Jean-Luc Dion

Fatma Abid

Gaël Chevallier

Hugo Festjens

Nicolas Peyret

Franck Renaud

Moustafa Seifeddine

Cyrille Stephan

COMPACT MODEL SYNTHESIS FOR PARTIALLY OBSERVED OPERATIONAL SYSTEMS

Keywords:

This work proposes a Compact Model Synthesis (CMS) for Partially Observed Operational Systems (POOS) without using the complete knowledge of models. Series of "grey boxes" fed with partial observations are built in order to synthesize target variables with compact models. The recursive process for real time computation is based on Kalman Filters (KF). This stochastic approach allows to converge in line toward deterministic models with estimated uncertainties and without intrusion on the complete model process. Mathematical context is described first and illustrated secondly with two examples.

INTRODUCTION

Reduced Order Models (ROM) are generally built in order to make simulation processes faster.

These methods are very useful as part of an optimization process using a High Fidelity Finite Element Model parameterized by design variables or as part of the acceleration process of simulations with a high number of Degrees Of Freedom. Both objectives are quite different insofar as, for optimization, it is interesting to build parametric models of small size; while for acceleration, design parameters are useless but accuracy is the main goal. We focus on the framework of optimization even if this paper refers only to the modeling step.

Numerous methods are available to achieve this goal. We can classify the methods of Model Reduction into two families: the first ones aims to provide a basis, spanning a subspace which belongs to the space of solutions, to link the variables of the Full Order Models (FOM) to a smaller number of variables; Such methods are often called Ritz methods or "kinematic methods" (KM); The second methods aims to identify a meta-model able to provide accurate results for selected degrees of freedom of the FOM; In the following, we will call this methods "Identification Methods" (IM). Sometimes, in an optimization framework, both kinds of methods are used: the first one to reduce the number of variables, the second one to build a parametric reduced order model, see for example R. Filomeno et al. [START_REF] Coelho | Model reduction for multidisciplinary optimization -Application to a 2D wing[END_REF]. To reduce the complex systems constituted with several components, it is rather commonplace to reduce each component with KM and to make the synthesis of the system, see for example Craig [START_REF] Craig | A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods[END_REF] or Guyan [START_REF] Craig | A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods[END_REF]. The Way to Build Ritz base has been widely discussed in the previous work and we can consider two methods: modal approaches and optimal approaches based on proper orthogonal decomposition, see for example Balmès [START_REF] Balmès | Optimal Ritz vectors for component mode synthesis using the singular value decomposition[END_REF]. The non-linear problems involve specific Model Order Reduction Techniques, see for example Kerschen et al. [START_REF] Kerschen | The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview[END_REF]. The final challenge is to build ROM parameterized by design variables, see for instance Amsallem et al. [START_REF] Amsallem | A Method for Interpolating on Manifolds Structural Dynamics Reduced-order Models[END_REF] or Balmès [START_REF] Balmès | Parametric families of reduced finite element models. Theory and applications[END_REF].

This work purpose is to build ROM based on observation of operating systems. "Operating Systems" could be models during simulations such as thermal Finite Element Models (FEM) [START_REF] Abid | System identification using Kalman Filters[END_REF] or operating experimental devices such as jointed structures in free vibration [START_REF] Quinn | Modal analysis of jointed structures[END_REF] and [START_REF] Festjens | Model Order Reduction of assembled Structures for dynamic analysis[END_REF] or motion picture of mechanical system. Proposed ROM are based on Series of Elementary Generic Models (SEGM) considered as Parametric Grey Boxes (PGB) which had to be identified during operating process. A priori, the length of SEGM is unknown but the nature of PGBs is chosen regarding the observed phenomenon. For example, thermic study are described with first order Ordinary Differential Equations (ODE) in PGBs whereas dynamics are described with second order ODE. Operating conditions are often imposed for several problems: Operating Modal Analysis (OMA) of helicopters or Energy Production Power Supply is performed during operating conditions. Survey and diagnostic for help to decision in automated manufacturing are sometimes carried out in real time in the process. Motion picture of mechanical scenario or FEM are composed of large number of observable variables. In that case, ROMs built in real time allow saving computational time and allocated memory. For these reasons KF are retained in the recursive parametric identification process of PGBs.

