V_{FB}]\Psi\right\rangle|$
$\displaystyle\leqslant\sum_{n=0}^{\ell-1}\binom{\ell}{n}(-2)^{\ell-n}\int_{\Lambda^{*}}|\hat{V}(k)|\
|\left\langle\Psi,D_{k}^{*}(t)\mathcal{N}^{n}b_{k}(t)\Psi\right\rangle|\mathrm{d}k$
(5.16)
$\displaystyle\leqslant\sum_{n=0}^{\ell-1}\binom{\ell}{n}(-2)^{\ell-n}\int_{\Lambda^{*}}|\hat{V}(k)|\
\|\mathcal{N}^{\frac{n-1}{2}}D_{k}(t)\Psi\|\,\|\mathcal{N}^{\frac{n+1}{2}}b_{k}(t)\Psi\|\mathrm{d}k\
.$

We can now combine Lemma 4.3, the Type-I estimate (4.21) and the norm bound
(4.20) to find that there exists a constant $C>0$ such that

$\|\mathcal{N}^{\frac{n-1}{2}}D_{k}(t)\Psi\|\,\|\mathcal{N}^{\frac{n+1}{2}}b_{k}(t)\Psi\|\leqslant
CR\|\mathcal{N}^{\frac{n+1}{2}}\Psi\|^{2}\ ,\qquad\forall n\geqslant 0\ .$
(5.17)

Finally, we put the two above estimates together and use the elementary fact
$\mathcal{N}^{\frac{n+1}{2}}\lesssim\mathcal{N}^{\ell}+1$ (valid for
$n\leqslant\ell-1$) to find that for some $C>0$ there holds

$|\left\langle\Psi,[\mathcal{N}^{\ell},V_{FB}]\Psi\right\rangle|\leqslant
CR\|\hat{V}\|_{\ell^{1}}\|(\mathcal{N}^{\ell}+1)\Psi\|^{2}\ ,\qquad\forall
t\geqslant 0\ $ (5.18)

which gives the desired estimate.

Proof of (3). First, we note that $[\mathcal{N},b_{k}^{*}(t)b_{k}(t)]=0$ for
all $t\in\mathbb{R}$ and $k\in\Lambda^{*}$. Hence, we can readily check that

$\left\langle\Psi,[\mathcal{N}^{\ell},V_{B}(t)]\Psi\right\rangle=\mathrm{Im}\int\hat{V}(k)\left\langle\Psi,[\mathcal{N}^{\ell},b_{k}(t)b_{-k}(t)]\Psi\right\rangle\mathrm{d}k\qquad\forall
t\in\mathbb{R}\ .$ (5.19)

In view of the commutation relation
$\mathcal{N}b_{k}(t)b_{-k}(t)=b_{k}(t)b_{-k}(t)(\mathcal{N}-4)$ we can
calculate using the pull-through formula for $f(x)=x^{\ell}$ that

$\displaystyle[\mathcal{N}^{\ell},b_{k}(t)b_{-k}(t)]=\sum_{n=0}^{\ell-1}\binom{\ell}{n}4^{\ell-n}(\mathcal{N}+4)^{\frac{n-1}{2}}b_{k}(t)b_{-k}(t)\mathcal{N}^{\frac{n+1}{2}}\
.$ (5.20)

Consequently, putting the last two displayed equations together one finds that
for all $t\in\mathbb{R}$

$\displaystyle|\left\langle\Psi,[\mathcal{N}^{\ell},V_{B}(t)]\Psi\right\rangle|$
$\displaystyle\leqslant\sum_{n=0}^{\ell-1}\binom{\ell}{n}4^{\ell-n}\int_{\Lambda^{*}}|\hat{V}(k)|\|(\mathcal{N}+4)^{\frac{n+1}{2}}\Psi\|\
\|b_{k}(t)b_{-k}(t)\mathcal{N}^{\frac{n-1}{2}}\Psi\|\mathrm{d}k\ .$ (5.21)

We estimate the right hand side as follows. First, we note that
$\|(\mathcal{N}+4)^{\frac{n+1}{2}}\Psi\|\leqslant
C(\ell)\|(\mathcal{N}+1)^{\ell/2}\Psi\|$ for all $0\leqslant
n\leqslant\ell-1$. Secondly, we use the Type-II estimate (4.24) and the
commutation relation (4.16) for $f\equiv 1$ to find that

$\displaystyle\|b_{k}(t)b_{-k}(t)\mathcal{N}^{\frac{n-1}{2}}\Psi\|$
$\displaystyle\ \lesssim\
R^{\frac{1}{2}}\,\|(\mathcal{N}+2)^{\frac{1}{2}}b_{-k}(t)\mathcal{N}^{\frac{n-1}{2}}\Psi\|$
$\displaystyle\ =\
R^{\frac{1}{2}}\,\|b_{-k}(t)\mathcal{N}^{\frac{1}{2}}\mathcal{N}^{\frac{n-1}{2}}\Psi\|$
$\displaystyle\ \lesssim\
R\,\|\mathcal{N}^{\frac{1}{2}}\mathcal{N}^{\frac{1}{2}}\mathcal{N}^{\frac{n-1}{2}}\Psi\|$
$\displaystyle\ \lesssim\ R\,\|(\mathcal{N}+1)^{\frac{\ell}{2}}\Psi\|$ (5.22)

where again we used the fact that $n\leqslant\ell-1$. The proof of the Lemma
is easily finished after we put together the last two displayed estimates. ∎

###  5.2. Surface-localized Number Operator Estimates

The main purpose of this section is proving Proposition 5.2. In order to
control the time evolution of $\mathcal{N}_{\mathcal{S}}$ with respect to
$\mathfrak{h}_{I}(t)$, we establish the following commutator estimates. Recall
that $R=|\Lambda|p_{F}^{d-1}$.

######  Lemma 5.2.

There exists a constant $C>0$ such that the following estimates hold true

  1. (1)

For all $\Psi\in\mathscr{F}$

$|\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{F}(t)]\Psi\right\rangle_{\mathscr{F}}|\leqslant
C\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|_{\mathscr{F}}\|\mathcal{N}^{3/2}\Psi\|_{\mathscr{F}}\
.$ (5.23)

  2. (2)

For all $\Psi\in\mathscr{F}$

$\displaystyle|\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{FB}(t)]\Psi\right\rangle_{\mathscr{F}}|$
$\displaystyle\leqslant
CR^{1/2}\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|_{\mathscr{F}}\|\mathcal{N}\Psi\|_{\mathscr{F}}\
.$

  3. (3)

For all $\Psi\in\mathscr{F}$

$|\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{B}(t)]\Psi\right\rangle_{\mathscr{F}}|\leqslant
CR\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|^{2}_{\mathscr{F}}+CR\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|_{\mathscr{F}}\|\Psi\|_{\mathscr{F}}\
.$

We shall defer the proof of Lemma 5.2 to next subsubsection. Now we turn our
attention to the proof of Proposition 5.2.

###### Proof of Proposition 5.2.

Throughout the proof, $C>0$ is a constant whose value may change from line to
line. First, in view of the decomposition of $\mathfrak{h}_{I}(t)$ given in
(5.4), Lemma 5.2 and Remark 5.1, there holds for all $t\in\mathbb{R}$

$\displaystyle\frac{d}{dt}\nu_{t}(\mathcal{N}_{\mathcal{S}})=\nu_{t}(i[\mathfrak{h}_{I}(t),\mathcal{N}_{\mathcal{S}}])$
$\displaystyle\leqslant
C\lambda[\nu_{t}(\mathcal{N}_{S})]^{\frac{1}{2}}[\nu_{t}(\mathcal{N}^{3})]^{\frac{1}{2}}$
$\displaystyle\ +C\lambda
R^{\frac{1}{2}}[\nu_{t}(\mathcal{N}_{S})]^{\frac{1}{2}}[\nu_{t}(\mathcal{N}^{2})]^{\frac{1}{2}}$
$\displaystyle\ +C\lambda R[\nu_{t}(\mathcal{N}_{S})]$ $\displaystyle\
+C\lambda
R[\nu_{t}(\mathcal{N}_{S})]^{\frac{1}{2}}[\nu_{t}(\mathds{1})]^{\frac{1}{2}}\
.$ (5.24)

