Theory of the electron structure.

Résumé

A theory of the electron which is based on the covariant derivative of the potential is developed. It is shown that the particle is represented by a tensor field which groups all the measured quantities mass, charge, spin and magnetic momentum of the electron. The theory gives the equations which relate these global quantities to the local elements of the tensors.

octobre 2019

1 Introduction.

Up to now the electron is represented as a point-like particle of matter and an unit of unbreakable electrical charge associated with a wave. These paradigms are very powerful to describe atomic, molecular and solid state phenomena. However they become useless to study the distribution of mass or electric charge inside an electron. The theory presented here proposes a new solution to the electron structure together with its associated characteristics. The long quest to understand the electron structure began at the end of the XIXth century [START_REF] Lorentz | La theorie electromagnetique de Maxwell et son application aux corps mouvants[END_REF]. In his book published in 1909 [START_REF] Lorentz | The theory of electrons[END_REF] Hendrik A. Lorentz "ascribes to each electron certain finite dimensions". However the history[3] of this particle shows that the different models ( [START_REF] Jauch | The theory of photons and electrons Springer[END_REF][START_REF] Weisskopf | Recent Developments in the Theory of the Electron[END_REF][START_REF] Heuson | A Structural Model of Quarks and Leptons[END_REF][START_REF] Sandhu | Fundamental Nature of Matter and Fields[END_REF]) to describe the inside of an electron were unsuccessful. The use of Maxwell's equations was the basis of the early works, but a turning point was the introduction by Max Born and Leopold Infeld [START_REF] Born | Proc. R. Soc[END_REF] of the idea of non-linear electromagnetism in the year 1934. Instead of starting from Maxwell's equations, they built a tensor [a ik ] which was the sum of the (antisymmetric) electromagnetic tensor [F ik ] and the (symmetric) metric tensor [g ik ]. Then they developed the properties of [a ik ] to describe the electron. However the elements F ik have the dimensions of a magnetic field while g ik has no dimension. Even if [F ik ] is properly normalized one can question the validity of adding it to [g ik ].

The new tensor [a ik ] should be the sum of [F ik ] and another symmetric tensor of a similar physical nature. But what is this tensor ? The classical electromagnetic potential [START_REF]The concept of electromagnetic potential has been used by Lorentz[END_REF][START_REF] Landau | Théorie des champs, Mir-Ellipses[END_REF][START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Griffiths | Introduction to Electrodynamics[END_REF] is a 4-components vector in Minkowski's spacetime and its 16 partial derivatives at an event M are the components of a tensor D(A i ). The antisymmetric part of D(A i ) is the widely used electromagnetic tensor [F ik ]. By contrast the symmetric part [S ik ] does not appear in classical textbooks nor in specialized literature. The answer to the preceding question is evidently [S ik ] and the tensor whose properties have to be analyzed is [a ik ] = D(A i ). We imagine yoctoscopic observers at M where the scale is small as compared to the size of the particle. Such observers use the potential as the base paradigm and D(A i ) as the fundamental object. Then they apply standard classical notions to develop their properties. In the course of the theory we will distinguish between local properties (accessible to our yoctoscopic observers) and global properties accessible to us. [S ik ] has two fundamental characteristics which do not appear in [F ik ] : [START_REF] Lorentz | La theorie electromagnetique de Maxwell et son application aux corps mouvants[END_REF] being symmetric, it can be diagonalized, and (2) the corresponding mixed tensor S i k has a trace which is invariant in a coordinate transformation. These two properties result [START_REF]Electromagnetic Particles[END_REF] in a Helmholtz equation whose solutions show concentrations of potential (without any divergence) around the origin. These solutions are characterized by three quantum (integer) numbers n, , m starting at n = = m = 0. When replacing the potential in D(A i ) by these solutions, one obtains a tensor at each event M in spacetime. The field of these tensors describes a particle (n, , m). As the determinant of S i k is also an invariant, we have associated it with a local Lagrangian density L in the elementary volume around M . This quantity can be integrated over the whole volume to give a global invariant which allows the calculation of the potential energy, or the total mass of the particle. The determinant of [a i k ] can also be used in Lagrange's equations which shows that the electric charge can be deduced from the potential. One finds that only the solution n = 1, = m = 0 can be globally electrically charged. We naturally named it "the electron". This particle is the most known among elementary particles. It can be isolated and trapped [START_REF] Ekstrom | The isolated electron[END_REF]. It is characterized by four measurable quantities which are its mass, its electric charge, its angular momentum (spin) and its magnetic momentum. Its Compton wavelength is a fifth characteristic quantity which appears when it diffuses Gamma rays. Up to now, these quantities seem to be independent of each other. The aim of the present study is essentially to show that they are all linked together and to express the relations between them and the potential. We first briefly describe the tensors, the way to obtain the basic Helmholtz equation and its solutions. We write the tensor field of the electron in the static (non rotating) case. The description of the spatial distribution of energy and electric density is given. Then the spinning particle is described as a vortex with a spatial distribution of angular frequency. The angular momentum and the magnetic moment are obtained. Finally the theoretical parameters are numerically computed. The main results of the following classical theory are the description of the structure of the "point-like particle", the explanation of the de Broglie's pilot wave, the agreement with Wheeler-Feynman's absorber theory and the numerical calculation of the electron characteristics.. 2 Generalities.

Notations.

The standard notation of the 4-potential vector is A i = (φ/c, A x , A y , A z ) in the cartesian coordinates frame of Minkowski's spacetime spanned by the normalized basis vectors ( e t , ẽx , e y , e z ). φ/c is the scalar potential and the set -→ A = (A x , A y , A z ) is the potential vector at an event M defined by its coordinates (x i = ct, x, y, z). The covector will be denoted by A = (A x , A y , A z ). We choose the convention (+, -, -, -) for the metric g mn . It is important to realize that the elementary volume surrounding M is small as compared to the electron volume : The particle will thus be characterized by a tensor field. The covariant derivative D(A i ) at point M is the particle tensor D(A i ) = [a ik ] = ∂A i /∂x k . D(A i ) is divided into its symmetric and antisymmetric parts [START_REF] Lovelock | Tensors, Differential forms and variationnal principles[END_REF] :

[s ik ] = 1 2 ([a ik ] + [a ki ]) , (1)
[f ik ] = 1 2 ([a ik ] -[a ki ]) . (2)
i and k represent respectively the line and the column indices in the square table representation. [f ik ] is the usual electromagnetic tensor [16]. [s ik ] can be named the matter (or mass) tensor. The particle tensor is the sum of the matter and the field tensors. They keep their symmetric or antisymmetric properties in a coordinate change. It is this association which explains the matter-field duality.

Helmholtz equation.

[s ik ] is a symmetric tensor : it can be diagonalized provided its determinant is not zero. It follows that an eigenframe exists where elements s 1k = s k1 = 0 with k = 2, 3, 4. Some details are given in the appendix. This is expressed by the equation :

-

-→ gradφ = ∂ A ∂t = - ∂ - → A ∂t . (3) Now the mixed tensors [a i k ], [s i k ], [f i k ] are introduced.
They are characterized by invariants in coordinates changes. Invariants are the coefficients of their characteristic polynomials. The most well known are the trace and the determinant. The trace [START_REF]Einstein's summation convention is adopted[END_REF] of [s i k ] remains the same in a time translation :

∂s i i c∂t = ∂ c∂t ∂(φ/c) c∂t + ∂A x ∂x + ∂A y ∂y + ∂A z ∂z = 0 . (4)
Associating eqs. (3) and ( 4) gives the Helmholtz equation :

∂ 2 φ/c c 2 ∂t 2 -φ/c = 0 , (5)
where represents the Laplacian. The permanent solutions oscillating at an angular freqency ω obey :

ω 2 c 2 φ/c + ∆φ/c = 0 , (6)
This equation is a tensor equation which remains the same in any system of coordinates and we will use it in the spherical reference frame attached to M . This frame is defined by its proper time t and the geometrical coordinates (r, θ, ϕ) such that :

x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ , (7)
The advantage of the (r, θ, ϕ) system is that it makes use of the spherical symmetry of the electron. Its disadvantage with respect to the cartesian system is that the formulation of D(A i ) is more complicated in the general case. Solutions of eq.( 6) have been studied in the context of the hydrogenoid atoms [START_REF] Demtröder | Atoms, molecules and photons[END_REF]. Coupled radial-angular solutions are found from the ansatz :

φ/c = R(r) Φ(ϕ) Θ(θ) . (8)
Decoupled solutions are obtained from the ansatz : φ/c = R(r) + Φ(ϕ) + Θ(θ).

In the present case, there is no need to do any approximation to write the radial function R(r). Now we introduce the normalized distance to the center : x= ωr/c, This distance will thus be measured in units of the reference length c/ω. The solutions are :

φ n, ,m (x, θ, φ) = A n, ,m J n (x) Y m (θ, ϕ) cos(ωt + α) . (9)
A n, ,m is the amplitude of the solution. We will choose α = 0 or -π/2 for reasons which will appear later.

J n (x) is the spherical Bessel function of order n and Y m (θ, ϕ) is the , m spherical harmonic.

2.3

The tensor field of the electron.

The solution corresponding to the electron is characterized by n = 1, = m = 0. The reason is that the electric charge density which is associated to a solution vanishes for all of them except for this special one( [START_REF]Electromagnetic Particles[END_REF]). This point is developed in section 4. The following developments will concern this case only. The scalar potential is expressed as an even or an odd function of ωt (resp. upper and lower line). One passes from the first to the second solution by removing π/2 from ωt :

φ 1,0,0 = A √ 4π J 1 cos ωt -sin ωt . ( 10
)
where J 1 is the spherical Bessel function of order 1. This solution is spherically symmetric. The potential vector has only one radial component given by eq.(3) :

A r = A c √ 4π J 1 sin ωt cos ωt (11)
where J 1 stands for the derivative of J 1 with respect to x. The tensor which will represent the electron at point M is explicitly written in its eigenframe for the even solution :

[a ik ] =     ∂(φ/c) c∂t ∂(φ/c) ∂r 0 0 -∂Ar c∂t -∂Ar ∂r 0 0 0 0 -Ar r 0 0 0 0 -Ar r     = 1 √ 4π Aω c 2     -sin ωt J 1 cos ωt J 1 0 0 -cos ωt J 1 -sin ωt J" 1 0 0 0 0 -sin ωt J 1 x 0 0 0 0 -sin ωt J 1 x     (12)
Terms A r /r = (Aω)/( √ 4πc 2 ) sin ωt J 1

x originate from Christoffel's coefficients which appear when shifting from the cartesian to the spherical system of coordinates [START_REF] Lovelock | Tensors, Differential forms and variationnal principles[END_REF]. The field tensor is :

[f ik ] = 1 √ 4π Aω c 2     0 cos ωt J 1 0 0 -cos ωt J 1 0 0 0 0 0 0 0 0 0 0 0     (13)
The matter tensor is :

[s ik ] = - 1 √ 4π Aω c 2 sin ωt     J 1 0 0 0 0 J" 1 0 0 0 0 J 1 x 0 0 0 0 J 1 x     (14)
These tensors are written in the frame spanned by normalized basis vectors where the metric tensor is also 1, -1, -1, -1 along the diagonal. It follows that the mixed field tensor is :

f i k = 1 √ 4π Aω c 2     0 cos ωt J 1 0 0 cos ωt J 1 0 0 0 0 0 0 0 0 0 0 0     =     0 -E 1 2 /c 0 0 -E 1 2 /c 0 0 0 0 0 0 0 0 0 0 0     (15) (16)
The three components of the electric field are associated to a pseudo vector

- → E . The radial component E 1 2 = E 2
1 is related to the potential by the relation :

E 1 2 c := 1 2 - ∂(φ/c) ∂r - ∂A r c∂t = - ∂(φ/c) ∂r = - ∂A r c∂t = - 1 √ 4π Aω c 2 cos ωt J 1 (17)
The factor 1/2 follows directly from eq.( 2). The standard relation [START_REF] Jackson | Classical Electrodynamics[END_REF] does not include this factor. Note that the magnetic field vanishes in the eigenframe of coordinates. The mixed matter tensor is :

s i k = 1 √ 4π Aω c 2 sin ωt ×     -J 1 0 0 0 0 J" 1 0 0 0 0 J 1 x 0 0 0 0 J 1 x     (18)
2.4 The far-field tensor.

