Effective Dynamics of Extended Fermi Gases in the High-Density Regime

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.

In the last years there has been substantial progress in the derivation of effective equations for interacting fermions in the mean-field regime.In this scaling limit, we consider systems of N particles, confined in a region Λ ⊂ R 3 with volume of order one, interacting through a weak two-body potential with range comparable to the size of Λ. Denoting by V ext the trapping potential and by V the interaction, the Hamilton operator takes the form and, in accordance with the Pauli principle, it acts on L 2 a (R 3N ), the subspace of L 2 (R 3N ) consisting of functions that are antisymmetric with respect to permutations.In (1.1), we set ε = N −1/3 .Together with the factor N −1 in front of the potential energy, this choice guarantees that all terms in the Hamilton operator are, typically, of order N .In fact, because of the fermionic statistics, the expectation of N j=1 −∆ x j on states trapped in a volume of order one is at least of order N 5/3 ; this can be verified with the Lieb-Thirring inequality, see e.g.[23,Chapter 4].
To describe low-energy properties of (1.1), we introduce Hartree-Fock theory, defined by restricting (1.1) to Slater determinants, i.e., to wave functions of the form where {f j } N j=1 is an orthonormal family in the one-particle space L 2 (R 3 ).Slater determinants are an example of quasi-free states: they are completely characterized by their one-particle reduced density matrix ω N = N tr 2,...,N |ψ Slater ψ Slater | = N j=1 |f j f j | , coinciding with the orthogonal projection onto the N -dimensional subspace of L 2 (R 3 ), spanned by the orbitals {f j } N j=1 .In particular, the energy of the Slater determinant (1.2) is given by the Hartree-Fock energy functional (1. 3) The interaction contributes to (1.3) through the direct term, proportional to the product of the particle densities ω N (x; x) and ω N (y; y) and through the exchange term, proportional to |ω N (x; y)| 2 .The Hartree-Fock energy E HF N , obtained minimizing (1.3) over all rank-N orthogonal projections ω N , provides a good approximation for the ground state energy of (1.1) as N ≫ 1; see [2,19] for the case of Coulomb systems.Recently, the large N asymptotics of the correlation energy, defined as the difference between the many-body ground state energy minus the Hartree-Fock ground state energy, has been determined in [20,6,7,8,14].
It is natural to ask what happens when the external traps are switched off; does the Hartree-Fock theory also describe the resulting many-body evolution ψ mf N,t = e −iH mf N (0)t/ε ψ N , generated by the translation invariant Hamiltonian H mf N (0)?Here, the presence of the parameter ε = N −1/3 guarantees that ψ mf N,t undergoes macroscopic changes for times t of order one.The convergence of the one-particle reduced density matrix γ associated with ψ mf N,t towards the solution of the time-dependent Hartree-Fock equation has been established in [17], for analytic potentials and for short times.Here ρ t (x) = N −1 ω N,t (x; x) is the density associated with ω N,t and X t is the exchange operator, defined by its integral kernel X t (x; y) = N −1 V (x − y)ω t (x; y).More recently, in [11] (and later in [28], following a different approach), this convergence has been generalized to a much larger class of interaction potentials and to all times.This result and the techniques that were used to derive it provide the starting point for the present work.The initial data considered in [11] are assumed to satisfy suitable semiclassical estimates, which appear as a natural characterization of trapped equilibrium states, in the mean-field regime.Furthermore, in [11] it is also shown that, for bounded potentials, the exchange term in (1.5) is subleading, compared with the direct term, and that the many-body dynamics can also be approximated by the Hartree equation Notice that the result of [11] holds in the sense of convergence of density matrices; convergence in L 2 -norm for homogeneous Fermi gases has been recently obtained in [9], via the rigorous bosonization techniques developed in [6,7,8] (in this case, ω N is translation invariant which implies, in particular, that ω N,t = ω N is stationary).The result of [11] has been extended to fermions with relativistic dispersion (known as pseudo-relativistic fermions) in [12] and to quasi-free mixed states in [5].See also [13] for a review.All these works consider bounded interaction potentials.As for unbounded potentials, the time-dependent Hartree-Fock equation for particles interacting through a Coulomb potential has been derived in [29], under the assumption that a suitable semiclassical structure of the initial datum propagates along the flow of the Hartree-Fock equation.Recently, the propagation of the semiclassical structure has been proven in [15], for mixed states and for a class of singular potentials that includes a suitably regularized version of the Coulomb interaction.In the absence of semiclassical scaling, that is, setting ε = 1 in the previous discussion, convergence to the time-dependent Hartree-Fock equation has been shown in [4] for bounded potentials, and then extended to Coulomb potentials in [18] (see also [3]).
Notice that both the Hartree-Fock equation (1.5) and the Hartree equation (1.6) still depend on the number of particles N .In the limit N → ∞, the Hartree-Fock and the Hartree dynamics are known to converge to the Vlasov equation, a classical effective evolution equation.The first proof of convergence from the quantum many-body dynamics to the Vlasov dynamics has been obtained in [27] for analytic potentials, and then extended in [32] to a much larger class of interactions.Next, convergence from the Hartree-Fock to the Vlasov equation has been proved in [24,25].All these results hold in a weak sense.Bounds on the rate of convergence from the Hartree-Fock equation to the Vlasov equation have been first obtained in [1], and more recently in [10] for a larger class of initial data and of interaction potentials.Unbounded interaction potentials, including the Coulomb interaction, have been considered in [21].Finally, let us mention the result [22], where convergence from the Hartree equation to the Vlasov equation is proven for local perturbations of the equilibrium state of extended Fermi gases at fixed density, in the high density regime.This last setting will be related to the one considered in the present work.
The results described above (with the exception of [22]) apply to the mean-field limit, where particles are initially trapped in a volume of order one.To describe the physically important case of extended gases, let us now consider N fermions moving in a large region Λ ⊂ R 3 , at high density ̺ = N/|Λ| ≫ 1.If the potential has range of order one, each particle interacts, at any given time, with order ̺ other particles.Furthermore, the kinetic energy of the N particles is now of the order ̺ 2/3 N .Therefore, to describe an extended Fermi gas at high density, we consider the Hamilton operator where we set ε = ̺ −1/3 to make sure that both terms are of order N .In contrast with the mean-field regime (where we had ε = N −1/3 ), ε is now small but independent of N .We will be interested in the many-body evolution governed by the Schrödinger equation for initial data ψ N,0 = ψ N that are close (in an appropriate sense) to a Slater determinant with reduced density matrix ω N , describing a quasi-free state of N fermions in a large domain Λ ⊂ R 3 , with density of particles bounded everywhere by ε −3 (up to a multiplicative constant), in the sense that sup (1.9) Additionally, we will assume the initial data to exhibit a local semiclassical structure, captured by localized commutator bounds for ω N with the position operator and with the momentum operator, that are expected to hold true for equilibrium states of confined systems at (or close to) zero temperature and at density of order ε −3 .The present setting can also be viewed as a Kac regime, see [30] for a review.In this situation, the density of particles is order 1, the range of the potential is γ ≫ 1 uniformly in the size of the system, and the strength of the potential is fixed so that its L 1 -norm is order 1.Therefore, the many-body Hamiltonian is: We look at the evolution of the system for times τ = O(γ), so that every particle covers a distance of the order of the range of the interaction potential.Thus, writing τ = γt, the time evolution of the system is described by the Schrödinger equation: Let us denote by U γ the unitary operator on L 2 (R 3N ) implementing the space rescaling: Applying the transformation to both sides of (1.10), we have: where now U γ ψ N,t is a wave function describing a quantum system with density O(γ 3 ).Thus, the dynamics (1.11) is equivalent to the one generated by (1.7), after setting ε = γ −1 .