Thus, we divide111Technically, one should introduce a regularization
$u_{\delta}(t)\equiv(\delta+\nu_{t}(\mathcal{N}_{\mathcal{S}}))^{1/2}$ in
order to avoid possible singularities whenever
$\nu_{t}(\mathcal{N}_{\mathcal{S}})=0$. One should then close the estimates
after taking the limit $\delta\downarrow 0$. We leave the details to the
reader by $\nu_{t}(\mathcal{N}_{\mathcal{S}})^{1/2}$ to find that thanks to
Proposition 5.1

$\displaystyle\frac{d}{dt}\nu_{t}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}$
$\displaystyle\leqslant C\lambda
R\nu_{t}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}+C\lambda
R\Big{(}\nu_{t}(\mathcal{N}^{3})^{\frac{1}{2}}/R+\nu_{t}(\mathcal{N}^{2})^{\frac{1}{2}}/R^{\frac{1}{2}}+1\Big{)}$
$\displaystyle\leqslant C\lambda
R\nu_{t}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}+C\lambda R\exp(\lambda
Rt)\Big{(}n^{\frac{3}{2}}/R+n/R^{\frac{1}{2}}+1\Big{)}\ .$ (5.25)

The Grönwall inequality now implies that for all $t\geqslant 0$

$\displaystyle\nu_{t}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}\leqslant
C\exp(C\lambda
Rt)\Big{(}\nu_{0}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}+\lambda
Rt\,(n^{\frac{3}{2}}/R+n/R^{\frac{1}{2}}+1)\Big{)}\ .$ (5.26)

Finally, we notice that in view of Condition 2 we have
$\nu_{0}(\mathcal{N}_{\mathcal{S}})\lesssim(\lambda R)^{2}$. The proof is then
finished once we simplify the right hand side using the bound $n\lesssim
R^{1/2}$, and take squares on both sides of the inequality. ∎

####  5.2.1. Commutator Estimates for $\mathcal{N}_{\mathcal{S}}$

In order to prove Lemma 5.2, we shall first establish the following useful
lemma. Here and in the sequel, $\mathds{1}_{\mathcal{S}}$ denotes the
characteristic function of the Fermi surface $\mathcal{S}$.

######  Lemma 5.3.

For all $k\in\Lambda^{*}$ and $g\in\ell^{\infty}$ the operator

$\mathcal{O}(k):=\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(p)g(p)a_{p+k}^{*}a_{p}\mathrm{d}p$
(5.27)

satisfies the following estimate

$\displaystyle\big{|}\left\langle\Phi,\mathcal{O}(k)\Psi\right\rangle_{\mathscr{F}}\big{|}\leqslant\|g\|_{\ell^{\infty}}\|\mathcal{N}^{1/2}\Phi\|\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|\
,\qquad\forall\Phi,\,\Psi\in\mathscr{F}.$ (5.28)

###### Proof.

Let $\Phi,\Psi\in\mathscr{F}$, $k\in\Lambda^{*}$ and $g\in\ell^{\infty}$.
Then, we calculate

$\displaystyle|\left\langle\Phi,\mathcal{O}(k)\Psi\right\rangle_{\mathscr{F}}|$
$\displaystyle=\bigg{|}\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(p)g(p)\left\langle
a_{p+k}\Phi,a_{p}\Psi\right\rangle_{\mathscr{F}}\mathrm{d}p\bigg{|}$
$\displaystyle\leqslant\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(p)|g(p)|\|a_{p+k}\Phi\|_{\mathscr{F}}\|a_{p}\Psi\|_{\mathscr{F}}\mathrm{d}p$
$\displaystyle\leqslant\|g\|_{\ell^{\infty}}\biggl{(}\int_{\Lambda^{*}}\|a_{p+k}\Phi\|_{\mathscr{F}}^{2}\mathrm{d}p\biggr{)}^{\frac{1}{2}}\biggl{(}\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(p)\|a_{p}\Psi\|_{\mathscr{F}}^{2}\mathrm{d}p\biggr{)}^{\frac{1}{2}}$
$\displaystyle=\|g\|_{\ell^{\infty}}\|\mathcal{N}^{\frac{1}{2}}\Phi\|_{\mathscr{F}}\|\mathcal{N}_{\mathcal{S}}^{\frac{1}{2}}\Psi\|_{\mathscr{F}}\
.$ (5.29)

In the last line we used the fact that
$\|a_{p}\Phi\|^{2}_{\mathscr{F}}=\left\langle\Phi,a_{p}^{*}a_{p}\Phi\right\rangle_{\mathscr{F}}$
for all $p\in\Lambda^{*}$, plus a change of variables $p\mapsto p-k$. A
similar argument holds for the term containing $\Psi$. This finishes the
proof. ∎

###### Proof of Lemma 5.2.

Throughout this proof, $\Psi\in\mathscr{F}$ is fixed. In addition, in order to
ease the notation, we shall drop the explicit time dependence in our estimates
– since the estimates are uniform in $t\in\mathbb{R}$, there is no risk of
confusion. Let us now fix $\ell\in\mathbb{N}.$

Proof of (1). Starting from (3.8) we can first calculate that

$\displaystyle\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{F}]\Psi\right\rangle$
$\displaystyle=2\mathrm{i}\
\int_{\Lambda^{*}}\hat{V}(k)\mathrm{Im}\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},D^{*}(k)]D(k)\Psi\right\rangle\mathrm{d}k\
.$ (5.30)

We now put the above commutator in an appropiate form. Using the explicit
expression of $D^{*}(k)$ in terms of creation- and annihilation- operators
(see Def. 7) together with the CAR, we find that for all $k\in\Lambda^{*}$
there holds

$\displaystyle[\mathcal{N}_{\mathcal{S}},D^{*}(k)]$
$\displaystyle=\int_{\Lambda^{*}}\Big{(}\mathds{1}_{\mathcal{S}}(p)-\mathds{1}_{\mathcal{S}}(p-k)\Big{)}\chi^{\perp}(p)\chi^{\perp}(p-k)a_{p}^{*}a_{p-k}\,\mathrm{d}p$
$\displaystyle-\int_{\Lambda^{*}}\Big{(}\mathds{1}_{\mathcal{S}}(h)-\mathds{1}_{\mathcal{S}}(h+k)\Big{)}\chi(h)\chi(h+k)a_{h}^{*}a_{h+k}\,\mathrm{d}h$
$\displaystyle\equiv\mathcal{O}_{1}(k)+\mathcal{O}_{2}(k)$ (5.31)

where we introduce the two following auxiliary operators (notice the change of
variables $p\mapsto p+k$ and $h\mapsto h-k$ in the second operator)

$\displaystyle\mathcal{O}_{1}(k)$
$\displaystyle:=\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(p)\chi^{\perp}(p,p-k)a_{p}^{*}a_{p-k}\,\mathrm{d}p-\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(h)\chi(h,h+k)a_{h}^{*}a_{h+k}\,\mathrm{d}h$
(5.32) $\displaystyle\mathcal{O}_{2}(k)$
$\displaystyle:=-\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(p)\chi^{\perp}(p,p+k)a_{p+k}^{*}a_{p}\,\mathrm{d}p+\int_{\Lambda^{*}}\mathds{1}_{\mathcal{S}}(h)\chi(h,h-k)a_{h-k}^{*}a_{h}\,\mathrm{d}h$
(5.33)

where for simplicity we denote
$\chi^{\perp}(p,p-k)\equiv\chi^{\perp}(p)\chi^{\perp}(p-k)$ and similarly for
$\chi(h,h+k)$. We are now able to write

$\displaystyle\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{F}]\Psi\right\rangle$
$\displaystyle=2\mathrm{i}\
\int_{\Lambda^{*}}\hat{V}(k)\mathrm{Im}\left\langle\Psi,\mathcal{O}_{1}(k)D(k)\Psi\right\rangle\mathrm{d}k$
(5.34) $\displaystyle+2\mathrm{i}\
\int_{\Lambda^{*}}\hat{V}(k)\mathrm{Im}\left\langle\Psi,D(k)\mathcal{O}_{2}(k)\Psi\right\rangle\mathrm{d}k$
(5.35) $\displaystyle+2\mathrm{i}\
\int_{\Lambda^{*}}\hat{V}(k)\mathrm{Im}\left\langle\Psi,[\mathcal{O}_{2}(k),D(k)]\Psi\right\rangle\mathrm{d}k\
.$ (5.36)

The first term in the above equation can be estimated using Lemma 5.3 for
$\mathcal{O}(k)=\mathcal{O}^{*}_{1}(k)$. Namely,

