Emmanuel Frénod

Mathieu Lutz

The Gyro-Kinetic Approximation An attempt at explaining the method based on Darboux Algorithm and Lie Transform

This Proceeding presents the method that allows us to get the Gyro-Kinetic Approximation of the Dynamical System satisfied by the trajectory of a particle submitted to a Strong Magnetic Field. The goal of the method is to build a change of coordinates in order to make the dynamic of two components of the trajectory to disappear. This change of coordinates is based on a Darboux mathematical Algorithm and on a Lie Transform

Introduction

Scientific framework -At the end of the 70', Littlejohn [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF][START_REF] Littlejohn | Hamiltonian formulation of guiding center motion[END_REF][START_REF] Littlejohn | Hamiltonian perturbation theory in noncanonical coordinates[END_REF] shed new light on what is called the Guiding Center Approximation. His approach was based on mathematical theories -Hamiltonian Mechanics, Differential Geometry, Symplectic Geometry -in order to clarify what has been done for years in the domain (see Kruskal [START_REF]Plasma Physics, chapter Elementary Orbit and Drift Theory[END_REF], Gardner [START_REF] Gardner | Adiabatic invariants of periodic classical systems[END_REF], Northrop [START_REF] Northrop | The guiding center approximation to charged particle motion[END_REF], Northrop & Rome [START_REF] Northrop | Extensions of guiding center motion to higher order[END_REF]). His papers claim that it has been for him an enormous effort to reach this goal, since he had to incorporate into a physical affordable theory high level mathematics. Sure, this theory is a nice success. It has been being widely used by physicists to deduce related models (Finite Larmor Radius Approximation, Drift Kinetic Model, Quasi-Neutral Gyro-kinetic Model, etc., see for instance Koseleff [9], Brizard [START_REF] Brizard | Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks[END_REF], Dubin et al. [START_REF] Dubin | Nonlinear gyrokinetic equations[END_REF], Frieman & Chen [START_REF] Frieman | Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria[END_REF], Hahm [START_REF] Hahm | Nonlinear gyrokinetic equations for tokamak microturbulence[END_REF], Hahm, Lee & Brizard [START_REF] Hahm | Nonlinear gyrokinetic theory for finitebeta plasmas[END_REF], Parra & Catto [START_REF] Parra | Limitations of gyrokinetics on transport time scales[END_REF][START_REF] Parra | Gyrokinetic equivalence[END_REF][START_REF] Parra | Turbulent transport of toroidal angular momentum in low flow gyrokinetics[END_REF]) making up the Gyro-Kinetic Approximation Theory, which is the basis of all kinetic codes used to simulate Plasma Turbulence emergence and evolution in Tokamak (see for instance Grandgirard et al. [START_REF] Grandgirard | A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation[END_REF][START_REF] Grandgirard | Global full-f gyrokinetic simulations of plasma turbulence[END_REF]).

The incorporation of mathematical concepts into a physical theory has been done so nicely that the resulting Gyro-Kinetic Approximation Theory is now very difficult for mathematicians. We begin by considering a charged particle within the vessel of a Tokamak in service. In Usual Coordinates (x, v) = (x 1 , x 2 , x 3 , v 1 , v 2 , v 3 ), where x stands for the position variable and v for the velocity variable, the position and the velocity at time t (X(t; x, v, s), V(t; x, v, s)) of the particle which is in x with velocity v at time s is the solution to

∂X ∂t = V, (1)
∂V ∂t = q m (E(X) + V × B(X)). (2)
This dynamical system means nothing but that time derivative of the position is the velocity and that time derivative of the velocity, which is the acceleration, in linked with Lorentz force by Newton's law. In this dynamical system magnetic field B is composed of a strong applied piece, of a strong self induced piece and of a self induced perturbations. The strong applied piece is generated by the electromagnets and wind around the axis of symmetry of the tokamak. Charged particles winding, with a relatively large velocity, around the axis of symmetry within the vessel generate the indispensable vertical component of the large magnetic field. This is what is called here the strong self induced piece. The part of the self induced magnetic field which is not vertical is called here the self induced perturbations and is, from now, forgotten. Electric field E is induced by the particles of the vessel. Without loss of generality, we can consider that E is the opposite of the gradient of an electric potential, and B the curl of vector potential, i.e.:

E = -∇Φ, B = ∇ × A. (3)
Helicoidal trajectories -Larmor Radius -It is well known that the trajectory of a charged particle submitted to a magnetic field is a helix. The helix axis is the magnetic field direction and its radius, which is called in the context of Tokamak Plasma the Larmor Radius, equals the norm of the projection of the velocity on the plan orthogonal to the magnetic field divided by the particle's mass time the norm of the magnetic field. As illustrated in left picture of Figure 2, because of the mass ratio (Source: S. Jardin's Lectures at Cemracs '10). Right: Screen effect and Debye Length in a plasma.

between ions and electrons, the order of magnitude of electron Larmor Radius is much smaller than that of ions. In Tokamak, those order of magnitude are ∼ 5 • 10 -4 m for electrons and ∼ 10 -2 m for ions. Those two lengths are two important scales of Tokamak Plasma Physics.

To simplify the purpose, from now we will forget the scale of electron Larmor Radius and only take ions under consideration.

Debye Length -We now explain an important phenomenon of Plasma Physics which is the Screen Effect, with its related length which is the Debye Length.

It is well known that a charged particle generates around it an electric potential which depends on the inverse of the distance to it, say ∼ 1/r. Since this function is the Green Kernel of the opposite to the Laplace operator, when following the Mean Field routine to deduce the model taking under consideration interactions of many particles, it is gotten that the electric field that fills the space is the solution to -∆Φ = Cst Charge Density. Because, of the long range of ∼ (1/r)-potential, the regularization properties of minus Laplacian are such that space variations of the electric potential (and consequently of the electric field) occure at a large scale. Now, if the density of particles is relatively large, as it happens in Plasma, and if we consider a test particle (which is drawn in blue in right picture of Figure 2), the other particles (in the picture, orange particles are positively charged and green particles negatively charged) which are beyond a given length λ, which is the Debye Length (the black circle in the picture is censed being of radius λ), may be gathered into subsets (the two grey ellipses in the picture are such subsets) which are such that the resulting action of pairs of those subsets on the test particle is negligible. This has the following consequence: it can be considered that the potential ∼ 1/r generated by a particle may be approximated by a ∼ e -r/λ /r potential. Then, following the Mean Field routine with this new assumption leads an electric field that fills the space which is the solution to -DΦ = Cst Charge Density. Operator -D is the pseudo-differential operator which Green Kernel is e -r/λ /r whose regularization properties are much less than that of minus Laplacian. Then the regularity of the electric field is not as high as we could think before this formal analysis. In particular the electric field can be prone to variations over lengths ranging from several Debye Lengths to several hundreds of Debye Lengths. Those variations are maybe not extremely large, but as no principle fends them off, they are stable.

What Ions see -In a very simplified slant, Electric field in a plasma is a bed with variations which are not so large but not negligible, and, more than anything, stable at scales ranging from several Debye Lengths to several hundreds of Debye Lengths. In 3 we symbolically represent this bed as a charcoal grey surface. (It is symbolic because a surface is drawn while, in real world, Electric Field fills a 3D space.) This surface shows variations at two scales which are cenced to be about 100 times the Debye Lengths and 1.000 times the Debye Lengths. Those variations are not of large amplitude but are stable. Now, ion motion, symbolically represented in red in the figure, shows an oscillation of amplitude which is about 100 times the Debye Length (ion Larmor Radius ∼ 10 -2 m, Debye Length : ∼ 10 -4 m). Hence, because of the stability of bed variations, its resulting effect can be important on ion trajectory after several periods of oscillation. To take into account this resulting effect, the concept of gyro-average was introduced.

Dimensionless Dynamical System -Here, we do not present the scaling routine, we only mention that two small parameters may be introduced. The first one is

ε ∼ Ion Larmor Radius Tokamak size ∼ 10 -2 m 10m ∼ 10 -3 (4)
The second one is η the Debye Length linked variations of the Electric Field and Potential. It is defined as η ∼ Characteristic scale of variations of the Electric Field Tokamak size .

If we follow a scaling routine, we will get that the dimensionless Electric Potential and Field write

Φ(x) = Φ 0 (x) + ηΦ 1 ( x η ) and E(x) = E 0 (x) + E 1 ( x η ), (6)
with

E(x) = -(∇Φ)(x), (7)
and that the dimensionless Magnetic Field write

B(x) ε = (∇ × A ε )(x). (8)
Then, the dimensionless trajectory, is solution to the following dynamical system:

∂X ∂t = V (9)
∂V ∂t = E 0 (X) + E 1 ( X η ) + V × B(X) ε . ( 10
)
We guess that anybody agree with the fact that this system is a good model to describe motion of charged particle within Tokamak.

Gyro-kinetic Model -Yet, there is the Gyro-Kinetic Model. It claims that a trajectory (r, ψ, ϕ, W , J, Γ) in a coordinates system (r, ψ, ϕ, w , j, γ) we will discuss on later on, is solution to:

∂r ∂t = (ED r + M CD r ), ∂ψ ∂t = W q(r)R + ED ψ + M CD ψ r , ∂ϕ ∂t = W R , ∂W ∂t = (E 0 + E 1 ) - J ε ∇ |B| + W |B| ED • (∇|B|), (11)
where Ψ Cemracs '10).

R = R 0 + r cos(ψ), ED = ε B × (E 0 + E 1 ) |B| 2 , M CD = ε B × (∇|B|) |B| 3 (W 2 + J |B| ε ), (12)
and

∇ = 1 R ∂ ∂ϕ + 1 q(r) ∂ ∂ψ . (13)
We now explain how to read equations ( 11) - [START_REF] Littlejohn | Hamiltonian perturbation theory in noncanonical coordinates[END_REF]. A Tokamak is the interior of a torus with a vertical symmetry axis. (A Tokamak and its associated position coordinate system (r, ψ, ϕ) is given in Figure 4.) Hence, it is thought as a small disc carried by a large circle of radius R 0 . Then, for a given point of the Tokamak, r is (almost) its distance from the center of the small disk, ψ is (almost) the angle that the point makes in the small disk and ϕ is (almost) the angle that it makes on the large circle. In what concerns the velocity variable system (w , j, γ), if a moving point is in (r, ψ, ϕ), the w -component of its velocity stands (almost) for the projection of its velocity on the direction the Magnetic Field has in (r, ψ, ϕ) (which is not far from being tangent to the large circle but which has a small component tangent to the small circle), j is (almost) half the square of the norm of its velocity on the plan orthogonal to the Magnetic Field direction, and γ is (almost) the angle that this projection makes in this plan.

Above, precision "almost" is essential, and constitutes the core of the Gyro-Kinetic Approximation which consists, as we will see, in building infinitesimal changes of coordinates to set out a dynamical system shaped like [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF].

We now comment system [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF], forgetting that the variables are only almost what they are supposed to be. The first equation in [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF] says that the r-component of a particle trajectory varies with a velocity which is the projection on the direction of the vector joining the center of the small disk and the point of the Electric Drift velocity ED and the Magnetic Curvature Drift velocity M CD, both defined in [START_REF] Littlejohn | Hamiltonian formulation of guiding center motion[END_REF]. The Electric Drift velocity ED involves the cross product of the Magnetic Field and what is called the gyro-average of the Electric Field, which is indicated by symbol . The Magnetic Curvature Drift velocity M CD involves the cross product of the Magnetic Field with the gradient of the Magnetic Field norm. In the second equation of [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF], q(r) is called the quality factor. Essentially, it is the number of revolutions in the small disk a particle makes while making one revolution along the large circle. Then, term W /(q(r)R), where R is defined in [START_REF] Littlejohn | Hamiltonian formulation of guiding center motion[END_REF], generates the winding of the particle trajectory in the small disk (this is illustrated by the red curves drawn on the left of the figure). The second term of the second equation of ( 11) is linked with the action of the Electric Drift velocity and Magnetic Curvature Drift velocity. The third equation involves the contribution W /R of the w -component of the particle velocity to the variation of the ϕ-component of the particle position. The last equation describes the variation of w -component of the particle velocity. It involves, among other terms, the projection of the gyro-average of the Electric Field on the direction of the Magnetic Field and the variation of the Magnetic Field norm in the direction of the Magnetic Field, materialized by operator ∇ defined by [START_REF] Littlejohn | Hamiltonian perturbation theory in noncanonical coordinates[END_REF].