The behavior of D(A i ) is very different in the regions close to or far from the origin of coordinates which is the center of the particle. Expressions of J 1 , J 1 and J" 1 are :

J 1 = - cos x x + sin x x 2 J 1 = sin x x + 2 cos x x 2 -2 sin x x 3 J" 1 = cos x x -3 sin x x 2 -6 cos x x 3 + 6 sin x x 4 (19)
At large distances from the origin, only terms in 1/x survive and J 1 , J 1 and J" 1 reduce to :

J 1 - cos x x J 1 sin x x J" 1 cos x x ( 20
)
The field becomes :

E 2 1 = - 1 √ 4π Aω c cos ωt sin x x = - 1 2 1 √ 4π Aω c 1 x × (sin(ωt + x) -sin(ωt -x)) (21)
The two remaining terms in s i k are :

s 1 1 = -s 2 2 = 1 √ 4π Aω c 2 sin ωt cos x x = 1 √ 4π Aω c 2 1 x (sin(ωt + x) + sin(ωt -x)) (22)
These expressions show travelling advanced waves sin(x + ωt) and retarded waves sin(x -ωt) which correspond exactly to those described in Wheeler-Feynman's absorber theory [START_REF] Wheeler | Interaction with the Absorber as the Mechanism of Radiation[END_REF]. We have thus adopted their interpretation which is based on causality : the outgoing waves are emitted by the particle and are absorbed by the surrounding medium which acts also as the emitter of waves which are absorbed by the particle. The equilibrium is obtained when both incoming and outgoing waves have the same energy. This is the condition for the stability of the particle and existence of permanent solutions. It follows that the amplitude A is fixed by this condition. The two terms s 1 1 and s 2 2 are interpreted as "matter waves" and should correspond to the de Broglie's pilot wave [START_REF] De | la mécanique ondulatoire et la structure atomique de la matière et du rayonnement[END_REF]. They intervene in self-interference effects in particle diffraction experiments. Their physical interpretation is different : -s 1 1 originates from the derivative of the scalar potential φ/c. The standing wave is scalar. -s 2 1 originates from the derivative of the radial potential A r . The standing wave is radially polarized. For future use, we write now the tensor( 12) at point M in the far field region (subscript " ff " stands for far field ) :

a i k ff =     ∂(φ/c) c∂t -E 2 1 /c 0 0 -E 2 1 /c ∂Ar ∂r 0 0 0 0 0 0 0 0 0 0     (23)
One is led to the conclusion that an electron (or more generally, any particle) is not localized : it manifests itself everywhere in the universe. It follows that one should make the distinction between a theoretical vacuum where the potential and its derivatives cancel everywhere and the real vacuum which is a superposition of electromagnetic and matter waves originating from the multitude of the particles of our universe. This state can be understood as the fundamental noise which is characterized by the quantities 0 , µ 0 and its impedance Z = 377Ω. In a region far from the center of any particle the contribution of one electron is very weak, while inside the electron, the noise can be neglected. We will come back on this subject later.

3 Potential energy.

In this section we will compute the local energy densities associated to a i k , s i k and f i k . The Lagrangian density δL at each point M is computed first : It is necessarily associated to an invariant of each of these tensors. The hamiltonian density can then be computed and its summation in spacetime will give the total potential energy which is one of the macroscopic measurable quantities which characterize the particle. Let us consider successively the three tensors.

Particle tensor.

Leaving aside the factor (Aω)/( √ 4πc 2 ), the characteristic polynomial of a i k is :

P (λ) = (sin ωt J 1 x + λ) 2 (λ + sin ωt J 1 ) (λ + sin ωt J" 1 ) -cos 2 ωt J 2 1 (24)
and the 4 invariants I k are :

I 0 = -cos 2 ωt sin 2 ωt J 2 1 J 2 1 x 2 + sin 4 ωt J 1 J" 1 J 2 1
x 2 (25)

I 1 = sin ωt J 1 + J" 1 + 2 J 1 x ( 26
)
I 2 = -cos 2 ωt J 2 1 + sin 2 ωt J 1 J" 1 + 2 sin 2 ωt J 1 J 1 x + 2 sin 2 ωt J" 1 J 1 x + sin 2 ωt J 2 1 x 2 (27)
I 3 = -2 cos 2 ωt sin ωt J 2 1 J 1 x + 2 sin 3 ωt J 1 J" 1 J 1 x + sin 3 ωt J 1 J 2 1 x 2 + sin 3 ωt J" 1 J 2 1 x 2 (28)
We associate [START_REF]Electromagnetic Particles[END_REF] the Lagrangian density [21] to the determinant δL a ∝ I 0 . We write :

δL a = -C 1 √ 4π Aω c 2 4 I 0 . ( 29
)
δL a has the dimensions of an energy density in the 4-volume (ct x y z). The proportionality constant C is a physical quantity which has the dimensions

[C] = M -3 L -2 T 2 Q 4 .
The minus sign is introduced to have a negative potential energy. The trace I 1 is the invariant which leads to the Lorentz gauge [START_REF]Electromagnetic Particles[END_REF]. I 2 and I 3 are not used here. One should note that any power I n 0 is also an invariant. We keep n = 1 which will give an energy proportional to the fourth power of the amplitude A. This property will explain the wide energy range of elementary particles [22]. The Hamiltonian is linked to the Lagrangian through a Legendre transform[23] :

δH = k ∂δL a ∂u k u k -δL a ( 30
)
where the u k 's are the independent terms which appear in δL a or in a i k . There are only four such terms :

u 1 = a 1 1 = 1 √ 4π Aω c 2 J 1 -sin ωt -cos ωt (31) u 2 = a 2 2 = 1 √ 4π Aω c 2 J" 1 sin ωt cos ωt (32)
u 3 = a 2 1 = a 1 2 = 1 √ 4π Aω c 2 J 1 cos ωt -sin ωt (33) u 4 = a 3 3 = a 4 4 = 1 √ 4π Aω c 2 J 1 x sin ωt cos ωt (34)
The upper and lower lines in these expressions correspond respectively to the even and the odd solutions.

The derivatives ∂δL a /∂u k which appear in eq.( 30) are the inductions corresponding to each u k .

If we use the expression for the determinant of [a i k ], one has :

δL a = -C (a 3 3 ) 2 a 1 1 a 2 2 -(a 1 2 )
2 , and eq.(30), we find :

δH = 3 δL a (35)
The Hamiltonian density is now easily obtained :

δH = 3 C 1 (4π) 2 A 4 ω 4 c 8    sin 2 ωt J 2 1 x 2 sin 2 ωt J 1 J" 1 + cos 2 ωt J 2 1 cos 2 ωt J 2 1 x 2 cos 2 ωt J 1 J" 1 + sin 2 ωt J 2 1 ( 36
)
δH is the energy included in the 4-volume element (d(ct) dv) where dv = r 2 dr sin θ dθ dϕ. The energy density in a geometrical volume dv is obtained by integrating δH over the length 2πc/ω on the time axis :

dH = 3 C (4π) 2 A 4 ω 4 c 8 π 4 c ω 3 J 1 J" 1 J 2 1 x 2 + J 2 1 J 2 1 x 2 (37)
The total energy associated to the electron is obtained from an integration over the whole geometrical volume :

H (total) = 2π 0 dϕ π 0 sin θ dθ ∞ 0 r 2 dr dH = 4π c 3 ω 3 ∞ 0 x 2 dH dx (38)
One obtains :

H (total) = C 3 16 A 4 c 4 × ∞ 0 dx J 2 1 3 J 1 J" 1 + J 2 1 = C 3 16 A 4 c 4 ∞ 0 dx d dx J 1 J 3 1 = 0. ( 39
)
The total energy of the particle is identically 0 : The integral vanishes because J 1(x=0) = 0 and the product J 1 J 3 1 ∝ 1/x 4 → 0 when x→ ∞. Its axial distribution includes a positive and a negative part which cancel each other after integration. This energy is the same for both even and odd solutions ; Fig. [START_REF] Lorentz | La theorie electromagnetique de Maxwell et son application aux corps mouvants[END_REF] represents the integrant d(J 1 J 3 1 )/dx as a function of x. The (essentially) negative part 3J 2 1 J 1 J" 1 will represent the mass energy while the positive part J 4 1 is linked to the field energy. We will keep in mind that δH in eq.(37) includes two terms : -The first term -C (a 3 3 ) 2 a 1 1 a 2 2 is an invariant of the mass tensor, it will represent the potential energy W p = m 0 c 2 that is associated to the mass.

-The second term C (a 3 3 ) 2 (a 1 2 ) 2 is the difference between two invariants. Is is also an invariant which will represent the potential energy of the field. These invariants will be used to study the spinning electron (Section 5) where mass and field momenta will appear.

Figure 1 -Radial distribution of the total electron rest energy vs. x, distance to the center (arbitrary units).

Matter tensor.

The determinant of the matter tensor (eq.( 18)) gives the potential energy density of the particle which is the first term in eq.(37) :

dW p = 9 C 64π Aω c 2 4 c ω J 1 J" 1 J 2 1 x 2 (40)
Applying Poincaré-Einstein relation dW p = dm 0 c 2 , we obtain the element of mass density dm 0 (in the geometrical volume) which we write explicitly below for future use :

dm 0 = 9 C 64π Aω c 2 4 1 ω c J 1 J" 1 J 2 1 x 2 (41)
Note that the potential energy density writes in the 4-volume :

4 δW p = 3 C (4π) 2 A 4 ω 4 c 8 sin 4 ωt J 2 1 x 2 J 1 J" 1 cos 4 ωt J 2 1 x 2 J 1 J" 1 (42)
Integrating dW p over the whole geometrical volume, one obtains the potential energy :

W p = C 3 16 A 4 c 4 ∞ 0 dx 3 J 2 1 J 1 J" 1 (43)
One notes that ω does not appear in this relation.