$\displaystyle\big{|}2\mathrm{i}\
\int_{\Lambda^{*}}\hat{V}(k)\mathrm{Im}\left\langle\Psi,\mathcal{O}_{1}(k)D(k)\Psi\right\rangle\mathrm{d}k\big{|}\leqslant
2\|\hat{V}\|_{\ell^{1}}\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|\|\mathcal{N}^{3/2}\Psi\|$
(5.37)

The second term in the above equation is estimates using Lemma 5.3 for
$\mathcal{O}(k)=\mathcal{O}_{2}(k)$. We get

$\displaystyle\big{|}2\mathrm{i}\
\int_{\Lambda^{*}}\hat{V}(k)\mathrm{Im}\left\langle\Psi,D(k)\mathcal{O}_{2}\Psi\right\rangle\mathrm{d}k\big{|}\leqslant
2\|\hat{V}\|_{\ell^{1}}\|\mathcal{N}^{3/2}\Psi\|\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|$
(5.38)

The third term in the above equation is actually zero. This comes from the
fact that the commutator between $\mathcal{O}_{2}(k)$ and $D(k)$ is self-
adjoint. More precisely, we can calculate using the CAR

$\displaystyle[\mathcal{O}_{2}(k),D(k)]$
$\displaystyle=\int_{\Lambda^{*}}\Big{(}\mathds{1}_{\mathcal{S}}(p+k)-\mathds{1}_{\mathcal{S}}(p)\Big{)}\chi^{\perp}(p,p+k)a_{p}^{*}a_{p}\mathrm{d}p$
(5.39)
$\displaystyle-\int_{\Lambda^{*}}\Big{(}\mathds{1}_{\mathcal{S}}(h-k)-\mathds{1}_{\mathcal{S}}(h)\Big{)}\chi(h,h-k)a_{h}^{*}a_{h}\mathrm{d}h\
.$ (5.40)

We put our results together to find that

$\displaystyle\big{|}\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{F}]\Psi\right\rangle\big{|}$
$\displaystyle\leqslant
4\|\hat{V}\|_{\ell^{1}}\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|\|\mathcal{N}^{3/2}\Psi\|\
.$ (5.41)

Proof of (2). Starting form (3.9) we can calculate that

$\displaystyle|\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{FB}]\Psi\right\rangle|$
$\displaystyle\leqslant
2\int_{\Lambda^{*}}|\hat{V}(k)|\,|\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},D^{*}(k)b(k)]\Psi\right\rangle|\mathrm{d}k\
,$ $\displaystyle\leqslant
2\int_{\Lambda^{*}}|\hat{V}(k)|\,\|[\mathcal{N}_{\mathcal{S}},D(k)]\Psi\|\,\|b(k)\Psi\|\mathrm{d}k$
$\displaystyle\quad+2\int_{\Lambda^{*}}|\hat{V}(k)|\,\|[\mathcal{N}_{\mathcal{S}},b(k)]\Psi\|\,\|D(k)\Psi\|\mathrm{d}k\
.$ (5.42)

Let us estimate the first term contained in the right hand side of (5.42). In
view of $D^{*}(k)=D(-k)$ and (5.31) we have that
$[\mathcal{N}_{\mathcal{S}},D(k)]=\mathcal{O}_{1}(-k)+\mathcal{O}_{2}(-k)$.
Each $\mathcal{O}_{i}(k)$ can be estimated using (4.18) –we conclude that
$\|[\mathcal{N}_{\mathcal{S}},D(k)]\Psi\|\lesssim\|\mathcal{N}\Psi\|\ .$ On
the other hand, we use the Type-II estimate (4.24) on $b(k)$. We conclude that

$\displaystyle\int_{\Lambda^{*}}|\hat{V}(k)|\,\|[\mathcal{N}_{\mathcal{S}},D(k)]\Psi\|\,\|b(k)\Psi\|\mathrm{d}k\
\lesssim\
R^{\frac{1}{2}}\|\mathcal{N}\Psi\|\,\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|\
.$ (5.43)

Let us now look at the second term contained in (5.42). First, we recall that
for $k\in\textnormal{supp}\hat{V}$ there holds
$[\mathcal{N}_{\mathcal{S}},b(k)]=-2b(k)$, see Lemma 4.4. Consequently, using
the Type-II estimate (4.24) we see that
$\|[\mathcal{N}_{\mathcal{S}},b(k)]\Psi\|\lesssim
R^{1/2}\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|$. On the other hand, we can use
the Type-I estimate (4.21) to find $\|D(k)\Psi\|\lesssim\|\mathcal{N}\Psi\|$.
These upper bounds can be put together to find that

$\displaystyle\int_{\Lambda^{*}}|\hat{V}(k)|\,\|[\mathcal{N}_{\mathcal{S}},b(k)]\Psi\|\,\|D(k)\Psi\|\mathrm{d}k$
$\displaystyle\ \lesssim\
R^{1/2}\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|\|\mathcal{N}\Psi\|\ .$ (5.44)

A direct combination of the last three displayed estimates finish the proof of
(2).

Proof of (3). Starting from (3.10) we decompose the boson-boson interaction
into a diagonal, and off-diagonal part. Namely, we write $V_{B}=V_{1}+V_{2}$,
where we set

$\displaystyle
V_{1}\equiv\int_{\Lambda^{*}}\hat{V}(k)b^{*}(k)b(k)\mathrm{d}k\quad\textnormal{and}\quad
V_{2}\equiv\frac{1}{2}\int_{\Lambda^{*}}\hat{V}(k)\Big{(}b(k)b(-k)+\mathrm{h.c}\Big{)}\mathrm{d}k\
.$ (5.45)

For $V_{1}$ we can quickly verify that its commutator with
$\mathcal{N}_{\mathcal{S}}$ vanishes. Indeed, thanks to Lemma 4.4 we find that
$[\mathcal{N}_{S},b^{*}(k)b(k)]=+2b^{*}(k)b(k)-2b^{*}(k)b(k)=0$ for all
$k\in\textnormal{supp}\hat{V}$. Hence, $[\mathcal{N}_{\mathcal{S}},V_{1}]=0$
upon summing over $k\in\Lambda^{*}$.

For $V_{2}$, we have the preliminary upper bound as our starting point

$\displaystyle|\left\langle\Psi,[\mathcal{N}_{\mathcal{S}},V_{2}]\Psi\right\rangle|$
$\displaystyle\leqslant
2\int_{\Lambda^{*}}|\hat{V}(k)||\left\langle\Psi,b(k)b(-k)\Psi\right\rangle|\mathrm{d}k\
.$ (5.46)

We estimate the integrand of the right hand side as follows –let us fix
$k\in\textnormal{supp}\hat{V}$. First, recalling that
$[\mathcal{N}_{\mathcal{S}},b_{k}]=0$ (see Lemma 4.4) we find that for any
measurable function $\varphi:\mathbb{R}\rightarrow\mathbb{C}$ the following
pull-through formula holds true

$\displaystyle\varphi(\mathcal{N}_{\mathcal{S}})b(k)=b(k)\varphi(\mathcal{N}_{\mathcal{S}}-2)$
(5.47)

Thus, using $\varphi(x)=(x+5)^{1/2}$ we find

$\displaystyle|\left\langle\Psi,b(k)b(-k)\Psi\right\rangle|$
$\displaystyle=|\left\langle(\mathcal{N}_{\mathcal{S}}+5)^{1/2}\Psi,b(k)b(-k)(\mathcal{N}_{\mathcal{S}}+1)^{-1/2}\Psi\right\rangle|$
$\displaystyle\leqslant\|(\mathcal{N}_{\mathcal{S}}+5)^{1/2}\Psi\|\,\|b(k)b(-k)(\mathcal{N}_{\mathcal{S}}+1)^{-1/2}\Psi\|\
.$ (5.48)

We use again the commutation relation $[\mathcal{N}_{\mathcal{S}},b_{k}]=0$
and the Type-II estimate (4.24) for $b$-operators to find that

$\displaystyle\|b(k)b(-k)(\mathcal{N}_{\mathcal{S}}+1)^{-1/2}\Psi\|$
$\displaystyle\,\lesssim\,R^{1/2}\|\mathcal{N}_{\mathcal{S}}^{1/2}b(-k)(\mathcal{N}_{\mathcal{S}}+1)^{-1/2}\Psi\|$
$\displaystyle\,\leqslant\,R^{1/2}\|(\mathcal{N}_{\mathcal{S}}+2)^{1/2}b(-k)(\mathcal{N}_{\mathcal{S}}+1)^{-1/2}\Psi\|$
$\displaystyle\,=\,R^{1/2}\|b(-k)\mathcal{N}_{\mathcal{S}}^{1/2}(\mathcal{N}_{\mathcal{S}}+1)^{-1/2}\Psi\|$
$\displaystyle\,\lesssim\,R\|\mathcal{N}_{\mathcal{S}}^{1/2}\mathcal{N}_{\mathcal{S}}^{1/2}(\mathcal{N}_{\mathcal{S}}+1)^{-1/2}\Psi\|$
$\displaystyle\,\leqslant\,R\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|\ .$ (5.49)