In system [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF], their is neither equation for J (the j-component of the particle velocity) nor for Γ (the γ-component of the particle velocity). This may be considered as a strange fact. The reason why this happens is that the dynamics of r, ψ, ϕ, W , the r-, ψ-, ϕ-, w -components of the particle trajectory and velocity do not depend on the γ-component of the particle velocity and that the its j-component is invariant. This is summarized as:

∂J ∂t = 0, ∂Γ ∂t = Does not matter. ( 14
)
What we explain and what we do not explain -In this document we give the key ideas that bring the deduction of the Gyro-Kinetic Dynamical System ( 12) from the original one ( 9) - [START_REF]Plasma Physics, chapter Elementary Orbit and Drift Theory[END_REF]. Among thing that we do not tackle, there is the question of the specific position coordinates (r, ψ, ϕ). We will remain with a generic position coordinate system. We will not talk about Gyro-average either. The question of the coupling of Gyro-Kinetic Approximation with computation of Electric and Magnetic Fields, using Quasi-Neutral Poisson Equation or Maxwell Equation is also not addressed here.

To end this paragraph we mention that the method only works in case when η is larger than ε, or in other words, when η = ε 1-κ for κ > 0. This is a important limitation in the theory.

Methode summarize

Key result -The method allowing us to go from system ( 9) -(10) to a system of the kind of (11), i.e. to make disappear two equations, has nothing to do with magic. It is based on the following result: Theorem 2.1. If, in a given coordinate system r = (r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ), a Hamiltonian Dynamical System writes:

∂R ∂t = P(R)∇ r H(R), P(R) =      M 0 : 0 0 : 0 0 • • • 0 0 1 0 • • • 0 -1 0      , (15)
(for a trajectory R = (R 1 , R 2 , R 3 , R 4 , R 5 , R 6 ) T
)) with a Hamiltonian Function that does not depend on the last variable, i.e.

∂H ∂r 6

= 0,

Then, submatrix M does not depend on the two last variables, i.e.

∂M ∂r 5

= 0 and ∂M ∂r 6

= 0,

and consequently, time-evolution of the four first components R 1 , R 2 , R 3 , R 4 is independent of the last component R 6 ; and, the penultimate component R 5 of the trajectory in not time-evolving, i.e.

∂R 5 ∂t = 0. ( 18
)
The proof of property ( 17) is relatively straightforward by a calculation. Equation ( 18) is a direct consequence of ( 16) that yields a sixth component of ∇ r H(R) with worth 0 and consequently, because of the form of P(R) a fifth line of Dynmical System [START_REF] Northrop | Extensions of guiding center motion to higher order[END_REF] which is exactly [START_REF] Parra | Turbulent transport of toroidal angular momentum in low flow gyrokinetics[END_REF].

Essentially, the Gyro-Kinetic version of Dynamical System (9) -( 10) exhibit essentially a property like [START_REF] Parra | Turbulent transport of toroidal angular momentum in low flow gyrokinetics[END_REF] and the property that the four first components of the trajectory are independent of the last component.

As a matter of fact, building the Gyro-Kinetic Approximation consists in building a change of coordinates that bring a system of coordinates in which P(R) has the same form as in [START_REF] Northrop | Extensions of guiding center motion to higher order[END_REF] and in which ( 16) is true.

Usual Coordinates

(x, v)

∂X ∂t = V ∂V ∂t = E 0 (X)+E 1 ( X η )+V× B(X) ε
Canonical Coordinates (q, p)

Hε η = Hε η (q, p): ∂Q ∂t = ∇p Hε η ∂P ∂t = -∇q Hε η 2 1: Hamiltonian? Cylindrical Coordinates (x, v , v ⊥ , θ) 3 Darboux Almost Canonical Coordinates (y, u , k, θ) 4: Darboux Method
Lie Coordinates (z, w , j, γ)

5: Lie Method Usual Coordinates (x, v) H ε η (x, v), P ε η (x, v) s.t:     ∂X ∂t ∂V ∂t     = P ε η ∇x,vH ε η Canonical Coordinates (q, p) Hε η (q, p), Pε η (q, p)=S s.t:     ∂Q ∂t ∂P ∂t     = S∇q,p Hε η 2 1: Hamiltonian? Cylindrical Coordinates (x, v , v ⊥ , θ) H ε η (x, v , v ⊥ , θ), P ε η (x, v , v ⊥ , θ) 3 Darboux Almost Canonical Coordinates (y, u , k, θ) H ε η (y, u , k, θ), P ε η (y, u , k, θ) 4: Darboux Method Lie Coordinates (z, w , j, γ) H ε η (z, w , j), P ε η (z, w , j, γ) = P ε η (z, w , j, γ)
5: Lie Method Panorama -As illustrated in Figure 5, the method to build the desired change of coordinates is made of 5 steps. The first one consists in checking that the Dynamical System ( 9) -( 10) is well Hamiltonian. This is symbolized by arrow 1 in the top picture of the Figure . Once this is done, we can go back into the Usual Coordinate System but knowing that the system writes as in the square which is in the top-left of the bottom picture, i.e. involving a Poisson Matrix P ε η and the gradient ∇ x,v H ε η of a Hamiltonian Function. It may be written in that form in any Coordinate System, and formula give how to transform the Hamiltonian Function and the Poisson Matrix while changing of Coordinates. The goal of the third step is to introduce a Cylindrical in Velocity Coordinate system which is known has being close to the Gyro-Kinetic Coordinate System. Writing the system in this coordinate system uses the formula giving how the Poisson Matrix and the Hamiltonian Function are transformed by change of coordinates. (At this level, it would be also possible to make a position change of coordinates fitting Tokamak geometry.) In the fourth step, we make another change of coordinates in order to have the Poisson Matrix form allowing the application of the Key Result (Theorem 2.1). In the fifth step, we make a last change of coordinates, leaving the Poisson Matrix form unchanged and leading to a Hamiltonian independent of the last variable.

Hamiltonian System

We make here the first step: we check that Dynamical System ( 9) -( 10) is Hamiltonian. Essentially, this means nothing but that there exists a coordinate system in which it writes:

∂Q ∂t = ∇ p Hε η , (19)
∂P ∂t = -∇ q Hε η , (20)
or

    ∂Q ∂t ∂P ∂t     = S∇ q,p Hε η , with S = 0 I 3 -I 3 0 , (21)
where Hε η is a function.

The Canonical Coordinates -If we use ( 7) and ( 8) in ( 9) -( 10), we get that in

the Usual Coordinates (x, v) = (x 1 , x 2 , x 3 , v 1 , v 2 , v 3 ), trajectory : (X(t; x, v, s), V(t; x, v, s)) is solution to ∂X ∂t = V, (22)
∂V ∂t = -∇ Φ 0 (X) + ηΦ 1 ( X η ) + V × ∇ × A(X) ε . (23)
We recall that (X,

V) = (X 1 , X 2 , X 3 , V 1 , V 2 , V 3 ).
We will show, in the next paragraph, that in coordinate system (q, p) = (q 1 , q 2 , q 3 , p 1 , p 2 , p 3 ) defined by

q = x, p = v + A(x) ε , (24)
with reverse transformation given by

x = q, v = p - A(q) ε , (25)
the trajectory (Q(t; q, p, s), P(t; q, p, s))

((Q, P) = (Q 1 , Q 2 , Q 3 , P 1 , P 2 , P 3
)) which is given by

Q = X, P = V + A(X) ε , X = Q, V = P - A(Q) ε , (26)
is solution of a Dynamical System of the form ( 19) -( 20) or (21), with

Hε η (q, p) = 1 2 p - A(q) ε 2 + Φ 0 (q) + ηΦ 1 ( q η ). (27)
Those (q, p) coordinates will be called Canonical Coordinates.

Check of Canonical nature of Canonical Coordinates -Making calculations suggested by ( 19) -( 20) with Hε η given by ( 27) gives

∂Q ∂t = ∇ p Hε η (Q, P) = P - A(Q) ε , (28)
∂P ∂t = -∇ q Hε η (Q, P) = (∇A(Q)) T ε P - A(Q) ε -∇ Φ 0 (Q) + ηΦ 1 ( Q η ) , (29)
which applying the following formula

(∇A) T (p -A) = (∇A)(p -A) + (p -A) × (∇ × A), (30)
yields

∂Q ∂t = P - A(Q) ε , (31)
∂P ∂t - (∇A(Q)) ε P - A(Q) ε = P - A(Q) ε × ∇ × A(Q) ε -∇ Φ 0 (Q) + ηΦ 1 ( Q η ) . (32)
Now,using (26) we get

∂X ∂t = V, (33)
∂P ∂t - (∇A(Q)) ε ∂Q ∂t = ∂ P - A(Q) ε ∂t = ∂V ∂t = P - A(Q) ε × ∇ × A(Q) ε -∇ Φ 0 (Q) + ηΦ 1 ( Q η ) (34) = V × ∇ × A(X) ε -∇ Φ 0 (X) + ηΦ 1 ( X η ) .
which is ( 22) -( 23), and then proving the equivalence between the two systems and leads to the conclusion that the Dynamical System we take interest in is Hamiltonian.

As by-products : Poisson Matrix, Poisson Bracket, Change of Coordinates Formula -To end the first step we now list some properties we can moreover deduce from the fact that Dynamical System ( 9) -( 10) is Hamiltonian.

In any coordinate system r = (r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ), the Dynamical System writes:

∂R ∂t = P(R)∇ r H(R), (35)
for a Poisson Matrix P, which is antisymmetric, and a Hamiltonian Function H.

We can consider the Poisson Bracket which is defined for two regular functions f and g : R 6 → R by {f, g}(r) = (∇ r f (r)) • (P(r)(∇ r g(r))).

(36)

If one of the two function is vector-valued, i.e. if for instance f : R 6 → R 6 and g : R 6 → R, then it is defined, for any component i = 1, . . . , 6, by

({f , g}(r)) i = (∇ r f i (r)) • (P(r)(∇ r g(r))). (37)
Then, if we introduce the Coordinate Function I, defined by I(r) = r, then (35) reads also:

∂R ∂t = {I, H}(R). (38)
Remark 3.1. In papers of physicists, formula (38) is generally read ∂r ∂t

= {r, H}.

In another coordinate system r = (r 1 , r2 , r3 , r4 , r5 , r6 ) which is tied with the first one by change-of-coordinates formula r = ρ(r) and r = ρ(r) = ρ -1 (r), the system writes

∂ R ∂t = P( R)∇ r H( R), (39)
where R = ρ(R), with Hamiltonian Function and Poisson Matrix given by H(r) = H(ρ(r)) and ( P(r)

) ij = ρ i , ρ j (ρ(r)). (40)
Remark 3.2. From (27) and applying (40), in Usual Coordinates, Hamiltonian Function expression is

H ε η (x, v) = 1 2 v 2 + Φ 0 (x) + ηΦ 1 ( x η ), (41)
leading to ∇ x,v H ε η = ∇ Φ 0 (x) + ηΦ 1 ( x η ) v , (42)
and Poisson Matrix expression is

P ε η (x, v) = 0 I 3 -I 3 (∇A(x)) T -(∇A(x)) ε . ( 43
)
This can be easily checked calculating

P ε η (x, v)∇ x,v H ε η =   v -∇ Φ 0 (x) + ηΦ 1 ( x η ) + (∇A(x)) T -(∇A(x)) ε v   , (44)
using (29),and comparing with (22) -(23). Expression (41) is consistent with (40), let us check that it is the same for (43). To do this, let us write the change of coordinates in the following way: (x, v) = υ(q, p) and (q, p) = π(x, v). Then, for instance: υ 5 (q, p) = p 2 -A2(q) ε , υ 6 (q, p) = p 3 -A3(q) ε and

∇υ 5 (q, p) =          1 ε ∂A2 ∂q1 (q) 1 ε ∂A2 ∂q2 (q) 1 ε ∂A2 ∂q3 (q) 0 1 0          , S = 0 I 3 -I 3 0 , ∇υ 6 (q, p) =          1 ε ∂A3 ∂q1 (q) 1 ε ∂A3 ∂q2 (q) 1 ε ∂A3 ∂q3 (q) 0 0 1          . ( 45
)
Hence, since {υ 5 , υ 6 } = (∇υ 5 ) • (S∇υ 6 ), expression of (P ε η ) 56 , obtained applying (40), is

{υ 5 , υ 6 }(π(x, v)) = 1 ε ∂A 3 ∂q 2 (x) - ∂A 2 ∂q 3 (x) , (46)
which is also the expression of (P ε η ) 56 obtained applying (43). (This may of course be led for the other entries of the matrix also.)