Although (1.7) does not describe a mean-field regime (particles typically interact with ε −3 other particles and the size of the potential is of the order ε 3 , with ε now independent of N ), for small ε > 0 we can still expect a local averaging mechanism to take place and thus that the many-body dynamics (1.8) can be approximated by the time-dependent Hartree equation with ω N,0 = ω N and where now Motivated by the results of [11], we are neglecting here the exchange term, appearing on the r.h.s. of (1.5), since it is expected to be small for the class of smooth interaction potentials that we are going to consider.In our main theorem we compare the one-particle reduced density matrix γ N,t associated with the solution of (1.8) with the solution of (1.12) and we show that γ (1) for short macroscopic times t of order 1 in ε, and for a constant C independent of ε and of N, |Λ|.This result should be compared with the trivial estimates γ It is important to notice that the rate of convergence, on the r.h.s. of (1.13),only depends on the parameter ε, and not on the volume of the system: in particular, our theorem applies to the setting in which the limit |Λ| → ∞ is taken before the limit ε → 0. To the best of our knowledge, this is the first derivation of the time-dependent Hartree equation for an interacting, extended Fermi gas (notice, however, that the dynamics of tracer particles or impurities moving through an extended ideal gas has been considered in [16,26]).Furthermore, we also consider the case of massive pseudo-relativistic fermions, evolving with Hamiltonian: .14)In this case, the relevant effective evolution equation is the pseudo-relativistic Hartree equation, For mean-field fermions, the validity of the pseudo-relativistic Hartree equation has been proved in [12].Here, we show the validity of the bound (1.13) for extended pseudo-relativistic fermions, for all times.
Technically, the main challenge we have to face consists in showing that the nonlinear Hartree equation, both in the non-relativistic and in the relativistic case, preserves the local semiclassical structure assumed on the initial data ω N .The fact that the commutator bounds for ω N,t are localized in space makes the proof of propagation of the semiclassical structure much more involved than in the mean-field regime.In particular, to achieve this we need to propagate the bound (1.9) on the density of particles along the solution of the Hartree equation.For non-relativistic fermions we are able to propagate this estimate for small times of order 1 in ε.For the pseudo-relativistic case, on the other hand, we can take advantage of the boundedness of the group velocity of the particles to show that (1.9) remains true for all times (of order 1 in ε).
The article is organized as follows.In Section 2 we state our main results, Theorem 2.2 (non-relativistic case) and Theorem 2.4 (relativistic case).Both theorems are stated for initial data satisfying the estimates of Assumption 2.1.These assumptions are verified for the free Fermi gas and for coherent states in Appendix A. In Section 3 we introduce the basic tools of our analysis, namely the fermionic Fock space and Bogoliubov transformations.In Section 4 we prove our main result; the proof is based on the adaptation of the method of [11] to extended systems, which crucially relies on the propagation of the local semiclassical structure along the flow of the Hartree equation, as stated in Theorem 4.1 for the nonrelativistic case.The proof of Theorem 4.1 is given in Section 5. Finally, in Section 6 we extend the propagation of the local semiclassical structure to the pseudo-relativistic case.
Let us denote by γ N the reduced one-particle density matrix of ψ N , with the integral kernel γ (1) In the high-density regime, the ground state of the system is expected to be well approximated by a Slater determinant for a suitable choice of orthonormal functions f i .The closeness of ψ N to a Slater determinant is expressed in terms of closeness of the reduced one-particle density matrix.The reduced one-particle density matrix of the Slater determinant (2.2) is given by a rank-N orthogonal projector and we shall suppose that 1 N γ (1) We shall now introduce some important assumptions on ω N , which are expected to hold for a wide class of confined systems at equilibrium.We shall check them in Appendix A, for the free Fermi gas and for coherent states.Strictly speaking, coherent states do not really fit our setting, since they are not projections.However, in Appendix A we will consider a family of coherent states that can be viewed as approximate projections.
For any z ∈ R 3 , any t ∈ R and any n ∈ N, let us define the localization operator as: where x(t) denotes the free evolution of the position operator Let us also introduce the weight function: with dist(x, Λ) = inf y∈Λ |x − y|.Next, we collect the assumptions we shall make on the reference Slater determinant.
Assumption 2.1 (Assumptions on the Initial Datum).There exist n ∈ N and T 1 ≥ 0 such that the following holds true: and sup and sup (2.9) We shall refer to the first two estimates in Assumption 2.1 as the local semiclassical structure of the initial datum.
We are now ready to state our main result.We shall separate the cases of non-relativistic and pseudo-relativistic fermions.Theorem 2.2 (Main Result: Non-Relativistic Case).Let ω N be a rank-N orthogonal projector on L 2 (R 3 ), satisfying Assumption 2.1 for some n ∈ N and T 1 > 0. Suppose that N − ω N tr ≤ Cε δ N , for some δ > 0. (2.11) Let ψ N,t = e −iH N t/ε ψ N , with H N given by (1.7), and let γ N,t be the reduced one-particle density matrix of ψ N,t .Let ω N,t be the solution of the time-dependent Hartree equation: ω N,0 = ω N . (2.12) Then, there exist 0 < T < T 1 and C > 0, independent of ε and N such that, for all t ∈ [0, T ]: The N -dependence of the estimates (2.11), (2.13) is the natural one.
In particular, these estimates allow to quantify the closeness of the expectation values of extensive operators.
(ii) As discussed in the introduction, the result should be compared with the trivial estimates γ N,t HS ≤ N 1/2 , ω N,t HS = N 1/2 .Normalizing the Hilbert-Schmidt norm by |Λ| 1/2 , and recalling that N = ̺|Λ| with ̺ = O(ε −3 ), our main theorem proves convergence of the many-body evolution towards the nonlinear Hartree equation as the density goes to infinity, uniformly in the system size |Λ|.
(iii) We expect the Hartree-Fock equation to give a better approximation of the many-body quantum dynamics.However, the difference between the Hartree and the Hartree-Fock dynamics is smaller than the error on the r.h.s. of (2.13) and thus it cannot be resolved with our present techniques.
The time T > 0 appearing in Theorem 2.2 is related to the validity of a suitable nonconcentration estimate for the solution of the time-dependent Hartree equation, which quantifies the number of particles in bounded regions of space.We prove this bound in Proposition 5.2 for short macroscopic times of order 1 in ε.
Next, let us consider pseudo-relativistic fermions.In this case, we only need assumptions on norms of the initial projector ω N and of its commutators, multiplied with the multiplication operator W (n) z (in the non-relativistic case, the assumptions involved instead the free evolution For this reason, in the next theorem we will only require Assumption 2.1 to hold with T 1 = 0 (which implies t = 0 in (2.6) -(2.9)).The other important difference, compared with the non-relativistic case, is that, thanks to the boundedness of the group velocity of the particles, we can establish convergence towards Hartree dynamics, for all fixed times t ∈ R (rather than only for short times).
Theorem 2.4 (Main Result: Pseudo-Relativistic Case).Let ω N be a rank-N orthogonal projector on L 2 (R 3 ), satisfying Assumption 2.1 for some n ∈ N and for Let ψ N be as in Theorem 2.2, let ψ N,t = e −iH rel N t/ε ψ N , with H rel N given by Eq. (1.14), and let γ (1) N,t be the reduced one-particle density matrix of ψ N,t .Let ω N,t be the solution of the time-dependent pseudo-relativistic Hartree equation: Then, for all t ∈ R: The rest of the paper is devoted to the proof of Theorems 2.2, 2.4.The proof of the two theorems are very similar, except for the propagation of the local semiclassical structure.To fix notations, we work with non-relativistic particles.Only in Section 6 we come back to the pseudo-relativistic case.In Section 3 we introduce the Fock space formalism, which will allow us to efficiently describe the fluctuation of the many-body evolution around the nonlinear effective dynamics as the creation and annihilation of particles around a suitable time-dependent state.The proof of Theorem 2.2 is given in Section 4. It relies on a bound for the growth of the number of fluctuations, proven in Proposition 4.4.In turn, this result crucially relies on the propagation of the local semiclassical structure along the flow of the Hartree equation.This is established in Section 5 for non-relativistic fermions and in Section 6 for pseudo-relativistic fermions.

