Marcel Urban

François Couchot

Xavier Sarazin

A model of quantum vacuum as the origin of the speed of light

Keywords:

We show that the vacuum permeability µ0 and permittivity ǫ0 may originate from the magnetization and the polarization of continuously appearing and disappearing fermion pairs. We then show that if we simply model the propagation of the photon in vacuum as a series of transient captures within these ephemeral pairs, we can derive a finite photon velocity. Requiring that this velocity is equal to the speed of light constraints our model of vacuum. Within this approach, the propagation of a photon is a statistical process at scales much larger than the Planck scale. Therefore we expect its time of flight to fluctuate. We propose an experimental test of this prediction.

Introduction

The vacuum permeability µ 0 , the vacuum permittivity ǫ 0 , and the speed of light in vacuum c are widely considered as being fundamental constants and their values, escaping any physical explanation, are commonly assumed to be invariant in space and time. In this paper, we propose a mechanism based upon a "natural" quantum vacuum description which leads to sensible estimations of these three electromagnetic constants. A consequence of this description is that µ 0 , ǫ 0 and c are not fundamental constants but observable parameters of the quantum vacuum: they can vary if the vacuum properties vary in space or in time. A similar analysis of the quantum vacuum, as the physical origin of the electromagnetism constants, has been proposed independently by Leuchs, Villar and Sanchez-Soto [1]. Although the two mechanisms are different, the original idea is the same: the physical electromagnetic constants emerge naturally from the quantum theory.

The paper is organized as follows. First we describe our model of the quantum vacuum filled with continuously appearing and disappearing fermion pairs. We show how µ 0 and ǫ 0 originate respectively from the magnetization and the electric polarization of these pairs. We then derive the photon velocity in vacuum by modeling its propagation as a series of interactions with the pairs. Finally, we predict statistical fluctuations of the transit time of photons across a fixed vacuum path.

An effective description of the quantum vacuum

The vacuum is assumed to be filled with continuously appearing and disappearing charged fermion pairs (ephemeral particle-antiparticle pairs). We consider neither intermediate bosons nor supersymmetric particles. All known species of charged fermions are taken into account: the three families of charged leptons e, µ and τ and the three families of quarks (u, d), (c, s) and (t, b), including their three color states. This gives a total of 21 pair species, noted i.

An ephemeral fermion pair is assumed to be the product of the fusion of two virtual photons of the vacuum. Thus its total electric charge and total color are null. We suppose also that the spins of the two fermions of a pair are antiparallel, and that they are on their mass shell. The only quantity which is not conserved is therefore the energy and this is the reason for the limited lifetime of the pairs. We assume that first order properties can be deduced assuming that pairs are created with an average energy, not taking into account a full probability density of the pairs kinetic energy. Likewise, we will neglect the total momentum of the pair.

The average energy W i of a pair is taken proportional to its rest mass energy 2m i c 2 :

W i = K W 2m i c 2 (1)
where K W is a constant, assumed to be independent from the fermion type. We take K W as a free parameter, greater than unity. The value of K W could be calculated if we knew the energy spectrum of the virtual photons together with their probability to create fermion pairs. The pairs lifetime τ i is given by the Heisenberg uncertainty principle (W i τ i = h/2). So

τ i = 1 K W h 4m i c 2 (2)
We assume that the ephemeral fermion pairs densities N i are driven by the Pauli Exclusion Principle. Two pairs containing two identical fermions in the same spin state cannot show up at the same time at the same place. However at a given location we may find 21 charged fermion pairs since different fermions can superpose spatially. In solid state physics the successful determination of Fermi energies [START_REF] Ch | Elementary Solid State Physics[END_REF] implies that one electron spin state occupies a hyper volume h 3 . We assume that concerning the Pauli principle, the ephemeral fermions are similar to the real ones. Noting ∆x i the spacing between identical i-type fermions and p i their average momentum, the one dimension hyper volume is p i ∆x i and dividing by h should give the number of states which we take as one per spin degree of freedom. The relation between p i and ∆x i reads

p i ∆x i /h = 1, or ∆x i = 2πh/p i .
We can express ∆x i as a function of W i if we suppose the relativity to hold for the ephemeral pairs

∆x i = 2πhc (W i /2) 2 -(m i c 2 ) 2 = λ Ci K 2 W -1 (3)
where λ Ci = h/(m i c) is the Compton length associated to fermion i.

The pair density is defined as:

N i ≈ 1 ∆x 3 i = K 2 W -1 λ Ci 3 (4)
Each pair can be produced only in the two fermionantifermion spin combinations up-down and down-up. We define N i as the density of pairs for a given spin combination.

Finally, we use the notation Q i = q i /e, where q i is the i-type fermion electric charge and e the modulus of the electron charge.

The vacuum permeability

When a torus of a material is energized through a winding carrying a current I, the resulting magnetic flux density B is expressed as:

B = µ 0 nI + µ 0 M. ( 5
)
where n is the number of turns per unit of length and nI is the magnetic intensity in A/m. M is the corresponding magnetization induced in the material and is the sum of the induced magnetic moments divided by the corresponding volume. In an experiment where the current I is kept a constant and where we lower the quantity of matter in the torus, B decreases. As we remove all matter, B gets to a non zero value: B = µ 0 nI showing experimentally that the vacuum is paramagnetic with a vacuum permeability

µ 0 = 4π 10 -7 N/A 2 .
We propose a physical mechanism to produce the vacuum permeability from the elementary magnetization of the charged fermion pairs under a magnetic stress. Each charged ephemeral fermion carries a magnetic moment proportional to the Bohr magneton

µ i = e Q i h 2m i = e Q i c λ Ci 4π . (6)
Since the total spin of the pair is zero, and since fermion and antifermion have opposite charges, each pair carries twice the magnetic moment of one fermion.

If no external magnetic field is present, the magnetic moments point randomly in any direction resulting in a null global average magnetic moment. In the presence of an external magnetic field B, the coupling energy of the i-type pair to this field is -2µ i B cos θ, where θ is the angle between the magnetic moment and the magnetic field B. The energy of the pair is modified by this term and the pair lifetime is therefore a function of the orientation of its magnetic moment with respect to the applied magnetic field:

τ i (θ) = h/2 W i -2µ i B cos θ . (7)
The pairs having their magnetic moment aligned with the field last a bit longer than the anti-aligned pairs. The resulting average magnetic moment M i of a pair is therefore different from zero. It is aligned with the applied field. Its value is obtained integrating over θ with a weight proportional to the pairs lifetime:

M i = π 0 2µ i cos θ τ i (θ) 2π sin θ dθ π 0 τ i (θ) 2π sin θ dθ . (8)
To first order in B, one gets:

M i ≃ 4µ 2 i 3W i B. (9)
The magnetic moment per unit volume produced by the i-type fermions is M i = 2N i M i , since one takes into account the two spin states per cell. The contribution μ0,i of the i-type fermions to the vacuum permeability is thus given by B = μ0,i M i or 1/μ 0,i = M i /B. Each species of fermions increases the induced magnetization and therefore the magnetic moment. By summing over all pair species, one gets the estimation of the vacuum permeability:

1 μ0 = i M i B = i 8N i µ 2 i 3W i (10)
Using Eq. ( 1), ( 4) and ( 6) and summing over all pair types, one obtains

μ0 = K W (K 2 W -1) 3/2 24π 3 h c e 2 i Q 2 i (11)
The sum i Q 2 i is taken over all pair types. Within a generation the absolute values of the electric charges are 1, 2/3 and 1/3 in units of the positron charge. Thus for one generation the sum writes (1 + 3 × (4/9 + 1/9)). The factor 3 is the number of colours. Each generation contributes equally, hence for the three families of the standard model i

Q 2 i = 8 (12)
One obtains:

μ0 = K W (K 2 W -1) 3/2 3π 3 h c e 2 (13)
The calculated vacuum permeability μ0 is equal to the observed value µ 0 when

K W (K 2 W -1) 3/2 = µ 0 c e 2 3π 3 h = 4 3 α π 2 (14)
which is obtained for K W ≈ 31.9 , greater than one as required. In this model, we have ignored the charged loop contribution.

The vacuum permittivity

Consider a parallel-plate capacitor with a gas inside. When the pressure of the gas decreases, the capacitance decreases too until there is no more molecules in between the plates. The strange thing is that the capacitance is not zero when we hit the vacuum. In fact the capacitance has a very sizeable value as if the vacuum were a usual material body.

The dielectric constant of a medium is coming from the existence of opposite electric charges that can be separated under the influence of an applied electric field E. Furthermore the opposite charges separation stays finite because they are bound in a molecule. These opposite translations result in opposite charges appearing on the dielectric surfaces in regard to the metallic plates. This leads to a decrease of the effective charge, which implies a decrease of the voltage across the dielectric slab and finally to an increase of the capacitance. In our model of the vacuum the ephemeral charged fermion pairs are the pairs of opposite charge and the separation stays finite because the electric field acts only during the lifetime of the pairs. In an absolute empty vacuum, the induced charges would be null because there would be no charges to be separated and the capacitance of a parallel-plate capacitor would go to zero when one removes all molecules from the gas. We show here that our vacuum filled by ephemeral fermions causes its electric charges to be separated and to appear at the level of 5.10 7 electron charges per square meter under an electric stress E = 1 V /m. The mechanism is similar to the one proposed for the permeability. However, we must assume here that every fermion-antifermion ephemeral pair of the i-type bears a mean electric dipole d i given by:

d i = Q i eδ i ( 15
)
where δ i is the average separation between the two fermions of the pair. We assume that it does not depend upon the fermion momentum and we use the reduced Compton wavelength of the fermion λ Ci /(2π) as this scale:

δ i ≃ λ Ci 2π (16)
If no external electric field is present, the dipoles point randomly in any direction and their resulting average field is zero. In presence of an external electric field E, the mean polarization of these ephemeral fermion pairs produce the observed vacuum permittivity ǫ 0 . This polarization shows up due to the dipole lifetime dependence on the electrostatic coupling energy of the dipole to the field. In a field homogeneous at the δ i scale, this energy is d i E cos θ where θ is the angle between the ephemeral dipole and the electric field E. The electric field modifies the pairs lifetimes according to their orientation:

τ i (θ) = h/2 W i -d i E cos θ (17)
As in the magnetostatic case, pairs with a dipole moment aligned with the field last a bit longer than the others. This leads to a non zero average dipole D i , which is aligned with the electric field E and given, to first order in E, by:

D i ≃ d 2 i 3W i E (18)
We estimate the permittivity ǫ0,i due to i-type fermions using the relation P i = ǫ0,i E, where the polarization P i is equal to the dipole density P i = 2N i D i , since the two spin combinations contribute. Thus:

ǫ0,i = 2N i D i E = 2N i e 2 Q 2 i δ 2 i 3W i (19)
Each species of fermion increases the induced polarization and therefore the vacuum permittivity. By summing over all pair species, one gets the general expression of the vacuum permittivity:

ǫ0 = 2e 2 3 i N i Q 2 i δ 2 i W i (20)
Expressing the model parameters from Eq. ( 1), ( 4), ( 12) and ( 16), one gets:

ǫ0 = (K 2 W -1) 3/2 K W e 2 3π 3 hc (21)
If we now use the value K W given in Eq. ( 14) obtained from the derivation of the permeabilitty, one gets the right numerical value for the permittivity: ǫ0 = 8.85 10 -12 F/m.

We notice that the permeability and the permittivity do not depend upon the masses of the fermions. The electric charges and the number of species are the only important parameters. This is in opposition to the common idea that the energy density of the vacuum is the dominant factor [START_REF] Latorre | [END_REF].

The propagation of a photon in vacuum

When a real photon propagates in vacuum, it interacts with and is temporarily captured by an ephemeral pair. As soon as the pair disappears, it releases the photon to its initial energy and momentum state. The photon continues to propagate with an infinite velocity. Then the photon interacts again with another ephemeral pair and so on. The delay on the photon propagation produced by these successive interactions implies that the velocity of light is finite.