The electron rest mass m 0 is an invariant, function of the amplitude A, the constant C and the speed of light :

m 0 = C A 4 c 6 I mo . ( 44
)
where the integral is exactly obtained :

I mo = ∞ 0 J 1 J" 1 J 2 1 dx = 17 π 18480 . (45)
In the region close to the origin, x 1 and a series expansion gives :

J 1 = x 3 - x 3 30 + O(x 4 ) J 1 = 1 3 - x 2 10 + O(x 4 ) J" 1 = - x 5 + x 3 42 + O(x 4 ) ( 46
)
The product J 2 1 J 1 J" 1 converges at x=0 :

J 2 1 J 1 J" 1 = - x 2 135 + 86x 4 14175 + O(x 5 ) (47)
Fig.

(2) represents the distribution of the mass energy along x. The total potential energy which is stored in the particle is the surface under this curve. If we substract H (mass) from H (total) , we obtain the field energy which is stored in the particle. The function J 4 1 represents the radial distribution of this energy. It is illustrated in fig.

(3).

Field tensor.

The characteristic polynomial of f i k is :

P (λ) = λ 2 (λ 2 -(E 2 1 ) 2 /c 2 ) (48)
The field invariants are thus 0 and (E 2 1 ) 2 . In the general case where the magnetic field

- → H = 0, they are H 2 -E 2 = inv. and - → E - → H = inv.
The local energy density of the field is obtained from the product of the field E 2 1 and the induction After integrating over time, it leads to the field term in eq.(39).

D 1 2 = ∂L a /∂E 2 1 : dH = E 2 1 D 1 2 = 2 C 1 √ 4π Aω c 2 4 sin 2 ωt cos 2 ωt J 4 1 x 2 (49)
As the particle manifests itself at large distances by terms in 1/x, the potential, the field or the energy stored in an element dv around a point M are very small in this region as compared to the noncoherent sum of the corresponding quantities originating from the multitude of particles of the universe. One can thus characterize this region (the "vacuum") by a noise tensor whose components n i k are incoherent and with a modulus large as compared to those originating from a single test-particle. We write the local tensor as a sum of this noise tensor and the test-particle tensor(eq.((23)). The Lagrangian density is the determinant :

L ff = -C n i k + a i kff (50)
The local induction due to the field (E 2 1 ) is the derivative of L ff with respect to E 2 1 /c = a 2 1 . Developing the determinant and performing the derivation gives the electric induction :

D 1 2 ff = ∂L ff ∂a 1 2ff = -2C (n 3 3 n 4 4 -n 3 4 n 4 3 ) (E 2 1 )/c (51)
This relation will be used in the next section.

Electrical charge.

This section deals with the electric charge Q associated to the electron. In traditional electrostatics, Q is the time-independent source for the field. However we have seen in section (2.4) that at large distances, the field is a longitudinally-oscillating vector (which prevents instantaneous action). In the present theory, Q should thus be an oscillating quantity : we will write Q = Q sin ωt (or cos ωt) to display this time dependance. The two fundamental equations which are related to Q in classical electromagnetism are : -Maxwell's equation : div -→ D = ρ ( -→ D : dielectric induction , ρ : charge density) -The Lorentz force F = Q E or, equivalently the electrostatic energy W el = QV acquired by a charge Q embedded in an external potential V . We study now the relation between the first of these equations and the tensor field of the electron.

Maxwell equation.

Maxwell's equation of classical electrostatics :

div - → D = ρ (52)
relates the charge density ρ in the small geometrical volume dv around a point M to the electric displacement

- → D with - → D = 0 - → E .
It is tempting to use this relation to define ρ as a function of the tensor components. The integral of this equation over the whole volume would give the total electric charge associated to the particle. However eq.( 52) is established for the far field region and makes reference to vacuum properties through 0 . It follows that the tensor field of the particle alone cannot lead to eq.( 52). In other words, while -→ E is created by the particle, 0 originates from the vacuum fields (the "noise") and -→ D contains the properties of both components. The situation is completely different inside the particle (close to the origin) where the amplitude of the vacuum fields becomes small and negligible as compared to the a i j s . We are thus led to use expression (51) in the far field and to introduce the vacuum permittivity 0 in the next paragraph.

Lagrange equation.

Lagrange's equations are applied to the potential components A j which are considered to be the generalized coordinates. For each component

A j : i   ∂L a ∂A j - ∂ ∂x i   ∂L a ∂ ∂A j ∂x i     = 0 . ( 53
)
L a is the Lagrangian density corresponding to the particle embedded in noise, where the components n i j are negligible inside the particle and preponderant outside. Terms ∂L a /∂ ∂A j /∂x i are the components of the 4-vector induction 4 D. This equation is again a tensor equation which we use in the spherical frame of coordinates. Its dimensions are QL -3 T -1 which shows that an electric charge can be recovered after an integration over time and geometrical space. Dimensions of -→ D (defined in the 3-dimensional space) in eq.( 52) are QL -2 . When eq.( 53) is written for the scalar potential A 0 = φ/c, the first term cancels because L a does not depend on φ/c explicitly. The second term expresses the 4-divergence of the 4-vector induction 4 D which can be split into its temporal and spatial parts :

4 D =   ∂L a ∂ ∂(φ/c) ∂(ct) , - → D   . ( 54
)
The spatial term has only one (radial) component :

- → D =   ∂L a ∂ ∂(φ/c) ∂r , 0 , 0   . (55)
Eq.(53) gives :

∂ ∂(ct) ∂L a ∂ ∂(φ/c) ∂(ct) + div - → D = 0 . ( 56
)
Here the divergence reduces to a single term : div

- → D = ∂ ∂r ∂(φ/c) ∂r = ∂ D r ∂r .
(57)

Now expression (56) is integrated over a sphere with a radius R large as compared to the particle size. Applying Gauss's theorem, one obtains :

dv ∂ ∂(ct) ∂L a ∂ ∂(φ/c) ∂(ct) + 4π R 2 D r (R) = 0 . ( 58
)
The electric induction in the far field region is given by eq.( 51)

D r (R) = D r ff = -2C (n 3 3 n 4 4 -n 3 4 n 4 3 ) E 2 1 (R) c = 2C 1 √ 4π Aω c 2 (n 3 3 n 4 4 -n 3 4 n 4 3 ) J 1 cos ωt -sin ωt (59)
The first term in eq.( 56) is computed now :

∂ ∂(ct) ∂L a ∂ ∂(φ/c) ∂(ct) = ∂ ∂(ct) ∂L a ∂a 1 1 = -C ∂ ∂(ct) (a 3 3 ) 2 a 2 2 = -3 C ω c 1 √ 4π Aω c 2 3 J 2 1 x 2 J 1 " sin 2 ωt cos ωt -cos 2 ωt sin ωt (60)
Integration over a spherical volume with R c/ω is equivalent to an integration over an infinite volume as in eq.( 38) :

dv ∂ ∂(ct) ∂L a ∂ ∂(φ/c) ∂(ct) ∼ -12π C c 2 ω 2 1 √ 4π Aω c 2 3 ∞ 0 J 2 1 J 1 " dx sin 2 ωt cos ωt -cos 2 ωt sin ωt (61)
The integral in this equation is :

∞ 0 J 2 1 J" 1 dx = 1 3 J 3 1 ∞ 0 = 1 3 4 (62)
Now one uses eqs.( 59) and (61) in eq.( 58), simplifies both terms by sin ωt or cos ωt ( D r reduces to D r ), and integrates over a period of time 2π/ω to obtain :

2π/ω 0 4π R 2 |D r (R)| dt = 12π 2 3 4 C c 2 ω 3 1 √ 4π Aω c 2 3 . (63)
Maxwell's equation (52) gives the interpretation of these results : the l.h.s. gives the electric induction in the geometrical volume and the r.h.s. is the electric charge :

e = ∓ 12π 2 3 4 C c 2 ω 3 1 √ 4π Aω c 2 3 ( 64
)
One notes the ∓ signs which appear in eq.( 61) for the different parities of the solutions. They correspond to the electron and the positron. The l.h.s. in eq.( 56) relates the components of the noise tensor to the vacuum permittivity 0 :

0 = 2 C c 2π/ω 0 n 3 4 n 4 3 -n 3 3 n 4 4 dt (65)
Dimensions of this equation are :

M -1 Q 2 L -3 T 2 .
In passing, one notes that the integral which appears in eq.( 62) nullifies if n = 1 ( J n(x=0) = 0 if n = 1). The space integral vanishes also for solutions containing Y m with m and n = 0. It follows that particles corresponding to solutions other than that for the electron are not electrically charged. However, they are still characterized by a field (term a 1 2 of the tensor). The conclusion is that it is the gravitational field and that the potential A i contains also the gravitational potential.

The fundamental unit charge e of the electron can thus be expressed in terms of the constant C, the potential A and the speed of light :

e = 12 π 2 3 4 c 2 ω 3 C 1 √ 4π Aω c 2 3 = √ π 54 1 c 4 C A 3 (66)
Note that ω disappears again in this formula. Note also that the term

∂ ∂(ct) ∂La ∂( ∂(φ/c) ∂(ct) )
in eq.( 60) is not an invariant. It follows that the charge e is not the same for the spinning electron. The integrand J 2 1 J" 1 represents the radial distribution of the charge. It is represented in fig. 4. A series expansion in the vicinity of x=0 shows the convergence of this expression around the origin :

The absolute value of the local charge density at point M is explicitly written below for future use :

ρ = 3π C 1 c 1 √ 4π Aω c 2 3 J 2 1 x 2 J" 1 (68)
It is obtained from (60) after the removal of sin ωt or cos ωt, and the integration over a period of time 2π/ω.

Spinning electron.

In this section we will consider the particle in rotation around the z axis in the inertial frame of the laboratory. We first compute the tensor in this frame. Each volume element around a point M rotates with a tangential velocity which we will deduce from the principle of energy conservation. In a preliminary study, we have studied a Maxwell-Boltzman distribution function for the angular velocity. However this is only a phenomenological model. The theory should rest on more fundamental laws : we use below the principle of energy conservation which allows an harmonious transition with the preceding sections.

Tensors in the laboratory frame.