Cylindrical Coordinates

Cylindrical Coordinates in velocity -We now turn to the second step which consists in setting the expression of Dynamical System ( 22) -(23) (or ( 9) -( 10) or ( 19), (20), ( 27)) in a Cylindrical in velocity Coordinate System (x, v , v ⊥ , θ) which is such that

B v ⊥ v v θ  
= v • B |B| , v ⊥ = v -v • B |B| B |B| , θ s.t. v -v • B |B| B |B| = v ⊥ (cos θ, sin θ). (47)
This means, as shown in Figure 6 where a helicoidal trajectory induced by a Magnetic Field pointing upward, that v is the projection of the velocity on the direction of the Magnetic Field, v ⊥ is the norm of the projection of the velocity on the plan orthogonal to the Magnetic Field and θ is the angle that this projection makes in this plan.

Hamiltonian Function and Poisson Matrix in Cylindrical Coordinates -

Applying formula (41), we get the expression of the Hamiltonian Function in this System:

H ε η (x, v , v ⊥ , θ) = 1 2 v 2 + v 2 ⊥ + Φ 0 (x) + ηΦ 1 ( x η ). ( 48
)
We can also get the expression of the Poisson Matrix, which is heavy, applying formula (41).

Here, we give its expression in the case when

B(x) =   b(x) 0 0   , b > 0, (49)
only. It is:

P ε η (x, v , v ⊥ , θ) =           0 0 0 b(x) ε 0 0 0 0 0 0 -sin(θ) -cos(θ) v ⊥ 0 0 0 0 -cos(θ) sin(θ) v ⊥ -b(x) ε 0 0 0 $$ $ 0 sin(θ) cos(θ) -$$ 0 -b(x) εv ⊥ 0 cos(θ) v ⊥ -sin(θ) v ⊥ -$ b(x) εv ⊥ 0           , ( 50
)
where

$ = v ε ( ∂b ∂x 2 (x) + ∂b ∂x 3 (x)) and $$ = v εv ⊥ (sin(θ) ∂b ∂x 2 (x) + 1 + sin 2 (θ) cos(θ) ∂b ∂x 3 (x)). (51)
In more general cases, we do not give the Poisson Matrix expression. Yet we mention a very important fact which is that

( P ε η (x, v , v ⊥ , θ)) 56 = ( P ε η (x, v , v ⊥ , θ)) v ⊥ θ = |B(x)| εv ⊥ > 0, (52)
is always true. This is Important for the Darboux Method which will be reached in a few lines. We will denote:

|B(x)| εv ⊥ = ω(x, v ⊥ ). ( 53
)

Darboux Algorithm

Darboux Algorithm Target -The third step is the application of a mathematical algorithm, so called the Darboux Algorithm, to build a Coordinate System (y, u , k, θ) in which the Poisson Matrix has the required form, given by ( 15), to apply the Key Result. In fact, in order to manage the small parameter ε, we will build the Coordinate System (y, u , k, θ) in order to get P ε η with the following form:

     M 0 : 0 0 : 0 0 • • • 0 0 1 ε 0 • • • 0 -1 ε 0      . ( 54
)
We introduce the following notations to manage change-of-coordinates mappings:

(y, u , k, θ) = Υ(x, v , v ⊥ , θ) and (x, v , v ⊥ , θ) = ξ(y, u , k, θ), (ξ = Υ -1
).

An important and constitutive fact in the Darboux Algorithm is that the θ-variable is left unchanged.

Now, Since

(P ε η (y, v , k, θ)) ij = {Υ i , Υ j }(ξ(y, v , k, θ)), {Υ i , Υ j } = (∇Υ i ) • ( P ε (∇Υ j )), (55)
the bottom-right of form given in (54), results from:

{Υ 6 , Υ 5 } = - 1 ε . (56)
Remark 5.1. If we write Υ 6 = Υ θ and Υ 5 = Υ k , (56) may be also read {Υ θ , Υ k } = -1 ε . In articles of physicists, this last equation reads {θ, k} = -1 ε . In the same way, the fact that the two last lines (or columns) contain only zeros results from:

{Υ 1 , Υ 5 } = 0, {Υ 1 , Υ 6 } = 0, {Υ 2 , Υ 5 } = 0, {Υ 2 , Υ 6 } = 0, {Υ 3 , Υ 5 } = 0, {Υ 3 , Υ 6 } = 0, {Υ 4 , Υ 5 } = 0, {Υ 4 , Υ 6 } = 0. (57)
Remark 5.2. Using the same conventions as in Remark 5.1, (57) may also read

{Υ y1 , Υ k } = 0, or {y 1 , k} = 0, {Υ y1 ,Υ θ } = 0, or {y 1 , θ} = 0, {Υ y2 ,Υ k } = 0, or {y 2 , k} = 0, {Υ y2 ,Υ θ } = 0, or {y 2 , θ} = 0, {Υ y3 ,Υ k } = 0, or {y 3 , k} = 0, {Υ y3 ,Υ θ } = 0, or {y 3 , θ} = 0, {Υ y4 ,Υ k } = 0, or v , k = 0, Υ v ,Υ θ = 0, or v , θ = 0. (58)
Equations ( 56) and (57) are hyperbolic PDEs that need to be solve to get changeof-coordinates mapping Υ.

First equation processing -Since (and this is a consequence of the fact that the θ-variable is left unchanged in the sought change-of-coordinates)

∇Υ 6 (= ∇Υ θ ) = (0, 0, 0, 0, 0, 1) T , (59)
we deduce {Υ 6 , Υ 5 } = (∇Υ 6 ) • ( P ε (∇Υ 5 )) is the last component of ( P ε (∇Υ 5 )). Hence, equation (57) reads

F 1 ∂Υ 5 ∂x 1 + F 2 ∂Υ 5 ∂x 2 + F 3 ∂Υ 5 ∂x 3 + F ∂Υ 5 ∂v + ω ∂Υ 5 ∂v ⊥ = - 1 ε , (60)
where F n and ω are functions of (x, v , v ⊥ , θ). Function ω is given by (53) and is positive. Solving (60) will give Υ 5 (x, v , v ⊥ , θ) which is the expression of component k of the Darboux Coordinates in terms of the Cylindrical in Velocity Coordinates (x, v , v ⊥ , θ).

To solve this equation we will use the Method of Characteristics.

Method of Characteristics -Dividing by ω, which is possible since ω is positive, (60) gives

∂Υ 5 ∂v ⊥ + ε v ⊥ F 1 |B| ∂Υ 5 ∂x 1 + ε v ⊥ F 2 |B| ∂Υ 5 ∂x 2 + ε v ⊥ F 3 |B| ∂Υ 5 ∂x 3 + ε v ⊥ F |B| ∂Υ 5 ∂v = v ⊥ |B| . (61)
In order to get a solution to this equation, we need to add a boundary (or initial) condition in a point where the involved vector field (εv ⊥ /|B|)(F 1 , F 2 , F 3 , F ) does not vanish. Hence we set

Υ 5|v ⊥ =ν = 0 for a small ν > 0. (62)
We consider the following characteristics (X , V ) = (X 1 , X 2 , X 3 , V ):

X 1 (v ⊥ ; x, v , u ⊥ ) such that ∂X 1 ∂v ⊥ = ε v ⊥ F 1 (X 1 , X 2 , X 3 , V , v ⊥ , θ) |B|(X 1 , X 2 , X 3 ) , X 1 (u ⊥ ) = x 1 , X 2 (v ⊥ ; x, v , u ⊥ ) such that ∂X 2 ∂v ⊥ = ε v ⊥ F 2 (X 1 , X 2 , X 3 , V , v ⊥ , θ) |B|(X 1 , X 2 , X 3 ) , X 2 (u ⊥ ) = x 2 , X 3 (v ⊥ ; x, v , u ⊥ ) such that ∂X 3 ∂v ⊥ = ε v ⊥ F 3 (X 1 , X 2 , X 3 , V , v ⊥ , θ) |B|(X 1 , X 2 , X 3 ) , X 3 (u ⊥ ) = x 3 , V (v ⊥ ; x, v , u ⊥ ) such that ∂V ∂v ⊥ = ε v ⊥ F (X 1 , X 2 , X 3 , V , v ⊥ , θ) |B(X 1 , X 2 , X 3 ) , V (u ⊥ ) = v , (63)
and the solution to (61) -( 62) is given by

Υ 5 (x, v , v ⊥ , θ) = Υ 5 (X (ν; x, v , v ⊥ ), V (ν; x, v , v ⊥ ), ν, θ)+ v ⊥ ν s |B(X (s; x, v , v ⊥ ))| ds = v ⊥ ν s |B(X (s; x, v , v ⊥ ))| ds. ( 64
)
(The last equality is gotten because of (62).) Rewriting system (63) in the following compact form,

∂ X V ∂v ⊥ = εF(X 1 , V , v ⊥ , θ), X V (u ⊥ ) = x v , (65)
with F = (v ⊥ /|B|)(F 1 , F 2 , F 3 , F ) and applying the formula that gives the expansion of the solution of a dynamical system in terms of its parameter, we get that

X V = X V (v ⊥ ; x, v , u ⊥ ), (66)
writes

X V = x v + v ⊥ 1! εF(x, v , v ⊥ , θ) + v 2 ⊥ 2! L εF (εF)(x 1 , v , v ⊥ , θ) + v 3 ⊥ 3! L 2 εF (εF)(x 1 , v , v ⊥ , θ) + . . . = x v + εv ⊥ 1! F(x, v , v ⊥ , θ) + ε 2 v 2 ⊥ 2! L F (F)(x 1 , v , v ⊥ , θ) + ε 3 v 3 ⊥ 3! L 2 F (F)(x 1 , v , v ⊥ , θ) + . . . ( 67
)
where L F (F) is the the Lie Derivative of vector field F in the direction of vector field

F, L 2 F (F) = L F (L F (F)) and L 3 F (F) = L F (L 2 F (F))
, and so on. Rewriting this expansion, we get

X = x + εv ⊥ X 1 + ε 2 v 2 ⊥ X 2 + ε 3 v 3 ⊥ X 3 + . . . (68)
V = v + εv ⊥ V 1 + ε 2 v 2 ⊥ V 2 + ε 3 v 3 ⊥ V 3 + . . . (69)
with

X 1 V 1 = F(x, v , v ⊥ , θ), X 2 V 2 = 1 2! L F (F)(x, v , v ⊥ , θ), X 3 V 3 = 1 3! L 2 F (F)(x, v , v ⊥ , θ), . . . (70)
Now, injecting (69) in expression (64) on Υ 5 , we get:

Υ 5 (x, v , v ⊥ , θ) = v ⊥ ν s |B(X (s; x, v , v ⊥ ))| ds = v ⊥ ν s |B(x)| ds + ε v ⊥ ν s 2 T 1 ( 1 |B(x)| )X 1 ds+ ε 2 v ⊥ ν s 3 T 2 ( 1 |B(x)| )X 1 + T 1 ( 1 |B(x)| )X 2 ds + . . . = (v ⊥ -ν) 2 2|B(x)| + ε v ⊥ ν s 2 T 1 ( 1 |B(x)| )X 1 ds + ε 2 v ⊥ ν s 3 T 2 ( 1 |B(x)| )X 1 + T 1 ( 1 |B(x)| )X 2 ds + . . . , (71)
where T i (1/|B(x)|) is the i th coefficient of the Taylor expansion of 1/|B(x)|.

Formula (71) gives the expression of Υ 5 , i.e. the expression of new variable k in terms of Cynlindrical Coordinates (x, v , v ⊥ , θ), as an expansion in ε.

Remark 5.3. In (71), it is possible to choose ν as small as we want. Hence, the first term of the expansion is essentially the Magnetic Moment v ⊥ /|B(x)|.