By defining σ i as the cross-section for a real photon to interact and to be trapped by an ephemeral i-type pair of fermions, the mean free path of the photon between two successive such interactions is given by:

Λ i = 1 σ i N i ( 22
)
where N i is the numerical density of virtual i-type pairs. Travelling a distance L in vacuum leads on average to N stop,i interactions on the i-type pairs, given by:

N stop,i = L Λ = Lσ i N i ( 23
)
The photon may encounter the pair any time between its appearence and disappearence. The life time of a pair being τ i , the photon will be stopped for an average time τ i /2. Each type of fermion pair contributes in increasing the propagation time of the photon. So, the total mean time T for a photon to cross a length L is:

T = i N stop,i τ i 2 (24)
Using Eq. ( 23), we obtain the photon velocity c as a function of three parameters of the vacuum model:

c = L T = 1 i σ i N i τ i /2 (25)
Using Eq. ( 2) and ( 4), we get the expression

c = K W (K 2 W -1) 3/2 16π i (σ i /λ 2 Ci ) c (26)
We now have to define the expression of the cross section σ i . We know that it should not depend on the photon energy, otherwise the vacuum would become a dispersive medium. Also the interaction of a real photon with a pair must not exchange energy or momentum with the vacuum (for instance, Compton scattering is not possible). We assume the cross-section to be proportional to the geometrical cross-section of the pair λ 2

Ci , and to the square of the electric charge Q 2 i . The cross-section is thus expressed as:

σ i = k σ Q 2 i λ 2 Ci ( 27
)
where k σ is a constant which does not depend on the type of fermions. The calculated photon velocity becomes:

c = K W (K 2 W -1) 3/2 16π k σ i Q 2 i c (28)
Using Eq. ( 12) and ( 14), one finally get:

c = 8α 3πk σ c (29)
The calculated velocity c of a photon in vacuum is equal to the observed value c when

k σ = 8 3π α (30)
It corresponds to a cross-section of 4 10 -26 m 2 on an ephemeral electron-positron pair, of the same order than the geometric transversal area of the pair, whose size is given in Eq. ( 16). We note that the photon velocity depends only on the electrical charge units Q i of the ephemeral charged fermions present in vacuum. It depends neither upon their masses, nor upon the vacuum energy density. We also remark that the average speed of the photon in our medium being c, the photon propagates, on average, along the light cone. As such, the effective average speed of the photon is independent of the inertial frame as demanded by relativity. This mechanism relies on the notion of an absolute frame for the vacuum at rest. It satisfies special relativity in the Lorentz-Poincaré sense.

Transit time fluctuations

An important consequence of our model is that stochastic fluctuations of the propagation time of photons in vacuum are expected, due to the fluctuations of the number of interactions of the photon with the virtual pairs and to the capture time fluctuations. Quantum gravity theories, which include stochastic fluctuations of the metric of compactified dimensions, predict also fluctuations of the propagation time of photons [4]. However observable effects are expected to be too small to be experimentally testable. It has been also recently predicted that the non commutative geometry at the Planck scale should produce a spatially coherent space-time jitter [5]. We show here that our effective model of photon propagation predicts fluctuations at a higher scale, which makes it experimentally testable with femtosecond pulses.

The propagation time T of a photon which crosses a distance L of vacuum is:

T = i=1 Nstop,i k=1 t i,k (31)
where t i,k is the duration of the k th interaction on i-type pairs and N stop,i the number of such interactions. The variance of T , due to the statistical fluctuations of the number of interactions and the fluctuation of the capture time is given by:

σ 2 T = i σ 2 Nstop,i t 2 stop,i + N stop,i σ 2 t,i (32)
where t stop,i = τ i /2 is the average stop time on a i-type pair, σ 2 t,i = τ 2 i /12 its variance, and σ 2 Nstop,i = N stop,i the variance of the number of interactions. Hence:

σ 2 T = 1 3 i N 2 stop,i τ 2 i = L 3 i σ i N i τ 2 i ( 33
)
Once reduced, the current term of the sum is proportional to λ Ci . Therefore the fluctuations of the propagation time are dominated by virtual e + e -pairs. Neglecting the other fermion species, and using σ e N e τ e /2 = 1/(8c), one gets

σ 2 T = τ e L 12c = λ Ce L 96πK W c 2 (34)
So

σ T = L c λ Ce c 1 √ 96πK W (35)
In our simple model where K W = 31.9, the predicted fluctuation is:

σ T ≈ 5 10 -2 fs.m -1/2 (36)
A way to search for these fluctuations is to measure a possible time broadening of a light pulse travelling a distance L of vacuum. This may be done using observations of brief astrophysical events, or dedicated laboratory experiments.

The strongest direct constraint from astrophysical observations is obtained with the very bright GRB 090510, detected by the Fermi Gamma-ray Space Telescope [6], at MeV and GeV energy scale. It presents short spikes in the 8 keV-5 MeV energy range, with the narrowest widths of the order of 4 ms (rms). Observation of the optical after glow, a few days later by ground based spectroscopic telescopes gives a common redshift of z = 0.9. This corresponds to a distance, using standard cosmological parameters, of about 2 10 26 m. Assuming that the observed width is correlated to the emission properties, this sets a limit for transit time fluctuations σ T of about 0.3 fs.m -1/2 . It is important to notice that there is no expected dispersion of the bursts in the interstellar medium at this energy scale. If we move six orders of magnitude down in distances we arrive to kpc and pulsars. Short microbursts contained in main pulses from the Crab pulsar have been recently observed at the Arecibo Observatory telescope at 5 GHz [7]. The frequency-dependent delay caused by dispersive propagation through the interstellar plasma is corrected using a coherent dispersion removal technique. The mean time width of these microbursts after dedispersion is about 1 µs, much larger than the expected broadening caused by interstellar scattering. Assuming again that the observed width is correlated to the emission properties, this sets a limit for transit time fluctuations of about 0.2 fs.m -1/2 .

The very fact that the predicted statistical fluctuations should go like the square root of the distance implies the exciting idea that experiments on Earth do compete with astrophysical constraints since we expect fluctuations in the femtosecond range at the kilometer scale.

An experimental setup using femtosecond laser pulses sent to a 100 m long multi-pass vacuum cavity equipped with metallic mirrors could be able to detect this phenomenon. With appropriate mirrors with no dispersion on the reflections, a pulse with an initial time width of 9 fs (FWHM) would be broadened after 30 round trips in the cavity, to an output time width of ∼ 13 fs (FWHM). An accurate autocorrelation measurement could detect this effect.

Conclusions

We describe the ground state of the unperturbed vacuum as containing a finite density of charged ephemeral fermions antifermions pairs. Within this framework, ǫ 0 and µ 0 originate simply from the electric polarization and from the magnetization of these pairs when the vacuum is stressed by an electrostatic or a magnetostatic field respectively. The finite speed of light is due to successive transient captures of the photon by these virtual particles. Our calculated values for ǫ 0 , µ 0 and c are equal to the measured values when the fermion pairs are produced with an average energy of about 30 times their rest mass. This model is self consistent and it proposes a quantum origin to the three electromagnetic constants. The propagation of a photon being a statistical process, we predict fluctuations of the speed of light. This could be within the grasp of modern experimental techniques and we plan to assemble such an experiment.

CNRS INSU/INP program with CNES & ONERA participations (Action Spécifique "Gravitation, Références, Astronomie, Métrologie")

Acknowledgments

It is a pleasure acknowledging helpful discussions with Gerd Leuchs. The authors thank also N. Bhat, J.P. Chambaret , I. Cognard, J. Haïssinski, P. Indelicato, J. Kaplan, P. Wolf and F. Zomer for fruitful discussions, and J. Degert, E. Freysz, J. Oberlé and M. Tondusson for their collaboration on the experimental aspects. This work has benefited from a GRAM 1 funding.


Mathieu Lewin

A nonlinear variational problem in relativistic quantum mechanics

Keywords: 2010 Mathematics Subject Classification. Primary 35Q40; Secondary 81V10. Keywords. Nonlinear analysis, variational methods, quantum mechanics, Dirac operator, renormalization, quantum electrodynamics, vacuum polarization

We describe several recent results obtained in collaboration with P. Gravejat, C. Hainzl, É. Séré and J.P. Solovej, concerning a nonlinear model for the relativistic quantum vacuum in interaction with a classical electromagnetic field.

Introduction

Quantum Electrodynamics (QED) is one of the most successful and precise theory in physics. Already studied in the early years of quantum mechanics by Dirac, Pauli, Heisenberg, Fermi, Weisskopf and others, it was finally formulated in its definite form by Bethe, Dyson, Feynman, Schwinger andTomonaga between 1946 and1950. This gave Feynman, Schwinger and Tomonaga a Nobel prize in 1965.

The theory is the combination of quantum mechanics and Einstein's special relativity. It aims at describing the interactions between matter and light, at the microscopic scale where quantum effects are dominant. It allows to determine the time-dependent behavior of charged particles (like the electrons in an atom or a molecule) when they are coupled to photons (the quanta of light). Quantum Electrodynamics has an important symmetry called charge conjugation. The latter means that the behavior of positively charged and negatively charged particles is described in a similar manner. More importantly, the theory predicts that any charged particle automatically has an anti-particle with the same mass but an opposite charge (for the electron, this particle is called the positron), and that it is possible to create a particle/anti-particle pair by providing a sufficient amount of energy to the vacuum. From Einstein's famous relation, this energy must be at least 2 × mc 2 . Because one can create matter from energy, the vacuum cannot be seen anymore as an empty and inert object as it is considered in everyday life. At the microscopic scale, the quantum vacuum is a fluctuating complicated system which participates to any physical phenomenon.

Quantum Electrodynamics is an extremely accurate theory. Its predictions are the most precise ever obtained from a physical model, when compared with experiments. The most famous successful predictions are the Lamb shift in the spectrum of the hydrogen atom and the electron anomalous magnetic dipole moment. The agreement with experiment is within a window of about 10 -8 .

In more technical terms, Quantum Electrodynamics is an abelian gauge field theory. The ideas of Dyson, Feynman, Schwinger and Tomonaga were later used to invent more complicated non-abelian gauge field (a.k.a. Yang-Mills) theories, like those describing the strong and weak forces. In units where the speed of light is c = 1 and Plank's constant is = 1, QED only depends on two parameters whose value has to be determined by experiment: the mass m of the electron, and the Sommerfeld coupling constant α which is the square of the electron charge, α = e 2 . Of course, there might also be external fields which are applied to the system. The physical value of the coupling constant α is small (about 1/137) and, in the physical literature, Quantum Electrodynamics is always formulated as a perturbative theory. This means that the interesting physical quantities are formally expanded as a power series in α, and that only the coefficients of the series are computed explicitly, order by order. But there are divergences occurring and these coefficients are all infinite! In order to solve this problem, a regularization parameter Λ has to be introduced in the model. Then, the divergences are absorbed by making a change of variable for m and α, in order to obtain a well-defined formal series in the limit Λ → ∞. This procedure is called renormalization.

It is fair to say that this perturbative formulation of QED is perfectly rigorous in the sense that there is a unique and well defined way to get the final answer (which, in most cases, is then in a surprising agreement with experiment). But, on the other hand, the perturbative nature is certainly frustrating from a mathematical point of view. In a famous quotation, Feynman himself said in 1985:

The shell game that we play (...) is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate. [START_REF] Feynman | QED: The Strange Theory of Light and Matter[END_REF] There has been no dramatic change since Feynman's quotation, and renormalization has become a common (and somehow accepted) tool.

On the mathematical side, there have been several works on models originating from QED, but not so many on the true theory itself. We should recall that one of the famous Millenium Prize Problems of the Clay Mathematics Institute concerns the construction of a well-defined Yang-Mills theory as well as the understanding of its low energy excitation spectrum. Yang-Mills theory gathers all non-abelian gauge field theories as alluded above, and any deeper understanding of the (abelian) theory of Quantum Electrodynamics is therefore desirable.