The point M is subject to a rotation around the z axis as seen by an observer O i in the inertial system of the laboratory. We compute now the expressions for the components of [a ij ] in the system of fixed (overlined) coordinates of O i . We consider the local infinitesimal coordinates c dt, dr, r dθ, r sin θ dϕ which define the volume element dv around M and which are measured by the observer attached to M . These coordinates become c dt, dr, r dθ, r sin θ dϕ for O (69)

The standard notations for the relative tangential velocity β , and for the Lorentz factor γ are used :

β = v c γ 1 1 -β 2 (70)
We will keep the spherical symmetry for β and γ and write β = β(x) and γ = γ(x).

Note that the temporal phase ωt (which is a true scalar) is invariant in a Lorentz transformation : ωt = ωt . Note also that the factor c/ω which appears in the expression of the potential or its derivatives is the normalization parameter which transforms the radial coordinate r into the non-dimensioned quantity x. This parameter is not modified here. The Jacobian of the transformation is the Lorentz matrix :

J L =     γ 0 0 γ β 0 1 0 0 0 0 1 0 γ β 0 0 γ     (71)
which gives the components bpm of the tensor in the lab. frame : bpm = (J L ) hp (J L ) km a hk (72)

Now we write expression [START_REF] Griffiths | Introduction to Electrodynamics[END_REF] for [a hk ] under the compressed form :

[a hk ] =     a 11 E r /c 0 0 -E r /c a 22 0 0 0 0 a 33 0 0 0 0 a 33     (73)
Applying transformation (72) gives the tensor expressed at M in the laboratory frame : The antisymmetric part is :

[a ik ] lab =     γ 2 (
[f ik ] lab =     0 γ E r /c 0 0 -γ E r /c 0 0 -β γ E r /c 0 0 0 0 0 β γ E r /c 0 0     (75)
As expected one notes the appearance of the magnetic field B θ = β γ E r /c. It is oriented along the local axis -→ θ . The behavior of B θ is governed by β which is studied in the next paragraph. The symmetric part is :

[s ik ] lab =     γ 2 (a 11 + a 33 β 2 ) 0 0 β γ 2 (a 11 + a 33 ) 0 a 22 0 0 0 0 a 33 0 β γ 2 (a 11 + a 33 ) 0 0 γ 2 (a 33 + a 11 β 2 )     (76)
Expression (80) can be used in the far field region together with the noise tensor to compute the magnetic induction and the magnetic permeability µ 0 . The calculation is similar to the computation of 0 (eq.(65).

Energy conservation.

The aim of this section is to apply the principle of energy conservation to the system in rotation. The idea is to consider a first, or initial state in the laboratory frame where the potential is flat everywhere. We associate a zero energy to this state. The second, or final state, corresponds to the distribution of potential of the solution with creation of potential wells. The first and the second state should have the same energy : the decrease of potential energy is used to create kinetic energy. We develop this idea below.

The theory of relativity tells us that mass and field are characterized each by a momentum-energy 4-vector. The time component is the energy divided by c and the spatial components are those of the momentum. Momenta are null in the eigenframe and appear in the laboratory frame. The Lorentz transformation changes the energy term and brings elementary momenta associated (1) to the mass element and (2) to the field (Poynting vector). The difference between the new energies and the proper energies (which are invariants) is the kinetic energy for the mass and a change of the field energy as well. The energy-momentum 4-vectors are separately written for the mass and the field parts because there is no energy exchanges between both parts of the tensor [a ij ]. After integrations the local changes will result in global quantities which can be compared with experimental figures.

We apply the principle of energy conservation in the following way. We consider an initial state (1) in the laboratory frame where the potential A i is flat which implies that the (coherent) potential energy is zero everywhere. This situation is ideal in the sense that we neglect the noise. Now A i evolves toward a final state (2) which is a solution of the Helmholtz equation. This state displays potential wells as illustrated above (fig. 2) in the rest frame. The initial and the final states should have the same total mechanical energy :

(W k +W m ) initial = (W k +W m ) f inal = 0.
The kinetic energy should thus compensate for the loss of potential energy. The potential energy δH stored in the volume element around M has been computed in the preceding sections (eq.( 36)) in the rest frame of the particle where the 4-momentum density vector -→ 4 P for the mass and the field is :

-

→ 4 P = (δH/c, 0, 0, 0) (77)
Applying a Lorentz transformation to this vector gives the energies and the momenta in the laboratory frame. The kinetic energy dW k is the difference between energies in the eigenframe and the lab. frame.

After integration over time it writes :

dW k = γ -1 c 3 C (4π) 2 A 4 ω 4 c 8 π 4 c ω 3 J 1 J" 1 J 2 1 x 2 + J 2 1 J 2 1 x 2 (78)
The mass part corresponds to the well-know formula : dW k (mass) = dm 0 c 2 (γ -1). This kinetic energy is acquired at the expense of potential energy as indicated above. The principle of conservation of energy for the whole particle results in the equations for the mass and the field parts (see eq.39) :

∞ 0 dx J 2 1 (3 J 1 J" 1 ) = ∞ 0 (γ -1) J 2 1 (3 J 1 J" 1 ) dx (79a) ∞ 0 dx J 2 1 3 J 1 J" 1 + J 2 1 = ∞ 0 (γ -1) J 2 1 3 J 1 J" 1 + J 2 1 dx (79b)
The simplest solution to these equations is γ = 2. More generally , γ could be a function of x. A previous study used a Maxwell-Boltzman distribution for the angular velocity. However this model gives a magnetic field which vanishes in the far field which contradicts the existence of the vacuum magnetic permeability µ 0 , We have thus abandoned this model. The ideal situation which is studied above deals with a lone electron in a perfect vacuum with no noise, i.e., without any interaction with other particles. This interaction can change γ a little bit : we will thus keep γ in the equations and consider it as an adjustable parameter whose exact value will be obtained from a comparison with experimental values.

Let us write now the expressions for the spinning charge ē , the angular momentum L and magnetic µ moment as functions of the theoretical parameters C, A, ω and γ. These expressions will be written in the geometrical space. The experimental values of m, e, L and µ should give the numerical values of the parameters.

Spinning charge.

The density of charge ρ has been computed in section (4.2) in the eigenframe of the tensor. For this purpose, Lagrange's equation was written for the scalar potential and the temporal derivative was shown to give the charge density ρ. However we cannot use this expression for ρ in the case of the spinning electron because ρ is not invariant in a coordinates change. One should write Maxwell's equation in the form : div -→ D = ρ where the notation for the charge density in the laboratory frame is ρ. In this frame the potential is expressed as A i = (γ φ/c, A r , 0, γβ φ/c) and the elementary coordinates are given by eqs. (69) : du i = (c dt, r, rθ, r sin θ dϕ) . The components of the potential derivative are ∂A i /∂u j and the mixed tensor can be expressed in terms of the components of a i k in the eigenframe :

a i k lab =     γ 2 (a 11 + a 33 β 2 ) γ Er/c 0 -β γ 2 (a 11 + a 33 ) γ Er/c -a 22 0 -β γ Er/c 0 0 -a 33 0 β γ 2 (a 11 + a 33 ) β γ Er/c 0 -γ 2 (a 33 + a 11 β 2 )     (80)
The determinant of a i k lab being the same as that of a i k it follows that the derivative of the Lagrangian with respect to c t is the minor relative to a 1 1 lab . Lagrange's equation gives :

div 4 -→ D = ∂ c∂t -a 22 0 -β γ Er/c 0 -a 33 0 β γ Er/c 0 -γ 2 (a 33 + a 11 β 2 ) = -γ 2 ∂ c∂t a 33 (a 33 (a 22 + β 2 (a 11 a 22 + Er/c) 2 ) (81)
It remains to replace the (a ij ) s and E r /c by their expressions, simplify both members by sin ωt or cos ωt to have the modulus and to integrate over a period of time T = 2π/ω to obtain the electrical density ρ. We recopy below the expressions of the (a ij ) s for the even solution (from eq.12) :

a 11 = - 1 √ 4π Aω c 2 sin ωt J 1 a 12 = E r /c = 1 √ 4π Aω c 2 cos ωt J 1 a 22 = - 1 √ 4π Aω c 2 sin ωt J" 1 a 33 = - 1 √ 4π Aω c 2 sin ωt J 1 x (82)
The temporal derivative is :

∂ c∂t    ∂La ∂ ∂φ/c c∂t    = γ 2 A 3 C ω 3 ω J 1 cos(ω t) 16c 7 π 3/2 x 2 × 3J 1 J" 1 + (J 2 1 -3J 1 J" 1 )x β 2 + 3(J 1 J" 1 + (-J 2 1 + J 1 J" 1 ) xβ 2 ) cos(2ω t) (83)
The integral of the modulus over a period of time 2π/ω gives the charge density ;

ρ = - γ 2 ω 3 A 3 C 8c 7 √ π x 2 J 1 -3J 1 J" 1 + (J 2 1 -3J 1 J" 1 ) x β 2 (84)
The total electric charge of the electron in the laboratory frame is obtained after replacing J 1 , J 1 , and J 1 by their expressions [START_REF] Wheeler | Interaction with the Absorber as the Mechanism of Radiation[END_REF] and integrating over the whole volume. The radial integrals are exactly computed :

I e1 = √ π 2 ∞ 0 J 1 (-3J 1 J" 1 ) dx = √ π 54 I e2 = √ π 2 ∞ 0 J 1 x J 2 1 -3J 1 J" 1 dx = 3 √ π 160 81(-28 + 23Log[3]) (85)
and the total spining charge is :

ē = (I e1 + I e2 β 2 ) γ 2 A 3 C c 4 = √ πγ 2 80 + 81β 2 (-28 + 23Log[3]) 4320 A 3 C c 4 (86)
For γ = 2 one finds : ē = 0.1410807...

A 3 C c 4 (87)
5.4 Angular momentum.

The angular momentum L with respect to O is computed now. It is obtained from the sum of the elementary contributions L(M ) at points M . It is a pseudovector in the geometrical space.The local velocity is oriented along -→ ϕ 0 . We keep the component dL θ which is oriented along the axis -→ θ 0 :

dL θ = r (γ dm 0 ) (β c)
The other component dL r which is oriented along the axis -→ r 0 will give a null contribution after the volume integration. The total angular momentum is oriented along the z axis. It is the sum of the projection on this axis of all elementary contributions dL θ sin θ. It is obtained after an integration over the geometrical volume in the laboratory frame where the volume element is :

dv = dr (r dθ) (r sin θ dϕ) (88)
Integration gives :

L z = γ β C A 4 c 4 1 ω π(43 -76Log[2]) 6400 (89)
where we have included the numerical value of the radial integral :

I L = ∞ 0 x J 1 J" 1 J 2 1 dx = 43 -76Log[2] 900
For γ = 2 one finds :

L z = -0.00822942... A 4 C ω c 4 (90)
5.5 Magnetic moment.