On other equations and on Poisson Matrix in Darboux Coordinates

Equation (56) was processed and gave expression of k. Equations (57) can be processed using similar Methods of Characteristics. A special attention needs to be given to the fact that Υ 1 , . . . Υ 4 are solutions of a PDE involving Υ 5 and of another one involving Υ 6 that need to be tackled together. They will give y and u as expansions in ε. Once this done, we will have

Υ = Υ 0 + εΥ 1 + ε 2 Υ 2 + ε 3 Υ 3 + . . . , (72)
or, in other words, the expression of (y, u , k) in terms of (x, v , v ⊥ , θ) and as an expansion in ε.

The other terms of the new Poisson matrix P ε η (y, u , k, θ) are given computing:

( P ε η ) 12 = {Υ 1 ,Υ 2 }(={Υ y1 ,Υ y2 }={y 1 , y 2 }), ( P ε η ) 13 = {Υ 1 ,Υ 3 }(={Υ y1 ,Υ y3 }={y 1 , y 3 }), . . . .

Hamiltonian Function in Darboux

Coordinates -We now have to compute the Hamiltonian Function in the Darboux Coordinates. For this, we build from expansion (72) of Υ, an asymptotic expansion of ξ = Υ -1 :

ξ = ξ 0 + εξ 1 + ε 2 ξ 2 + ε 3 ξ 3 + . . . . (73)
Then, we use the expression of the Hamiltonian Function in Cylindrical in Velocity Coordinates:

H ε η (x, v , v ⊥ , θ) = 1 2 v 2 + v 2 ⊥ + Φ 0 (x) + ηΦ 1 ( x η ), (74)
and the following formula, which is gotten from (40),

H ε η (y, u , k, θ) = H ε η (ξ(y, u , k, θ)) = H ε η (ξ 0 + εξ 1 + ε 2 ξ 2 + ε 3 ξ 3 + . . . ) = H ε η (ξ 0 ) + εT 1 ( H ε η )(ξ 0 ) • ξ 1 + • • • = u 2 2 + |B(y)|k + Φ 0 (y) + ηΦ 1 ( y η ) + εH 1,η (y, u , k, θ) + ε 2 H 2,η (y, u , k, θ) + . . . . (75)
Notice that for this expansion to be valid, it is necessary to have the property we mention on page 7: η = ε 1-κ for κ > 0.

In expression (75), there is an important fact for the setting out of the to come Lie Transform based Method: the first term is independent of θ.

Lie Transform based Method

Lie Transform based Method Target -As a result of the Darboux Algorithm, we obtained a Poisson Matrix P ε η (y, u , k, θ) with the required form to apply the Key Result (Theorem 2.1), but the resulting Hamiltonian Function

H ε η (y, u , k, θ) = H 0,η (y, u , k) + εH 1,η (y, u , k, θ) + ε 2 H 2,η (y, u , k, θ) + . . . , (76)
depends on θ. And we need to make this dependency to vanish.

Let us notice two important facts, which are in fact linked: The first term in the asymptotic expansion (75) does not depend on θ and the Hamiltonian Function expressed in the Cylindrical in Velocity Coordinates does not depend on θ. The consequence of those facts is that it is certainly possible to make the last variable to vanish using a mapping parametrized by ε and close to identity for small ε. Moreover, we need to build this sought mapping in such a way that it does not change the Poisson Matrix expression. This means that, as viewed as functions, P ε η and P ε η must be the same, i.e.:

P ε η (z, w , j, γ) = P ε η (z, w , j, γ) or P ε η (y, u , k, θ) = P ε η (y, u , k, θ), (77)
for any z, w , j, γ or y, u , k, θ. Yet, among changes of variables, the symplectic ones do not change the Poisson Matrix expression. And among symplectic changes of variables, are the flows of Hamiltonian Vector Fields, which are moreover close to identity for small values of their parameter.

Hence, the Lie Transform based Method consists in building a change of variables, parametrized by ε, reading

(y, u , k, θ) → (z, w , j, γ) = ζ(ε; y, u , k, θ), (78) with ζ
(ε; y, u , k, θ) = (y, u , k, θ) + εζ 1 (y, u , k, θ), +ε 2 ζ 2 (y, u , k, θ) + . . . , (79)
which is the flow of parameter ε of a Hamiltonian Vector Fields.

The way to do -Looking for ζ as the flow of a Hamiltonian Vector Field means looking for a Hamiltonian Function:

G(ε; y, u , k, θ) = G 0 (y, u , k, θ) + εG 1 (y, u , k, θ) + ε 2 G 2 (y, u , k, θ) + . . . , (80)
such that ζ(ε; y, u , k, θ) is solution to:

∂ζ ∂ε = P ε η ∇G, ζ(ε = 0; y, u , k, θ) = (y, u , k, θ). (81)
Hence the target becomes to find functions G 0 , G 1 , G 2 , . . . , such that:

H ε η (z, w , j) = H ε η (λ(z, w , j, γ)) where λ = ζ -1 . (82)
A result -We will use the following theorem Theorem 6.1. For any two Hamiltonian Functions:

H ε η (y, u , k, θ) = H 0,η (y, u , k) + εH 1,η (y, u , k, θ) + ε 2 H 2,η (y, u , k, θ) + . . . , (83)
and

H ε η (z, w , j) = H 0,η (z, w , j) + ε H 1,η (z, w , j) + ε 2 H 2,η (z, w , j) + . . . , (84)
expressed in two different variable systems (y, u , k, θ and z, w , j, γ) , with the property that H 0,η (z, w , j) = H 0,η (z, w , j),

writes g(ε; z, w , j, γ) = g 0 (z, w , j, γ) + εg 1 (z, w , j, γ) + ε 2 g 2 (z, w , j, γ) + . . . ,

where g 0 , g 1 , g 2 , . . . are given by g 0 , H 0,η = O 0 (H 0,η ), g 1 , H 0,η = O 1 (H 0,η , H Remark 6.1. Despite we do not explicitly write the dependency of λ, g 0 , g 1 , g 2 , . . . , λ and G 0 , G 1 , G 2 , . . . with respect η to all those functions do depend on η. Here again, to be able to write the expansions with respect to ε, it is necessary to have the property mentioned on page 7: η = ε 1-κ for κ > 0.

The Lie Transform based Method -As in the case of the Darboux Algorithm, we do not do the computations that give ζ and λ. We only give the steps that allows us to get them. The Lie Transform based Method may be summarized as:

1. Fix : H 1,η (z, w , j), H 2,η (z, w , j), . . . . They can be fixed, a priory, with no restriction. Nevertheless those functions are involved in the PDEs of (91). Hence, they need to be chosen in a way that leads to PDEs which are as simple as possible to solve.

2. Solve PDE in (91) to get g 0 , g 1 , g 2 recursively.

They are quite complex but we can solve them using the Method of Characteristics.

Get ζ solving (87).

It is gotten as an expansion in ε.

4. Compute expression of the Lie Variables: (z, w , j, γ) = ζ(y, u , k, θ)

5. Since H 1,η , H 2,η , . . . were fixed in the begining of the process, it is not necessary to compute λ to get: H ε η (z, w , j) = H ε η (λ(z, w , j, γ)). Nonetheless, λ may be gotten as an expansion in ε by inverting expansion of ζ.

Brief conclusion

At the end of the day, in the Lie Coordinates System, the trajectory (Z, W , J, Γ) of a charged particle is solution to a Dynamical System which has the following form:

∂Γ ∂t = Something complicated, (92)
∂J ∂t = 0, (93)
∂ Z W ∂t = Something independent of Γ. ( 94
)
This form is essentially the same as the one of [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF].

As a conclusion, we just say that the method which is summarized here is the one that allows us to get [START_REF] Littlejohn | A guiding center Hamiltonian: A new approach[END_REF] or any other Gyro-Kinetic Approximation Model.

Insisting one more time, the important facts in system (92) -( 94) is that the evolution of the components Z and W of the trajectory and of its γ-component Γ are uncoupled and that component J does not evolve and then becomes a parameter. Hence, even if the γ-component Γ is not computed, it does not preclude the computation of J, Z and W solving a collection, parametrized by J, of four-dimensional Dynamical Systems.

Figure 1 :

 1 Figure 1: Artist vision of Iter.

Figure 2 :

 2 Figure 2: Left: Helicoidal trajectories of ions and electrons submitted to a magnetic field. (Source: S. Jardin's Lectures at Cemracs'10). Right: Screen effect and Debye Length in a plasma.

Figure 3 :

 3 Figure 3: Symbolic picture of oscillating trajectory of ion in a electric field with variations with typical size comparable with trajectory oscillation.

Figure

  Figure3we symbolically represent this bed as a charcoal grey surface. (It is symbolic because a surface is drawn while, in real world, Electric Field fills a 3D space.) This surface shows variations at two scales which are cenced to be about 100 times the Debye Lengths and 1.000 times the Debye Lengths. Those variations are not of large amplitude but are stable. Now, ion motion, symbolically represented in red in the figure, shows an oscillation of amplitude which is about 100 times the Debye Length (ion Larmor Radius ∼ 10 -2 m, Debye Length : ∼ 10 -4 m). Hence, because of the stability of bed variations, its resulting effect can be important on ion trajectory after several periods of oscillation. To take into account this resulting effect, the concept of gyro-average was introduced.

Figure 4 :

 4 Figure 4: Position coordinates in use in a Tokamak. (Source: V. Grandgirard's Lectures at

Figure 5 : 4 :

 54 Figure 5: The method is made of 5 steps. 1: Check that the Dynamical System (9)-(10) is well Hamiltonian. 2: Write the system using Hamiltonian Function and Poisson Matrix. 3: Write the system in a cylindrical in velocity coordinate system using the formula giving how the Poisson Matrix and the Hamiltonian Function is transformed by change of coordinates. 4: Make another change of coordinates in order to have the Poisson Matrix form allowing the application of the Key Result (Theorem 2.1). 5: Make a last change of coordinates, leaving the Poisson Matrix form unchanged and leading to a Hamiltonian independent of the last variable.

Figure 6 :

 6 Figure 6: Helicoidal trajectory and Cylindrical Coordinates for the velocity variable.

  v

There exists a

  Hamitonian Function writingG(ε; y, u , k, θ) = G 0 (y, u , k, θ) + εG 1 (y, u , k, θ) + ε 2 G 2 (y, u , k, θ) + . . . ,(86)such that the solution ζ of the Hamiltonian Dynamical System associated with G and with parameter ε, i.e. solution to:∂ζ ∂ε = P ε η ∇G, ζ(ε = 0; z, w , j, γ) = (y, u , k, θ),(87)is such thatH ε η (z, w , j) = H ε η (λ(z, w , j, γ)) where λ = ζ -1 .(88)Moreover, the Hamiltonian Function g(ε; z, w , j, γ) such that λ is solution to ∂λ ∂ε = P ε η ∇g, λ(ε = 0; y, u , k, θ) = (z, w , j, γ),

  1,η , H 1,η , g 0 ), g 2 , H 0,η = O 2 (H 0,η , H 1,η , H 1,η , H 2,η , H 2,η , g 0 , g 1 ), . . . . (91)for differential operators O 0 , O 1 , O 2 , . . . defined by recursive formula.


Flavio Mercati

Diego Mazón

Giovanni Amelino-Camelia

José Manuel Carmona

José Luis Cortés

Javier Induráin

Claus Lämmerzahl

Guglielmo M Tino

Diego Maz Ón

José Luis

Cort És

Javier Indur Áin

Claus L Ämmerzahl

Probing the quantum-gravity realm with slow atoms

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Over the last decade there has been growing interest in the possibility to investigate experimentally some candidate effects of quantum gravity. The development of this "quantum-gravity phenomenology" [START_REF] Amelino-Camelia | Quantum Gravity Phenomenology[END_REF] of course focuses on rare contexts in which the minute effects induced by the ultra-high "Planck scale" M P (≡ c/G 1.2 • 10 28 eV) are not completely negligible. Several contexts of this sort have been found particularly in the study of quantum-gravity/quantum-spacetime effects for the propagation of ultrarelativistic particles (see, e.g., Refs. [START_REF] Amelino-Camelia | astro-ph/9712103[END_REF]3,4,[START_REF] Amelino-Camelia | gr-qc/9808029[END_REF]6,7,[START_REF] Amelino-Camelia | gr-qc/0306019[END_REF]9,10]), and often specifically for cases in which the ultrarelativistic on-shell condition §, E p + m 2 /(2p), is modified by Planck-scale effects.