In the last few years, mathematicians have been particularly interested in studying the interaction of light with non relativistic matter. In this simplified model the electrons are quantum but they are assumed to have a speed which is much smaller than the speed of light, such that relativistic effects can be neglected. Their description then involves the Laplacian instead of the Dirac operator. Events like the creation of electron-positron pairs are encoded in the Dirac operator and they cannot be described within such a non-relativistic theory. But it is already a very important and fundamental problem to understand the effect of quantized light on non-relativistic particles. Some of the recent works concern for instance the existence of a lower eigenvalue for the underlying Hamiltonian [START_REF]Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field[END_REF][START_REF] Griesemer | Ground states in non-relativistic quantum electrodynamics[END_REF][START_REF] Hiroshima | Ground states of a model in nonrelativistic quantum electrodynamics. I[END_REF], resonances and the relaxation to the ground state [START_REF] Abou Salem | On the theory of resonances in non-relativistic quantum electrodynamics and related models[END_REF][START_REF] Bach | Renormalization group analysis of spectral problems in quantum field theory[END_REF][START_REF]Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field[END_REF], problems related to divergences and mass renormalization [START_REF] Bach | The renormalized electron mass in non-relativistic quantum electrodynamics[END_REF][START_REF] Bach | Infrared-finite algorithms in QED. II. The expansion of the groundstate of an atom interacting with the quantized radiation field[END_REF][START_REF] Hainzl | Mass renormalization and energy level shift in nonrelativistic QED[END_REF][START_REF] Hasler | Smoothness and analyticity of perturbation expansions in qed[END_REF][START_REF] Lieb | A bound on binding energies and mass renormalization in models of quantum electrodynamics[END_REF], and the stability of large systems [START_REF] Fefferman | Stability of ultraviolet-cutoff quantum electrodynamics with non-relativistic matter[END_REF][START_REF] Lieb | The stability of matter and quantum electrodynamics[END_REF][START_REF]The thermodynamic limit for matter interacting with Coulomb forces and with the quantized electromagnetic field. I. The lower bound[END_REF][START_REF] Lieb | Stability of matter in magnetic fields[END_REF].

Coming back to relativistic particles, some authors have considered Lattice QED [START_REF] Osterwalder | Gauge field theories on a lattice[END_REF][START_REF] Seiler | Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics[END_REF]. Other studied linear and nonlinear models based on the Dirac operator for finitely many particles in an 'inert' vacuum and with a classical electromagnetic field, see e.g. [START_REF] Dolbeault | Relativistic hydrogenic atoms in strong magnetic fields[END_REF][START_REF] Esteban | Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations[END_REF][START_REF] Esteban | Solutions of the Dirac-Fock equations for atoms and molecules[END_REF][START_REF] Flato | Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations[END_REF][START_REF] Georgiev | Small amplitude solutions of the Maxwell-Dirac equations[END_REF][START_REF] Huber | Solutions of the Dirac-Fock equations and the energy of the electron-positron field[END_REF] and the references in [START_REF] Esteban | Variational methods in relativistic quantum mechanics[END_REF]. Finally, the quantum vacuum and the process of pair creation was investigated in a noninteracting setting (meaning without any light at all), e.g. in [START_REF] Klaus | The regular external field problem in quantum electrodynamics[END_REF][START_REF]Vacuum polarization in Fock space[END_REF][START_REF] Nenciu | Existence of the spontaneous pair creation in the external field approximation of Q.E.D[END_REF][START_REF] Nenciu | On regular external fields in quantum electrodynamics[END_REF][START_REF] Pickl | On adiabatic pair creation[END_REF].

With P. Gravejat, C. Hainzl, É. Séré and J.-P. Solovej, we followed another route in a series of works [START_REF] Gravejat | Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields[END_REF][START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF][START_REF]Renormalization and asymptotic expansion of Dirac's polarized vacuum[END_REF][START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF]Self-consistent solution for the polarized vacuum in a no-photon QED model[END_REF][START_REF]Existence of atoms and molecules in the mean-field approximation of nophoton quantum electrodynamics[END_REF][START_REF] Hainzl | A minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics[END_REF][START_REF] Hainzl | The mean-field approximation in quantum electrodynamics: the no-photon case[END_REF] which originated from a fundamental paper of Chaix and Iracane [START_REF] Chaix | From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism[END_REF][START_REF] Chaix | From quantum electrodynamics to mean field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation[END_REF], and which was stimulated by the previous works [START_REF] Bach | On the stability of the relativistic electron-positron field[END_REF][START_REF] Hainzl | Non-perturbative mass and charge renormalization in relativistic no-photon quantum electrodynamics[END_REF]. We considered relativistic particles in a fluctuating vacuum, both described by the Dirac equation, and in interaction with a classical electromagnetic field. That light is not quantized in this model prevents us from describing important physical effects. But, on the other hand, these seem to be the first mathematical results dealing with the quantum vacuum in interaction with light. Also, as we will explain, we are able to construct the associated model in a fully non-perturbative fashion, in a simplified setting.

The purpose of this article is to review these recent results. We will particularly insist on the uncommon mathematical aspects, like those related to renormalization, and which are related to important physical effects. The reader interested in knowing more details can also read the last section of the review [START_REF] Esteban | Variational methods in relativistic quantum mechanics[END_REF].

The quantum vacuum in classical electromagnetic fields

In this section we present a model for quantum relativistic particles evolving in a fluctuating quantum vacuum, and interacting with a classical electromagnetic field. We will not discuss too much here the mathematical meaning of the equations, which will be explained in the next sections. Actually, most of the terms that we write here are infinite quantities if no special care is employed. We will concentrate on the time-independent model and look for stationary states.

Our system is composed of particles (electrons or positrons) together with the quantum vacuum on the one side, and of a classical electromagnetic field (E, B) describing light on the other side, all evolving in the physical space R 3 . It is useful to use electromagnetic potentials V and A which are such that E = -∇V and B = ∇ ∧ A. We might also consider fixed external fields E ext = -∇V ext and B ext = ∇ ∧ A ext which are applied to the system. The particles and the vacuum are described by the Dirac operator

D m,eAtot := 3 k=1 α k -i ∂ ∂x k -eA tot (x) k + mβ -eV tot (x), (1)
where A tot := (V tot , A tot ) with V tot = V + V ext and A tot = A + A ext are the total electromagnetic fields, and e is the (bare) electron charge. The four Dirac matrices α = (α 1 , α 2 , α 3 ) and β are equal to

α k := 0 σ k σ k 0 and β := I 2 0 0 -I 2 ,
the Pauli matrices σ 1 , σ 2 and σ 3 being defined by

σ 1 := 0 1 1 0 , σ 2 := 0 -i i 0 and σ 3 := 1 0 0 -1 .
These matrices are chosen to ensure that D m,0,0

2 = -∆ + m 2 which is the quantum equivalent of Einstein's fundamental relation E 2 = c 2 p 2 + m 2 c 4
for the relativistic classical energy E in terms of the momentum p, with c = 1 in our case.

The operator in (1) acts on the Hilbert space H := L 2 (R 3 , C 4 ). Under reasonable assumptions on V tot and A tot , it is self-adjoint on the Sobolev space H 1 (R 3 , C 4 ), see [START_REF] Thaller | The Dirac equation[END_REF]. The charge conjugation is the anti-unitary operator defined on H by C f := iβα 2 f , and which satisfies C D m,eAtot C -1 = -D m,-eAtot . This relation implies that the spectrum of D m,0,0 is symmetric with respect to 0,

σ D m,0,0 = (-∞, -m] ∪ [m, ∞).
When V tot and A tot decay at infinity, the essential spectrum stays the same by the Rellich-Kato theorem, σ ess D m,eAtot = (-∞, -m] ∪ [m, ∞). Isolated eigenvalues of finite multiplicity can however appear in the gap (-m, m) when V tot = 0. The state of one particle is described by a normalized wave function ϕ ∈ L 2 (R 3 , C 4 ), R 3 |ϕ| 2 = 1, and its corresponding energy is ϕ, D m,eAtot ϕ (which has to be understood in the form sense when ϕ ∈ H 1/2 (R 3 , C 4 )). The fact that the Dirac operator is unbounded from below, hence that the energy can be arbitrarily negative was very surprising for his inventor, P.A.M. Dirac [START_REF] Dirac | The quantum theory of the electron[END_REF][START_REF] Dirac | Note on exchange phenomena in the Thomas atom[END_REF]. In quantum mechanics it is always assumed that the most stable state of the system is the one with the lowest energy, and there does not seem to be any here. From a mathematical point of view this makes a huge difference as compared to non-relativistic models based on the Laplace operator -∆. That the energy is unbounded from below means that minimization methods cannot be employed and that one has to resort to complicated min-max techniques to construct solutions [START_REF] Esteban | Variational methods in relativistic quantum mechanics[END_REF].

Dirac did not want to renounce the physical picture that states with lower energy are more stable. So, in 1930 he suggested to reinterpret the problem by changing the role of the vacuum:

We make the assumption that, in the world as we know it, nearly all the states of negative energy for the electrons are occupied, with just one electron in each state, and that a uniform filling of all the negative-energy states is completely unobservable to us. [START_REF]Théorie du positron[END_REF] Physically, one therefore has to imagine that the vacuum (called the Dirac sea) is filled with infinitely many virtual particles occupying the negative energy states. With this conjecture, a real electron cannot be in a negative state. This is because electrons are fermions and that two such particles can never be in the same quantum state (Pauli principle). With this interpretation, Dirac was able to conjecture the existence of "holes" in the vacuum, interpreted as anti-electrons or positrons, and which were later experimentally discovered by Anderson [START_REF] Anderson | The positive electron[END_REF]. Dirac also predicted the phenomenon of vacuum polarization: In an external electric field, the virtual electrons are displaced, and the vacuum acquires a non constant density of charge. The idea of a fluctuating quantum vacuum was born.

Let us now explain Dirac's idea in more mathematical terms. The state of N electrons can be represented by N wave functions ϕ 1 , ..., ϕ N ∈ H 1 (R 3 , C 4 ), such that ϕ i , ϕ j L 2 = δ ij , the latter constraint being the mathematical formulation of the Pauli principle. The total energy is the sum of the energies of the individual electrons, which can be written as

N j=1 ϕ j , D m,eAtot ϕ j = Tr D m,eAtot P ,
where P is the orthogonal projector onto the N -dimensional space spanned by the ϕ j 's. Using the bra-ket notation we have P = N j=1 |ϕ j ϕ j |. Now if we allow any number N of particles and look for the state of lowest energy, the formal solution is the negative spectral projector

P = P - m,eAtot := 1 (-∞,0) D m,eAtot . (2)
Think of a finite hermitian matrix M , then the solution to the minimization problem inf{Tr(M P ), P 2 = P = P * } is the negative spectral projector P = 1 (-∞,0) (M ) of M , and the corresponding energy is -Tr M -with x -:= -min(x, 0) denoting the negative part. The minimizer is unique when ker(M ) = {0}. In our case the state in (2) corresponds to filling with particles all the negative energies of D m,eAtot as suggested by Dirac. This state P - m,eAtot is the quantum vacuum in the presence of the electromagnetic potential A tot . It has infinitely many particles and the corresponding total energy is also infinite.

Remark 2.1. There are several ways to give a mathematical meaning to the assertion that P - m,eAtot minimizes P → Tr(D m,eAtot P ). The first is to use a socalled thermodynamic limit. The ambient space H = L 2 (R 3 , C 4 ) is approximated by a sequence of finite-dimensional spaces in which everything makes sense and it is proved that in the limit the minimizer converges to P - m,eAtot

+ P - m,eAtot (1 -P )P - m,eAtot |D m,eAtot | 1/2 ≥ 0. ( 3
)
We have used here the commutativity of the trace and that D m,eAtot = |D m,eAtot | P + m,eAtot -P - m,eAtot , by definition of the spectral projections. The last term in (3) is the trace of a non-negative operator which is always well-defined in [0, ∞], and which can be taken as a definition for the trace on the left side. See [26, Sec. 2.1] for a systematic and abstract theory of traces of this form.