The tensor field in the fixed inertial frame of the laboratory displays a magnetic field, showing that the spinning electron is a little magnet. An elementary electric current di is associated to each volume dv around M . This current is the quantity of charges which crosses the elementary surface r dθ dr during the unit of time : di = ρ r dθ dr βc .

and the modulus of the current density is di s = ρ βc. Its direction is along the axis -→ ϕ . One applies the definition of the differential magnetic moment :

-→ dµ = 1 2 - → r ∧ - → j s dv = 1 2 c ω x ρ βc - → θ . (92)
The component dµ z along the z axis writes :

dµ z = 1 2 c ω x sin θ ρ βc. (93)
Integrating over the whole volume gives the total magnetic moment :

µ e = V ol dµ z c 3 ω 3 x 2 dx sin θ dϕ (94)
Using expression (84) in dµ z , one finally obtains :

µ e = π 5/2 β γ 2 (-3 + 140β 2 ) 15360 C A 3 ω c 2 (95)
For γ = 2 one finds :

µ e = 0.402415... A 3 C ω c 2 (96)
6 Physical constants.

The parameters which we have introduced in the theory are : -the amplitude A and the frequency ω of the scalar potential φ/c, -the constant C which relates the energy and the Lagrangian densities, -the Lorentz parameter γ which we found to be close to 2 to obey the principle of energy conservation. We have obtained the expressions of the mass (eq.( 45)), the electric charge(eq.( 87)), the angular momentum(eq.( 90)) and the magnetic moment (eq.( 96)) as functions of these parameters.We group below these formula (written with γ = 2) together with the experimental values[25], written with the index exp. We first compute the dimensionless Landé factor g of the electron. The definition of g is the ratio of (µ e /L z )/(ē/2m 0 ). For γ = 2 one finds :

g = µ e /L z ē/2m 0 = -2.00338317 (98)
This value of the Landé factor is a major result of the theory. The experimental value is -2.002319304362. The difference between both figures is probably due to the noise, or to the particle environment effect. It is interesting to note that Paul Dirac found also g = 2 and that the difference was computed from the properties of the vacuum in the model standard . The value of γ giving the experimental figure g exp is the solution of the equation :

µ e /L z ē/2m 0 = g exp = -2.002319304362 (99)
One finds γ = 2.0006639663 which is very close to 2. For this value, the relative speed β = 0.86612. The following numbers are computed with this corrected value of γ.

Solving (eq.(97d )) and (eq.(97b) for A and C gives :

A = 2.497264943 10 7 (M T -2 L 2 Q -1 ) (100a) C = 5.88380001 10 -7 (M -3 T 2 L -2 Q 4 ) (100b)
Dimensions of A and C are given in parenthesis. The remaining symbol to be evaluated is the normalization factor ω. Inserting the preceding values of γ, A and C in the expressions (97c) of L z or (97d) of µ e gives : ω = 4.423325945 10 21 rad/s ; It follows that the unit on the x axis of the figures is c/ω = 6.777510 -14 m which gives an idea of the electron size. We note that the Compton wavelength corresponds to an angular frequency ω c = 7.76...10 20 rad/s. It is outside the scope of the present work to study the Compton diffusion process which leads to the difference between ω and ω c . This calculation achieves the primary objective of this theory which was to demonstrate that the physical constants which characterize the electron are not independent. The vortex model based on energy conservation leads to satisfying results.

Conclusion.

We have developed a theory of the electron which is based on well-known and well-proved classical notions and principles : -The base paradigm is the 4-dimensional electromagnetic potential in Minkowski's spacetime ; -The second notion is the covariant derivative of this potential which groups the 16 partial derivatives in a tensor ; -The principles of symmetry conservation and invariance of some quantities belonging to the tensors in a coordinate change are applied ; -The association of a Lagrangian density to one invariant allows the calculation of the mass and charge densities ; -The principle of least action leads to Lagrange's equations ; -The principle of relativity and the Lorentz transformation are applied to find the local tensors in the laboratory frame ; -The principle of energy conservation is used to find the rotation speed of the spinning particle ; -The usual standard definitions of the angular momentum and the magnetic moment are used.

The theory is essentially based on the study of the symmetric part [s ik ] of the covariant derivative [a ik ] = ∂A i /∂x k of the 4-potential A i in spacetime x k . [s ik ] can be diagonalized and is characterized by an eigenframe of coordinates. A relation between the derivatives of the scalar and the vector potential occurs in this frame. When this relation is associated to the conservation of the trace of [s i k ] in a time translation, a Helmholtz equation results. Among its solutions, one of them is electrically charged and represents the electron. The tensor field is obtained by replacing the potential by this solution at each point. Each tensor describes local properties such as the electromagnetic field. A Lagrangian density is associated to the determinant of [s i k ] which allows the calculation of the potential energy, or the mass of the electron. The Lagrangian of [a i k ] is defined in the same way and used in Lagrange's equations to compute the electric charge. The spinning electron is studied. The principle of energy conservation is applied to find the tangential speed of rotation. This makes the electron looks like a vortex. A Lorentz transformation of local coordinates from the tensor eigenframe to the laboratory frame allows us to find the spinning electric charge, the angular momentum and the magnetic moment. Comparing theoretical formulas with experimental figures gives the numerical values of the four parameters used in the equations. Neglecting noise effects, a value of the Lorentz parameter is found to be 2. This value leads to a Landé factor g = 2.0033... which we consider to be a very satisfactory result. The fundamental peculiarity of the theory is that it starts from local characteristics of the electron at each point of its structure. Integration over the whole volume leads to the measurable global properties. The different parameters which are used in the equations are (1) the amplitude, (2) the frequency of the basic solution, (3) the proportionality constant between the determinant of the tensor and the energy density and (4) the tangential speed of rotation, or the Lorentz factor γ. We have given the equations which link these parameters to the known characteristics of the electron : mass, electrical charge, angular momentum, magnetic moment. We have found that its fundamental frequency of vibration is close to the Compton frequency. A complete coherency between theoretical symbols and measured values is obtained. A conclusion is that the electron structure as it is computed here fits reasonably the intuitive image of a rotating body with a circulating current. However, it is rather a vortex than a rigid body like a top : the angular frequency decreases from the center to the outside of the particle. Its representation needs a tensor field whose local properties include electromagnetic and matter waves. These are sums of incoming and outgoing waves which extend over the whole universe. The electron is not a superposition of disconnected quantities : waves, electric and magnetic fields, mass and electrical charge. These are all included in D(A i ), the covariant derivative of the potential. A classical description of any experiment (like diffraction experiments) with electrons should thus be done with D(A i ) as the primary tool. Understanding the electron structure would allow to use its properties in practical devices : for instance mastering its fundamental frequency would lead to extraordinary precise clocks. But maybe the most important conclusion of this theory is that it offers a new way to study the yoctoscopic world of elementary particles. This way is that of a "bottom-up" approach which is different (and complementary) to ordinary quantum physics which is a "top-down" theory.

[22] We have made a preliminary study of the energies of some elementary particles by adding the potentials corresponding to an association of different solutions. This study neglects planetary effects arising from rotation of particles around each other (such as in the positronium) but gives a very good evaluation of the energies of various particles ranging from 1 for the electron to 11000 for the Λ 0 b .

[23] R.K.P. Zia, E.F. Redish and S.R. McKay, "Making Sense of the Legendre Transform", Am. J. Phys., 77, pp. 614-622 (2009). 8 Annex.

The 16 partial derivatives ∂A i /∂x k of the 4-potential A i = (φ/c, A x , A y , A z ) in spacetime x i = (ct, x, y, z) define the components of the mixed tensor D(A i ) in the cartesian frame : The metric tensor is taken to be :

a i k =      ∂(φ/
[g ij ] =     1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1    
It follows that the covariant tensor is :

[s ik ] or s i k can be diagonalized which means that a proper time and a eigen-geometrical frame of coordinates exist where all elements but the diagonal cancel. Let us describe the series of transformations which will lead to a tensor with elements s 1 2 = s 1 3 = s 1 4 = 0 . One starts with s i k and the first operation is a diagonalization of the lower bottom-right 3X3 block of this tensor in the geometrical space. Only the spatial derivatives of the potential vector are concerned by this transformation which gives :

[ s i j ] = M s i k M -1 = =     

Figure 2 -

 2 Figure 2 -Radial distribution of potential (mass) energy for the electron vs. x, normalized distance to the center (arbitrary units).

Figure 3 -

 3 Figure 3 -Radial distribution of field energy for the electron vs. x, normalized distance to the center (arbitrary units).

Figure 4 -

 4 Figure 4 -Radial distribution of electric charge for the electron vs. x, normalized distance to the center (arbitrary units).

  i . Both sets of coordinates are linked by a local, tangential, Lorentz transformation. The motion is along the -→ ϕ axis and the two other local axis -→ r and -→ θ are perpendicular to it. It follows that the coordinates dr and r dθ are not affected by the rotation. Only the length element r sin θ dϕ and the time element c dt at event M are subject to the Lorentz transformation : c dt = γ c dt + γ β r sin θ dϕ , r = r , θ = θ , r sin θ dϕ = γ r sin θ dϕ + γ β c dt.

4 C ω c 4 L zexp = 2 = 5 .

 4425 m 0 = 0.00289... C A 4 c 6 m 0exp = 9.1093837015 10 -31 kg (97a) ē = 0.1410807... A 3 C c 4 ēexp = 1.602176634 10 -19 C (97b) L z = -0.00822942... A 2728610 -35 Js (97c) µ e = 0.402415... A 3 C ω c 2 µ eexp = 9.284764704310 -24 J/T (97d) These expressions are used now to compute the values of the different theoretical parameters.

  [24] See ref 10, §14 . [25] "Codata recommended values of the fundamental physical constants 2014", P.J. Mohr, D.B. Newel, B.N. Taylor, Arkiv :7.07956v1[physics.atom-ph]21 jul 2015.

  This tensor is divided into its symmetric and antisymmetric parts (eqs. (1) and (2)). Elements of the first line of the symmetric part are :

  a 11 + a 33 β 2 )

	γ Er/c	0	β γ 2 (a 11 + a 33 )	
	-γ Er/c 0 β γ 2 (a 11 + a 33 ) β γ Er/c a 22 0	0 a 33 0	-β γ Er/c 0 γ 2 (a 33 + a 11 β 2 )	  
				(74)

where the diagonalizing matrix M has the form :

The relative speed of the tensor with respect to the observer along the x axis is noted v x in units of c and

x . The transformed tensor is :

Now we choose v x to be a solution of the equation :

i.e. ,

Inserting one of these value of v x in [ ŝi j ], elements of the first line become : ŝ11 = (s

One notes also that this manipulation ( block diagonalization followed by a Lorentz transformation along x) result in the multiplication of the original s 1 3 and s 1 4 by γ x . It is clear now that the same manipulation done on the y and then on the z axis will allow to cancel terms s 1 3 and s 1 4 in the same way. After the last Lorentz transformation, the form of the tensor is such that its elements in the first line and the first column are null, with the exception of s 1 1 . Without any further developments we will accept the theorem following which the tensor and the observer share the same proper time if the terms s

Let us apply this rule to the tensor D(A i ) when it is written in the spherical frame of coordinates :

We have used the compressed notation for the derivatives, for instance ; ∂A r /c∂t ≡ A r ,t . The elements of the mixed symmetric tensor which should cancel in the proper time reference are :

adding s 1 2 , s 1 3 and s 1 4 together gives :

or :

This is the first formula which is used to obtain the fundamental Helmholtz equation for the elementary particles. The second equation is obtained from the trace of [a i k ]. This trace is invariant in a coordinates change :

The expression of the divergence of the vector

and thus :

T race[a i k ] = φ ,t + div

The invariance of T race[a i k ] in a time translation leads to the second formula which is used to obtain the fundamental Helmholtz equation.