Fock Space Representation
To prove Theorem 2.2 we switch to a Fock space formulation of the problem.
Given ψ ∈ F, ψ = (ψ (0) , ψ (1) , . . ., ψ (n) , . ..), we define the scalar product: Equipped with this natural inner product, the Fock space F is a Hilbert space.We henceforth denote by • the norm induced by this inner product, and with a slight abuse of notation we shall use the same notation to denote the operator norm of linear operators acting on F.
It is convenient to introduce creation and annihilation operators, acting on F. Let f ∈ L 2 (R 3 ).We define the creation operator a * (f ) and the annihilation operator a(f ) as for any ψ ∈ F. From a physics viewpoint, the creation operator creates a particle with wave function f , while the annihilation operator annihilates a particle with wave function f .These definitions are supplemented by the requirement a(f )Ω = 0.It is not difficult to see that a * (f ) = a(f ) * .Also, the creation and annihilation operators satisfy the canonical anticommutation relations: These relations imply that a(f ) ≤ f 2 , a * (f ) ≤ f 2 (it is not difficult to see that these bounds are sharp, that is the norms of the operators are actually equal to f 2 ).In the following sections, we will also make use of operator-valued distributions, e.g., a * x and a x for x ∈ R 3 , such that The creation and annihilation operators can be used to define the second quantization of observables.For instance, consider the number operator N , acting on a given Fock space vector as (N ψ) (n) = nψ (n) .In terms of the operator-distributions, it can be written as: More generally, for a given one-particle operator J on L 2 (R 3 ) we define its second quantization dΓ(J) as the operator on the Fock space acting as follows: where 1) .If J has the integral kernel J(x; y), we can write dΓ(J) as: In the next lemma we collect some bounds for the second quantization of one-particle operators, that will play an important role in the proof of our main result.Their proofs can be found in [11,Lemma 3.1].
Lemma 3.1.Let J be a bounded operator on L 2 (R 3 ).We have, for any ψ ∈ F: Let J be a Hilbert-Schmidt operator.We then have, for any ψ ∈ F: Let J be a trace class operator.We then have, for any ψ ∈ F: Given a Fock space vector ψ ∈ F, we define its reduced one-particle density matrix as the non-negative trace class operator γ (1) If ψ is an N -particle state, it is not difficult to check that this definition agrees with (2.1).Furthermore, given a one-particle observable J, we have: an identity which motivates the definition (3.1).In particular, tr γ (1) ψ = ψ, N ψ is the expected number of particles in ψ.
Next, we lift the many-body Hamiltonian to the Fock space, as follows.We define the second quantization of H N as In terms of the operator-valued distributions, we can write: The time evolution of a state in the Fock space is defined as ψ t = e −iH N t/ε ψ.On N -particle states, this coincides with the solution of the Schrödinger equation (1.8).