When A tot = 0, then P - m,0 is nothing but the negative spectral projector of the free Dirac operator D m,0 , which represents the free vacuum. This operator is translation invariant and, even if the corresponding charge density is infinite, it is somehow constant. This is the 'uniformity' alluded to in Dirac's quotation. When eA tot = 0, the state of the vacuum changes. Later we will want to optimize over A with A ext fixed, but before we have to modify a bit our theory in order to make it charge-conjugation invariant. If we take an arbitrary state P = P 2 , then we get Tr D m,eAtot P = -Tr D m,-eAtot C P C -1 . This is not very satisfactory because -C P C -1 is not a fermionic state and we cannot reinterpret this as the energy of something. It is better to subtract half the identity to the operator P , which amounts to adding a (infinite) constant. So we consider instead the energy Tr D m,eAtot P -1 2 = Tr D m,eAtot P -P ⊥ 2 with P ⊥ := 1-P . Now we see that Tr D m,eAtot (P -P ⊥ ) = Tr D m,-eAtot (P ′ -P ′ ⊥ ) with P ′ := C P ⊥ C -1 . The subtraction of half the identity to P is a common technique for systems at half filling like ours. It is explained using the formalism of second quantization in [START_REF] Hainzl | The mean-field approximation in quantum electrodynamics: the no-photon case[END_REF].

We are now able to write the total Lagrangian of our system, which is the (formal) sum of the particle energy and Maxwell's classical Lagrangian:

L m,e (P, A ; A ext ) := Tr D m,e(A+Aext) P - 1 2 + 1 8π R 3 |B| 2 -|E| 2 ,
where we recall that E = -∇V and B = ∇ ∧ A. Our purpose is to look for critical points of this Lagrangian, obtained by minimizing over P and A and maximizing over V , as is usually done in classical electrodynamics. We have already explained that the minimum over P with A fixed gives P = P - m,eAtot . It is also possible to optimize over A, with P fixed. The optimal potentials solve Gauss' equation -∆V = 4π e ρ P -1/2 , -∆A = 4π e j P -1/2

where ρ P -1/2 and j P -1/2 are the (formal) density of charge and density of current of the vacuum in the state P . These are defined by ρ M (x) = Tr C 4 M (x, x) and j M (x) k = Tr C 4 α k M (x, x), for any locally trace-class operator M with integral kernel M (x, y). Note that these are zero for the free vacuum:

ρ P - m,0 -1/2 ≡ 0, j P - m,0 -1/2 ≡ 0, as is shown using the commutation properties of the matrices α k . This emphasizes the importance of subtracting half the identity to our vacuum state.

Our task is to optimize over both P and A and it is reasonable to think that the free vacuum P = P - m,0 is the optimal state when A ext = 0. But, the functional L m,e is infinite and so we cannot easily optimize it. We can however subtract the universal (infinite) constant L m,e (P - m,0 , 0 ; 0) and consider the relative Lagrangian

L rel m,e (P, A ; A ext ) = Tr D m,0 P -P - m,0 -e R 3 j P -P - m,0 • (A + A ext ) -e R 3 ρ P -P - m,0 (V + V ext ) + 1 8π R 3 |B| 2 -|E| 2 . ( 4
)
Because the difference P -P - m,0 is a much better behaved operator than P -1/2, we will be able to give a clear meaning to this Lagrangian. The purely electrostatic case A = A ext = 0 is much easier to deal with than the general case, and we discuss it first in the next section.

The purely electrostatic case

In the relative Lagrangian (4), we now assume A = A ext = 0 and we maximize with respect to V . We find the following energy functional

Tr D m,0 Q -e R 3 ρ Q V ext + α 2 D(ρ Q , ρ Q )
with α = e 2 (the Sommerfeld coupling constant), Q := P -P - m,0 and

D(f, g) := R 3 R 3 f (x) g(y) |x -y| dx dy = 4π R 3 f (k) ĝ(k)
|k| 2 dk which is the Coulomb (i.e. Ḣ-1 (R 3 )) scalar product. The associated space will be denoted by

C := {f ∈ S ′ (R 3 ) : D(f, f ) < ∞}.
In our setting we are interested in taking for V ext the external field induced by some localized density of charge ν ext , describing for instance the nuclei of a molecule. So we take V ext = eν ext * |x| -1 and get the energy

E νext m,α (Q) = Tr D m,0 Q -αD(ρ Q , ν ext ) + α 2 D(ρ Q , ρ Q ) (5)
which is called the reduced Bogoliubov-Dirac-Fock energy [START_REF] Chaix | From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism[END_REF][START_REF]Self-consistent solution for the polarized vacuum in a no-photon QED model[END_REF].

Under suitable assumptions on ν ext , the minimization of this functional in Q makes perfect sense. First we remark that, by the Cauchy-Schwarz inequality,

-αD(ρ Q , ν ext ) + α 2 D(ρ Q , ρ Q ) ≥ - α 2 D(ν ext , ν ext )
which is finite when ν ext ∈ C (for instance for ν ext ∈ L 6/5 (R 3 ), by the Hardy-Littlewood-Sobolev inequality [41]). Then we recall that Tr D m,0 Q ≥ 0 for any operator Q = P -P - m,0 , provided the trace is defined like in (3). This is reminiscent of the fact that P - m,0 formally minimizes P → Tr D m,0 P . From all this we conclude that E νext (Q) ≥ -αD(ν ext , ν ext )/2 and the energy is bounded from below.

However we are unlucky that the infimum of E νext m,α is never attained [27, Thm 2], except when ν ext ≡ 0 where Q = 0 is the optimizer. The clear mathematical statement is inf

-P - m,0 ≤Q≤P + m,0 Tr D m,0 Q -αD(ρ Q , ν) + α 2 D(ρ Q , ρ Q ) = - α 2 D(ν, ν).
In the infimum one can restrict to finite-rank smooth operators Q, which ensures that all the terms make sense. We see that there is never a minimizer when ν ext = 0, because it should satisfy both ρ Q = ν and Q = 0. As explained in [START_REF]Self-consistent solution for the polarized vacuum in a no-photon QED model[END_REF], this issue is due to certain divergences in Fourier space at infinity. In order to give a meaning to our minimization problem, we have to impose a cut-off for the large Fourier frequencies. There are several ways to do this, some being more natural than others. In the purely electric case, we can consider a simple sharp cut-off consisting in replacing our ambient Hilbert space H by

H Λ := f ∈ L 2 (R 3 , C 4 ) : f (k) = 0 for |k| ≥ Λ .
Note that D m,0 maps H Λ into itself. For simplicity we use the same notation D m,0 and P ± m,0 for the associated restrictions to H Λ . The operator Q is defined on H Λ and its energy has the same expression as in [START_REF] Bach | Infrared-finite algorithms in QED. II. The expansion of the groundstate of an atom interacting with the quantized radiation field[END_REF]. It now has minimizers. Theorem 3.1 (Existence of the polarized vacuum [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF]Self-consistent solution for the polarized vacuum in a no-photon QED model[END_REF]). Let m > 0, 0 < Λ < ∞ and α ≥ 0. For any fixed

ν ext ∈ C = Ḣ-1 (R 3 ), the minimization problem inf E νext m,α (Q) : Q : H Λ → H Λ , -P - m,0 ≤ Q ≤ P + m,0 , ∓ Tr P ± m,0 QP ± m,0 < ∞ (6)
admits at least one minimizer Q * . All these minimizers share the same density

ρ Q * . Any minimizer Q * is a solution of the nonlinear equation    Q * = 1 (-∞,0) (D * ) -P - m,0 + δ D * = Π Λ D m,0 + α(ρ Q * -ν ext ) * |x| -1 Π Λ (7)
where Π Λ is the projection onto H Λ , and 0 ≤ δ ≤ 1 {0} (D * ).

If απ 1/6 2 11/6 D(ν, ν) 1/2 < √ m, then ker (D * ) = {0} and Tr(P + m,0 Q * P + m,0 + P - m,0 Q * P - m,0 ) = 0. Hence δ ≡ 0 in [START_REF]Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field[END_REF], and the minimizer Q * is unique.

The equation ( 7) on the infinite rank operator P * = Q * + P - m,0 can be interpreted as an infinite system of coupled nonlinear partial differential equations. As we have said we always use the convention

Tr D m,0 Q := Tr |D m,0 | Q ++ -Q --
where Q ++ := P + m,0 QP + m,0 ≥ 0 and Q --:= P - m,0 QP - m,0 ≤ 0 when Q satisfies the constraint -P - m,0 ≤ Q ≤ P + m,0 . Since with the cut-off Λ the operator D m,0 is bounded, it is sufficient to ask that Q ++ and Q --are trace-class in order to properly define the trace, as required in [START_REF] Bach | Renormalization group analysis of spectral problems in quantum field theory[END_REF]. On the other hand simple algebraic manipulations show that the constraint is equivalent to

-P - m,0 ≤ Q ≤ P + m,0 ⇐⇒ Q 2 ≤ Q ++ -Q -- (8)
as remarked in [START_REF] Bach | On the stability of the relativistic electron-positron field[END_REF]. So we see that we must also have Tr Q 2 < ∞. In other words, for the first term in the energy to be finite, Q must be a Hilbert-Schmidt operator whose diagonal blocks Q ±± are both trace-class. Under these conditions and with the ultraviolet cut-off Λ, it was proved in [START_REF]Existence of atoms and molecules in the mean-field approximation of nophoton quantum electrodynamics[END_REF] that

ρ Q ∈ L 2 (R 3 ) ∩ C
. This is sufficient to prove Theorem 3.1 using simple arguments from convex analysis.

In [START_REF] Hainzl | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation[END_REF][START_REF]Self-consistent solution for the polarized vacuum in a no-photon QED model[END_REF] a more complicated model with an additional so-called exchange term is considered and this term makes the analysis much more involved.

Let us emphasize that Theorem 3.1 is valid for any value of the coupling constant α and any value of the ultraviolet cut-off Λ. This is therefore a non-perturbative result. Because no smallness assumption is made on the external density ν ext , the model is appropriate for the description of non-perturbative events like the spontaneous creation of electron-positron pairs in strong external fields [START_REF] Sabin | Static Electron-Positron Pair Creation in Strong Fields for a Nonlinear Dirac model[END_REF]. Note that for simplicity we have minimized the energy with the relaxed constraint (8) instead of the original constraint that P := Q + P - m,0 = P 2 . However, for a small external density ν ext , the minimizer Q * is unique and the corresponding state P * is an orthogonal projection, as expected.

In Theorem 3.1 we have constructed the polarized quantum vacuum in the presence of the external density ν ext , which is by definition a global minimizer of the energy. It is also possible to construct states having a certain number N of 'real' electrons (or positrons). More precisely, we can minimize the energy under a charge constraint of the form " Tr Q" := Tr(Q ++ + Q --) = N , see [START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF][START_REF]Existence of atoms and molecules in the mean-field approximation of nophoton quantum electrodynamics[END_REF]. Minimizers do not always exist, depending on the value of N and on the strength of the density ν ext . Estimates on the maximal number of electrons that can be bound by ν ext were provided in [START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF], following ideas of Lieb [START_REF] Lieb | Bound on the maximum negative ionization of atoms and molecules[END_REF]. Any minimizer with charge constraint solves a nonlinear equation similar to [START_REF]Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field[END_REF], with the energy level 0 replaced by a Lagrange multiplier µ ∈ (-m, m):

Q * = 1 (-∞,µ) (D * ) -P - m,0 + δ (9)
where now 0

≤ δ ≤ 1 {µ} (D * ). If µ > 0, splitting 1 (-∞,µ) = 1 (-∞,0) + 1 (0,µ)
gives the corresponding states for the polarized vacuum and for the electrons. If µ < 0, we split 1 (-∞,µ) = 1 (-∞,0) -1 (µ,0) and get the state of the polarized vacuum and of the positrons. In the rest of this article we will not discuss further this constrained minimization problem, and we will instead concentrate on the pure vacuum case considered in Theorem 3.1.