Pascale Charpin

Pascale Charpin Crooked

Hal

Crooked functions

Keywords: Vectorial function, Boolean function, derivative, differential set, plateaued function, partially-bent functions, bent functions, APN function, AB function, permutation

come

Introduction

The crooked functions have been introduced by Bending and Fon-Der-Flaass in 1998, as combinatorial objects of great interest [START_REF] Bending | Crooked functions, bent functions, and distance regular graphs[END_REF]. Such a function has been defined from V to W , two n-dimensional vector spaces over F 2 , by the following property: the image set of any of its derivatives is the complement of a hyperplane. This characterization implies that a crooked function is bijective, and allows, in particular, to construct distance regular graphs. Later, several authors have developed this work, and have generalised the

Definitions, basic properties

Thoughout this paper, |E| denotes the cardinality of the set E, and E * = E \ {0}. Let F be a mapping, from the finite field F 2 n to itself. Such a function is called a vectorial function, while a function f , from F 2 n to F 2 is, as usually, a Boolean function. We denote by Im(ξ) the image set of any function ξ.

A vectorial function F , from F 2 n to itself, is said to be an almost perfect nonlinear (APN) function if and only if all the equations,

F (x) + F (x + a) = c, a, c ∈ F 2 n , a ̸ = 0 , (1)
have zero or two solutions in F 2 n , say x and x + a. Throughout this paper, we call derivative of F , with respect to a, the function from F 2 n to V :

x → D a F (x) = F (x) + F (x + a), a ∈ F * 2 n .
where V will be a subfield of F 2 n , usually F 2 n or F 2 . We call differential set, in point a, the image set of D a F :

Im(D a F ) = { F (x) + F (x + a) | x ∈ F 2 n }. ( 2
)
Clearly, when F is APN, we have for any a ∈ F * 2 n : D a F (x) = D a F (x + a) = c, for some c and for only one x.

This means that D a F is a 2-to-1 function. Thus, one can formulate the APN property as follows.

Proposition 1 Any vectorial function F , over F 2 n , is APN if and only if all their differential sets have cardinality 2 n-1 .

The 2 n , so-called, components of F are the Boolean functions

f λ : x -→ T r(λF (x)) , λ ∈ F 2 n ,
where f 0 is the null function, by convention. They are linearly defined by means of the absolute trace on F 2 n :

x → T r(x) = x + x 2 + . . . , x 2 n-1 .

The dual V ⊥ , of any subspace V of F 2 n , is the subspace of those y such that T r(yx) = 0, for all x ∈ V . TheWalsh transform of a Boolean function f , is defined as

a ∈ F 2 n → W f (a) = ∑ x∈F 2 n (-1) f (x)+T r(ax) .
Recall the Parseval's relation:

∑ a∈F 2 n (W f (a)) 2 = 2 2n .
We will need the following result:

Lemma 1 [6, Lemma V.2] Let f be a Boolean function over F 2 n , and let V be a subspace of (x+u) .

F 2 n of dimension k. Then ∑ v∈V (W f (v)) 2 = 2 k ∑ u∈V ⊥ ∑ x∈F 2 n (-1) f (x)+f
We now define particular APN functions, which exist for odd n only.

Definition 1

The function F is said to be an almost bent

(AB) function if the numbers µ F (a, λ) = ∑ x∈F 2 n (-1) T r(λF (x)+ax) , ( 3
)
are equal to 0 or ±2 n+1 2 only, when a ∈ F 2 n and λ ∈ F * 2 n . W f (u) number of u ∈ F 2 n 0 2 n -2 n-s 2 (n+s)/2 2 n-s-1 + (-1) f (0) 2 (n-s)/2-1 -2 (n+s)/2 2 n-s-1 -(-1) f (0) 2 (n-s)/2-1 Table 1: Walsh spectrum of the Boolean s-plateaued function f Note that µ F (a, λ) = W f λ (a) for any fixed λ.
A Boolean function f , over F 2 n , is said to be bent when W f takes two values {±2 n/2 } only, so that n must be even. It is said to be s-plateaued when either it is bent or W f takes three values, {0, ±2 (n+s)/2 }, with 1 ≤ s ≤ n -2 and n + s even.

By convention, a bent function is 0-plateaud. The value 2 (n+s)/2 is the amplitude of f . A plateaued vectorial function is a vectorial function whose components are plateaued Boolean functions. It is said that F is plateaued with single amplitude, when all components of F have the same amplitude.

The sum-of-square indicator of f is defined by

ν(f ) = ∑ a∈F 2 n W 2 Daf (0) = 2 -n ∑ b∈F 2 n W 4 f (b). ( 4
) If f is s-plateaued then ν(f ) = 2 2n+s . Moreover, the vectorial function F is APN if and only if ∑ λ∈F * 2 n ν(f ) = 2 2n+1 (2 n -1) (5)
(see [START_REF] Berger | On almost perfect nonlinear functions[END_REF]Corollary 1]). A Boolean function f is said to be balanced if it takes the values 0 and 1 the same number of times. Recall a well-known result:

Theorem 1 A vectorial function F is a permutation if and only if all its components f λ , λ ∈ F * 2 n , are balanced.

Brief record

We use our terminology, of the previous section, rather than of the initial works on crooked functions. The next definition was proposed by Bending and Fon-Der-Flaass in [START_REF] Bending | Crooked functions, bent functions, and distance regular graphs[END_REF], twenty years ago.

Definition 2 Let F be a function from F 2 n to itself. This function is called crooked if it satisfies the following three properties:

(i) F (0) = 0;
(ii) F (x) + F (y) + F (z) + F (x + y + z) ̸ = 0, for any three distinct x, y, z;

(iii) D a F (x) + D a F (y) + D a F (z) ̸ = 0, for arbitrary x, y, z and any a ̸ = 0.

This definition implies that such a function F is a bijection over F 2 n , where n must be odd. Note that the condition (ii) means that F is APN (see [START_REF] Bending | Crooked functions, bent functions, and distance regular graphs[END_REF]). Also, F is crooked if and only if any of its differential sets is a complement of a hyperplane. Further, other properties are studied, in [START_REF] Bending | Crooked functions, bent functions, and distance regular graphs[END_REF], such as some relations of crooked permutations with bent functions of dimension n -1, and with the so-called Kerdock sets.

Crooked permutations allow to construct some distance regular graphs. This was shown in [START_REF] Bending | Crooked functions, bent functions, and distance regular graphs[END_REF], generalizing previous constructions. Later, van Dam and Fon-Der-Flaass proposed another construction, and then another generalisation (see, in particular, Thorem 3 of [START_REF] Van Dam | Uniformly packed codes and more distance regular graphs from crooked functions[END_REF]). Conversely, Godsil and Roy have shown that crooked permutations can be fully characterized by Preparata codes of minimum distance 5. Similarly, some distance-regular graphs provide crooked permutations [START_REF] Godsil | Two characterization of crooked functions[END_REF]Theorems 3,[START_REF] Byrne | On the non-existence of quadratic APN and crooked functions on finite fields[END_REF].

Since the high interest for APN functions in cryptography and coding theory [START_REF] Carlet | Codes, bent functions and permutations suitable for DES-like cryptosystems[END_REF], the existence of crooked functions was later the core of the research on crooked functions. Kyureghyan proposed another definition of crooked functions, identifying all APN functions which are such that their differential sets are affine hyperplanes. She established the basic properties of such functions. She notably proved that the monomial crooked functions are quadratic [START_REF] Kyureghyan | The only crooked power functions are x 2 k +2 l[END_REF][START_REF] Kyureghyan | Crooked maps in F 2 n[END_REF].

The APN quadratic functions are crooked. We do not know if crooked functions of higher degree do exist. This is a recurring question, about which only negative results have been obtained. An important result was obtained by Bierbrauer and Kyureghyan: binomial crooked functions are quadratic [START_REF] Bierbrauer | Crooked binomials[END_REF].

Structure of crooked functions

A hyperplane of F 2 n can be seen as a subspace of F 2 n , as follows defined:

H λ = { y | T r(λy) = 0 }, λ ∈ F * 2 n . ( 6
)
We will denote by H λ the complement of H λ . The dual of H λ is obviously

H ⊥ λ = {0, λ}.
The next definition of crooked functions is due to Kyureghan [START_REF] Kyureghyan | Crooked maps in F 2 n[END_REF]. The corpus of such functions includes the crooked permutations, but also a large variety of non bijective crooked functions, especially in even dimension.

Definition 3 A function F , from F 2 n to itself, is called crooked when it is such that, for every a, its differential set S a = { F (x) + F (x + a) | x ∈ F 2 n },
is an affine hyperplane.

We directly deduce from Proposition 1:

Claim 1 Any crooked function is an APN function.

Assuming that S a = H λ , for some λ, we get

T r (λ(D a F (x))) = c, for all x, where c ∈ F 2 .
This means that the derivative of the component f λ of F , in point a, is a constant function. In this case, a is said to be a linear structure of f λ . The linear space of f λ is the subspace, including a = 0, of their linear structures. We will see that crooked functions have always components with nonzero linear structures.

Crooked and partially-bent functions

Let f be any Boolean function over F 2 n . Denote by N d , the number of balanced derivatives of f , and by N f , the size of the set {a|W f (a) = 0}. Partially-bent functions were introduced in [START_REF] Carlet | Partially-bent functions[END_REF], as functions satisfying

(2 n -N d )(2 n -N f ) = 2 n . ( 7
)
There is another characterization of partially-bent functions, which allows to determine their Walsh spectrum.

Theorem 2 [9, Theorem] A Boolean function f is partially-bent if and only if its derivatives are either constant or balanced.

This leads to the precise description of the Walsh spectrum of any partiallybent function. Note that bent functions are particular partially-bent functions, with N d = 2 n -1 and N f = 0. Moreover, a partially-bent function is, in a certain sense, obtained by concatenating the same bent function, several times. The next result is partly given by [9, Proposition 2]. See also [START_REF] Kyureghyan | Crooked maps in F 2 n[END_REF]Theorem 1], which concerns crooked functions but, actually, holds for any partially-bent function. For the Walsh spectrum, see a proof in [7, Proposition 4].

Corollary 1 Let f be a partially-bent Boolean function. Assume that f has a linear space V , of dimension s > 0.