Bogoliubov Transformations
The Fock-space representation of the problem is particularly convenient also in view of the representation of Slater determinants via Bogoliubov transformations, defined here.Given an orthonormal family (f i ) N i=1 , the corresponding Slater determinant can be represented in Fock space as follows: where the only nontrivial entry is the N -th.A crucial fact for our analysis, as was the case in previous work starting from [11], is the existence of a unitary operator R ω N : F → F with the following two properties: where Equivalently, consider an orthonormal basis (f i ) i≥1 of L 2 (R 3 ) obtained completing the orthonormal family f 1 , . . ., f N in an arbitrary way.We have: The map R ω N is known as Bogoliubov transformation, and it acts as a particle-hole transformation.It allows us to switch to a new representation of the system, where the new vacuum is the Slater determinant associated to the reduced density ω N .The new creation operators, given by (3.5), create excitations around the Slater determinant, which are either particles outside the determinant or holes in it.The proof of the existence of the unitary operator R ω N with the properties listed above can be found, for example, in [31].
More generally, for every t ∈ R, we can associate a Bogoliubov transformation R ω N,t to the solution of the time-dependent Hartree equation (2.12).Then R ω N,t Ω is the Slater determinant with reduced one-particle density matrix ω N,t and

Proof of the Main Result
Here we shall prove our main result, Theorem 2.2.It will be a corollary of an estimate for the growth of the number operator evolved with a suitable fluctuation dynamics, Proposition 4.1, proven in the next section.

Bound on the Growth of Fluctuations
Let ω N,t be the solution of the time-dependent Hartree equation.We introduce the fluctuation dynamics Given an N -particle state ψ, we define the corresponding fluctuation vector ξ = R * ω N ψ.Then, we rewrite the many-body evolution of ψ as To show our main theorem we need to prove that, for N -particle initial data ψ close to the Slater determinant with reduced one-particle density matrix ω N , the evolution ψ t remains close to the Slater determinant with reduced one-particle density matrix ω N,t .This will follow, if we can control the growth of the expectation of the number of particles To reach this goal, a key ingredient is the propagation of the semiclassical structure, introduced in Assumption 2.1, along the flow of the Hartree equation.This is the content of the next theorem. and Remark 4.2.(i) The constant C is a priori explicit.It is independent of |Λ| and of ε, and it depends on T .
(ii) The requirement that ω N is a rank-N projection is not needed in this theorem.It could be replaced by 0 The proof of Theorem 4.1 is postponed to Section 5. From Theorem 4.1, we obtain the following corollary, which will be used to control the growth of the expectation (4.2) and thus to prove our main result, Theorem 2.2.