In the recent Ref. [11] some of us observed that experiments involving cold (slow, nonrelativistic) atoms, and particularly measurements of the atom-recoil frequency, can provide valuable insight on certain types of modifications of the dispersion relation which had been previously considered in quantum-gravity literature. We here extend the scopes of the analysis briefly reported in Ref. [11], also adopting a style of presentation that allows to comment in more detail the derivation of the result. Concerning the conceptual perspective that guides this recent research proposal, we here expose some previously unnoticed aspects of complementarity between the nonrelativistic and the ultrarelativistic regimes in the study of Planck-scale modifications of the dispersion relation. And we offer several observations on how the insight gained from studies of slow atoms might translate into limits of different strength depending on some details of the overall framework within which the modifications of the dispersion relation are introduced. We also report a preliminary exploration of the relativistic issues involved in these studies, which have been already well appreciated in the ultrarelativistic regime but appear to provide novel challenges when the focus is instead on the nonrelativistic regime.

Complementarity of nonrelativistic and ultrarelativistic regimes

Results in support of the possibility of modifications of the energy/momentum ("dispersion") relation have been reported in studies of several approaches to the quantum-gravity problem, and perhaps most notably in analyses inspired by Loop Quantum Gravity [6,12], and in studies that assumed a "noncommutativity" of spacetime coordinates [START_REF] Amelino-Camelia | hep-th/9907110[END_REF][START_REF] Kowalski-Glikman | astro-ph/0006250[END_REF]15]. The analyses of these quantum-gravity approaches that provide encouragement for the presence of corrections to the dispersion relation have become increasingly robust over the last decade [12,[START_REF] Smolin | Three roads to quantum gravity[END_REF][START_REF] Amelino-Camelia | hep-th/9907110[END_REF][START_REF] Kowalski-Glikman | astro-ph/0006250[END_REF]15], but in the majority of cases they are still unable to establish robustly the functional dependence of the correction on momentum. This has led to the proposal that perhaps on this occasion experiments might take the lead by establishing some experimental facts (at least amounting to constraints on the form of the dispersion relation) that may provide guidance for the ongoing investigations on the theory side. From this perspective the fact that presently-available results on the theory side are insufficient to provide narrowly defined phenomenological models is not viewed as sufficient reason for being discouraged: the alternative is giving up on any experimental guidance in the search for quantum gravity, and instead even the constraints produced by a phenomenology of rather broad scopes can be of some value on the theory side, hopefully in turn allowing theorists to provide sharper indications to the phenomenologists.

In light of these considerations the majority of phenomenological studies of Planckscale corrections to the dispersion relation have assumed a rather general ansatz,

E 2 = p 2 + m 2 + ∆ QG (p, m, M P ) , (1)
denoting with E the energy of the particle and with ∆ QG a model-dependent function of the Planck mass M P and of the spatial momentum p and of the mass m of the particle. Different models do give (more or less detailed) guidance on the form of ∆ QG , and we shall consider this below, but even at a model-independent level a few characteristics can be assumed with reasonable robustness . As most authors in the field, we shall here focus our analysis on cases in which the mass m still is the rest-energy and the dispersion relation regains its ordinary special-relativistic form in the limit where the Planck scale is removed (M P → ∞):

∆ QG (p, m, M P ) --→ p→0 0 , ∆ QG (p, m, M P ) ----→ M P →∞ 0 . (2)
And, since the relevant phenomenology clearly can at best hope to gain insight on the leading terms of a small-M -1 P expansion, it is natural to focus on a power-series expansion,

E 2 = p 2 + m 2 + 1 M P ∆ (1) QG (p, m) + 1 M 2 P ∆ (2)
QG (p, m) + . . . ,

where the terms in the power series are subject to the condition ∆

QG (p, m)

p=0 = 0 = ∆ (2) QG (p, m) p=0 .
This past decade of vigorous investigations of these modifications of the dispersion relation focused primarily (but not exclusively) on terms linear in M -1 P and reached its most noteworthy results in analyses of observational astrophysics data, which of course concern the ultrarelativistic (p m) regime of particle kinematics [START_REF] Amelino-Camelia | astro-ph/9712103[END_REF]3,4,[START_REF] Kifune | [END_REF][START_REF] Amelino-Camelia | gr-qc/0012049[END_REF]7]. For these applications the function ∆

(1) QG (p, m) can of course be usefully parametrized in such a way that the relation between energy and spatial momentum takes the following form:

E p + m 2 2p + 1 2M P η 1 p 2 + η 2 m p + η 3 m 2 , (4)
We should stress however that, while the perspective schematized in our Eqs. ( 2)-( 3) is by far the most studied in the relevant quantum-gravity-inspired literature, in principle more general possibilities may well deserve investigation. For example, one might contemplate non-integer powers of M P to appear, and this would not be too surprising, especially in light of the rather common expectation that the correct description of quantum gravity might require sizable nonlocality.

where, considering the large value of M P , we only included correction terms that are linear in 1/M P , and, considering that this formula concerns the ultrarelativistic regime of p m, the labels on the parameters η 1 , η 2 , η 3 reflect the fact that in that regime p 2 /M P is the leading correction, mp/M P is next-to-leading, and so on.

Evidence that at least some of these η 1 , η 2 , η 3 parameters have nonzero values is indeed found in studies inspired by the Loop-Quantum-Gravity approach and by the approach based on spacetime noncommutativity, and most importantly some of these studies [6,12,[START_REF] Amelino-Camelia | hep-th/9907110[END_REF][START_REF] Kowalski-Glikman | astro-ph/0006250[END_REF]15] provide encouragement for the presence of the strongest imaginable ultrarelativistic correction, the leading-order term η 1 p 2 /(2M P ).

Unfortunately, as usual in quantum-gravity research, even the most optimistic estimates represent a gigantic challenge from the perspective of phenomenology. This is because, if the Planck scale is indeed roughly the characteristic scale of quantumgravity effects then correspondingly parameters such as η 1 , η 2 , η 3 should take (positive or negative) values that are within no more than 1 or 2 orders of magnitude of 1. And this in turn implies that, for example, all effects induced by Eq. ( 4) could only affect the running of our present particle-physics colliders at the level [START_REF] Amelino-Camelia | Quantum Gravity Phenomenology[END_REF] of at best 1 part in 10 14 . In recent years certain semi-heuristic renormalization-group arguments (see, e.g., Refs. [START_REF] Amelino-Camelia | Quantum Gravity Phenomenology[END_REF]19] and references therein), have encouraged the intuition that the quantumgravity scale might be plausibly even 3 orders of magnitude smaller than the Planck scale (so that it could coincide [19] with the "grand unification scale" which appears to play a role in particle physics). But even assuming for η 1 , η 2 , η 3 values plausibly as "high" as 10 3 is not enough help at traditional high-energy particle-collider experiments.

It was therefore rather exciting for many quantum-gravity researchers when it started to emerge that some observations in astrophysics could be sensitive to manifestations of the parameter η 1 all the way down to |η 1 | ∼ 1 and even below [START_REF] Amelino-Camelia | astro-ph/9712103[END_REF]3,4,[START_REF] Kifune | [END_REF][START_REF] Amelino-Camelia | gr-qc/0012049[END_REF]7], thereby providing for that parameter the ability to explore the full range of values that could be motivated from a quantum-gravity perspective. These studies are presently being conducted at the Fermi Space Telescope [20,21,22,23,24], and other astrophysics observatories.

In the recent Ref. [11] some of us observed that it would be very valuable to combine to these astrophysics studies of the ultrarelativistic regime of the dispersion relation also a complementary phenomenology program of investigation of the nonrelativistic regime of p m (which of course is not accessible to massless particles). When p m the 3 largest contributions to ∆

E m + p 2 2m + 1 2M P ξ 1 mp + ξ 2 p 2 + ξ 3 p 3 m , (5)
¶ Note that a contribution of form m 3 (i.e. momentum-independent) to ∆

QG (p, m) cannot be included in the nonrelativistic regime because of the requirement ∆ QG (p = 0, m) = 0 of course is not relevant), but we ignored it since m 3 is too small with respect to p 3 , mp 2 and m 2 p in the nonrelativistic regime.

where, again, ξ 1 , ξ 2 , ξ 3 are dimensionless parameters.

Evidence that at least some of these dimensionless parameters ξ 1 , ξ 2 , ξ 3 should be non-zero has been found for example in the much-studied framework introduced in Refs. [6,25], which was inspired by Loop Quantum Gravity, and produces a term linear in p in the nonrelativistic limit (the effect here parametrized by ξ 1 ). And for the purposes of this Section, which we are devoting to the complementarity of the nonrelativistic and ultrarelativistic regimes of the dispersion relation, it is particularly insightful to consider two of the most studied scenarios that have emerged in the literature on noncommutative-geometry-inspired deformations of Poincaré symmetries. These are the scenarios proposed in Refs. [START_REF] Amelino-Camelia | gr-qc/0012051[END_REF][START_REF] Amelino-Camelia | hep-th/0012238[END_REF] and in Ref. [28], which respectively produce the following proposals for the exact form of the dispersion relation:

2M P η 2 sinh 2 ηE 2M P = 2M P η 2 sinh 2 ηm 2M P + e -η E M P p 2 , (6)
and

m 2 (1 -η m M P ) 2 = E 2 -p 2 (1 -η E M P ) 2 , (7)
Both of these proposals have the same description in the nonrelativistic regime

E m + p 2 2m -η p 2 2M P , (8)
i.e. the type of correction term in the nonrelativistic regime that we are here parameterizing with ξ 2 . But these proposals have significantly different behavior in the ultrarelativistic regime. From Eq. ( 6) in the ultrarelativistic regime one finds

E p + m 2 2p -η p 2 2M P , (9)
whereas from Eq. (7) in the ultrarelativistic regime one finds

E p + m 2 2p -η m 2 M P . ( 10
)
Therefore the example of these two much studied deformed-symmetry proposals is such that by focusing exclusively on the nonrelativistic regime one could not (not at the leading order at least) distinguish between them, but one could discriminate between the two proposals using data on the ultrarelativistic regime. The opposite is of course also possible: different candidate dispersion relations with the same ultrarelativistic limit, but with different leading-order form in the nonrelativistic regime. And in general it would be clearly very valuable to constrain the form of the dispersion relation both using experimental information on the leading nonrelativistic behavior and experimental information on the leading ultrarelativistic behavior.

Probing the nonrelativistic regime with cold atoms

Our main objective here is to show that cold-atom experiments can be valuable for the study of Planck-scale effects. We illustrate this point mainly by considering the possibility, already preliminarily characterized in Ref. [11], to use cold-atom studies for the derivation of meaningful bounds on the parameters ξ 1 and ξ 2 , i.e. the leading and next-to-leading terms in (5) for the nonrelativistic limit:

E m + p 2 2m + 1 2M P ξ 1 mp + ξ 2 p 2 . ( 11
)
In this Section we work exclusively from a laboratory-frame perspective, as done in Ref. [11], but, as for most relativistic studies, it is valuable to also perform the analysis in one or more frames that are boosted with respect to the laboratory frame, and we shall discuss this in Sec. 5.