The minimizers obtained in Theorem 3.1 are very singular mathematical objects. The following says that Q * is never a trace-class operator. Theorem 3.2 (Renormalized charge [START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF]). Let Q * be a minimizer as in Theorem 3.1, for some

ν ext ∈ L 1 (R 3 ) ∩ C. Then we have ρ Q * ∈ L 1 (R 3 ) and R 3 ρ Q * -ν ext = Tr(Q ++ * + Q -- * ) - R 3 ν ext 1 + αB Λ/m ( 10
)
where

B Λ/m = 1 π Λ √ m 2 +Λ 2 0 z 2 -z 4 /3 1 -z 2 dz = 2 3π log Λ m - 5 9π + 2 log 2 3π + O(m 2 /Λ 2 ). ( 11
)
In particular, if απ 1/6 2 11/6 D(ν, ν)

1/2 < √ m, then Q * is not trace-class.
Let us recall that a compact operator B is trace-class when Tr |B| = Tr √ B * B < ∞. This is equivalent to saying that j ϕ j , |B|ϕ j < ∞ for one (hence for all) orthonormal basis {ϕ j } of the ambient Hilbert space. By the spectral theorem, this is also the same as saying that the eigenvalues of B are summable, j |λ j | < ∞. If the ambient space is an L 2 space, we can write by the spectral theorem B = j λ j |ψ j ψ j | and we get that the corresponding density ρ B = j λ j |ψ j | 2 is in L 1 and that ρ B = Tr B.

For a non positive operator B, it can happen that j ϕ j , Bϕ j is convergent for one basis and not for another. This is exactly what is happening for our operator Q * . The two diagonal blocks Q ++ * and Q -- * are trace-class, which means that j ϕ + j , Q * ϕ + j + ϕ - j , Q * ϕ - j converges for any orthonormal basis {ϕ ± j } of H ± Λ := P ± m,0 H Λ . The surprise is that ρ Q * is always in L 1 (R 3 ) but that in general R 3 ρ Q * = Tr(Q ++ + Q --). The discrepancy between these two quantities is universal and it is given by the relation [START_REF] Dirac | The quantum theory of the electron[END_REF]. The problem comes from the offdiagonal densities ρ Q ±∓ * which are in L 1 (R 3 ) but do not have a vanishing integral. More precisely, only the first order term when ρ Q ±∓ * are expanded in a power series in α contribute to R 3 ρ Q * . The proof of Theorem 3.2 consists in studying this term in details [START_REF] Gravejat | Ground state and charge renormalization in a nonlinear model of relativistic atoms[END_REF].

Theorem 3.2 has a natural interpretation in terms of charge renormalization. Imagine that we put a nucleus of charge Z = R 3 ν ext in the vacuum, which is weak in the sense that D(ν ext , ν ext ) is small, and let P * = Q * + P - m,0 be the corresponding unique polarized vacuum obtained by Theorem 3.1. This vacuum is neutral, Tr(Q ++ * + Q -- * ) = 0, which means that the external field is not strong enough to create electron-positron pairs. In reality we never measure the charge of the nucleus alone, but we always also observe the corresponding vacuum polarization. Hence we do not see Z, but rather Z/(1 + αB Λ/m ). This corresponds to having a physical coupling constant α ph given by the renormalization formula

α ph = α 1 + αB Λ/m ⇐⇒ α = α ph 1 -α ph B Λ/m . (12)
In our theory we must take α ph ≃ 1/137, the 'bare' α can never been observed. So using the change of variable [START_REF] Dirac | Note on exchange phenomena in the Thomas atom[END_REF] we should express any other physical quantity predicted by our model in terms of α ph , m and Λ only.

The natural question arises whether it is possible to remove the ultraviolet cutoff Λ, by keeping α ph and m fixed (note that the mass m is not renormalized in this model). The answer is clearly no! We immediately get from (12) that α ph B Λ/m < 1 and therefore α ph → 0 whenever we try to take Λ → ∞. This phenomenon is called the Landau pole [START_REF] Landau | On the quantum theory of fields[END_REF] and one has to look for a weaker definition of renormalizability. The cut-off Λ which was first introduced as a mathematical trick to regularize the model has actually a physical meaning. A natural scale occurs beyond which the model does not make sense. Fortunately, this corresponds to momenta of the order Λ ∼ me 3π/2α ph , a huge number for α ph ≃ 1/137.

In [START_REF]Renormalization and asymptotic expansion of Dirac's polarized vacuum[END_REF], the regime α ph ≪ 1, Λ ≫ 1 with α ph log Λ fixed was studied. It was proved that the dressed density of the nucleus admits a Taylor expansion

α(ν ext -ρ Q * ) = α ph ν ext + K k=1 ν k α k+1 ph + O(α K+2 ph ) (13)
where the terms ν k in the expansion are independent of the value of α ph log Λ. The first correction ν 1 gives rise to the famous Uehling potential [START_REF] Serber | Linear modifications in the Maxwell field equations[END_REF][START_REF] Uehling | Polarization effects in the positron theory[END_REF]. The relation [START_REF] Dolbeault | Relativistic hydrogenic atoms in strong magnetic fields[END_REF] shows that the density of the nucleus can be renormalized order by order. It is believed that the series in ( 13) is divergent [START_REF] Dyson | Divergence of Perturbation Theory in Quantum Electrodynamics[END_REF], but no mathematical argument has been provided so far.

To summarize, in the purely electrostatic case we have a well defined nonperturbative theory for all m > 0, all α ph ≥ 0 and all cut-off Λ such that α ph B Λ/m < 1. In the regime α ph ≪ 1, Λ ≫ 1 with α ph log Λ fixed, the perturbation series computed by the physicists is recovered.

The electromagnetic case

Let us go back to our relative Lagrangian (4), including now the electromagnetic fields B tot = B + B ext . Optimizing with respect to P we find a formal Lagrangian depending only on the classical fields

L rel m,e (A ; A ext ) = 1 2 Tr |D m,0 | -|D m,e(A+Aext) | + 1 8π R 3 |B| 2 -|E| 2 . ( 14
)
This Lagrangian is not really well-defined because of the ultraviolet divergences which have to be taken care of. On the contrary to the pure electrostatic case studied in the last section, we cannot impose a sharp cut-off here, as it is extremely important to keep the magnetic gauge invariance corresponding to replacing A by A+∇ϕ. In [START_REF] Gravejat | Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields[END_REF] we used the Pauli-Villars method [START_REF] Pauli | On the invariant regularization in relativistic quantum theory[END_REF], which consists in introducing two fictitious particle fields of very high masses m 1 , m 2 ≫ 1 which play the role of ultraviolet cut-offs. The Pauli-Villars-regulated Lagrangian is defined by

L PV m,e (
) 16
The main result of [START_REF] Gravejat | Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields[END_REF] is the following

for some radius r > 0 which only depends on

(iii) If A ext ≡ 0, then A * ≡ 0 as well.

This result completely settles the problem to construct Dirac's vacuum in small external fields. It is an open question to do the same in large external fields (which could be useful to understand non-perturbative events like the creation of electronpositron pairs). It is also not known whether our solution is global. For the free vacuum, A ext ≡ 0, we conjecture that the saddle point A ≡ 0 is the unique solution to [START_REF] Esteban | Solutions of the Dirac-Fock equations for atoms and molecules[END_REF] with r ≡ ∞. Even if this was not done in [START_REF] Gravejat | Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields[END_REF], it is possible to renormalize the coupling constant α in the same fashion as in the previous section.

  . The second possibility is to directly argue in the whole space H that Tr D m,eAtot (P -P - m,eAtot ) ≥ 0 for all projection P . Under appropriate assumptions on P (for instance P -P - m,0 finite rank and smooth), we can write Tr D m,eAtot (P -P - m,eAtot ) = Tr |D m,eAtot | P + m,eAtot (P -P - m,eAtot )P + m,eAtot -P - m,eAtot (P -P - m,eAtot )P - m,eAtot = Tr |D m,eAtot | 1/2 P + m,eAtot P P + m,eAtot

R 3 |B| 2 - 2 j=0 c j = 2 j=0 c j m 2 j

 32222 |D mj,0 |-|D mj,e(A+Aext) | + 1 8π |E| 2 (15)where c 0 = 1, m 0 = m and the fictitious fields are chosen such that = 0. (

Theorem 4 . 1 (

 41 The Pauli-Villars-regulated vacuum in electromagnetic fields[START_REF] Gravejat | Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields[END_REF]). Let e ∈ R, m > 0 and c 1 , c 2 , m 1 , m 2 satisfying[START_REF] Esteban | Variational methods in relativistic quantum mechanics[END_REF].(i) The functional A → Tr Tr C 4 2 j=0 c j |D mj ,0 | -|D mj ,A | is well-defined for A ∈ C ∞ c (R 3 , R 4) and it admits a unique continuous extension toḢ1 div (R 3 ) = (V, A) ∈ L 6 (R 3 , R 4 ) : divA = 0, (-∇V, ∇ ∧ A) ∈ L 2 (R 3 , R 6 ) .So the Pauli-Villars-regulated Lagrangian (15) is well-defined and continuous onḢ1 div (R 3 ) × Ḣ1 div (R 3 ). (ii) For any A ext ∈ Ḣ1 div (R 3 ) with e A ext Ḣ1 div (R 3 ) < r √ m/2, there exists a unique solution A * = (V * , A * ) ∈ Ḣ1 div (R 3 ) to the min-max problem L PV m,e (A * , A ext ) = max ∇V L 2 < r √ (A ; A ext ).

2 j=0= 2 j=0 c j 1 (

 221 |c j |(m/m j ) and provided that e stays in a bounded set.(ii) The four-potential A * = (V * , A * ) is a solution to the nonlinear equations-∆V * = 4πe ρ A * +Aext , -∆A * = 4πe j A * +Aext , (18)where ρ A * +Aext and j A * +Aext refer to the charge and current densities of Q * -∞,0) D mj,e(A * +Aext) .

* Grants from the French Ministry of Research (ANR BLAN-10-0101) and from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013 Grant Agreement MNIQS 258023) are gratefully acknowledged.


Gilles Pagès
email: gilles.pages@upmc.fr

Harald Luschgy
email: luschgy@uni-trier.de

Critical dimension for quadratic functional quantization

Keywords: quadratic functional quantization, Karhunen-Loève expansion, Gaussian process, optimal quantizer, asymptotically optimal quantizer 2010 AMS Classification: 60G15, 60G99, 94A29

In this paper we tackle the asymptotics of the critical dimension for quadratic functional quantization of Gaussian stochastic processes as the quantization level goes to infinity, i.e. the smallest dimensional truncation of an optimal quantization of the process which is "fully" quantized. We first establish a lower bound for this critical dimension based on the regular variation index of the eigenvalues of the Karhunen-Loève expansion of the process. This lower bound is consistent with the commonly shared sharp rate conjecture (and supported by extensive numerical experiments). Moreover, we show that, conversely, constructive optimized quadratic functional quantizations based on this critical dimension rate are always asymptotically optimal (strong admissibility result).

Introduction

The aim of this paper is two-folded: first we aim at providing a constructive proof of the sharp rate of functional quantization in the quadratic case for (a wide class of) Gaussian processes or more generally of Gaussian random vectors X taking values in a separable Hilbert space H, (.|.) H . Secondly, we provide several results about the critical quantization dimension in this framework, especially a "sharp" asymptotic lower bound for the genuine critical dimension (sharp with respect to a conjecture supported by extensive numerical experiments carried out on the Brownian motion and the Brownian bridge, see [START_REF] Luschgy | Asymptotically optimal quantization schemes for Gaussian processes[END_REF]) and the sharp asymptotics of the "asymptotic" critical dimension.

By constructive proof we mean that we exhibit sequences of quantizers of size n which induces an asymptotically sharp rate quadratic mean quantization error (this sharp rate has been first established in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF]) as n goes to infinity. These quantizers live in finite dimensional subspaces spanned by the (first component of the) Karhunen-Loève expansion of the Gaussian process of interest. The genuine critical dimension at level n is the lowest dimension of such a subspace which contains an optimal n-quantizer whereas the "asymptotic" critical dimension corresponds to asymptotically optimal n-quantizers. Like the mean quantization rate, the asymptotics of the critical dimension is ruled by the rate of decay of the K-L eigenvalues (listed in a decreasing order).