On the other hand, the other term multiplying in (5.48) can be bounded as
follows
$\|(\mathcal{N}_{\mathcal{S}}+5)^{1/2}\Psi\|\lesssim\|\mathcal{N}_{\mathcal{S}}^{1/2}\Psi\|+\|\Psi\|\
.$ A straightforward combination of the estimates contained in (5.46), (5.48)
and (5.49) now finish the proof. ∎

##  6\. Leading Order Terms I: Emergence of $Q$

In Section 3 we considered a double commutator expansion (3.20) for the
momentum distribution of particles and holes, $f_{t}(p)$. This expansion is
expressed in terms of the nine quantities $\\{T_{\alpha,\beta}(t)\\}$ that
arise from the three different interacting potentials $V_{F}$, $V_{FB}$ and
$V_{B}$, respectively. The main goal of this section is analyzing the single
term $T_{F,F}$. In particular, we prove that one may extract the mollified
collision operator $Q_{t}$–originally introduced in Def. 2– up to reminder
terms that we have control of. A precise statement is given in the following
proposition. We remind the reader that $R=|\Lambda|p_{F}^{d-1}$

######  Proposition 6.1 (Analysis of $T_{F,F}$).

Let $T_{F,F}(t,p)$ be the quantity defined in Eq. (3.21) for $\alpha=\beta=F$,
and let $m>0$. Then, there exists a constant $C>0$ such that for all
$\varphi\in\ell_{m}^{1}$ and $t\geqslant 0$ the following inequality holds
true

$\displaystyle\big{|}T_{F,F}(t,\varphi)+\,|\Lambda|\,t\left\langle\varphi,Q_{t}[f_{0}]\right\rangle\big{|}\leqslant
C|\Lambda|\lambda
t^{3}\|\hat{V}\|_{\ell^{1}}^{3}\|\varphi\|_{\ell^{1}}\sup_{\tau\leqslant
t}\Big{(}R^{2}\nu_{\tau}(\mathcal{N}^{4})^{\frac{1}{2}}+\nu_{\tau}(\mathcal{N}^{4})\Big{)}$
(6.1)

where $T_{F,F}(t,\varphi)\equiv\left\langle\varphi,T_{F,F}(t)\right\rangle$
and $Q_{t}$ is given in Def. 2.

In order to prove Proposition 6.1 we shall perform an additional expansion of
$\nu_{t}$ with respect to the interaction Hamiltonian $\mathfrak{h}_{I}(t)$.
Namely, we consider

$\displaystyle T_{F,F}(t,\varphi)$
$\displaystyle=\int_{0}^{t}\int_{0}^{t_{1}}\nu\big{(}[[N(\varphi),V_{F}(t_{1})],V_{F}(t_{2})]\big{)}\mathrm{d}t_{1}\mathrm{d}t_{2}$
$\displaystyle\quad-i\int_{0}^{t}\int_{0}^{t_{1}}\int_{0}^{t_{2}}\nu_{t_{2}}\big{(}[[[N(\varphi),V_{F}(t_{1})],V_{F}(t_{2})],\mathfrak{h}_{I}(t_{3})]\big{)}\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}\
,$ (6.2)

where we recall
$N(\varphi)\equiv\int_{\Lambda^{*}}\overline{\varphi(p)}a_{p}^{*}a_{p}\,\mathrm{d}p$.
We then analyze the two terms of the right hand side of (6.2) separately.
Thus, we split the proof into two parts, which are contained in the following
two lemmas.

######  Lemma 6.1.

Let $\nu:B(\mathscr{F})\rightarrow\mathbb{C}$ be an initial state satisfying
Condition 2, and let $f_{0}(p)=|\Lambda|^{-1}\nu(a_{p}^{*}a_{p})$ for all
$p\in\Lambda^{*}$. Let $V_{F}(t)$ be the Heisenberg evolution of the fermion-
fermion interaction, defined in (3.19) for $\alpha=F$. Then, for all
$\varphi\in\ell^{1}$ and $t\geqslant 0$

$\int_{0}^{t}\int_{0}^{t_{1}}\nu\big{(}[[N(\varphi),V_{F}(t_{1})],V_{F}(t_{2})]\big{)}\mathrm{d}t_{1}\mathrm{d}t_{2}=-t|\Lambda|\left\langle\varphi,Q_{t}[f_{0}]\right\rangle\
.$ (6.3)

The proof of the identity contained in Lemma 6.1 will be heavily inspired by
the work of Erdős, Salmhofer and Yau [31], on a heuristic derivation of the
quantum Boltzmann equation. In fact, we shall make use of some of their
algebraic relations.

######  Lemma 6.2.

Let $(\nu_{t})_{t\in\mathbb{R}}$ be the interaction dynamics as given in Def.
5, with initial data $\nu=\nu_{0}$ satisfying Condition 2. Let $V_{F}(t)$ be
the Heisenberg evolution of the fermion-fermion interaction, defined in (3.19)
for $\alpha=F$. Then, there exists a constant $C>0$ such that for all
$\varphi\in\ell^{1}$ and $t\geqslant 0$

$\displaystyle\bigg{|}\int_{0}^{t}\int_{0}^{t_{1}}\int_{0}^{t_{2}}\nu_{t_{2}}\big{(}[[[N(\varphi),$
$\displaystyle
V_{F}(t_{1})],V_{F}(t_{2})],\mathfrak{h}_{I}(t_{3})]\big{)}\mathrm{d}t_{1}\mathrm{d}t_{2}\mathrm{d}t_{3}\bigg{|}$
$\displaystyle\leqslant C\lambda
t^{3}\|\hat{V}\|_{\ell^{1}}^{3}|\Lambda|\|\varphi\|_{\ell^{1}}\sup_{\tau\leqslant
t}\Big{(}\nu_{\tau}(\mathcal{N}^{4})+R^{2}\nu_{\tau}(\mathcal{N}^{4})^{\frac{1}{2}}\Big{)}\
.$ (6.4)

We remind the reader that the interaction Hamiltonian $\mathfrak{h}_{I}(t)$
admits the decomposition given in (5.4) in terms of the Heisenberg evolution
of $b$ and $D$-operators–see (5.5), (5.6) and (5.7).

###### Proof of Proposition 6.1.

It suffices to put together Eq. (6.2) and Lemmas 6.1 and 6.2. ∎

We dedicate the rest of this section to the proof of Lemmas 6.1 and 6.2.

###  6.1. Proof of Lemma 6.1

Before we jump into the proof of Lemma 6.1, we shall re-write the fermion-
fermion interaction term $V_{F}(t)$ in a form that will be suitable for our
analysis. This representation is recorded in Lemma 6.3, which we study in the
next subsubsection.

####  6.1.1. Normal ordering of $V_{F}(t)$

Let us fix the time label $t\in\mathbb{R}$. First, we see from (5.5) that
$V_{F}(t)=\int_{\Lambda^{*}}\hat{V}(k)D_{k}^{*}(t)D_{k}(t)\mathrm{d}k$ can be
written in terms of the Heisenberg evolution of the $D$-operators, as given in
Def. 7. These can be written explicitly in terms of creation- and
annihilation- in the following way

$\displaystyle
D_{k}(t)=\int_{(\Lambda^{*})^{2}}d_{t}(k,p,q)a_{p}^{*}a_{q}\mathrm{d}p\mathrm{d}q$
(6.5)

where the coefficients in the above expression are given as follows

$d_{k}(t,p,q)\,\equiv\,e^{it(E_{p}-E_{q})}\big{[}\chi^{\perp}(p)\chi^{\perp}(q)\delta(p-q+k)-\chi(p)\chi(q)\delta(p-q-k)\big{]}$
(6.6)

for all $k,p,q\in\Lambda^{*}$. Since $D_{k}^{*}(t)=D_{-k}(t)$ it readily
follows that we can write the fermion-fermion interaction in the following
form