The measurement strategy proposed in Ref. [11] is applicable to measurements of the "recoil frequency" of atoms with experimental setups involving one or more "twophoton Raman transitions" [29,30,31]. Let us initially set aside the possibility of Planck-scale effects, and discuss the recoil of an atom in a two-photon Raman transition from the perspective adopted in Ref. [31], which provides a convenient starting point for the Planck-scale generalization we shall discuss later. One can impart momentum to an atom through a process involving absorption of a photon of frequency ν and (stimulated [29,30,31]) emission, in the opposite direction, of a photon of frequency ν . The frequency ν is computed taking into account a resonance frequency ν * of the atom and the momentum the atom acquires, recoiling upon absorption of the photon: ν ν * + (hν * + p) 2 /(2m) -p 2 /(2m), where m is the mass of the atom (e.g. m Cs 124 GeV for Caesium), and p its initial momentum. The emission of the photon of frequency ν must be such to de-excite + the atom and impart to it additional momentum: ν +(2hν * + p) 2 /(2m) ν * +(hν * + p) 2 /(2m). Through this analysis one establishes that by measuring ∆ν ≡ ν -ν , in cases (not uncommon) where ν * and p can be accurately determined, one actually measures h/m for the atoms:

∆ν 2ν * (ν * + p/h) = h m . ( 12
)
This result has been confirmed experimentally with remarkable accuracy. A powerful way to illustrate this success is provided by comparing the results for atom-recoil measurements of ∆ν/[ν * (ν * + p/h)] and for measurements [32] of α 2 , the square of the fine structure constant. α 2 can be expressed in terms of the mass m of any given particle [31] through the Rydberg constant, R ∞ , and the mass of the electron, m e , in + We only give a schematic and simplified account of the process, which suffices for the scopes of our analysis. A more careful description requires taking into account that, rather than a single ground state, the relevant two-photon Raman transition involve hyperfine-splitted ground states [29,30,31]. And that, rather than tuning the two lasers exactly on some energy differences between levels, some detuning is needed [29,30,31].

the following way [31]:

α 2 = 2R ∞ m me h m .
Therefore according to Eq. ( 12) one should have

∆ν 2ν * (ν * + p/h) = α 2 2R ∞ m e m u m u m , (13)
where m u is the atomic mass unit and m is the mass of the atoms used in measuring ∆ν/[ν * (ν * + p/h)]. The outcomes of atom-recoil measurements, such as the ones with Caesium reported in Ref. [31], are consistent with Eq. ( 13) with the accuracy of a few parts in 10 9 .

The fact that Eq. ( 12) has been verified to such a high degree of accuracy proves to be very valuable for our purposes as we find that modifications of the dispersion relation require a modification of Eq. ( 12). Our derivation can be summarized briefly by observing that the logical steps described above for the derivation of Eq. ( 12) establish the following relationship

h∆ν E(p + hν + hν ) -E(p) E(2hν * + p) -E(p) , (14)
and therefore Planck-scale modifications of the dispersion relation, parametrized in Eq. ( 5), would affect ∆ν through the modification of E(2hν * +p)-E(p), which compares the energy of the atom when it carries momentum p and when it carries momentum p + 2hν * . Since our main objective here is to expose sensitivity to a meaningful range of values of the parameter ξ 1 , let us focus on the Planck-scale corrections with coefficient ξ 1 . In this case the relation ( 12) is replaced by

∆ν 2ν * (hν * + p) m + ξ 1 m M P ν * , (15)
and in turn in place of Eq. ( 13) one has

∆ν 2ν * (ν * +p/h) 1-ξ 1 m 2M P m hν * + p = α 2 2R ∞ m e m u m u m . (16)
We have arranged the left-hand side of this equation placing emphasis on the fact that our quantum-gravity correction is as usual penalized by the inevitable Planck-scale suppression (the ultrasmall factor m/M P ), but in this specific context it also receives a sizable boost by the large hierarchy of energy scales m/(hν * + p), which in typical experiments of the type here of interest can be [29,30,31] of order ∼ 10 9 . Our result [START_REF] Smolin | Three roads to quantum gravity[END_REF] for the case of modification of the dispersion relation by the term with coefficient ξ 1 can be straightforwardly generalized to the case of a modified dispersion relation of the form

E m + p 2 2m + ξ β 2 m 2-β M P p β (17)
which reproduces our terms with parameters ξ 1 and ξ 2 respectively when β = 1 and β = 2 (but in principle could be examined even for non-integer values of β).

One then finds

∆ν 2ν * (ν * + p/h) 1 -ξ β m 2-β (p + 2hν * ) β -p β 4M P hν * m hν * + p = α 2 2R ∞ m e m u m u m , (18)
which indeed reproduces ( 16) for β = 1 and gives [11]

∆ν 2ν * (ν * +p/h) 1-ξ 2 m M P = α 2 2R ∞ m e m u m u m (19)
for β = 2.

We have so far assumed that the only Planck-scale corrections to the analysis come from parameters such as ξ 1 and ξ 2 , characteristic of the nonrelativistic regime, for particles of nonzero mass. In the experimental setups we are considering all particles are indeed nonrelativistic with the exception of course of the photons involved. Clearly massless particles are inevitably ultrarelativistic and actually (at leading order in 1/M P ) there is a single possible modification of the dispersion relation for massless particles, the one with coefficient η 1 and quadratic dependence on momentum (see Eq. ( 4)). Of course, in a given quantum-gravity scenario one might have, say, ξ 1 = 0 and η 1 = 0, in which case the derivations we gave above would immediately apply. It is natural to also contemplate the possibility of cases in which both ξ 1 and η 1 are roughly of order 1. We find however that for the analysis of atom-recoil studies the effects produced by an η 1 of order 1 are completely negligible with respect to the effects produced by a ξ 1 of order 1. This is essentially due to the fact that photons enter the derivation of the recoil frequency through momentum transfers that never have a chance to pick up the "amplification" coming from the only large energy scale in the problem which is the atom mass. The amplification of the effects of ξ 1 which was underlined in the comments we offered just after Eq. ( 16) is not found for the effects of η 1 . Indeed by redoing all the steps of our derivation allowing for a nonzero η 1 one ends up replacing Eq. [START_REF] Kowalski-Glikman | astro-ph/0006250[END_REF] with

h∆ν E(2hν * + p -η 1 h 2 ν 2 * /M P ) -E(p) . (20)
And from this one arrives at a rather intelligible characterization of the different roles of ξ 1 and η 1 in atom-recoil analysis:

∆ν 2ν * (ν * +p/h) = h m + ξ 1 hm 2M P (p + hν * ) -η 1 hm 2M P (p + hν * ) hν * p + 2h 2 ν 2 * m 2 , (21)
which shows that the effects of η 1 are suppressed with respect to the ones of ξ 1 by a factor of order (hν * /m) 2 or hν * p/m 2 (notice the two powers of the mass in the denominator, and that the mass of the atoms in the setup here of interest is much larger than both p and ν * ). The balance of strengths changes a bit, but not enough, in the case of scenarios with ξ 1 = 0 but both ξ 2 and η 1 of order 1. In such cases one should compare the effects of ξ 2 (which we established to be smaller than the ones of ξ 1 ) to the effects of η 1 . What one finds is summarized by the formula

∆ν 2ν * (ν * +p/h) = h m + ξ 2 h M P -η 1 h M P hν * 2m 1 + hν * p + hν * , (22)
which shows that the effects of η 1 are significantly suppressed with respect to the ones of ξ 2 (here it is only one power of the mass in the denominator, but that is plenty of suppression, considering the large hierarchy between mass of the atoms and spatial momenta available in atom-recoil studies).

Limits on different models

From a phenomenological perspective the most noteworthy observation one can ground on the results reported in the previous Section is that the accuracies achievable in coldatom studies allow us to probe values of ξ 1 that are not distant from |ξ 1 | ∼ 1. This is rather meaningful since, as stressed in the previous Section, the quantum-gravity intuition for parameters such as ξ 1 is that they should be (in models where a nonzero value for them is allowed) within a few orders of magnitude of 1. Besides discussing this point, in this Section we also consider the case of the term with ξ 2 parameter and we comment on the relevance of these analyses from the perspective of a class of phenomenological proposals which is broader than the one here discussed in Section II.

The closing remarks of this Section are devoted to observations that may be relevant for attempts to further improve the relevant experimental limits.

Limits on ξ 1 and ξ 2

The fact that our analysis provides sensitivity to values of ξ 1 of order 1 is easily verified by examining our result for the case of the ξ 1 parameter, which we rewrite here for convenience ∆ν 2ν

* (ν * +p/h) 1-ξ 1 m 2M P m hν * + p = α 2 2R ∞ m e m u m u m , (23)
and taking into account some known experimental accuracies. Let us focus in particular on the Caesium-atom recoil measurements reported in Ref. [31], which were ideally structured for our purposes. Let us first notice that R ∞ , m e /m u and m u /m Cs are all known experimentally with accuracies of better than 1 part in 10 9 . When this is exploited in combination with the value of α -1 recently determined from electron-anomaly measurements [32], which is α -1 = 137.035999084 [START_REF] Heyman | gr-qc/0312089[END_REF], the results of Ref. [31,33] then allow us to use (23) to determine that ξ 1 = -1.8 ± 2.1. This amounts to the bound -6.0 < ξ 1 < 2.4, established at the 95% confidence level, and shows that indeed the cold-atom experiments we here considered can probe the form of the dispersion relation (at least in one of the directions of interest) with sensitivity that is meaningful from a Planck-scale perspective.

As mentioned in Section 2 among the models that could be here of interest there are some where, by construction, ξ 1 = 0 but ξ 2 = 0. In such cases it is then of interest to establish bounds on ξ 2 derived assuming ξ 1 = 0, for which one can easily adapt the derivation discussed above. These are therefore cases in which our result (19) is relevant, and one easily then finds that the atom-recoil results for Caesium atoms reported in Refs. [31,33] can be used to establish that -3.8 • 10 9 < ξ 2 < 1.5 • 10 9 . This bound is still some 6 orders of magnitude above even the most optimistic quantum-gravity estimates. But it is a bound that still carries some significance from the broader perspective of tests of Lorentz symmetry [11].

We should stress that, we relied on the results of Ref. [32], our noteworthy bounds on ξ 1 and ξ 2 could in principle be affected by the hypothesis of Planck-scale effects that happened to be relevant for the determination of α from electron-anomaly measurements. One could consider the possibility of a conspiracy between the "Planckscale-kinematics effects" that appear on the left-hand side of ( 23) and the "Planckscale gravity-interaction effects" that could be relevant for the determination of α from electron-anomaly measurements. At the present stage of understanding of the quantumgravity problem such a conspiracy appears implausible, since in the relevant models (see, e.g., Refs. [6,12]) anomalous behavior of gravitational interactions are only expected to start at order M -2 P . And we should also stress that electron-anomaly measurements are not the only way to determine accurately α (although they presently provide the most precise determination): one could for example use our strategy of analysis to obtain a bound weakened by not more than one order of magnitude without relying on electronanomaly measurements, but rather comparing the results of atom-recoil experiments with different types of atoms (e.g. Caesium and Rubidium).

Relevance for other quantum-gravity-inspired scenarios

Up to this point we have assumed "universal" effects, i.e. modifications of the dispersion relation that have the same form for all particles, independently of spin and compositeness, and with dependence on the mass of the particles rigidly inspired by the quantum-gravity arguments suggesting correction terms of the form m j p k /M P l (i.e.

with a characteristic dependence on momentum and with a momentum-independent coefficient written as a ratio of some power of the mass of the particle versus some power of the Planck-scale). While this universality is indeed assumed in the majority of studies of the fate of Poincaré symmetry at the Planck scale, alternatives have been considered by some authors [34] and there are good reasons to at least be open to the possibility of nonuniversality. One reason of concern toward universality originates from the fact that clearly modifications of the dispersion relation at the Planck scale are a small effect for microscopic particles (always with energies much below the Planck scale in our experiments), but would be a huge (and unobserved) effect for macroscopic bodies, such as planets and, say, soccer balls. Even the literature that assumes universality is well aware of this issue, and in fact the opening remarks of papers on this subject always specify a restriction to microscopic particles. With our present (so limited) understanding of the quantum-gravity realm we can indeed contemplate for example the possibility that such effects be confined to motions which admit description in terms of coherent quantum systems (by which we simply mean that the focus is on the type of particles whose quantum properties could also be studied in the relevant class of phenomena, unlike the motions of planets and soccer balls). This is clearly (at least at present) a plausible scenario that many authors are studying and for which atoms provide an extraordinary opportunity of investigation of the nonrelativistic regime. Let us compare for example our study to the popular studies of the ultrarelativistic regime with photons. The best limits on the ultrarelativistic side are obtained [23] through observations of photons with energies of a few tens of GeV's. The limit we here obtained in the nonrelativistic regime involves very small speeds ( c) but for particles, the atoms, with (rest) energies in the ∼ 100 GeV range.