Then f is a s-plateaued function, such that n + s is even, and W f takes three values, {0} and {±2 (n+s)/2 }. The Walsh spectrum of f is given by the Table 1 above.

The definition of vectorial partially-bent functions F is simply derived.

Definition 4 Let F : F 2 n → F 2 n .
The function F is said to be a partiallybent vectorial function if every component of F is partially-bent. When n is even, some components can be bent.

Theorem 3 Let F be a vectorial function over F 2 n with components f λ . Set, for a ∈ F * 2 n , Λ a = { λ ∈ F * 2 n | D a f λ is constant } ∪ {0}
, and denote by ℓ(a) the dimension of Λ a . Then we have:

• Assume that F is partially-bent. Then the differential sets of F are affine subspaces. For any a ∈ F * 2 n , this subspace is of codimension ℓ(a), with ℓ(a) ≥ 1 and D a F is a 2 ℓ(a) -to-1 function.

• If F is a crooked function then F is partially-bent. Proof. Recall that D a f λ (x) = T r(λD a F (x)), for all x.
Obviously, Λ a is a subspace of F 2 n . Now, fixing a and x, we compute

B(a, x) = ∑ λ∈F 2 n ∑ y∈F 2 n (-1) Daf λ (x)+Daf λ (y) = ∑ y∈F 2 n ∑ λ∈F 2 n (-1) T r(λ(DaF (x)+DaF (y))) = 2 n × |{ y | D a F (x) = D a F (y)}|.
On the other hand, we set for any λ ∈ F * 2 n :

B(λ) = ∑ y∈F 2 n (-1) Daf λ (x)+Daf λ (y) = (-1) Daf λ (x) ∑ y∈F 2 n (-1) Daf λ (y) .
We first assume that F is partially bent. Note that ℓ(a) ≥ 1, since otherwise we would have that the Boolean function D a f λ is balanced, for any λ ∈ F * 2 n , from Theorem 2. This is impossible since D a F cannot be a permutation (see Theorem 1).

Clearly, B(λ) = 0 if and only if the function D a f λ is balanced. If it is not balanced, then λ ∈ Λ a , and this function is constantly equal to either 0 or 1. In both cases, we get B(λ) = 2 n . Hence, we get

B(a, x) = ∑ λ∈Λa B(λ) = 2 n 2 ℓ(a) , for all x,
implying that the number of y such that

D a F (x) = D a F (y) equals 2 ℓ(a) , i.e., D a F is 2 ℓ(a) -to-1. Since Im(D a F ) is contained in an affine subspace of codimension ℓ(a), according to the definition of Λ a , Im(D a F ) is equal to this affine subspace. Now, we suppose that F is a crooked function. Consider any component f λ of F . Let a ∈ F * 2 n such that D a f λ is not constant. Set V = Im(D a F
). Then we have that any x satisfies:

T r(λD a F (x)) = 0 if and only if x ∈ V ∩ H λ .
There are 2 n-1 such x, since D a F is 2-to-1 and V is an affine hyperplane, which is neither H λ nor its complement. Hence D a f λ is balanced. We have proved that f λ is partially-bent, completing the proof. ⋄

Let F be any quadratic function:

F (x) = ∑ 0≤i<j<n u i,j x 2 i +2 j , u i,j ∈ F 2 n .
The derivatives of F are affine functions, say L a for any a ∈ F * 2 n . Thus, F is partially-bent; it is crooked if and only if every L a is an affine function with kernel of dimension 1.

Corollary 2 Any quadratic vectorial function is partially-bent. It is crooked if and only if it is APN.

Crooked of odd dimension

In this section, we study crooked functions of odd dimension, bijective or not. Theorem 4 (below) is the main result, describing the exceptional properties of such functions. These results are quite known, but were partially presented in several papers [START_REF] Van Dam | Codes, graphs, and schemes from nonlinear functions[END_REF][START_REF] Kyureghyan | The only crooked power functions are x 2 k +2 l[END_REF][START_REF] Kyureghyan | Crooked maps in F 2 n[END_REF]. First, it is easy to describe the set of crooked permutations.

Lemma 2 Let F be a crooked function such that F (0) = 0, with differential sets S a . Let a and λ be such that S a equals either H λ or H λ , where H λ is defined by [START_REF] Canteaut | On cryptographic properties of the cosets of R(1, m)[END_REF]. Then we have:

S a = H λ ⇐⇒ T r(λF (a)) = 1.
Besides, F is a permutation if and only if S a = H λ , for any such pair (λ, a). In this case, n is odd.

Proof. By hypothesis, we have T r(λD a F (x)) = c for all x, where c ∈ {0, 1}. In particular, T r(λF

(a)) = c; further, c = 0 means that Im(D a F ) = H λ .
The function F is not bijective if and only if F (x) = F (x + a), for some pair (x, a). Equivalently, there is (λ, a) such that Im(D a F ) = H λ , since 0 belongs to Im(D a F ).

When n is even, F cannot be a permutation, since it is an APN function, which is plateaued (see [START_REF] Berger | On almost perfect nonlinear functions[END_REF]Theorem 3]). ⋄

When n is odd, a crooked function may not always be bijective, as we show by the next example.

Example 1 Assume that n is odd. It is well-known that

F : x → x 2 t +1 , with gcd(t, n) = 1,
is an AB permutation. So, it is a crooked permutation. Consider now the function x → G(x) = x 2 t +1 + x, which is AB too, and then crooked. Since

G(0) = G(1) = 0, G is not a permutation.
Definition 5 A Boolean function is said to be near-bent if it is 1-plateaued, i.e., its Walsh transform takes the values 0 and ±2 (n+1)/2 only.

Theorem 4 Let F be a crooked function over F 2 n , n odd, such that F (0) = 0. Every differential set S a , a ∈ F * 2 n , is an affine hyperplane:

S a = β + H λ , β ∈ F 2 n , for some λ ∈ F * 2 n .
Then the following properties hold: Proof. Since F is crooked, it is partially-bent. Let f λ , λ ∈ F * 2 n , be the components of F . Thus, for any λ and for any a, the derivative of f λ , in point a, is either constant or balanced. Hence ∑

(i)
x∈F 2 n (-1) f λ (x)+f λ (x+a) ∈ {0, ±2 n }.
For any fixed λ, there is at least one a, say a(λ), such that D a f λ is constant, since otherwise the function f λ would be bent, which is impossible when n is odd. Thus, we get the set of the a(λ), whose size is at most 2 n -1. However, a(λ) = a(µ) = b, for some µ ̸ = λ, would mean the following: the Boolean functions x → T r(λD b F (x)) and x → T r(µD b F (x)) are both constant. This would imply that S b is of codimension at least 2, a contradiction. To each λ corresponds one and only one a, completing the proof of (i).

We deduce that any component f λ has only one nonzero linear structure, say a, i.e., its linear space has dimension 1. Obviously, if f λ is balanced, its derivative in point a is constantly equal to 1. When F is a permutation, all f λ are balanced, completing the proof of (ii).

From Corollary 1, every f λ is near-bent, providing

W f λ (a) ∈ { 0, ± 2 (n+1)/2 }, for all a ∈ F 2 n .
According to Definiton 1, F is an AB function, completing the proof.

⋄

From Table 1, we know that the support of the Walsh spectrum of any nearbent Boolean function on F 2 n has size 2 n-1 . It could be an affine subspace of codimension 1, as it holds for components of some crooked functions.

Proposition 2 Let F be a vectorial function over F 2 n , n odd, such that F (0) = 0. Assuming that F is crooked, the set

W λ = { a ∈ F 2 n | W f λ (a) = 0}
is an affine subspace of codimension 1, for all λ ∈ F * 2 n . Conversely, assume that the sets W λ are affine hyperplanes. In this case, if F is APN then F is crooked.

Proof. Assume that F is crooked and let λ ∈ F * 2 n . Since F is an AB function, f λ is partially bent, with linear space {0, b}, for some nonzero b. Obviously, the function g a : x → f λ (x) + T r(ax) is partially bent too, with linear space {0, b}, for any a ∈ F 2 n . We know that any partially-bent function is balanced if and only if it is not constant on its linear space (see [START_REF] Carlet | Partially-bent functions[END_REF]Proposition 2]). Thus, g a is balanced if and only if

D b g a (x) = 1, i.e., D b f λ (x) + T r(ab) = 1, for all x.
Hence, we get T r(ab) = 1 + c, c ∈ F 2 , since D b f λ is a constant function. We can suppose that c = 0. Thus, g a is balanced if and only if T r(ab) = 1, that is a ∈ H b . Thus

W λ = { a ∈ F 2 n | T r(ab) = 1} = H b .
Similarly, with c = 1 we obtain W λ = H b .

Conversely, suppose that W λ is an affine subspace of codimension 1, say u + H b , u ∈ F 2 n , for some b. So, H ⊥ b = {0, b} and we have from Lemma 1:

∑ a∈H b W f λ (a) 2 = 2 n-1 ( ∑ x∈F 2 n (-1) T r(λD 0 F (x)) + ∑ x∈F 2 n (-1) T r(λD b F (x))
) Since Perceval's relation, the sum above on the left is equal either to 0 or to 2 2n . We deduce that

D b f λ is a constant function. So, if D b F is 2-to1, then Im(D b F ) is equal to either H λ or H λ . ⋄

Crooked of even dimension

When n is even, crooked functions are partially-bent functions which have bent components. We recall this property below, by Theorem 5.

Theorem 5 Let F be a plateaued function over F 2 n , with n even and n > 4.

Let 2 (t λ +n)/2 be the amplitude of any component of F , namely of any

f λ , λ ∈ F * 2 n . Denote by B the number of bent components of F . Then F is APN if and only if B = ∑ λ∈F * 2 n ,t λ >0 (2 t λ -2). ( 8
)
This property holds, in particular, for crooked functions. Consequently, B satisfies

2(2 n -1) 3 ≤ B < 2 n -2 n/2 , ( 9
)
where the lower bound is reached if and only if t λ = 2 for all non zero t λ .

Proof. The equality [START_REF] Canteaut | Structural weaknesses of permutations with a low differential uniformility and generalized crooked functions[END_REF] has been proved by [START_REF] Charpin | Differential uniformity and the associated codes of cryptographic functions[END_REF]Proposition 6], but may be computed by using [START_REF] Byrne | On the non-existence of quadratic APN and crooked functions on finite fields[END_REF]. Indeed, since each f λ is plateaued, we have ν(f λ ) = 2 t λ +2n , for any λ, and then:

A = ∑ λ∈F * 2 n ν(f λ ) = 2 2n ∑ λ∈F * 2 n 2 t λ .
According to [START_REF] Byrne | On the non-existence of quadratic APN and crooked functions on finite fields[END_REF], F is APN if and only if A = 2 2n+1 (2 n -1), providing

B + ∑ λ∈F * 2 n ,t λ >0 2 t λ = 2(2 n -1) = 2(B + N ), (10)
since 2 n -1 = B + N with N = |{λ | t λ > 0}|.
Hence, the equality above is equivalent to [START_REF] Canteaut | Structural weaknesses of permutations with a low differential uniformility and generalized crooked functions[END_REF]. Note that t λ > 0 implies t λ ≥ 2, since n + t λ must be even. The lower bound of B, in (9), is known (see [START_REF] Berger | On almost perfect nonlinear functions[END_REF]Corollary 3]). The upper bound has been proved by [START_REF] Pott | On the maximum number of bent component of vectorial functions[END_REF]Theorem 3]. It cannot be reached here, since we must have B ≡ 2 (mod 4), from [START_REF] Carlet | Codes, bent functions and permutations suitable for DES-like cryptosystems[END_REF].