Corollary 4.3 (Bounds for Commutators with Regular Functions
).Under the same assumptions of Theorem 4.1, the following is true.Let: Then, the following bound holds true: Proof.Let χ(p) be a smooth, non-increasing, compactly supported function, equal to 1 for |p| ≤ ε −1 − 1 and equal to 0 for |p| > ε −1 .We write: ) Let us define, for ♯ =≤, >: together with f ♯ z (x) := f ♯ (x − z) and g ♯ z (x) := g ♯ (x − z).By the assumptions (4.5) and the smoothness of χ, Consider the first term on the r.h.s. of (4.8).We have: Consider the second term on the r.h.s. of (4.11).We estimate it as: Therefore, using (4.4), the contribution of this term to the final bound (4.6) is: where we used (4.10).Consider now the first term in the r.h.s. of (4.11).We estimate it as: , where we used that g (♯) is bounded, which follows from (4.10).We then have: where we denoted the Fourier transform of (1 + | • | 4n ) −1 by W (n) .The contribution of the first term to the final bound (4.6) is estimated exactly as before.We get: sup Consider now the second term in (4.14).We estimate it as: Using that and using (4.3) to estimate the last trace norm in (4.16), we get: (4.17) Putting together (4.13), (4.15), (4.17), we find: Together with (4.11), the bounds (4.13), (4.18) imply: This proves the desired estimate for the first term on the r.h.s. of (4.8).Consider now the second term on the r.h.s. of (4.8).By opening the commutator and by using the invariance of the norm under hermitian conjugation, we get: By (4.10), together with the fact that f (>) (p) = 0 for |p| < ε −1 − 1: 3), we easily get: Plugging this estimate in (4.20), we get: Combined with (4.19) and with (4.8), this concludes the proof of Corollary 4.3.
The next result is the key to control the distance between the many-body and the effective dynamics.It relies on the propagation of the local semiclassical structure, Theorem 4.1, and on its Corollary 4.3.Proposition 4.4 (Bound on the Growth of Fluctuations).Under the same assumptions of Theorem 2.2, the following is true.Let ψ be the Fock space vector associated to ψ N , and let ξ = R * ψ.Then, there exists C > 0 such that: The proof of Theorem 2.2 is a corollary of Proposition 4.4, and will be given in Section 4.2.Let us now prove Proposition 4.4 Proof of Proposition 4.4.The proof is based on a Gronwall-type argument, as in [11,12].For convenience, we shall use the following notations The starting point is the following identity, for any ξ ∈ F: with a natural identification of each term.The proof of this identity follows [11, Proof of Proposition 3.3].The difference with respect to [11] is that now the fluctuation dynamics is defined starting from the Hartree equation rather than the Hartree-Fock equation.This is the reason for the extra quadratic term in the right-hand side of (4.23), which in [11] is cancelled by the presence of the exchange term in the time-dependent Hartree-Fock equation.
Let us briefly sketch the arguments and refer to [11, Proof of Proposition 3.3] for further details.By using the definition of the Bogoliubov transformation, one can see that .24) Plugging the Hartree equation, and using that we easily get: where V is the second quantization of the many-body interaction.After conjugation with the Bogoliubov transformation, see (3.4), and normal ordering, we get: Bound for the term I.We rewrite the interaction potential as: where: The function V (2) is bounded, and its regularity can be inferred from the assumption (2.10) on the potential.Accordingly, the term I is re-written as dy V (2)  z (y)a(v t;y )a(u t;y )ξ t , where we used the notation f z (x) = f (x − z).Next, we notice that and that By the Cauchy-Schwarz inequality and by Lemma 3.1 we can bound the term I as: where in the third step we used that u N,t = 1−ω N,t together with the orthogonality condition u N,t v N,t = v N,t u N,t = 0, and that v N,t op = 1.The trace norm of the commutator can be estimated using Corollary 4.3.In fact, the function V (1) satisfies the assumptions of Corollary 4.3, and the same is true for the function V (2) , thanks to the assumptions on the potential V , Eq. (2.10).Therefore, we get: Using that R 3 dz X Λ (z) −2 ≤ C|Λ|, we find: Bound for the term II.We write: dr ds v N,t V x (x)u N,t (r, s)a r a s a(u t;x )ξ t , recall (4.28).By the Cauchy-Schwarz inequality and by Lemma 3.1 we obtain The trace norm of the commutator can be estimated using Corollary 4.3.The last term is estimated in terms of the number operator: where we used that u 2 N,t = u N,t .Thus, we get, for t ∈ [0, T ]: Bound for the term III.We write dr ds u N,t V x (x)v N,t (r; s)a r a s a(v t;x )ξ t .
Proceeding as we did for the term II, we find, for t ∈ [0; T ], where we used that v N,t v N,t ≤ 1.
Bound for the term IV.Finally, we consider the term IV, containing the quadratic contributions.We rewrite the potential as in (4.25).We then get: where we used that v N,t (s, x) = v N,t (x, s).By the Cauchy-Schwarz inequality and by Lemma 3.1 we obtain where we used that u N,t op = v N,t op = 1.Next, we estimate: where we used that By the Gronwall lemma, for all t ∈ [0, T ], for a constant K depending on T : which concludes the proof.