The L r -mean quantization error of a random variable X defined on a probability space (Ω, A, P) and having a finite r th moment taking value in a separable Hilbert space is defined by where | . | H denotes the norm on the Hilbert space H. Any random vector which achieves the above minimum (there is always at least one) is called an (L r -)optimal n-quantization. One can show that such an optimal quantization is always of the form Y * = π α * (X) where π is a Borel projection on α * = Y * (Ω) following the nearest neighbour rule. The subset α is called an optimal Voronoi nquantizer (or optimal Voronoi quantizer at level n). By extension any random vector of the form π α (X) is called a Voronoi quantization whereas α is often called an n-quantizer (if |α| = n). The term Voronoi refers to the nearest neighbour projection. A sequence (α n ) n≥1 of n-quantizers is called

asymptotically n-optimal if lim n min a∈αn |X -a| H r e n,r (X) = 1.
In the quadratic framework (r = 2), we will drop the subscript r for simplicity.

When H is an infinite dimensional separable Hilbert space (typically H is function space like L 2 ([0, T ], dt)), one often speaks of functional quantization. The first problem of interest (beyond the existence of optimal n-quantizers for every n ∈ N) is the rate of decay of the mean (quadratic) quantization error. This sharp rate problem in a Hilbert has been solved for a wide class of H-valued Gaussian random vectors/processes X in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF] (see Theorem 2.1 below). Namely when the eigenvalues of the Karhunen-Loève eigensystem of an H-valued random vector X ordered in a non-increasing order read λ n = ϕ(n) where ϕ is a non-increasing regularly varying function with index -b, b ≥ 1. Thus we know that for the Brownian motion (or any Gaussian processwhose parameter b is equal to 2),

lim n log n e n (W ) = √ π 2 T.
and the conjecture on the genuine critical dimension reads as follows

lim n (log n) -1 d W n = 2.
Rather unexpectedly, this proof in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF] is not constructive and provides no straightforward information on the critical dimension d X n as n grows. In this paper we fill this two gaps (only partially as concerns the second one). In particular the adopted approach relied on a block product quantization where we let the size of the blocks go to infinity. Other proofs based on self-similarity arguments have been proposed (in an L r -framework, see [START_REF] Dereich | Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces[END_REF]). Note however in [START_REF] Luschgy | Asymptotically optimal quantization schemes for Gaussian processes[END_REF] a first attempt of asymptotically optimal quantization grids has been carried out using varying block sizes.

The paper is organized as follows: in Section 2 we provide some more rigorous background on Karhunen-Loève expansions of Gaussian random vectors and functional quantization. Then, we state our main results by exhibiting a a sequence of asymptotically optimal quantization grids and provide an asymptotic lower bound for the genuine critical dimension. In Section 3 and 4 we establish some upper and lower bounds respectively for the mean quantization error. Section 5 is devoted to the proofs and constructive aspects. We conclude by few numerical illustrations which support the conjecture.

Our main tools, beyond discrete optimization techniques used for the upper bounds, are the Shannon's Shannon's source coding Theorem and the connection between mean quantization error and Shannon ε-entropy (or rate-distortion function, see [START_REF] Dembo | Large deviations techniques and applications[END_REF]).

Notations. • | . | canonical Euclidean norm on R d . • N * = {1, 2, 3, . . .} the set of positive integers. • L 2 T = L 2 ([0, T ], dt) equipped with its Hilbert norm |f | L 2 T = T 0 f 2 (t)dt 1 2 .
• Let X : (Ω, A) → H.

• Let (a n ) and (b n ) be two sequences of real numbers. a n ∼ b n if there exists a sequence (u n ) such that a n = u n b n and lim n u n = 1.

• o(1) denotes a sequence indexed by n ∈ N * going to 0 as n → ∞ (which may vary from line to line)

2 Background on optimal functional quantization and main result

dimK X = +∞,
where K X is the self-reproducing space of X.

A typical example is the case of a Gaussian stochastic process X = (X t ) t∈[0,T ] with continuous paths. Clearly, for such a process a.s. t → X t (ω) lies in L 2

T so that X can be see as a random vector taking values in

L 2 T , |.| L 2 T .
Let (λ X k , e X k ) k≥1 be the orthonormal eigensystem of the (positive trace) covariance operator of X, also known as the Karhunen-Loève (K-L) orthonormal system of X. Since the sequence (λ X n ) n≥1 has only one limiting value, 0, one may assume without loss of generality that K-L eigensystem is indexed so that the sequence of eigenvalues is non-increasing. To alleviate notations we will drop the dependency of the eigenvalues in X by simply noting λ n instead of λ X n . Throughout the paper, the main results are obtained under the following assumption about the K-L eigenvalues:

(R) ≡

There exists b ∈ [1, +∞) and a non-increasing function ϕ : (0, +∞) → (0, +∞) with regular variations at infinity of index -b (hence going to 0 at infinity) such that λ k = ϕ(k), k ≥ 1.

Then, the Karhunen-Loève decomposition of X reads

X = k≥1 λ k ξ k e X k where ξ k = X, e X k √ λ k , k ≥ 1,

Optimal quadratic functional quantization

The optimal quantization problem for X in L 2 H (P) is

e n (X) = inf min a∈α |X -a| H 2 , α ⊂ H, |α| ≤ n . (2.1)
For every integer d ≥ 1, we define the H-orthogonal projection X (d) of X on span{e X 1 , . . . , e X d }, namely

X (d) = d k=1 λ k ξ k e X k and e n (X (d) ) = inf min a∈α |X (d) -a| H 2 , α ⊂ ⊕ 1≤k≤d Re X k , |α| ≤ n (2.2) = inf min a∈α | λ k Z k ) 1≤k≤d -a| 2 , α ⊂ R d , |α| ≤ n where Z = (Z 1 , . . . , Z d ) d = N (0; I d ).
Finally we set,

e 2 n (X, d) = e 2 n (X (d) ) + k≥d+1 λ k and C(d) = sup k≥1 k 2 e 2 k N 0; I d .
We know from [START_REF] Luschgy | Functional quantization of stochastic processes[END_REF] (see Proposition 2.1) that, for every n ∈ N * , the infimum in (2.1) holds as a minimum: there exists at least one optimal quantizer α * ,n which turns out to have full size n. Furthermore α * ,n lies in a finite dimensional space spanned by finitely many elements of the K-L basis. We can define the (genuine) critical dimension as the smallest vector subspace of span{e X n , n ≥ 1} in which some optimal n-quantizer lies, namely:

d n = min d ∈ N * : ∃ α * ,n optimal n-quantizer s.t. α * ,n ⊂ span{e X k , 1 ≤ k ≤ d} .
The sequence (d n ) n≥1 makes up a sequence satisfying

e 2 n (X) = e 2 n (X, d n ).
It is clear that d n goes to infinity, otherwise one could extract a subsequence

d n ′ such that d n ′ ≤ d < +∞. If so, we would have e 2 n (X) ≥ k≥ d+1
λ k which contradicts the obvious fact that e 2 n (X) goes to zero as n goes to infinity (see e.g. [START_REF] Luschgy | Functional quantization of stochastic processes[END_REF]). This last claim is a consequence of the fact that, if (z n ) n≥1 is everywhere dense in H, then

e 2 n (X) ≤ E min 1≤i≤n |X -z i | 2 H → 0 as n → ∞.
Otherwise very little is known on the sequence (d n ) n≥1 , in particular we do not know whether this sequence is monotone.

We will use the following easy lemma

Lemma 2.1. Let n ≥ 1 be an integer. The sequence d → e 2 n (X, d) is non-increasing (and so is constant for d ≥ d n ). Proof. Let d ≤ d ′ . Let α * ,n (
d) be an optimal quadratic quantizer for X (d) of size (at most) n. It is clear that for every a ∈ α * ,n (d),

|X (d ′ ) -a| 2 H = |X (d) -a| 2 H + d ′ k=d+1 λ k since α * ,n (d) ⊂ span{e X 1 , . . . , e X d }. As a consequence e 2 n (X (d ′ ) ) ≤ e 2 n (X (d) ) + d ′ k=d+1
λ k and one concludes by adding the tail k≥d ′ +1 λ k .

It holds as a conjecture for a long time that, under Assumption (R),

lim n d n log n = 2 b
whereas the sharp rate of quadratic quantization has been elucidated for long in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF] (with several extension to more general Banach settings obtained ever since for L p ([0, T ], dt)-norms, 1 ≤ p ≤ +∞, see [START_REF] Dereich | Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces[END_REF], etc).

Extensive computations carried out in [START_REF] Luschgy | Asymptotically optimal quantization schemes for Gaussian processes[END_REF] provide strong evidence that in fact, as concerns the standard Brownian motion and the Brownian bridge (which corresponds to b = 2), we even have that

d n ∈ {⌊log n⌋, ⌈log n⌉}.
These conjecture are also supported by results obtained for optimal block quantization results either with constant size blocks or varying size blocks (see [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF][START_REF] Luschgy | Asymptotically optimal quantization schemes for Gaussian processes[END_REF]).

The aims of this paper can now be summed up as follows : firstly to provide a constructive proof of the sharp rate theorem recalled below, secondly to provide a partial answer to the above conjecture and finally to provide a complete answer to the "asymptotic" dimension problem.

Theorem 2.1. (see [START_REF] Luschgy | Functional quantization of stochastic processes[END_REF]) Assume (R). Let ψ : (0, +∞) → (0, +∞) be defined by

ψ(x) =      1 xϕ(x) if b > 1 so that ψ is a regularly varying function with index b -1 1 ∞
x ϕ(y)dy if b = 1 so that ψ is a slowly varying function.

(2.3)

Then lim n ψ(log n)e 2 n (X) =    b 2 b-1 b b -1 if b > 1 1 if b = 1.

Main result: critical dimension

Now we pass to results on the critical dimension, with an emphasis on the constructive aspects. While the genuine critical dimension is of considerable theoretical interest, for practical purposes the "asymptotic" critical dimension is more interesting. It corresponds to the smallest admissible dimension according to the definition below.

Definition 2.1. Assume (R). Let ψ be defined as in Theorem 2.1. Let (δ n ) n≥1 be a sequence of positive integers going to infinity.

(a) A sequence (δ n ) n≥1 is admissible for X if lim n e 2 n (X, δ n ) e 2 n (X) = 1. (b) If b > 1, an admissible sequence (δ n ) n≥1 is strongly admissible for X if lim n ψ(δ n )e 2 n (X (δn) ) = 1.
Admissibility suggests that such a sequence of "pseudo-critical dimensions" can be used to produce asymptotic optimal quantizers in practice.

Strong admissibility models the fact that, asymptotically no dimension, is useless in X (δn) so that the dimension δ n is (asymptotically) minimal in view of practical computations of optimal quadratic quantizers of a Gaussian process. For more insight on these numerical aspects, we refer to [START_REF] Luschgy | Asymptotically optimal quantization schemes for Gaussian processes[END_REF] and [START_REF] Pagès | Functional quantization for numerics with an application to option pricing[END_REF].

The following theorem provides a complete solution of the dimension problem in the asymptotic sense, at least for b > 1.

Theorem 2.2. Assume (R). Let ψ be defined as in Theorem 2.1.

(a) If b > 1, (δ n ) n≥1 is admissible ⇐⇒ lim inf n δ n log n ≥ 2 b and (δ n ) n≥1 is strongly admissible ⇐⇒ lim n δ n log n = 2 b . (b) If b = 1, (δ n ) n≥1 is admissible =⇒ lim inf n ψ(δ n ) ψ(log n) ≥ 1 and lim inf n δ n log n > 0 =⇒ (δ n ) n≥1 is admissible.
As for the genuine critical dimension, we thus obtain the following lower bounds.