⋄

By the study of bent components, one understand precisely the difference between odd and even cases. Let F be a crooked function on F 2 n , with components f λ . We have seen that, when n is odd, there is a one to one correspondence λ → a(λ), where a(λ) is the unique linear structure of f λ .

When n is even, the linear space of f λ has dimension s, where s is even. So, either f λ is bent (s = 0) or s ≥ 2. The number of hyperplanes involved in the structure of F is at most (2 n -1)/3. For any a, there is one and only one λ such that Im(D a F ) is equal to H λ or to H λ . But, the function D a f λ is constant for any a belonging to the linear space of f λ . There are at least three such nonzero a, when f λ is not bent.

Proposition 3 Let F be a crooked function on F 2 n , n even, with components f λ . For any a ∈ F * 2 n , there is a unique λ such that the derivative of f λ , in point a, is a constant function.

Conversely, any function f λ is either bent or with linear space V of dimension k ≥ 2.

Proof. By hypothesis, Im(D a F ) is an hyperplane H λ , or the complement of H λ , for any a ∈ F * 2 n . Hence T r(λD a F (x)) = c, for all x, where c ∈ F 2 . Such a λ is unique, because otherwise Im(D a F ) would be an affine subspace of codimension 2.

Since F is a plateaued APN function, at least 2(2 n -1)/3 components of F are bent. Let f λ be a non bent component. Its linear set has an even dimension, so this dimension must be greater than or equal to 2. ⋄

Bent from crooked functions

In this section, we consider crooked functions over F 2 n , where n is odd. The components of such a function are near-bent Boolean functions. Moreover, every component has (only) one derivative which is a constant function. (see Theorem 4). We will show that a set of 2 n -1 bent functions of n + 1 variables can be derived, using this strong property. Our main reference is the construction of Leander and McGuire [START_REF] Leander | Construction of bent functions from nearbent functions[END_REF], on the near-bent Boolean functions, that we apply to vectorial functions which are crooked. We first recall some facts which are more or less known, maybe not in this form.

Lemma 3 Let f be a near-bent Boolean function over F 2 n , where n is odd. Assume that f (0) = 0. Then (i) and (ii) are equivalent:

(i) f has a constant derivative in point a ∈ F * 2 n .
(ii) There exists a such that

{ u ∈ F 2 n | W f (u) = 0 } = { u ∈ F 2 n | T r(ua) = 1 + f (a) }.
Proof. Note that, f cannot have more than one linear structure, since it is near-bent. Assume that there is (a unique) a such that D a f (x) = c, for all x, where c ∈ F 2 . Thus, c = f (0) + f (a) = f (a). Now, we compute W f (u):

W f (u) = ∑ x∈F 2 n (-1) f (x)+T r(ux) = ∑ x∈F 2 n (-1) f (x+a)+T r(u(x+a)) = ∑ x∈F 2 n (-1) f (x)+f (a)+T r(u(x+a)) = (-1) f (a)+T r(ua) W f (u), since f (x) + f (x + a) = f (a). Clearly, if u is such that T r(ua) + f (a) = 1 then W f (u) = 0.
But, there are 2 n-1 such u, proving that (ii) holds. Now suppose that (ii) holds, for some a. Recall that

H a = { u ∈ F 2 n | T r(ua) = 0 } so that H ⊥ a = {0, a}.
Thus, the set of those u such that W f (u) = 0 is either equal to the hyperplane H a , or equal to its complement, according to either f (a) = 1 or f (a) = 0. So, we can apply Lemma 1:

∑ v∈Ha W f (v) 2 = 2 n-1 ( ∑ x∈F 2 n (-1) 0 + ∑ x∈F 2 n (-1) f (x)+f (x+a)
)

.

By hypothesis, this sum equals 0 if f (a) = 1 and 2 2n if f (a) = 0, since the Parseval's relation. This is possible only if D a f is a constant derivative of f . It is the only one such derivative, since f is near-bent. ⋄ Lemma 4 Let f be a near-bent Boolean function on F 2 n (n odd) such that f (0) = 0. Assume that f has a linear structure a. Let g be the function from

G = F 2 n × F 2 to F 2 : g(x, y) = (y + 1)f (x) + y ( f (x) + T r(a -1 x) ) ( 11
)
Then g is a bent function of n + 1 variables.

Proof. For any u, we denote by f u the Boolean function x → f (x) + T r(ux). Let b = a -1 . The restriction of g to F 2 n (y = 0) and to its complement in G (y = 1) are respectively f and f b which are both near-bent Boolean functions. Moreover

D a f (x) = f (a) and D a f b (x) = f (a) + T r(ba) = f (a) + 1 = f b (a), since T r(ba) = T r(1) = 1. Let u ∈ F * 2 n . Applying Lemma 3, f u is balanced if and only if T r(ua) = 1 + f (a) and f b+u is balanced if and only if T r(a(u + b)) = 1 + f (a), providing T r(ua) = 1 + 1 + f (a) = f (a).
Thus f u is balanced if and only if f b+u is not balanced. This is equivalent to say that g is bent (see [START_REF] Canteaut | On cryptographic properties of the cosets of R(1, m)[END_REF]Theorem V.3] or [START_REF] Leander | Construction of bent functions from nearbent functions[END_REF]Theorem 2]). ⋄ Now, we consider a crooked function F , from F 2 n to itself. According to Theorem 4, we know that all components f λ of F are near-bent with a (unique) constant derivative. This allows to derive a specific set of bent functions from any crooked function, in odd dimension.

Theorem 6 Let F be a crooked function over F 2 n where n is odd, such that F (0) = 0. Denote by a λ the linear structure of the component f λ of F . Then we get a set B(F

) of 2 n -1 bent functions g λ , each from G = F 2 n × F 2 to F 2 : B(F ) = { g λ (x, y) = (y + 1)f λ (x) + y ( f λ (x) + T r(a -1 λ x) ) | λ ∈ F * 2 n }. Proof.
Since F is a crooked function, there is a bijective correspondence between the λ ∈ F * 2 n , and then the functions f λ , and the linear structures a λ of every f λ . Thus, Lemma 4 applies for every λ, using a -1 λ , providing 2 n -1 distinct bent functions. ⋄

Remark 1

The construction, given by Theorem 6, could be of interest regarding the problem of the existence of crooked functions. It would be useful to exhibit some properties of B(F ), or to construct other sets of bent functions. For instance, g λ could have an higher degree, by replacing f λ by another Boolean function, affinely equivalent to f λ . The question is to know if such a set of bent functions does exist, for some degree greater than 2.

Note that a set like B(F ) is not a subspace, but could generate a linear code with few weights.

To derive effectively bent functions from a given crooked function, it is necessary to have an expression of the linear structures a λ . This is generally an open problem, even when F is a quadratic APN function. Now suppose that T r(µ(x + y)) = 1. Hence G a (x) = G a (y) + β. But this is impossible, since G a (x) and G a (y) belong to H λ while β ̸ ∈ H λ . Thus, we get again a contradiction and can conclude that R a is a permutation.

By construction, R a (H µ ) = H λ , since T r(µx) = 0 for all x ∈ H µ and G a (H µ ) = H λ . Indeed, G a is 2-to-1 and its image set is H λ . For any pair (x, x+a), we have x ∈ H µ if and only if x+a ∈ H µ (with G a (x) = G a (x+a)). ⋄

Comments to conclude

As a prelude of this conclusion, we want to recall the main conjecture concerning crooked functions: Conjecture: Let F : F 2 n → F 2 n be a crooked function, as defined by Definition 3. Then F is quadratic or, in other terms, has algebraic degree 2.

By the two previous sections, we want to exhibit specific constructions derived from crooked functions (only). Our purpose is to open new ways to study the existence of crooked functions. We are convinced that other properties have to be found, increasing the knowledge on crooked functions, especially on the existence of such functions.

On the other hand, the quadratic functions form a corpus of great interest about which many problems remain open. In particular, we are still far from understanding the different structures of quadratic APN functions of even dimension. The determination of the number of bent components of such a function would be of great interest. Negative answers, concerning the APN property of quadratic vectorial functions, have been obtained, for instance in [START_REF] Berger | On almost perfect nonlinear functions[END_REF] [START_REF] Byrne | On the non-existence of quadratic APN and crooked functions on finite fields[END_REF]. In [START_REF] Bierbrauer | A family of crooked funcions, Designs, Codes and Cryptography[END_REF], a description of the corpus of binomial crooked functions is presented. As a conclusion, Bierbrauer has conjectured that, up to equivalence, no other binomial crooked functions exist.

Crooked functions of codimension k, 1 ≤ k ≤ n -1, appeared during the cryptanalysis of a hash function, called MARACA [START_REF] Canteaut | Structural weaknesses of permutations with a low differential uniformility and generalized crooked functions[END_REF]. Here, the differential sets are affine subspaces of same codimension. The authors of this cryptanalysis, Canteaut and Naya-Placienta, introduced the crooked property. They have shown that this property could make a cryptographic primitive very weak. We will explore, in a more general context, the functions having the crooked property, in a forthcoming work.

  The differential sets S a are two by two distinct, corresponding to two by two distinct hyperplanes H λ . More precisely, to every a ∈ F * 2 n corresponds one and only one λ ∈ F * 2 n .

(ii) Any component f λ of F is a near-bent Boolean function with linear space of dimension 1, say {0, a}. When f λ is balanced, its derivative in point a is equal to 1. This holds for any f λ , when F is a permutation, (iii) F is an AB function.

Example 2 Let

Permutations from crooked functions

Any crooked function allows to construct a set of permutations, via their derivatives.

Let us define:

• λ ∈ F * 2 n , defining the hyperplane H λ , which is the image set of G a ;

• β be such that T r(λβ) = 1, i.e. β ̸ ∈ H λ ;

• H µ be any hyperplane such that T r(µa) = 1, i.e., a ̸ ∈ H µ .

Then, the function

If T r(µ(x + y)) = 0 then G a (x) + G a (y) = 0, which implies y = x + a, since G a is 2-to-1 and G a (x) = G a (x + a). In this case, we get T r(µ(x + y)) = T r(µa) = 0, a contradiction.