While it is therefore rather clear that atoms are excellent probes of scenarios with universality for "quantum-mechanically microscopic particles", their effectiveness can be sharply reduced in models with some forms of nonuniversality. In particular, one could consider the compositeness of particles as a possible source of nonuniversality [START_REF] Amelino-Camelia | gr-qc/0210063[END_REF]. And this would imply that in the study of processes involving, say, protons and pions one should adopt a "parton picture" with the number of partons acting in the direction of averaging out the effects: if quantum-spacetime effects affect primarily the partons then a particle composed of 3 partons could feel the net result of 3 such fundamental features, with a possible suppression (e.g. by a factor of √

3) of the effect for the particle with respect to the fundamental effect for partons. These ideas have not gained much attention, probably also because things might change only at the level of factors of order 1 if one was for example to devise ways to keep track of the different number of partons for nucleons and for pions. But in the case of atoms, that we are now bringing to the forefront of quantum-gravity phenomenology, clearly these concerns cannot be taken lightly: for the description of an atom one might have to consider hundreds of partons (or at least ∼ 100 nucleons). We therefore expect that our strategy to place limits on ξ 1 and ξ 2 will be less effective (limits more distant from the Planck scale) in scenarios based on one or another form of "parton model" for the implications of spacetime quantization on quantum-mechanical particles. We do not dwell much on this here at the quantitative level since the literature does not offer us definite models of this sort that we could compare to data.

Even assuming that the effect is essentially universal one could consider alternatives to the most common assumption that quantum-gravity corrections have the form m j p k /M l P . In particular, some authors (see, e.g., Refs. [START_REF] Hinterleitner | gr-qc/0409087[END_REF]37,38]) have argued that the density of energy (or mass) of a given particle (be it elementary or composite) should govern the magnitude of the effect, rather than simply the mass of the particle. This is another possibility which is also under investigation [START_REF] Hinterleitner | gr-qc/0409087[END_REF]37,38] as a mechanism for effectively confining the new effects to elementary particles. In the simplest scenarios this proposal might amount to replacing terms such as our ξ 1 mp/(2M P ) with terms of the general form ξ1 ρ 1/4 p/(2M P ), but of course the implications of such pictures depend crucially on exactly which density ρ one adopts. For different choices of ρ the limits derived from atom-recoil experiments can be more or less stringent than those derived in studies of lighter particles, such as electrons.

Another framework which can be used to illustrate the different weight that coldatom studies can carry in different scenarios for the deformation of the dispersion relation is the one already studied in Refs. [39,40], parameterized by a single scale λ such that E 2 = m 2 + p 2 + 2λp. Limits on this form of the dispersion relation have been obtained for neutrinos in Ref. [39], and for electrons, in Ref. [40]. Taking into account that from E 2 = m 2 + p 2 + 2λp it follows that in the nonrelativistic limit E = m + p 2 /(2m) + λp/m, one easily finds that the parametrization we introduced in Eq. ( 5) and the parametrization of Refs. [39,40] are related by ξ 1 m/M P ≡ 2λ/m. And in light of this one can quickly estimate that the study of atom-recoil measurements can provide access to |λ| ∼ 10 -6 eV. This shows that the cold-atom-based strategy is suitable also for studies of the λ-parameter picture of Refs. [39,40]. But, while, as some of us already stressed in Ref. [11], these atom-based studies on λ are more powerful (by roughly 6 orders of magnitude) than bounds previously obtained on λ using neutrino data [39], we should here notice that the best present bound on λ is the electron-based bound derived in Ref. [40], which is at the level |λ| ≤ 10 -7 eV. We stress that there is no contradiction between the remarks we offered above on the unique opportunities that cold-atom studies provide for setting bounds on the parameter ξ 1 , and the fact that instead for the λ parameter electron studies are competitive with (and actually still slightly more powerful than) atom-based studies: this difference between the strategies for bounding the ξ 1 parameter and the λ parameter is easily understood in light of the relation ξ 1 m/M P ↔ 2λ/m and of the large difference of masses between electrons and (Caesium or Rubidium) atoms.

Finally, in closing this Subsection on alternative models, let us mention the possibility of intrinsically non-universal modifications of the dispersion relation, i.e. phenomenological scenarios in which the modifications of the dispersion relation are assumed to be different for different particles without introducing any specific prescription linking these differences to the mass, the spin or other specific properties of the particles. For example, in Ref. [34], and references therein, the authors introduce a free parameter for each different type of particle. In such cases studies of Caesium and, say, Rubidium atoms could be used to set constraints on parameters that are specialized to those types of atoms. In essence, according to this (certainly legitimate) perspective, we might learn that for Caesium and Rubidium ξ 1 is small but without assuming any implications for the values of ξ 1 for other particles. And another noteworthy example is the one of Ref. [41], and references therein, where it is argued, within a specific scenario for quantum gravity, that the effects of modification of the dispersion relation should be confined to a single type of particle, the photon (in which case of course atoms cannot possibly be of any help).

Strategies for improving the limits

As a contribution toward the development of experimental setups which in some cases may be optimized for our proposal it is important for us to stress that, while essentially here we structured our analysis in a way that might appear to invite interpretation as "quantum-gravity corrections to h/m measurements", not all improvements in the sensitivity of measurements of h/m will translate into improved bounds on the parameters we here considered.

First we should notice that our result for the ξ 1 -dependent correction to ∆ν/[2ν * (ν * +p/h)] would not appear as a constant shift of h/m, identically applicable to all experimental setups. This is primarily due to the fact that, as shown in Eq. ( 23), our quantum-gravity correction factor has the form 1 -ξ 1 m 2 /[2M P (hν * + p)], and therefore at the very least should be viewed as a momentum-dependent shift of h/m. Different h/m measurements, even when relying on the same atoms (same m), are predicted to find different levels of inconsistency with the uncorrected relationship between h/m and α 2 . This is particularly important because the remarkable accuracy of some measurements of h/m relies crucially [31,42] on imparting to the atoms high values of momentum, but from our perspective one should notice that the magnitude of the ξ 1 -governed effect decreases with the magnitude of momentum. This is after all one of the reasons why the bound on ξ 1 which we discussed here relied on the determinations of h/m reported in Ref. [31,33]: a more accurate determination of h/m was actually obtained in the cold-atom (Rubidium) studies reported in Refs. [43,44], but those more accurate determinations of h/m relied on much higher values of momentum, thereby producing a bound on ξ 1 which is not competitive [11] with the one obtainable using the h/m determination of Ref. [31,33]. The challenge we propose is therefore the one of reaching higher accuracies in the measurement of h/m without increasing significantly the momentum imparted to the atoms.

Interestingly these concerns do not apply to our result for the ξ 2 parameter. In fact, our result for the ξ 2 -dependent correction to ∆ν/[2ν * (ν * + p/h)] would actually appear as a constant shift of h/m, a mismatch between h/m results and α 2 results of identical magnitude in all experimental setups using the same atoms (same m). This is due to the fact that, as shown in Eq. ( 19), our quantum-gravity correction factor has the form [1 -ξ 2 m/M P ], and therefore can indeed be viewed as a (mass-dependent but) momentum-independent shift of h/m.

Besides these issues connected with the role played by the momentum of the atoms in our analysis, there are clearly other issues that should be taken into consideration by colleagues possibly contemplating measurements of h/m that could improve the limits on our parameters. One of these clearly deserves mention here, and concerns the setup of h/m measurements as differential measurements. In this respect it is rather significant that our derivation of dependence of the measured ∆ν on the Planck-scale effects shows that the sign of the correction term depends on the "histories" (beamsplitting/beam-recombination histories) of the atoms whose interference is eventually measured. Even from this perspective our result is therefore not to be viewed simply as "a shift in h/m": often in the relevant cold-atom experiments one achieves a very accurate determination of h/m by comparing (in the sense of a differential measurement) two different values of ∆ν obtained by interference of different pairs of beams produced in the beam-splitting/beam-recombination sequence of a given experimental setup. We should therefore warn our readers that for some differential measurements the effect measured would be twice as large as the one we here computed (same effect but with opposite sign on the two sides of the differential measurement), but on the other hand it is not hard to arrange * for a differential measurement that is insensitive to the quantumgravity effects (if the "histories" are such that the correction carries the same sign on the two sides of the differential measurement).

Atom velocity, energy-momentum conservation and other relativistic issues

We have so far focused on schemes which assume that the only new relevant quantumgravity-induced law amounts to a modification of the energy-momentum dispersion relation. The main results here derived in Section 3 relied on a strategy of analysis that only requires a specification in the "laboratory frame" of the form of the dispersion relation (which is used to establish, for example, the energy gained by an atom when its spatial momentum is increased) and the law of energy-momentum conservation (which is used to establish, for example, the spatial momentum imparted to an atom upon absorption of a photon of known wavelength). Even within that scheme of analysis one clearly should consider also the possibility of modifications of the law of energy-momentum conservation, especially in light of the fact that certain quantum-gravity scenarios establish (see below) a direct link between modifications of the dispersion relation and some corresponding modifications of the law of energymomentum conservation.

Moreover, the laboratory-frame perspective is of course too narrow for the investigation of the relativistic issues that clearly must be involved in scenarios that introduce modifications of the dispersion relation. Also from this perspective the quantum-gravity literature offers significant motivation for a careful investigation, since modifications of the laws of transformation between reference frames have been very actively studied (see below). And, as we shall here stress, connected to this issue of boost transformations between reference frames one also finds intriguing challenges for what concerns the description of the velocity of particles.

In this Section we offer an exploratory discussion of these issues. Even in the quantum-gravity literature on ultrarelativistic modifications of the dispersion relation the study of these issues has proven very challenging, and many unsolved puzzles remain. So we shall not even attempt here to address fully these issues in the novel domain of the nonrelativistic limit, which we are here advocating. But we hope that the observations we report here may provide a valuable starting point for more detailed future studies.

Among the "exploratory aspects" of our discussion we should in particular stress that we assume here, as done in most of the related quantum-gravity-inspired literature, that concepts such as energy, spatial momentum and velocity can still be discussed in standard way, so that the novelty of the pictures resides in new laws linking symbols that admit a conventional/traditional physical interpretation. Of course, alternative possibilities also deserve investigation: a given quantum-gravity/quantum-spacetime picture might well (when fully understood) provide motivation not only for novel forms of, say, the dispersion relation but also impose upon us a novel description of the entities, such as the energy E, that appear in the dispersion relation. But we shall already highlight several challenges for the more conservative scenario (with traditional "interpretation of symbols"), and therefore we postpone to future works the investigation of alternative interpretations.

Velocity and boosted-frame analysis

As a partial remedy to the laboratory-frame limitation of the strategy of analysis discussed in Section 3, we take as our next task the one of obtaining the same result using a scheme of derivation involving boosting and the Doppler effect. The role played by transformation laws between different observer-frames motivates part of our interest for this calculation, since investigations of the fate of Poincaré symmetry in models with Planck-scale modifications of the dispersion relation must in general address the issue of whether the symmetries are "deformed", in the sense of the "Doubly Special Relativity" scenario [START_REF] Amelino-Camelia | gr-qc/0012051[END_REF][START_REF] Amelino-Camelia | hep-th/0012238[END_REF], or simply "broken". When the symmetry transformations are correspondingly "deformed" the dispersion relation will be exactly the same for all observers [START_REF] Amelino-Camelia | gr-qc/0012051[END_REF][START_REF] Amelino-Camelia | hep-th/0012238[END_REF]. In the symmetry-breaking alternative scenario the laws of boosting are unmodified and as a result one typically finds that the chosen form of the dispersion relation only holds for one class of observers (at the very least one must expect [START_REF] Jacob | [END_REF] observer dependence of the parameters that characterize the modification of the dispersion relation). And another aspect of interest for such analyses originates from the fact that the description of the Doppler effect requires a corresponding description of the velocity of the atoms, and therefore requires a specification of the law that fixes the dependence of speeds on momentum/energy at the Planck scale: this too is a debated issue, with many authors favoring v(p) = ∂E/∂p, but some support in the literature also for some alternatives, the most popular of which is v = p/E.

As stressed in the opening remarks of this Section, we are just aiming for a first exploratory characterization of these issues and their possible relevance for our atom-recoil studies. Consistently with these scopes we assume that the Doppler effect (boosting) is undeformed and that the dispersion relation is an invariant law. This of course is only one (and a particularly peculiar) example of combination of the possible formulations of the main issues here at stake, but it suffices for exposing the potentially strong implications that the choice of these formulations can have for the analysis.