Proof of Theorem 2.2
Proof of Theorem 2.2.We now prove our main result.It turns out that the distance between the many-body evolution and the Hartree equation can be quantified by the average number of particles in the fluctuation vector ξ t = U N (t; 0)ξ = R * ω N,t ψ N,t , associated with the solution ψ N,t = e −iH N t/ε ψ N of the Schrödinger equation (1.8), with initial data ψ N = R ω N ξ.In fact where we used that tr γ In the second step we used the cyclicity of the trace, while in the last step we used that γ  N,t = tr ω N,t = N .On the other hand, the quantity ξ t , N ξ t is controlled by the distance between γ In particular, the assumption (2.11) implies that ξ, N ξ ≤ Cε δ N . (4.37) Hence, thanks to Proposition 4.4, we have: plugging this bound in (4.36), the final claim (2.13) follows.This concludes the proof of Theorem 2.2.

Propagation of the Semiclassical Structure: Nonrelativistic case
Here we shall prove Theorem 4.1, which is the key technical ingredient to control the growth of the number of fluctuations around the Hartree dynamics, Proposition 4. 4. In what follows, we will denote multi-indices in N 3 by Greek letters, and we shall denote the length of a multiindex by |α| := j α j .Moreover, unless otherwise specified, we will denote generic constants, possibly depending on n and T , by C and K, with the understanding that these constants can be different on different lines.Throughout the section, we will make use of the following elementary lemma.
Lemma 5.1 (Monotonicity Properties of the Trace Norm).Let A, B, C be bounded operators on L 2 (R 3 ) such that |A| 2 ≤ |B| 2 .Suppose that AC and BC are trace class.Then: Proof.We write: where the inequality follows from the operator monotonicity of the square root.This concludes the proof of the lemma.

Evolution of the Localization Operators
With respect to previous work, [11,12,5,29], one important difference is the presence of the localization operators in the semiclassical structure defined in Assumption 2.1.To propagate these bounds, we need to control the behavior of the localization operators under the Hartree dynamics U (t; s), defined by where ρ t (x) = ε 3 ω N,t (x; x), ω N,t is the solution of the Hartree equation with initial datum ω N .To achieve this, a key role will be played by the following proposition.

Proposition 5.2 (No-Concentration Bound for Short Times).
Under the assumption on the potential V of Theorem 2.2, and under the assumption of Eq. (2.9) on the initial datum ω N , the following is true.There exists T * > 0 independent of ε such that: We will postpone the proof of Proposition 5.2 after the proof of the next proposition, where we control the propagation of the localization operators.Proposition 5.3 (Bounds on the Evolution of the Localization Operator).Under the same assumptions of Theorem 2.2, consider the modified Hartree generator U p (t; s), defined by Then, for all z ∈ R 3 , t 0 ∈ R, for any 0 ≤ s, t ≤ T and any 1 ≤ k ≤ 2n the following is true: (5.2) Remark 5.4.Using the inequalities: Eq. ( 5.2) also implies: These inequalities are of the form |A| 2 ≤ |B| 2 , and will be extensively used in combination with Lemma 5.1.
The proof of Proposition 5.3 relies on the following technical lemma.
Lemma 5.5.Let k ∈ N and let F : R 3 → C be such that: Then, the following is true: for some constants C k .
Proof of Lemma 5.5.We have: (5.5) Next, we write: Consider the last commutator.We have: Recalling that xi (t) = xi − i2tε∂ i , we get: Similarly, we can rewrite F (x), |x(t)| 2n as a sum of terms, involving 0 ≤ j ≤ 2k − 1 operators xi (t) on the left times a partial derivative of F (x) of order 2k − j, multiplied by a factor (2t) 2k−j ε 2k−j .It is not difficult to see that: This concludes the proof of (5.4).
Remark 5.6.As a consequence of the assumption in Eq. (2.9), the function V * ρ t satisfies the hypotheses of Lemma 5.5 for 0 ≤ t ≤ T , 1 ≤ k ≤ max(2n, 3).In fact, for j ≤ max(4n, 7) we have: where we used the non-negativity of the density ρ t and the assumption (2.10) on the potential V .Next, by Eq. (5.1): which proves the boundedness of D j V * ρ t ∞ for any j ≤ max(4n, 7).
We are now ready to prove Proposition 5.3.
Since g(0) = f (0), we have: for 0 ≤ t ≤ T * , choosing T * so that the denominator is positive.This bound, together with (5.24), proves the final claim with, e.g., T * = min{ 1  2 (f (0)K) −1/2 , T 1 }.Notice that the constant K depends on the potential V , and can be made arbitrarily small by taking V small enough.