Corollary 2.1. Assume (R). (a) If b > 1, lim inf n d n log n ≥ 2 b . (b) If b = 1, lim inf n ψ(d n ) ψ(log n) ≥ 1.
where this time ψ

(x) = 1 ∞ x ϕ(y)dy , x > 0, is a slowly varying function.
Furthermore, it follows from Theorem 2.2 that the above conjecture is true if and only if

lim n→+∞ ψ(d n )e 2 n (X (dn) ) = 1.

Upper bound

Since we are trying to provide a fair new constructive proof of the sharp rate for quadratic functional quantization, we are not yet in position at this stage to claim that lim n ψ(log n)e 2 n (X) does exist. This is the reason why the claims in the proposition below involve lim sup n ψ(log n)e 2 n (X) which always exists (the same will be true with Proposition 4.1 in the next section). Proposition 3.1. Assume (R). Let (δ n ) n≥1 be a sequence of integers going to infinity.

(a) If b > 1 and lim inf n δ n log n ≥ 2 b , then lim sup n ψ(log n) e 2 n (X) ≤ lim sup n ψ(log n) e 2 n (X, δ n ) ≤ b 2 b-1 b b -1 . Furthermore, if lim n δ n log n = 2 b , then lim sup n ψ(log n) e 2 n (X (δn) ) ≤ b 2 b-1 . (b) If b = 1 and lim inf n δn log n = κ ∈ (0, +∞), then lim sup n ψ(log n)e 2 n (X) ≤ lim sup n ψ(log n) e 2 n (X, δ n ) ≤ 1.
First we need two lemmas devoted two block quantization and their critical dimension which are the key of the proof. For every integer d ≥ 1, we define set λ

(d) k = λ (k-1)d+1 . Lemma 3.1 (Block quantization). Let d, d 0 ∈ N * , d > d 0 . Then, for every k ∈ N * , k ≤ d d 0 , we have e 2 n (X (d) ) ≤ C(d 0 ) min k ℓ=1 λ (d 0 ) ℓ n -2 d 0 ℓ , n 1 , . . . , n k ∈ N * , k ℓ=1 n ℓ ≤ n + d i=kd 0 +1 λ i
Proof. We introduce the (sub-optimal) d 0 -block product quantizer defined as follows

X (d,d 0 ,k) = k ℓ=1 d 0 i=1 λ (ℓ-1)d 0 +i Proj α (ℓ) (ξ j ) (ℓ-1)d 0 +1≤j≤ℓd 0 i e X (ℓ-1)d 0 +i
where α (ℓ) ⊂ R d 0 is an optimal quadratic quantizer of size (at level) n ℓ of N 0; I d 0 and Proj α (ℓ) : R d 0 → α (ℓ) a Borel nearest neighbour projection on α (ℓ) . Elementary computations based on Pythagorus's theorem (see Lemma 4.2) in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF]) show that

X -X (d) 2 = k ℓ=1 d 0 i=1 λ (ℓ-1)d 0 +i E Proj α (ℓ) (ξ j ) (ℓ-1)d 0 +1≤j≤ℓd 0 i -ξ (ℓ-1)d 0 +i 2 + d i=kd 0 +1 λ i , ≤ k ℓ=1 λ (ℓ-1)d 0 +1 E Proj α (ℓ) (ξ j ) (ℓ-1)d 0 +1≤j≤ℓd 0 i -ξ (ℓ-1)d 0 +i 2 + d i=kd 0 +1 λ i , ≤ k ℓ=1 λ (ℓ-1)d 0 +1 e 2 n ℓ N 0; I d 0 + d i=kd 0 +1 λ i .
The definition of C(d 0 ) completes the proof. This optimal integer bit allocation has a formal almost optimal solution given by

n ℓ = ⌊x ℓ ⌋ with x ℓ = λ (d 0 ) ℓ d 0 2   k j=1 λ (d 0 ) j   - d 0 2k n 1 k , ℓ = 1, . . . , k,
as suggested by considering the problem on (R + ) k instead of (N * ) k . This solution is admissible as soon as all the n ℓ s are nonzero or equivalently since they are non-increasing in ℓ as soon as n k ≥ 1.

Lemma 3.2 (Critical dimension for block quantization). Let d, d 0 and k be like in Lemma 3.1.

A(n, d 0 ) =      k ≥ 1 : λ (d 0 ) k d 0 2   k j=1 λ (d 0 ) j   - d 0 2k n 1 k ≥ 1      . (a) A(n, d 0 ) = {1, . . . , k n (d 0 )}. (b) Assume (R) Then k n (d 0 ) ∼ 2 bd 0 log n. (c) For every integer k ≤ min k n (d 0 ), d d 0 , e 2 n (X (d) ) ≤ 4 1 d 0 C(d 0 )kλ (d 0 ) k + d i=kd 0 +1 λ i .
Proof. (a) This follows from the fact that the sequence

a k = a (d 0 ) k := d 0 2 k ℓ=1 log λ (d 0 ) ℓ -k log λ (d 0 ) k (3.4) is non-decreasing since A(n, d 0 ) = {k : a k ≤ log n}. (b) First note that λ (d 0 ) j
= ϕ((j -1)d 0 + 1) and that ϕ((. -1)d 0 + 1) is regularly varying still with index b if ϕ is. As a consequence standard arguments on regularly varying functions show that e 2 n (X (δn) ) ≤ 4

2a k d 0 ∼ bk or equivalently a k ∼ bd 0 k 2 as k → ∞ which in turn implies that k n (d 0 ) ∼ 2 bd 0 log n as n → ∞. (c) It is straightforward that k ℓ=1 λ (d 0 ) n -2 d 0 ℓ ≤ 2 2 d 0 k ℓ=1 λ (d 0 ) ℓ (n ℓ + 1) -2 d 0 ≤ 4 1 d 0 k ℓ=1 λ (d 0 ) ℓ x -2 d 0 ℓ ≤ 4 1 d 0 kλ (d 0 ) k x -2 d 0 k ≤ 4 1 d 0 kλ (d 0 ) k since λ (d 0 ) ℓ x -2 d 0 ℓ =   k j=1 λ (d 0 ) j   - d 0 2k
1 d 0 C(d 0 )k n λ (d 0 ) kn + δn i=knd 0 +1 λ i ≤ 4 1 d 0 C(d 0 )k n λ (d 0 ) kn + (δ n -d 0 k n )λ (d 0 ) kn+1 . (3.5)
Now, mimicking arguments in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF] involving regularly varying functions, namely ϕ, we get

d 0 k n λ (d 0 ) kn = d 0 k n ϕ(k n d 0 + 1) ∼ 2 b log n 2 b -b ϕ(log n) = 2 b 1-b 1 ψ(log n) as n → ∞. Moreover (δ n -d 0 k n )λ (d 0 ) kn+1 = δ n k n d 0 -1 k n d 0 λ (d 0 ) kn+1 = o 1 ψ(log n) since δ n ∼ k n d 0 ∼ 2 b log n. Consequently, by letting d 0 go to infinity, we get lim sup n ψ(log n)e 2 n (X (δn) ) ≤ 2 b 1-b lim sup d 0 C(d 0 ) d 0 .
One concludes by using (see [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF]) that owing to the converse Shannon theorem

lim d C(d) d = 1.
On the other hand

i≥knd 0 +1 λ (d 0 ) i ∼ k n d 0 ϕ(k n d 0 ) b -1 ∼ 1 (b -1)ψ 2 b log n ∼ 2 b 1-b 1 (b -1)ψ log n .
which yields the announced result by sub-additivity of lim sup n .

If lim inf

n δ n log n ≥ 2 b , then set δ ′ n = δ n ∧ 2 log n b , n ≥ 1. Then lim n δ ′ n log n = 2 b whereas by Lemma 3.1 e 2 n (X, δ n ) ≤ e 2 n (X, δ ′ n ) which implies lim sup n ψ(log n)e 2 n (X, δ n ) ≤ lim sup n ψ(log n)e 2 n (X, δ ′ n ) ≤ 2 b 1-b b b -1 .
(b) Assume first that lim n δn log n = κ ∈ (0, +∞). Owing to Lemma 3.1, we can assume as above that k n defined like in (a) satisfies

k n ∼ κ ′ log n where κ ′ = κ ∧ 2 d 0 . First we derive that i≥δn+1 λ (d 0 ) i ∼ 1 ψ(δ n ) ∼ 1 ψ(κ log n) ∼ 1 ψ(log n)
since ψ has slow variation defined in Theorem 2.1 satisfying xϕ(x) = o(1/ψ(x)) (see [START_REF] Bingham | Regular Variation[END_REF], Proposition 1.5.9 b). On the other hand,

d 0 k n λ (d 0 ) kn = d 0 k n ϕ(d 0 k n + 1) = o 1 ψ(d 0 k n + 1) = o 1 ψ(log n)
since ψ is slowly varying, and

δn i=d 0 kn+1 λ i ≤ δ n d 0 k n -1 d 0 k n λ (d 0 ) kn = o 1 ψ(log n) since δn d 0 kn -1 has a finite limit κ d 0 κ ′ -1. As a consequence lim sup n ψ(log n)e 2 n (X, δ n ) ≤ 1.
The extension to the general case lim inf n δn log n = κ ∈ (0, +∞) is straightforward up to the extraction of a subsequence.

Remark. Note that when b = 1, we do not need to let d 0 go to infinity. Since this rate is optimal (in view of Theorem 2.1) i.e., this means in particular that scalar product quantization (i.e. block quantization with blocks of size d 0 = 1) is asymptotically optimal.

Lower bound

We will rely on the famous notion in Information Theory, the Shannon ε-entropy (or rate-distortion function) of P (see [START_REF] Shannon | The Mathematical Theory of Communication[END_REF]). Let P be a probability measure on H. For ε > 0, it is defined by

R P (ε) = inf H(Q|P ⊗ Q 2 ) : Q probability measure on H × H with first marginal Q 1 = P and H×H x -y 2 dQ(x, y) ≤ ε 2 ,
where H(Q|P ⊗ Q 2 ) classically denotes the relative entropy (mutual information)

H(Q|P ⊗ Q 2 ) =        H log dQ dP ⊗ Q 2 dQ if Q is absolutely continuous with respect to P ⊗ Q 2 +∞ otherwise.
The simple converse part of the source coding theorem (see [START_REF] Berger | Rate Distortion Theory[END_REF] Theorem 3.2.2; [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF], p.163) says that the minimal number N (ε) of codewords needed in a codebook α such that E min a∈α Xa 2 ≤ ε 2 satisfies log N (ε) ≥ R(ε) so that, in particular R(e n (X)) ≤ log n.

We rely here on the closed form for Shannon's entropy of Gaussian vectors known as Kolmogorov-Ihara's formula (see [START_REF] Kolmogorov | On the Shannon entropy of information transmission in the case of continuous signals[END_REF][START_REF] Ihara | On ε-entropy of equivalent Gaussian processes[END_REF]) that we will apply to P = L(X (d) ) and H = Of course, the eigenvalues are still supposed to be ordered in a non-increasing way.

Theorem 4.1 (Kolmogorov-Ihara, see [START_REF] Kolmogorov | On the Shannon entropy of information transmission in the case of continuous signals[END_REF][START_REF] Ihara | On ε-entropy of equivalent Gaussian processes[END_REF]). Let d ≥ 1 and let P = N 0; Diag(λ 1 , . . . , λ d ) where

λ 1 ≥ • • • ≥ λ d . For every ε > 0 such that ε 2 ∈ (0, λ 1 + • • • + λ d ), R(ε) = 1 2 r(ε) k=1 log λ k θ(ε) = log   r(ε) k=1 λ k θ(ε)   1 2 10
where r(ε) = max{k ∈ {1, . . . , d} : λd

k > ε 2 }, with λd k = kλ k + λ k+1 + • • • + λ d , k = 1, . . . , d and λd k = 0, k ≥ d + 1
, and θ(ε) is the unique solution to the equation

ε 2 = r(ε)θ(ε) + d k=r(ε)+1 λ k .
Note that the definition of r(ε) is consistent since ( λd k ) 1≤k≤d is non-increasing; furthermore by construction θ(ε) ∈ [λ r(ε)+1 , λ r(ε) ).