Let us start by reanalyzing the recoil of atoms in terms of a Doppler effect, neglecting initially the possible Planck-scale effects (which we shall reintroduce later in this Section). When an atom absorbs a photon whose frequency is ν in the laboratory frame, in the rest frame of the atom the photon has frequency ν = ν(1 -v), where v is the speed of the atom in the lab frame (and for definiteness we are considering the case of photon velocity parallel to the atom velocity). Then in the rest frame, if the absorption of the photon takes the atom to an energy level hν * , energy conservation takes the form

ν ν * + hν 2 * 2m , (24)
which of course can also be equivalently rewritten in terms of the lab-frame frequency of the photon

ν ν * (1 + v) + hν 2 * 2m , (25)
also neglecting a contribution of order v hν 2 * /m which is indeed negligible in the nonrelativistic (v 1) regime. This photon absorption also takes the atom from velocity v to velocity v ,

v v + hν * /m , (26)
in the laboratory frame (where we also observed that the gain of momentum of the atom is approximately hν * ).

For the stage of (stimulated) emission of a second photon, whose frequency in the lab frame we denote with ν , the atom would then be moving at this speed v , and in the rest frame of the atom the frequency of this emitted photon is ν = ν (1 + v ) (also taking into account that if, in the lab frame, the absorbed photon moved in parallel with the atom, the emitted photon must then move in anti-parallel direction). In the case of photon emission, conservation of energy in the rest frame has a different sign with respect to Eq. ( 24), i.e.

ν ν * - hν 2 * 2m , (27)
which again one may prefer to re-express in terms of the lab-frame frequency of the photon

ν ν * (1 -v ) - hν 2 * 2m . ( 28
)
So the lab-frame frequency difference between the two photons is

∆ν = ν * (v + v ) + hν 2 * m 2vν * + 2hν 2
It is easy to verify that redoing this Doppler-effect-based derivation in presence of our Planck-scale corrections to the dispersion relation (but setting aside, at least for now, possible Planck-scale dependence of the Doppler effect) one ends up replacing (29) with

∆ν = ν * [v(p) + v(p + hν * )] + hν 2 * m + ξ 1 m M P ν * . (30)
This is the formula that should reproduce our main result (15). Indeed this is the point where one might encounter the necessity of Planck-scale modifications of the boost/Doppler-effect laws and/or of Planck-scale modifications of the law that fixes the dependence of speeds on momentum/energy. Concerning speeds, if one assumes (as done by most authors [START_REF] Amelino-Camelia | astro-ph/9712103[END_REF]4,6,9,12]) v = ∂E/∂p then in our context (nonrelativistic regime, with ξ 1 parameter) one finds v(p) = p/m + ξ 1 m/M P . If instead, as argued by other authors [START_REF] Kosinski | [END_REF]48,49], consistency of the Planck-scale laws requires that v = p/E should be enforced then in our nonrelativistic context one of course has v(p) = p/m. We find that the desirable agreement between ( 30) and ( 15) is found upon assuming v(p) = p/m, which indeed allows one to rewrite (30)

as ∆ν = 2ν * (p + hν * ) m + ξ 1 m M P ν * . (31)
If instead one insists on the alternative v(p) = ∂E/∂p = p/m + ξ 1 m/M P , then (30) takes the form

∆ν = 2ν * (p + hν * ) m + 2ξ 1 m M P ν * , (32)
which is sizably different from (15). Our observation that the law v = p/m is automatically consistent with a plausible symmetry-deformation perspective is intriguing, but might well be just a quantitative accident. We thought it might still be worth reporting just as a way to illustrate the complexity of the issues that come into play if our cold-atom studies are examined within a symmetry-deformation scenario, issues that we postpone to future studies. The Doppler effect in models with deformed Poincaré symmetries had not been previously studied, and there are several alternative "schools" on how to derive from the energymomentum dispersion relation a law giving the speed as a function of energy. In the specific case of the correction term we here parametrized with ξ 1 it would seem that v = p/m is a natural choice, at least in as much as the choice v(p) = ∂E/∂p appears to be rather pathological/paradoxical since it leads to v(p) = p/m + ξ 1 m, i.e. a law that assigns nonzero speed to the particle even when the spatial momentum vanishes.

Testing energy-momentum conservation

Up to this point our analysis has focused on tests of the Lorentz sector of Poincaré symmetry. But of course there is also interest in testing the translation sector, and indeed there has been a corresponding effort, particularly over the last decade. The aspect of the translation sector on which these studies have primarily focused is the law of energy-momentum conservation in particle-physics processes, and particularly noteworthy are some results [START_REF] Amelino-Camelia | astro-ph/0008107[END_REF][START_REF] Heyman | gr-qc/0312089[END_REF] which exposed "Planck-scale sensitivity" for the analysis of certain classes of "ultraviolet" (high-energy) modifications of the law of energy-momentum conservation. Even for these studies one can contemplate the alternative between breaking and deforming Poincaré symmetry, and from this perspective it is rather noteworthy that the scenarios in which one deforms Poincaré symmetry require [START_REF] Amelino-Camelia | gr-qc/0012051[END_REF][START_REF] Amelino-Camelia | gr-qc/0210063[END_REF] a consistency between the scheme of modification of the dispersion relation and the scheme of modification of the law of energy-momentum conservation. Instead of course if one is willing to break Poincaré symmetry one can consider independently (or in combination) both modifications of the dispersion relation and modifications of the law of energy-momentum conservation.

In this Section we want to point out that our cold-atom-based strategy also provides opportunities for studies of the form of the law of energy-momentum conservation in the nonrelativistic regime. The observations on cold-atom experiments that some of us reported in Ref. [11] already inspired the recent analysis of Ref. [52], which provides preliminary encouragement for the idea of using cold-atom experiments for the study of the form of the law of energy-momentum conservation in the nonrelativistic regime. The scopes of the analysis reported in Ref. [52] were rather limited, since it focused on one specific model, which in particular codifies no modifications of the dispersion relation: the only modification allowed in Ref. [52] appeared in the law of energy-momentum conservation and appeared only at subleading order (in the sense here introduced in Sections 2-3) in the nonrelativistic limit.

While maintaining the perspective of a first exploratory investigation of these issues, we shall here contemplate a more general scenario, with modifications of both energy-momentum conservation and dispersion relation, and with correction terms strong enough to appear even at the leading order in the nonrelativistic regime. Besides aiming for greater generality, our interest in this direction is also motivated by the desire of setting up future analysis which might consider in detail the interplay between modifications of the dispersion relation and modifications of energy-momentum conservation, particularly from the perspective of identifying scenarios with deformation (rather than breakdown) of Poincaré symmetries, for which, as mentioned, this interplay is in many instances required [START_REF] Amelino-Camelia | gr-qc/0012051[END_REF][START_REF] Amelino-Camelia | gr-qc/0210063[END_REF]. While we shall not here attempt to formulate a suitable deformed-symmetry scenario, the observations we here report are likely to be relevant for the possible future search of such a formulation.

In light of the exploratory nature of our investigation of this point we shall be satisfied illustrating the possible relevance of the interplay between dispersion relation and energy-momentum conservation for the specific case of modified laws of conservation These consistency requirements for a deformation of Poincaré symmetry are very restrictive but may not suffice to fully specify the form of the law of energy-momentum conservation by insisting on compatibility with a chosen form of the dispersion relation [START_REF] Amelino-Camelia | gr-qc/0012051[END_REF][START_REF] Amelino-Camelia | gr-qc/0210063[END_REF].

of spatial momentum (ordinary conservation of energy):

p 1 + p 2 -
We are focusing on the case of two incoming and two outgoing particles (relevant for processes in which a photon is absorbed and one is emitted by an atom), and we characterized the modification in terms of parameters ρ 1 and ρ 2 . As announced, we shall keep track of these parameters ρ 1 and ρ 2 together with the parameters ξ 1 and ξ 2 that parametrized the modifications of the dispersion relation in the nonrelativsitic limit † †. For a two-photon Raman transition our modified law of conservation of spatial momentum has significant implications along the common direction of the laser beams used to excite/de-excite the atoms: (35) While this is, as stressed, only an exploratory investigation of the role that could be played by modifications of energy-momentum conservation (in particular there is clearly a strong influence of the specific ansatz we adopted for the modified law of conservation of energy-momentum) it is still noteworthy that the parameter ρ 1 enters the final result at the same order as the parameter ξ 1 and similarly the parameter ρ 2 enters the final result at the same order as the parameter ξ 2 . In particular, this implies that even at the type of leading order we here mainly focused on (the order where ξ 1 appears) the possibility of modifications of the law of energy-momentum conservation may well be relevant, with nonnegligible effects even in cases where ξ 1 = 0 but ρ 1 = 0.

h| k| + | p|

Closing remarks

We have here used the noteworthy example of atom-recoil measurements to explore whether it is possible to setup a phenomenology for the nonrelativistic limit of the † † For simplicity we here simply assume that the photon dispersion relation (bound to be in the ultrarelativistic regime) is undeformed. As stressed at the end of Sec. 3, one could introduce for photons the parameter η 1 , characteristic of the ultrarelativistic regime, but in the analysis of atomrecoil studies η 1 turns out to have effects much smaller than the ones of ξ 1 and ξ 2 (as already discussed at the end of Sec. 3) and also of ρ 1 and ρ 2 .

energy-momentum dispersion relation that adopts the same spirit of a popular research program focusing instead on the corresponding ultrarelativistic regime. It appears that this is indeed possible and that on the one hand there is a strong complementarity of insight to be gained by combining studies of the nonrelativistic regime and of the ultrarelativistic regime, and on the other hand the nature of the conceptual issues that must be handled (particularly the relativistic issues associated with the possibility of breaking or deforming Poincaré symmetry) are closely analogous. We therefore argue that by adding the nonrelativistic limit to the relevant phenomenology agenda we could improve our ability to constrain certain scenarios, and we could also gain a powerful tool from the conceptual side, exploiting the possibility to view the same conceptual challenges within regimes that are otherwise very different.

For what concerns the phenomenology we here proposed it is noteworthy that, particularly considering the values of ξ 1 being probed, any improvement in sensitivity that will be achieved could also be viewed as a (slim but valuable) chance for a striking discovery. We therefore feel that our analysis should motivate experimentalists to tailor some of their plans in this direction (also using the remarks we offered in Subsection 4.3) and should motivate theorists toward a vigorous effort aimed at overcoming the technical difficulties on the quantum-gravity-theory side that presently obstruct the derivation of more detailed quantitative predictions.

( 1 )

 1 QG (p, m) have behavior ¶ m 2 p, mp 2 and p 3 , allowing to cast the relation between energy and spatial momentum in the following form:

( 1 )

 1 QG (p = 0, m) = 0. A contribution to ∆(1) QG (p, m) of form m 3 is instead admissible in the ultrarelativistic regime (since in that regime the requirement ∆[START_REF] Amelino-Camelia | Quantum Gravity Phenomenology[END_REF]

  (E 3 p 4 + E 4 p 3 ) .

	ρ 1 4M P	E 2 1 E 2	p 1 +	E 2 2 E 1	p 2 -	ρ 2 2M P	(E 1 p 2 + E 2 p 1 ) =
	= p 3 + p 4 -	ρ 1 4M P	E 2 3 E 4	p 3 +	E 2 4 E 3	p 4 -	ρ 2 2M P

  In Section 3 we used ordinary momentum conservation, h| k|+ | p| = -h| k | + | p |, but if instead one adopts(34) the following result is then straightforwardly obtained:

	-	ρ 1 4M P	h 2 ν 2 m	h| k| +	m 2 hν	| p| -	ρ 2 2M P	(hν| p| + Eh| k|) =	(34)
	-h| k | + | p | -2M ∆ν ρ 1 4M P -h 2 ν 2 m h| k | + m 2 hν ρ 2 | p | -2ν * (ν * + p/h) h m 1 + M

P (hν | p | -E h| k |) , P [m(ξ 1 -ρ 1 ) + (2ξ 2 -ρ 2 )p + 2(ξ 2 -ρ 2 )hν * ]

hν * 2ν * (hν * + p) .

Acknowledgments

We gratefully acknowledge valuable conversations with H. Müller and L. Urrutia. G. A.-C. is supported in part by grant RFP2-08-02 from The Foundational Questions Institute (fqxi.org). C. L. is supported in part by the German Research Foundation and the Centre for Quantum Engineering and Space-Time Research QUEST. D. M., J. M. C., J. L. C. and J. I. are supported by CICYT (grants FPA2006-02315 and FPA2009-09638) and DGIID-DGA (grant 2009-E24/2). J. I. acknowledges a FPU grant and D. M. a FPI grant from MICINN.