Proof of Theorem 4.1
In this section we shall consider initial data satisfying Assumptions 2.1, and we shall prove the stability of the local bounds in the assumptions (2.6) and (2.8) under the Hartree flow.The proof of (4.3) follows straightforwardly by application of Proposition 5.3 and of Lemma 5.1, in fact for 0 ≤ t ≤ T : where we used the invariance of the trace norm under unitary conjugation and (2.8).
Our strategy for proving (4.4) also relies on controlling the regularized Hartree evolution of the localization operators by their free evolution.The proof will be divided in a few steps.
Part 1: Setting up the Gronwall estimate.Our goal is to control the following quantity , ω N,t tr (5.27) using a Gronwall-type strategy.The claim in the theorem corresponds to the special case s = t.Let us introduce the modified Hartree dynamics U q (t; s) as in Proposition 5.3: Following [11, Section 5], we have: , ω N,t U −q (t; s) = iU q (t; s) * e iq•x , εq[ε∇, ω N,t ] U −q (t; s) .

.29)
We shall now plug this identity into (5.27), and estimate the various terms.Consider the one due to the first term on the r.h.s. of (5.29).Using Proposition 5.3, Eq. (5.2), Lemma 5.1 and the invariance of the trace under unitary conjugation, we have: We notice that for all t ≤ T Accordingly, we get: where the last step follows from assumption (2.6), since t ≤ s ≤ T .This bound, combined with (5.30), shows that: (5.32) Next, let us consider the contribution due to the second term in (5.29), namely: By Proposition 5.3 and Lemma 5.1, we have: The two contributions to the anticommutator are estimated in the same way.For instance, consider: where in the localization operator we used that e we have: A similar bound holds for the second contribution to the anticommutator in (5.33).Hence, combining (5.29), (5.32), (5.35), we obtain: (5.36) Part 2: Control of the convolution.Our last task is to estimate the argument of the integral in Eq. (5.36).As in [11], we start by writing: which gives the identity: We plug this identity in the integrand in (5.36).We get: Consider the first term on the r.h.s. of (5.38).We have, by Proposition 5.3, Lemma 5.1 and the invariance of the trace under unitary conjugation: Hence, the contribution to the integrand in (5.36) due to the first term on the r.h.s. of (5.38) is bounded by, for t ≤ s ≤ T : where in the last step we used the assumption (2.7).Let us consider now the second term in (5.38).Again using Proposition 5.3 and Lemma 5.1: Therefore, the integrand in (5.36) is bounded as: (5.39) To estimate the latter integrand, we write: y ]F y (x) tr . (5.40) where we let F y (x) := W (1) y −1 ∇V (x − y).We estimate the first term on the right-hand side as: where we set ρ p η (y) := ρ η (y)e −ip•y .To control the operator norm, we write where we set xz (s − η) := x(s − η) − z.We write the commutator by moving the powers of xz (s − η) to the left: (5.43) We then estimate the latter operator norm as follows, for 0 ≤ |α ′ | ≤ 4n: ad where we used Proposition 5.2 and that W (1) and ρ η are positive.All in all, putting together the bounds (5.41), (5.42), (5.43) and (5.44), we have: To control this integral, we split it into small and large momenta: x ω N,η tr . (5.46) The first term on the right-hand side is estimated as follows: where we used the assumptions on the interaction potential (2.10).To estimate the second term on the r.h.s. of (5.46), we follow the computations in (5.34) and (5.35), to write sup Then, by (5.26) and by the assumption on the potential (2.10), we obtain: (5.49) Therefore, putting together (5.41), (5.45), (5.46), (5.47), (5.49), we have: (5.50) To estimate the second term on the r.h.s. of (5.40), we proceed in a similar way: then estimate the operator norm as in (5.44) and the integral as in (5.46), using that W (1)  decays fast enough.
6 Propagation of the Semiclassical Structure: Pseudo-Relativistic case In this section, we show how to propagate the local semiclassical structure along the flow of the pseudo-relativistic Hartree equation with h rel (t) := √ 1 − ε 2 ∆ + V * ρ t , and ρ t (x) := ε 3 ω N,t (x; x).With respect to the nonrelativistic case, here we will be able to propagate the local semiclassical structure for all times.This allows us to prove the convergence of the many-body pseudo-relativistic dynamics to the pseudo-relativistic Hartree dynamics, Theorem 2.4.The following theorem is the analogue of Theorem 4.1.Theorem 6.1 (Propagation of the Local Semiclassical Structure.).Under the same assumptions of Theorem 2.4, we have 2) The main improvement with respect to the non-relativistic case is that in the pseudorelativistic case we are able to rule out excessive concentration of the density globally in time, thanks to the boundedness of the velocity of the particles.We start by adapting the propagation estimate for the localization operators, Proposition 5.3.Proposition 6.2 (Bounds for the Evolution of the Localization Operator).Under the same assumptions of Theorem 2.4, consider the pseudo-relativistic Hartree evolution generator U rel (t; s) defined by Then, for all k ≥ 1, there exists a constant C such that for all The reader should compare the result with Eq. (5.2).The reason why we are able to control U rel (t) * W z (x)U rel (t) with W z (x), for all times, is the fact that the velocity operator Proof.We compute the derivative so that, for any φ ∈ L 2 (R 3 ) we have where we used that W z .To control the operator norm, we write Since ad which implies the claim by the Gronwall lemma.
As a corollary, Proposition 6.2 immediately implies absence of excessive concentration for the density, for all times.tr W (1)  z ω N,t ≤ Ce Ct ε −3 .(6.8) Proof.We have: tr W (1)  z ω N,t = tr U rel (t; 0) * W (1)  z U rel (t; 0)ω N ≤ e Ct tr W (1)  z ω N ≤ Ce Ct ε −3 , (6.9) where the first inequality follows from Proposition 6.2 and from the positivity of ω N , while the last inequality follows from the assumption of Eq. (2.9).