$V_{F}(t)=\int_{\Lambda^{*4}}\bigg{[}\int_{\Lambda^{*}}\hat{V}(k)d_{t}(-k,p_{1},q_{1})\,d_{t}(k,p_{2},q_{2})\mathrm{d}k\bigg{]}a_{p_{1}}^{*}a_{q_{1}}a_{p_{2}}^{*}a_{q_{2}}\mathrm{d}p_{1}\mathrm{d}p_{2}\mathrm{d}q_{1}\mathrm{d}q_{2}\
.$ (6.7)

Clearly, the expression in (6.7) is not normally ordered. Our next goal is
then to put $V_{F}(t)$ in normal order, with explicit coefficients. To this
end, we introduce the following coefficient function

$\phi_{t}(\vec{p})\equiv\int_{\Lambda^{*}}\hat{V}(k)\,d_{t}(-k,p_{1},p_{4})\,d_{t}(k,p_{2},p_{3})\mathrm{d}k$
(6.8)

where $\vec{p}=(p_{1},p_{2},p_{3},p_{4})\in(\Lambda^{*})^{4}$. A
straightforward calculation using the CAR in Eq. (6.7) now yields

$\displaystyle V_{F}(t)$
$\displaystyle=\int_{\Lambda^{*4}}\phi_{t}(p_{1},p_{2},q_{2},q_{1})a_{p_{1}}^{*}a_{p_{2}}^{*}a_{q_{2}}a_{q_{1}}\mathrm{d}p_{1}\mathrm{d}p_{2}\mathrm{d}q_{1}\mathrm{d}q_{2}$
$\displaystyle+\int_{\Lambda^{*2}}\bigg{[}\
\int_{\Lambda^{*2}}\phi_{t}(p_{1},p_{2},q_{2},q_{1})\delta(q_{1}-p_{2})\mathrm{d}p_{2}\mathrm{d}q_{1}\bigg{]}a_{p_{1}}^{*}a_{q_{2}}\mathrm{d}p_{1}\mathrm{d}q_{2}\
.$ (6.9)

We shall denote by $:V_{F}(t):$ the normal ordering of $V_{F}(t)$, that is,
the first term in Eq. (6.9).

Next, we shall put the above normal order form in a more explicit
representation by calculating explictly the coefficient function $\phi_{t}$,
together with its contraction for $q_{1}=p_{2}$. Before we do so, let us
introduce some convenient notation:

$\square$ When $\vec{p}=(p_{1},p_{2},p_{3},p_{4})\in(\Lambda^{*})^{4}$ is
known from context, we let

$\chi_{1234}\equiv\chi(p_{1})\chi(p_{2})\chi(p_{3})\chi(p_{4})\qquad\textnormal{and}\qquad\chi_{1234}^{\perp}\equiv
1-\chi_{1234}$

and similarly for $\chi_{ij}$ and $\chi_{ij}^{\perp}$ for any combination of
$i,j\in\\{1,2,3,4\\}$.

$\square$ For any $\vec{p}=(p_{1},p_{2},p_{3},p_{4})\in(\Lambda^{*})^{4}$ we
let

$\displaystyle\Delta E(\vec{p})\equiv E_{p_{1}}+E_{p_{2}}-E_{p_{3}}-E_{p_{4}}$
(6.10)

where $E_{p}$ is the dispersion relation of the system–see (2.20).

Starting from (6.8) and using the definition of $d_{t}(k,p,q)$ we may
explicitly calculate that for all $\vec{p}\in(\Lambda^{*})^{4}$ there holds

$\displaystyle\phi_{t}(\vec{p})$ $\displaystyle=e^{it\Delta
E(\vec{p})}\delta(p_{1}+p_{2}-p_{3}-p_{4})\hat{V}(p_{1}-p_{4})\big{(}\chi_{1234}+\chi_{1234}^{\perp}\big{)}$
$\displaystyle-e^{it\Delta
E(\vec{p})}\delta(p_{1}-p_{2}+p_{3}-p_{4})\hat{V}(p_{1}-p_{4})\big{(}\chi_{13}\chi^{\perp}_{24}+\chi_{13}^{\perp}\chi_{24}\big{)}\
.$ (6.11)

In particular, a straightforward calculation using (6.1.1) shows that the
integrand of the quadratic term in (6.9) can be written as

$\int_{\Lambda^{*2}}\phi_{t}(p_{1},p_{2},q_{2},q_{1})\delta(q_{1}-p_{2})\mathrm{d}p_{2}\mathrm{d}q_{1}=\delta(p_{1}-p_{3})g(p_{1})$
(6.12)

where
$g(p)\equiv\chi(p)(\hat{V}*\chi)(p)+\chi^{\perp}(p)(\hat{V}*\chi^{\perp})(p)$–the
explicit form of $g(p)$ is not important, but the $\delta(p_{1}-p_{3})$
dependence in the last equation implies that the second term in (6.9) commutes
with $a_{p}^{*}a_{p}$. This fact we shall use in the proof of Lemma 6.1.

Finally, thanks to the CAR, the coefficients
$\phi_{t}(p_{1},p_{2},p_{3},p_{4})$ inside of $:V_{F}(t):$ can be
antisymmetrized with respect to the permutation of the variables
$(p_{1},p_{2})\mapsto(p_{2},p_{1})$ and $(p_{3},p_{4})\mapsto(p_{4},p_{3})$,
respectively. Namely, the coefficients $\phi_{t}$ in $:V_{F}(t):$ may be
replaced by

$\Phi_{t}(\vec{p})\equiv\frac{1}{4}\Big{(}\phi_{t}(p_{1},p_{2},p_{3},p_{4})-\phi_{t}(p_{2},p_{1},p_{3},p_{4})+\phi_{t}(p_{2},p_{1},p_{4},p_{3})-\phi_{t}(p_{1},p_{2},p_{3},p_{4})\Big{)}\
,$ (6.13)

which can be put in an explicit form, using (6.1.1). We record all these
results in the following lemma.

######  Lemma 6.3 (Normal ordering).

Let $t\in\mathbb{R}$ and $V_{F}(t)$ the Heisenberg evolution of the fermion-
fermion interaction. Then, the following identity holds

$V_{F}(t)\ =\ :\\!V_{F}(t)\\!:\ +\ N(g)\ .$ (6.14)

Here, $:\\!V_{F}(t)\\!:\ =\ \int_{\Lambda^{*4}}\Phi_{t}(p_{1}\cdots
p_{4})a_{p_{1}}^{*}a_{p_{2}}^{*}a_{p_{3}}a_{p_{4}}\mathrm{d}p_{1}\cdots\mathrm{d}p_{4}$
is the normal ordering of $V_{F}(t)$, and
$N(g)=\int_{\Lambda^{*}}g(p)a_{p}^{*}a_{p}\mathrm{d}p$, where
$g(p)\equiv\chi(p)(\hat{V}*\chi)(p)+\chi^{\perp}(p)(\hat{V}*\chi^{\perp})(p)$.

The coefficient function $\Phi_{t}:(\Lambda^{*})^{4}\rightarrow\mathbb{C}$ is
partially antisymmetric

$\Phi_{t}(p_{1},p_{2},p_{3},p_{4})=-\Phi_{t}(p_{2},p_{1},p_{3},p_{4})=+\Phi_{t}(p_{2},p_{1},p_{4},p_{3})=-\Phi_{t}(p_{1},p_{2},p_{3},p_{4})$
(6.15)

and admits the following decomposition

$\Phi_{t}=\Phi_{t}^{(1)}+\Phi_{t}^{(2)}$

where $\Phi_{t}^{(1)}$ is given by

$\Phi_{t}^{(1)}(\vec{p})=\,\frac{1}{2}\,e^{it\Delta
E(\vec{p})}\delta(p_{1}+p_{2}-p_{3}-p_{4})\big{(}\hat{V}(p_{1}-p_{4})-\hat{V}(p_{1}-p_{3})\big{)}\big{(}\chi_{1234}+\chi^{\perp}_{1234}\big{)}$
(6.16)

and $\Phi_{t}^{(2)}$ is given by

$\displaystyle\Phi_{t}^{(2)}(\vec{p})$
$\displaystyle=\,\frac{1}{2}\,e^{it\Delta
E(\vec{p})}\delta(p_{1}+p_{3}-p_{2}-p_{4})\hat{V}(p_{1}-p_{4})\big{(}\chi^{\perp}_{14}\chi_{23}+\chi^{\perp}_{23}\chi_{14}\big{)}$
$\displaystyle\ \ \ -\frac{1}{2}e^{it\Delta
E(\vec{p})}\delta(p_{1}+p_{4}-p_{2}-p_{3})\hat{V}(p_{1}-p_{3})\big{(}\chi^{\perp}_{13}\chi_{24}+\chi^{\perp}_{24}\chi_{13}\big{)}\
.$ (6.17)

####  6.1.2. Proof of Lemma 6.1

###### Proof.

We start with the normal ordering of $V_{F}(t)$ found in Lemma 6.3.