By the definition of optimal quantization at level n, we have, as recalled above (see also [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF]),

∀ n ≥ 1, R e n (X (d) ) ≤ log n. Lemma 4.1. Let d, n ∈ N * . Then e 2 n (X (d) ) ≥ min   n -2 d d d k=1 λ k 1 d , dλ d   . Proof. If e 2 n (X (d) ) =: ε 2 < λd d , then r(ε) = d and θ(ε) = ε 2 d so that R(ε) = log d k=1 λ k 1 2 - d 2 log ε 2 d ≤ log n iff e 2 n (X (d) ) = ε 2 ≥ n -2 d d d k=1 λ k 1 d
. Proposition 4.1 (Lower bound). Assume (R). Let δ n be a sequence of dimensions going to infinity.

lim inf n ψ(log n)e 2 n (X, δ n ) ≥ κ 1-b 1 b -1 + e -2( 1 κ -b 2 ) + . Furthermore, if lim sup n ψ(log n)e 2 n (X, δ n ) = lim sup n ψ(log n)e 2 n (X), then lim inf n δ n log n ≥ 2 b . (b) If b = 1, then lim inf n ψ(log n)e 2 n (X) ≥ lim inf n ψ(log n) ψ(δ n ) . Furthermore, if if lim sup n ψ(log n)e 2 n (X, δ n ) = lim sup n ψ(log n)e 2 n (X), then lim inf n ψ(δ n ) ψ(log n) ≥ 1.
Proof. (a) Having in mind that ψ(x) = 1/(xϕ(x)), it follows from Lemma 4.1 that

ψ(log n)e 2 n (X (δn) ) ≥ ψ(log n) min   n -2 δn δ n δn k=1 λ k 1 δn , 1 ψ(δ n )   = ψ(log n) ψ(δ n ) min e -2 log n δn 1 ϕ(δ n ) e 1 δn 1≤k≤δn log ϕ(k) , 1 .
The function ϕ being regularly varying with index -b, b > 1, one checks (see [START_REF] Bingham | Regular Variation[END_REF])

1 m m k=1 log ϕ(k) = b + log ϕ(m) + o(1) as m → ∞ so that, ψ(log n)e 2 n (X (δn) ) ≥ ψ(log n) ψ(δ n ) min e -2 log n δn +b+o(1) , 1
which in turn implies that

ψ(log n)e 2 n (X, δ n ) ≥ ψ(log n) ψ(δ n ) min e -2 log n δn +b+o(1) , 1 + 1 + o(1) b -1 .
At this stage we introduce the function g b defined on [0, +∞] (with the usual conventions) by

g b (u) := min e -2( 1 u -b 2 ) , 1 + 1 b -1 u 1-b = u 1-b 1 b -1 + e -2( 1 u -b 2 ) + .
The function g b is decreasing on [0, +∞] with

g b 2 b = b 2 b-1 b b -1 .
Let (n ′ ) be a subsequence such that which in turn implies c ≥ 2 b . (b) Using again standard results from [START_REF] Bingham | Regular Variation[END_REF] about regularly varying functions with index -1, we get

i≥δn+1 λ i ∼ 1 ψ(δ n ) still with ψ(x) = 1 +∞ x ϕ(y)dy .
Hence

ψ(log n)e 2 n (X, δ n ) ≥ ψ(log n) i=δn+1 λ i ∼ ψ(log n) ψ(δ n ) .
Using the same trick (based on the sequence ( δ n ) n≥1 ) as in the former case, we derive similarly that, if lim sup

n ψ(log n)e 2 n (X, δ n ) = lim sup n ψ(log n)e 2 n (X), then lim sup n ψ(log n)
ψ(δn) ≤ 1 i.e. the announced result.

Synthesis

Proof of Theorem 2.1

First we provide a proof of Theorem 2.1 based on the upper and lower bounds established in former sections and the following lemma (already established in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF] but reproduced here for the reader's convenience). Furthermore it has to be noticed that it provides an easily tractable (and asymptotically optimal) lower bound for the quadratic quantization error, keeping in mind that the sequence (k n (1)) n≥1 is defined in Lemma 3.2.

k n (1)λ kn(1)+1 ∼ 1 ψ(k n (1)) ∼ b 2 b-1 1 ψ(log n) and k≥kn(1)+1 λ k ∼ 1 (b -1)ψ(k n (1)) ∼ b 2 b-1 1 (b -1)ψ(log n) .
so that lim inf As for strong admissibility, the direct claim is as follows: from the definition of strong admissibility, we get e n (X, δ n ) 2 ∼ e n (X) 2 (by admissibility) and

e n (X, δ n ) 2 ∼ 1 ψ(δ n ) + 1 b -1 1 ψ(δ n ) = b b -1 1 ψ(δ n ) so that b b -1 1 ψ(δ n ) ∼ e n (X) 2 .
Then comparing with the sharp rate from Theorem ??, we get

1 ψ(δ n ) ∼ b 2 b-1 1 ψ(log n)
which finally implies, having in mind that ψ is regularly varying with index b -1, that This result is helpful from a numerical point of view since it shows, e.g. for the Brownian motion (for which b = 2, see below), that considering a truncation at δ n = ⌊log n⌋ or δ n = ⌈log n⌉ is at least asymptotically optimal whatever the future of the sharper conjecture d n ∈ {⌊log n⌋, ⌈log n⌉}. could be. Of course, such a choice is also asymptotically optimal for all processes whose K-L eigensystem has an varying index equal to 2 like the Brownian bridge, Ornstein-Uhlenbeck process, etc.

δ n ∼ b 2 log n.
For other examples of families of processes satisfying Assumptions (R) (including multi-parameters processes like the Brownian sheet, we refer to [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF]).

⊲ Numerical experiments on the Brownian motion.

We know that the K-L eigensystem of the standard Brownian motion W = (W t ) t∈[0,T ] over [0, T ] is given by

λ W k = T π(k -1 2 )
2 The graph in Figure 1 suggests, at this (low) range of the computation, that the limiting value for n → log(n)e 2 n (W ) is higher (≈ 0.22) than the theoretical one (≈ 0.2026). This impression is misleading since further computations not reproduced here show that the sequence n → log(n)e 2 n (W ) starts to be slowly decreasing beyond n ≥ 1000. The value 0.22 seems to be a local maximum. For further details on these (highly time consuming) computations we refer to [START_REF] Luschgy | Asymptotically optimal quantization schemes for Gaussian processes[END_REF].

The quantization grids computed at the occasion of these numerical experiments by stochastic optimization methods (randomized Lloyd's procedure, Competitive Learning Vector Quantization algorithm) for N = 1 up to 10 4 for the standard Brownian motion (when T = 1) can be downloaded from the website www.quantize.maths-fi.com

1 where

 1 e n,r (X) = inf min a∈α |X -a| H r , α ⊂ H, |α| ≤ n . |α| denotes the cardinality of the set α. It can also be characterized as e n,r (X) = min { |X -Y | H r , Y : (Ω, A, P) → H, |Y (Ω)| ≤ n}

n 1 kthat k n ∼ 2 b log n d 0

 10 does not depend on ℓ and x k ≥ 1. Proof of Proposition 3.1. (a) First assume that lim n δ n log n = 2 b . Let d 0 ∈ N * be a (temporarily) fixed integer. Set k n = k n (d 0 ) ∧ δn d 0 for n large enough to have δ n ≥ d 0 . It follows from Lemma 3.2(b) and k n ≤ k n (d 0 ) and k n ≤ δn d 0 so that by Lemma 3.2(c) we get as soon as n ≥ n d 0 ,

  equivalently to the d-dimensional normal distribution P = N 0; Diag(λ 1 , . . . , λ d ) on the canonical space H = R d ).

( a )

 a If b > 1 and κ = lim sup n δ n log n ∈ [0, +∞] then, with standard conventions,

  δ n ′ log n ′ → u ∈ [0,+∞]. Using that ψ is regularly varying with index b -1 we derive that lim infn ψ(log n ′ )e 2 n (X, δ n ′ ) ≥ g b (u) so that finally lim inf n ψ(log n)e 2 n (X, δ n ) ≥ sup u≤κ g b (u) = g b (κ) where κ = lim sup n δ n log n . Assume now that lim sup n ψ(log n)e 2 n (X, δ n ) = lim sup n ψ(log n)e 2 n (X). Let c =: lim inf n δ n log n ∈[0, +∞] and let (δ n ′ ) n≥1 be a subsequence such thatδ n ′ log n ′ → c. Let ( δ n ) n≥1be a sequence going to infinity and satisfying δ n ′ = δ n ′ and lim sup n δ n log n = c. Then one gets lim sup nψ(log n)e 2 n (X, δ n ) ≥ lim inf n ψ(log n ′ )e 2 n ′ (X, δ n ′ ) ≥ lim inf n ψ(log n)e 2 n (X, δ n ) ≥ g b (c). If c = 0,g b (0) = +∞ and we would have that ψ(log n)e 2 n (X, δ n ) → +∞ which is in contradiction with claim (a) in Proposition 3.1. If c ∈ (0, +∞], the upper bound obtained in Proposition 3.1 implies g b (c) ≤ g b 2 b

Lemma 5 . 1 . 2 b-1 b b- 1 .

 5121 For every n ∈ N * , e 2 n (X) ≥ k n (1)λ kn(1)+1 + k≥kn(1)+1 λ kProof. It follows from Kolmogorov-Ihara's formula that for every ε 2 ∈ (0,λ 1 + • • • + λ d ), R(ε) > a (1) r(ε) since θ(ε) < λ r(ε) (see Equation (3.4) for a definition of a (1) k ). As a consequence, a(1)r(en(X (d) )) ≤ log n. Consequently, it follows from Lemma 3.2(a) that r(e n (X(d) )) ≤ k n (1) which in turn implies that, for every d ∈ N * , λd kn(1)+1 ≤ e 2 n (X(d) ). Noting that e 2 n (X) ≥ e 2 n (X(d) ) and letting d go to infinity, we get, for every n ∈ N * , e2n (X) ≥ (k n (1) + 1)λ kn(1)+1 + k≥kn(1)+2 λ k . Proof of Theorem 2.1 Case b > 1. We know from Proposition 3.1 that lim sup n ψ(log n)e 2 n (X) ≤ b On the other hand, combining the fact that k n (1) ∼ 2 b log n and arguments based on regularly varying functions already used in Proposition 3.1 yield that

2 b- 1

 21 n ψ(log n)e 2 n (X) ≥ b b b -1 which completes the proof. Case b = 1. One concludes likewise since k n (1)λ kn(1)+1 ∼ 1 ψ(log n) and k≥kn(1)+1 λ k = o(1/ψ(log n)).

5. 2

 2 Proof of Theorem 2.2 Proof of Theorem 2.2. (a) When b > 1, the direct claim on admissibility is a consequence of Proposition 4.1 The converse claim follows from Proposition 3.1(a) and Theorem 2.1.

⊲

  The converse claim is a consequence of Proposition 3.1(a) and Theorem 2.1. Claim (b) follows the same lines and details are left to the reader. Back to the conjecture(s) (b > 1). As concerns the conjecture lim n critical dimension d n , strictly speaking, we only made half the way by proving that lim inf However the strong admissibility result in Theorem 2.2(a) can be seen as an answer in the asymptotic sense since it shows that if lim n resulting quadratic quantization error is asymptotically optimal and (asymptotically almost) all dimensions are used (strong admissibility).

T 2 Figure 1

 21 Figure 1 depicts the graph of the n → log(n)e 2 n (W ) (with T = 1) . One can see that it looks as a piecewise affine function with breaks in the slope. Note that the exponential function e x satisfies e 3 ≈ 20.09 ≈ 20 e 4 ≈ 54.59 ≈ 55 e 5 ≈ 148.41 ≈ 148.These values graphically fit with the monotony slope breaks.

Figure 1 :

 1 Figure 1: n → log(n)e 2 n (W ); green: d = 2 ; red: d = 3 ; cyan : d = 4 ; magenta : d = 5.