Proof of Theorem 6.1
The first bound is an immediate consequence of Proposition 6.2 and of the assumption (2.8) on the initial datum.Let us now prove the second inequality.By using the Jacobi identity, we write: iε∂ t e iq•x , ω N,t = h rel (t), e iq•x , ω N,t + ω N,t , h rel (t), e iq•x .(6.10) Consider the second term on the right-hand side.It can be rewritten as where we have introduced the operator
This allows us to write iε∂ t U rel (t; s) * e iq•x , ω N,t U rel;q (t; s) = U rel (t; s) * e iq•x ω N,t , εA(q) U rel;q (t; s) .(6.13) Writing this equation in integral form we get: We shall now plug this identity into W (n) z e iq•x , ω N,t tr , and estimate the various terms.The first term gives the contribution, where we used Proposition 6.2, Lemma 5.1 and the invariance of the trace under unitary conjugation.We then bound the term due to the integrand in (6.14) as follows: where C τ = C exp(Cτ ).Since (1 + ε 2 (−i∇ + sq) 2 ) −1/2 op ≤ 1, the first term on the last line of (6.16) is estimated by To bound the second term, we use the integral representation valid for any self-adjoint B > 0. Accordingly: Using the following bounds, for |α| ≤ 4n: we obtain and also Putting together (6.19), (6.20) and (6.21) and using (1+ε 2 (−i∇+sq) 2 +λ) −1 op ≤ (1+λ) −1 , we obtain the estimate which, combined with (6.17) implies All in all, we have thus proven that sup Proceeding as in the proof of Theorem 4.1, we find where we used Proposition 6.2, Lemma 5.1 and the invariance of the trace under unitary conjugation.With respect to the non-relativistic case, notice the simplification introduced by the fact that localization operator is not time-evolved.Letting F y (x) be as below (5.40), we have: compare with (5.40) and (5.41).By Corollary 6.3 we have that W (1)  * ρ p η op , F * ρ p η op ≤ C η uniformly in p, whereas the integral on the last line in (6.26) is controlled by splitting in |p| ≤ ε −1 and |p| > ε −1 as was done in (5.46).Accordingly, by using the assumptions on the potential we obtain sup (6.27) Putting together the bounds (6.24), (6.25) and (6.27), we finally get: The final claim, Eq. (6.2), follows by the application of Gronwall lemma.
A Check of Assumption 2.1 Here we shall discuss examples of fermionic states satisfying Assumptions 2.1.Specifically, we shall consider the case of the free Fermi gas on R 3 at positive density, and of coherent states.We expect these assumptions to hold true for a wide class of fermionic equilibrium states.

A.1 The Free Fermi Gas
For the sake of simplicity, here we prove the assumptions for the free Fermi on R 3 , at positive density.We expect similar estimates to hold true for a free Fermi gas in a large enough, finite periodic box Λ ⊂ R 3 .Let ω be the operator on L 2 (R 3 ) with integral kernel: This operator describes the homogeneous free Fermi gas in R 3 at density ω(x; x) = (1/6π 2 )ε −3 .Clearly, ω = ω 2 = ω * , and tr ω = ∞.
Check of (2.6).Being the state defined in Λ = R 3 , here we shall replace X Λ (z) with 1. Simple computations show that the kernel of the operator [e ip•x , ω] is given by 3 1 Sp e iq•(x−y) , with S p the following set: Let F p (q) be a C ∞ smoothing of the characteristic function χ Sp (q), such that: χ Sp (q) ≤ F p (q) , F p (q) ↾ Sp = 1 , F p (q) = 0 if dist(q, S p ) ≥ 1 . (A.1) One can check that the following holds: Let O p be the operator with integral kernel: O p (x; y) = R 3 dq (2π) 3 F p (q)e iq•(x−y) .
Integrating by parts, we get: where the last step follows from the assumptions in (A.10), recalling that k F (r) = κρ(r) 1/3 .This concludes the check of (2.7) for n ≥ 2.
To conclude, using that the integral is supported for |q| ≤ k F (r) ≤ Cε −1 : ≤ C(1 + t 4n )ε −3 , where the last step follows from the assumptions (A.10).This concludes the check of (2.8), and establishes the validity of the local semiclassical structure for coherent states for n ≥ 2.

Theorem 4 . 1 (
Propagation of the Local Semiclassical Structure).Under the same assumptions of Theorem 2.2, the following is true.There exists C > 0 and T > 0 such that:

( 1 )
N,t and ω N,t , in the trace norm topology.In fact:
gives the claim.Let us now bound the terms appearing on the r.h.s.of(4.23).For brevity, we set ξ t := U N (t; 0)ξ.