First, we observe that we may disregard the quadratic term
$N(g)\equiv\int_{\Lambda^{*}}g(t,p)a_{p}^{*}a_{p}\mathrm{d}p$. Indeed, since
$[a_{p}^{*}a_{p},N(g)]=0$ we find that for any $p\in\Lambda^{*}$

$\displaystyle\nu\big{(}[[a_{p}^{*}a_{p},V_{F}(t)],V_{F}(s)]\big{)}=\nu\big{(}[[a_{p}^{*}a_{p},:\\!V_{F}(t)\\!:],V_{F}(s)]\big{)}$
(6.18)

Furthermore, since $\nu$ is quasi-free and translation invariant, it verifies
the identities (3.16). Thus, since $[a_{p}^{*}a_{p},:V_{F}(t):]$ is quartic in
creation- and annihilation operators, we find that

$\displaystyle\nu\big{(}[[a_{p}^{*}a_{p},V_{F}(t)],V_{F}(s)]\big{)}$
$\displaystyle=\nu\big{(}[[a_{p}^{*}a_{p},:\\!V_{F}(t)\\!:],V_{F}(s)]\big{)}$
$\displaystyle=\nu\big{(}[[a_{p}^{*}a_{p},:\\!V_{F}(t)\\!:]\,,\,:\\!\\!V_{F}(s)\\!\\!:]\big{)}-\nu\big{(}[N(g),[a_{p}^{*}a_{p},:\\!V_{F}(t)\\!:]]\big{)}$
$\displaystyle=\nu\big{(}[[a_{p}^{*}a_{p},:\\!V_{F}(t)\\!:]\,,\,:\\!\\!V_{F}(s)\\!\\!:]\big{)}\
,$ (6.19)

for all $p\in\Lambda^{*}$.

Secondly, we note that a standard calculation using the CAR implies that

$\displaystyle\nu\big{(}[[a_{p}^{*}a_{p},V_{F}(t)],V_{F}(s)]\big{)}$
$\displaystyle=\int_{\Lambda^{*4}\times\Lambda^{*4}}\big{(}\delta(p-k_{1})+\delta(p-k_{2})-\delta(p-k_{3})-\delta(p-k_{4})\big{)}$
$\displaystyle\qquad\times\Phi_{t}(\vec{k})\Phi_{s}(\vec{\ell})\nu\big{(}[\,a_{k_{1}}^{*}a_{k_{2}}^{*}a_{k_{3}}a_{k_{4}}\,,\,a^{*}_{\ell_{1}}a^{*}_{\ell_{2}}a_{\ell_{3}}a_{\ell_{4}}\,]\big{)}\mathrm{d}\vec{k}\mathrm{d}\vec{\ell}$
$\displaystyle=\int_{\Lambda^{*4}\times\Lambda^{*4}}M_{p}(\vec{k},\vec{\ell})\,\nu(a_{k_{1}}^{*}a_{k_{2}}^{*}a_{k_{3}}a_{k_{4}}a^{*}_{\ell_{1}}a^{*}_{\ell_{2}}a_{\ell_{3}}a_{\ell_{4}})\mathrm{d}\vec{k}\mathrm{d}\vec{\ell}$
(6.20)

where (supressing the explicit $t,s\in\mathbb{R}$ dependence)

$\displaystyle M_{p}(\vec{k},\vec{\ell})$
$\displaystyle\equiv\Phi_{t}(\vec{k})\Phi_{s}(\vec{\ell})\big{(}\delta(p-k_{1})+\delta(p-k_{2})-\delta(p-k_{3})-\delta(p-k_{4})\big{)}$
$\displaystyle\ \
-\Phi_{t}(\vec{\ell})\Phi_{s}(\vec{k})\big{(}\delta(p-\ell_{1})+\delta(p-\ell_{2})-\delta(p-\ell_{3})-\delta(p-\ell_{4})\big{)}\
.$

A change of variables
$(k_{3},k_{4},\ell_{1},\ell_{2},\ell_{3},\ell_{4})\mapsto(\ell_{3},\ell_{4},k_{3},k_{4},\ell_{1},\ell_{2})$
now yields

$\displaystyle\nu\big{(}[[a_{p}^{*}a_{p},V_{F}(t)]$
$\displaystyle,V_{F}(s)]\big{)}$ (6.21)
$\displaystyle=\int_{\Lambda^{*4}\times\Lambda^{*4}}M_{p}(k_{1}k_{2}\ell_{3}\ell_{4},k_{3}k_{4}\ell_{1}\ell_{2})\,\nu(a_{k_{1}}^{*}a_{k_{2}}^{*}a_{\ell_{4}}a_{\ell_{3}}a^{*}_{k_{3}}a^{*}_{k_{4}}a_{\ell_{2}}a_{\ell_{1}})\mathrm{d}\vec{k}\,\mathrm{d}\vec{\ell}\
.$

It is important to note that the coefficient function
$M_{p}(\vec{k},\vec{\ell})$ is antisymmetric with respect to $k_{1}\mapsto
k_{2}$, $k_{3}\mapsto k_{4}$, $\ell_{1}\mapsto\ell_{2}$ and
$\ell_{3}\mapsto\ell_{4}$ respectively. In addition,
$M_{p}(\vec{k},\vec{\ell})=-M_{p}(\vec{\ell},\vec{k})$. Indeed, these
symmetries allow to simplify the right hand side of the last equation as
follows. First, quasi-freeness of the state $\nu$ implies that

$\displaystyle\nu(a_{k_{1}}^{*}a_{k_{2}}^{*}a_{\ell_{4}}a_{\ell_{3}}a^{*}_{k_{3}}a^{*}_{k_{4}}a_{\ell_{2}}a_{\ell_{1}})=\det\begin{pmatrix}\nu_{11}&\nu_{12}&\nu_{13}&\nu_{14}\\\
\nu_{21}&\tilde{\nu}_{22}&\tilde{\nu}_{23}&\tilde{\nu}_{24}\\\
\nu_{31}&\tilde{\nu}_{32}&\tilde{\nu}_{33}&\tilde{\nu}_{34}\\\
\nu_{41}&\tilde{\nu}_{42}&\tilde{\nu}_{43}&\tilde{\nu}_{44}\\\ \end{pmatrix}$
(6.22)

where we denote $\nu_{ij}\equiv\nu(a_{k_{i}}^{*}a_{\ell_{j}})$ and
$\tilde{\nu}_{ij}\equiv\delta(k_{i}-\ell_{j})-\nu_{ij}$. Secondly, based on
the symmetries of $M_{p}(\vec{k},\vec{\ell})$, we may follow the algebraic
analysis carried out in [31, pps 374–375] to find that

$\displaystyle\nu\big{(}[[a_{p}^{*}a_{p},$ $\displaystyle
V_{F}(t)],V_{F}(s)]\big{)}$ (6.23)
$\displaystyle=\int_{\Lambda^{*4}\times\Lambda^{*4}}M_{p}(k_{1}k_{2}\ell_{3}\ell_{4},k_{3}k_{4}\ell_{1}\ell_{2})\,4\big{(}\nu_{11}\nu_{22}\tilde{\nu}_{33}\tilde{\nu}_{44}+4\nu_{11}\nu_{23}\nu_{42}\tilde{\nu}_{34}\big{)}\mathrm{d}\vec{k}\mathrm{d}\vec{\ell}\
.$

Thirdly, translation invariance $\nu(a_{p}^{*}a_{q})=\delta(p-q)f_{0}(p)$ now
yields two terms

$\displaystyle\nu\big{(}[[a_{p}^{*}a_{p},V_{F}(t)],$ $\displaystyle
V_{F}(s)]\big{)}$
$\displaystyle=4\int_{\Lambda^{*}}M_{p}(k_{1}k_{2}k_{3}k_{4},k_{3}k_{4}k_{1}k_{2})f_{0}(k_{1})f_{0}(k_{2})\tilde{f}_{0}(k_{3})\tilde{f}_{0}(k_{4})\mathrm{d}\vec{k}$
$\displaystyle\quad+16\int_{\Lambda^{*}}M_{p}(k_{1}k_{2}k_{2}k_{3},k_{3}k_{4}k_{1}k_{4})f_{0}(k_{1})f_{0}(k_{2})f_{0}(k_{3})\tilde{f}_{0}(k_{4})\mathrm{d}\vec{k}.$
(6.24)

Similarly as in [31], we look at the two terms of the right hand side of
(6.1.2) by evaluating the function $M_{p}$ in the different cases.

The second term of (6.1.2). Let us show that the second term vanishes. Indeed,
we use the fact that
$\Phi_{t}(k_{3}k_{4}k_{1}k_{2})=\Phi_{-t}(k_{1}k_{2}k_{3}k_{4})$ together with
antisymmetry with respect to $k_{1}\mapsto k_{2}$ and $k_{3}\mapsto k_{4}$ to
find that

$\displaystyle M_{p}(k_{1}k_{2}k_{2}$ $\displaystyle
k_{3},k_{3}k_{4}k_{1}k_{4})$ (6.25)
$\displaystyle=2\cos\big{[}(t-s)(E_{1}-E_{3})\big{]}\big{(}\delta(p-k_{3})-\delta(p-k_{1})\big{)}\Phi(k_{1}k_{2}k_{3}k_{2})\Phi(k_{1}k_{4}k_{3}k_{4})$

where we denote $\Phi(\vec{k})\equiv\Phi_{0}(\vec{k})$. One may verify that
$\Phi(k_{1}k_{2}k_{3}k_{2})$ is proportional to $\delta(k_{1}-k_{3})$ and,
consequently, it holds that
$\big{(}\delta(p-k_{3})-\delta(p-k_{1})\big{)}\Phi(k_{1}k_{2}k_{3}k_{2})=0$.

The first term of (6.1.2). Using the fact that
$\Phi_{t}(k_{3}k_{4}k_{1}k_{2})=\Phi_{-t}(k_{1}k_{2}k_{3}k_{4})$ one finds

$\displaystyle M_{p}(k_{1}k_{2}$ $\displaystyle
k_{3}k_{4},k_{3}k_{4}k_{1}k_{2})$ (6.26)
$\displaystyle=2\cos\big{[}(t-s)\Delta
E(\vec{k})\big{]}|\Phi(\vec{k})|^{2}\big{(}\delta(p-k_{1})+\delta(p-k_{2})-\delta(p-k_{3})-\delta(p-k_{4})\big{)}\
.$

We plug this result back in (6.1.2) to find that after a change of variables
$(k_{1}k_{2})\mapsto(k_{3}k_{4})$,

$\displaystyle\nu\big{(}[[a_{p}^{*}a_{p},$ $\displaystyle
V_{F}(t)],V_{F}(s)]\big{)}$ (6.27)
$\displaystyle=4\int_{\Lambda^{*}}\big{(}\delta(p-k_{1})+\delta(p-k_{2})-\delta(p-k_{3})-\delta(p-k_{4})\big{)}\
|\Phi(\vec{k})|^{2}$ $\displaystyle\times\cos\big{[}(t-s)\Delta
E(\vec{k})\big{]}\Big{(}f_{0}(k_{1})f_{0}(k_{2})\tilde{f}_{0}(k_{3})\tilde{f}_{0}(k_{4})-f_{0}(k_{3})f_{0}(k_{4})\tilde{f}_{0}(k_{1})\tilde{f}_{0}(k_{2})\Big{)}\
.$

Finally, we integrate against time and a test function $\varphi(p)$ to find
that

$\displaystyle\int_{0}^{t}\int_{0}^{t_{1}}\nu\big{(}[[N(\varphi),V_{F}(t_{1})],V_{F}(t_{2})]\big{)}\mathrm{d}t_{1}\mathrm{d}t_{2}=-t|\Lambda|\int_{\Lambda^{*}}\varphi(p)Q_{t}[f_{0}](p)\mathrm{d}p\
$ (6.28)

where $Q_{t}[f_{0}]$ is the expression given by

$\displaystyle Q_{t}[f_{0}](p)$
$\displaystyle=4\pi\int_{\Lambda^{*4}}\,\frac{|\Phi(\vec{k})|^{2}}{|\Lambda|}\,\Big{[}\delta(p-k_{1})+\delta(p-k_{2})-\delta(p-k_{3})-\delta(p-k_{4})\Big{]}$
(6.29) $\displaystyle\times\delta_{t}[\Delta
E(\vec{k})]\Big{(}f(k_{3})f(k_{4})\tilde{f}(k_{1})\tilde{f}(k_{2})-f(k_{1})f(k_{2})\tilde{f}(k_{3})\tilde{f}(k_{4})\Big{)}\mathrm{d}\vec{k}\
.$

where we recall $\delta_{1}(x)=\frac{2}{\pi}\frac{\sin^{2}(x/2)}{x^{2}}$ and
$\delta_{t}(x)=t\delta_{1}(tx).$ Upon expanding $\Phi=\Phi^{(1)}+\Phi^{(2)}$
in the above expression with respect to the decomposition found in Lemma 6.3,
one may check that the formula is in agreement with the operator $Q_{t}$, as
given by Def. 2. This finishes the proof of the lemma. ∎

###  6.2. Proof of Lemma 6.2

###### Proof.

Let $\varphi\in\ell^{1}$ and $t,s\in\mathbb{R}$, let us introduce the
following notation for the fermion-fermion double commutator

$\displaystyle C_{F}(\varphi,t,s)$
$\displaystyle:=[[N(\varphi),V_{F}(t)],V_{F}(s)]$
$\displaystyle=\int_{\Lambda^{*2}}\hat{V}(k)\hat{V}(\ell)\Big{[}\Big{[}N(\varphi),D_{k}^{*}(t)D_{k}(t)\Big{]},D_{\ell}^{*}(s)D_{\ell}(s)\Big{]}\
\mathrm{d}k\mathrm{d}\ell$ (6.30)

where we have written $V_{F}(t)$ in terms of $D$-operators, see (5.5). For
simplicity, we shall assume that $\varphi$ is real-valued so that
$C_{F}(\varphi,t,s)$ is self-adjoint (in the general case, one may decompose
$\varphi=\mathrm{Re}\varphi+\mathrm{i}\mathrm{Im}\varphi$ and apply linearity
of the commutator). We claim that there exists a constant $C>0$ such that

$\|C_{F}(\varphi,t,s)\Psi\|\leqslant
C\|\hat{V}\|_{\ell^{1}}^{2}|\Lambda|\|\varphi\|_{\ell^{1}}\|\mathcal{N}^{2}\Psi\|\
,$ (6.31)

for all $\Psi\in\mathscr{F}$. To see this, we shall expand the double
commutator of the right hand side of (6.30) into eight terms. In order to ease
the notation, we shall drop the time labels $t,s\in\mathbb{R}$. Since our
estimates are uniform in time, there is no risk in doing so. In terms of the
contraction operators $D_{k}^{*}(\varphi)\equiv[N(\varphi),D_{k}^{*}]$ and
$D_{k}(\varphi)\equiv[N(\varphi),D_{k}]$ we find

$\displaystyle\Big{[}\Big{[}N(\varphi),D_{k}^{*}D_{k}\Big{]},D_{\ell}^{*}D_{\ell}\Big{]}$
$\displaystyle\ =\ D_{k}^{*}(\varphi)\Big{[}D_{k},D_{\ell}^{*}\Big{]}D_{\ell}\
+\ D_{k}^{*}(\varphi)D_{\ell}^{*}\Big{[}D_{k},D_{\ell}\Big{]}$ $\displaystyle\
+\ \Big{[}D_{k}^{*}(\varphi),D_{\ell}^{*}\Big{]}D_{\ell}D_{k}\ +\
D_{\ell}^{*}\Big{[}D_{k}^{*}(\varphi),D_{\ell}\Big{]}D_{k}$ $\displaystyle\ +\
D_{k}^{*}D_{\ell}^{*}\Big{[}D_{k}(\varphi),D_{\ell}\Big{]}\ +\
D_{k}^{*}\Big{[}D_{k}(\varphi),D_{\ell}^{*}\Big{]}D_{\ell}$ $\displaystyle\ +\
D_{\ell}^{*}\Big{[}D_{k}^{*},D_{\ell}\Big{]}D_{k}(\varphi)\ +\
\Big{[}D_{k}^{*},D_{\ell}^{*}\Big{]}D_{\ell}D_{k}(\varphi)\ .$ (6.32)

All these operators can be controlled using the Type-I and Type-IV estimates,
found in Lemma 4.5 and Lemma 4.8, respectively, together with the commutator
identities $[D_{k},\mathcal{N}]=[D_{k}(\varphi),\mathcal{N}]=0$, see Lemma
4.3. For instance, given $\Psi\in\mathscr{F}$ the first term can be estimated
as follows

$\displaystyle\|D_{k}^{*}(\varphi)\big{[}D_{k},D_{\ell}^{*}\big{]}D_{\ell}\Psi\|$
$\displaystyle\,\leqslant\,\|D_{k}^{*}(\varphi)\|\|\big{[}D_{k},D_{\ell}^{*}\big{]}D_{\ell}\Psi\|$
$\displaystyle\,\leqslant\,C\|\varphi\|_{\ell^{1}}|\Lambda|\|\mathcal{N}D_{\ell}\Psi\|$
$\displaystyle\,=\,C\|\varphi\|_{\ell^{1}}|\Lambda|\|D_{\ell}\mathcal{N}\Psi\|$
$\displaystyle\,\leqslant\,C\|\varphi\|_{\ell^{1}}|\Lambda|\|\mathcal{N}^{2}\Psi\|$
(6.33)

for a constant $C>0$. Every other term in the expansion (6.32) can be analyzed
in the same fashion, and satisfy the same bound –we leave the details to the
reader. Thus, we plug the estimate (6.33) back in the expansion (6.32) and
integrate over $k,\ell\in\Lambda^{*}$. One then obtains (6.31).

Let us now estimate the integral reminder term, we fix $0\leqslant
t_{3}\leqslant t_{2}\leqslant t_{1}$. As a first step, since $C_{F}$ and
$\mathfrak{h}_{I}$ are self-adjoint, we use the following rough upper bound

$\displaystyle\nu_{t_{3}}\big{(}$
$\displaystyle[[[N(\varphi),V_{F}(t_{1})]V_{F}(t_{2})],\mathfrak{h}_{I}(t_{3})]\big{)}\leqslant
2\nu_{t_{3}}\Big{(}C_{F}(\varphi,t_{1},t_{2})^{2}\Big{)}^{\frac{1}{2}}\nu_{t_{3}}\Big{(}\mathfrak{h}_{I}(t_{3})^{2}\Big{)}^{\frac{1}{2}}$
(6.34)

In view of Remark 5.1, we can turn the estimate (6.31) into the upper bound

$\nu_{t_{3}}\Big{(}C_{F}(\varphi,t_{1},t_{2})^{2}\Big{)}^{\frac{1}{2}}\leqslant
C\|\hat{V}\|_{\ell^{1}}^{2}|\Lambda|\|\varphi\|_{\ell^{1}}\nu_{t_{3}}\big{(}\mathcal{N}^{4}\big{)}^{\frac{1}{2}}\
.$ (6.35)

On the other hand, using the operator norm estimates (4.20), a simple but
rough estimate for the interaction Hamiltonian is found to be

$\displaystyle\|\mathfrak{h}_{I}(t)\Psi\|$ $\displaystyle\ \leqslant\
\lambda\|V_{F}(t)\Psi\|+\lambda\|V_{FB}(t)\Psi\|+\lambda\|V_{BB}(t)\Psi\|$
$\displaystyle\ \lesssim\
\lambda\|\hat{V}\|_{\ell^{1}}\|\mathcal{N}^{2}\Psi\|+\lambda\|\hat{V}\|_{\ell^{1}}R\|\mathcal{N}\Psi\|+\lambda\|\hat{V}\|_{\ell^{1}}R^{2}\|\Psi\|$
$\displaystyle\ \lesssim\
\lambda\|\hat{V}\|_{\ell^{1}}\big{(}\|\mathcal{N}^{2}\Psi\|+R^{2}\|\Psi\|\big{)}$

where we recall that $R=|\Lambda|p_{F}^{d-1}$. Consequently, in view of Remark
5.1 we find that

$\nu_{t_{3}}\Big{(}\mathfrak{h}_{I}(t_{3})^{2}\Big{)}^{\frac{1}{2}}\leqslant
C\lambda\|\hat{V}\|_{\ell^{1}}^{2}\Big{(}\nu_{t_{3}}\big{(}\mathcal{N}^{4}\big{)}^{\frac{1}{2}}+R^{2}\Big{)}$
(6.36)

where we used the fact that $\nu_{t}(\mathds{1})=1$ for all $t\in\mathbb{R}.$
The proof of the lemma is now finished once we combine Eqs. (6.34), (6.35) and
(6.36), and integrate over the time variables $0\leqslant t_{3}\leqslant
t_{2}\leqslant t_{1}\leqslant t$. ∎

##  7\. Leading Order Terms II: Emergence of $B$

The main purpose of this section is to analyze the term $T_{FB,FB}(t)$ found
in the double commutator expansion (3.20), introduced in Section 3. In
particular, we show that this term gives rise to the operator $B_{t}$, as
given in Def. 3, corresponding to the second leading order term describing the
dynamics of $f_{t}(p).$ It describes interactions between particles/holes as
mediated by virtual bosons around the Fermi surface. This is manifest in the
fact that, as we shall see, it contains the propagator of free bosons

$G_{k}(t-s)\equiv\left\langle\Omega,[b_{k}(t),b_{k}^{*}(s)]\Omega\right\rangle_{\mathscr{F}}$
(7.1)

defined for $k\in\Lambda^{*}$, and $t,s\in\mathbb{R}$.

We state the main result of this section in the following proposition, which
we prove in the remainder of the section.

######  Proposition 7.1 (Analysis of $T_{FB,FB}$).

Let $T_{FB,FB}(t,p)$ be the quantity defined in Eq. (3.21) for
$\alpha=\beta=FB$, and let $m>0$. Then, there exists a constant $C>0$ such
that for all $\varphi\in\ell_{m}^{1}$ and $t\geqslant 0$ the following
inequality holds true

$\displaystyle\big{|}T_{FB,FB}$
$\displaystyle(t,\varphi)+|\Lambda|t\left\langle\varphi,B_{t}[f_{0}]\right\rangle\big{|}$
$\displaystyle\leqslant
C|\Lambda|t^{2}\|\varphi\|_{\ell^{1}}\|\hat{V}\|_{\ell_{1}}^{2}\sup_{\tau\leqslant
t}\Big{(}R^{\frac{1}{2}}\nu_{\tau}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}\nu_{\tau}(\mathcal{N})^{\frac{1}{2}}+CR^{\frac{3}{2}}\nu_{\tau}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}+Rp_{F}^{-m}\nu_{\tau}(\mathcal{N}_{1}^{2})\Big{)}$
$\displaystyle\,+\,|\Lambda|t^{3}\lambda
R\|\varphi\|_{\ell^{1}}\|\hat{V}\|_{\ell_{1}}^{3}\sup_{\tau\leqslant
t}\Big{(}R^{\frac{3}{2}}\nu_{\tau}(\mathcal{N}_{\mathcal{S}})^{\frac{1}{2}}+R\nu_{\tau}(\mathcal{N}_{\mathcal{S}})+Rp_{F}^{-m}\nu_{\tau}(\mathcal{N})^{\frac{1}{2}}\Big{)}$
(7.2)

where
$T_{FB,FB}(t,\varphi)\equiv\left\langle\varphi,T_{FB,FB}(t)\right\rangle$ and
$B_{t}$ is given in Def. 3.

######  Remark 7.1.

In order to prove Proposition 7.1, we expand $T_{FB,FB}$ into several terms
and analyze each one separately. This expansion is based in the following two
observations:

(i) For any self-adjoint operators $N,T,S$ and state $\mu$, there holds:

$\mu\big{(}[[N,T+T^{*}],S]\big{)}=2\mathrm{Re}\,\mu\big{(}[[N,T],S]\big{)}\ .$
(7.3)

(ii) Thanks to the symmetries $D_{k}=D^{*}_{-k}$, $\hat{V}(-k)=\hat{V}(k)$ and
the vanishing commutator $[D^{*}_{k},b_{k}]=0$, starting from the
representation (5.6) we may re-write the fermion-boson interaction term as

$V_{FB}(t)=\int_{\Lambda^{*}}\hat{V}(k)B^{*}_{k}(t)D_{k}(t)\mathrm{d}k\qquad\textnormal{where}\qquad
B^{*}_{k}(t)\equiv b^{*}_{k}(t)+b_{-k}(t)\ .$ (7.4)

Starting from (3.21), based on these two observations we are able to re-write
the term $T_{FB,FB}$ for all $t\in\mathbb{R}$ and $\varphi\in\ell^{1}$ in the
following form

$\displaystyle T_{FB,FB}$ $\displaystyle(t,\varphi)$