KALMAN FILTERS AND SERIES OF ELEMENTARY GENERIC MODELS

CMS is performed with the help of three different kinds of KF [START_REF] Kalman | A new approach to linear filtering and prediction Problems[END_REF]

State Systems

Three kind of state systems are studied in this paper. The first one concerns DLSS which could be writen as :

1 . . + = + + n n n n n n X F X B U W (1) . = + n n n n Z H X V (2)
where : When the discrete system could not be written as a linear system (with invariant matrix) the DNLSS is assumed as :

X
( )

1 , + = Φ + n n n n X X U W (3) ( ) = + n n n Z H X V ( 4
)
with : φ Non linear state function H Non linear observation function

In the case of linear continuous time systems the equations are :

( ) ( ) ( ) . . ( ) = + + X t F X t B U t w t (5) ( ) ( ) ( ) . . ( ) = + + Z t H X t D U t v t (6)
For non linear systems, CSS is described in a general sense as follows :

( ) ( ) ( ) ( ) , ( ) = Φ + X t X t U t w t (7) ( ) ( ) ( ) ( ) , ( ) = + Z t H X t U t v t (8)
All these systems are suitable for KF, EKF or UKF.

Kalman Filters

The KF and EKF has been widely used over the past decades in the case of discrete systems. In this formulation the state variables are extended to the parameters of the PGBs that are to be identified. It allows to compute an estimation of the PGBs in the case where the shape of the solution of the problem is already kwown. For instance, in the case of oscillating systems, the time evolution is pseudo-harmonic (see section 2.3); in the case of relaxation system, such as thermodynamics systems, the evolution is pseudo-exponential (see section 2.4). The incrementation of the EKF requires the linearization of the estimated evolution which can lead to a strong approximation in the case of non-smooth non-linear systems. In order to avoid this linearization process Julier and Uhlmann (1997) have developed the "unscented kalman filter" UKF [START_REF] Dormand | A family of embedded runge-kutta formulae[END_REF]. In that method, the estimation of the state variables and their variances are directly incremented thanks to the unscented transformation (UT). This function can be more generally used to estimate the result of applying a given nonlinear transformation to a probability distribution. The two first moments of the distribution are simply propagated by applying the nonlinear function to a set of point referred to as sigma points. Eliminating the need for linearization also provides advantages independent of any improvement in estimation quality. One immediate advantage is that the UT can be applied to identify the parameters of a Non-linear ODE for which the solution cannot be analytically solved. Practically, the implementation of the UKF is also easier than EKF.

Series of Elementary Generic Models and bases : oscillating systems

Second order ODEs often leads to oscillating solutions. In the case of non linear systems oscillating solutions are amplitude and frequency modulated.

A sinusoid whose amplitude and frequency are modulated over time can be described in the complex domain as an analytic signal :

( ) ( ) ( ) ( ) φ = x t a t j t exp (9)
where a(t) is the instantaneous complex amplitude and Ф(t) is the instantaneous phase. The discrete form of x(t) at the time step t n =n∆t is x n =x(n∆t). The complex variable x n can be divided into its real and imaginary parts x n =x 1,n +jx 2,n . A sinusoid that slightly varies over time can be approximated by :

( ) ( ) π = ∆ n n n x a j f t n exp 2 (10)
where f n is the instantaneous frequency.

As the parameters fn and an of the sinusoid slightly vary over time, they are almost equal between two consecutive time steps. Then a transition formulation can be given from x n to x n+1 :

( ) ( ) ( ) ( ) π π + ≈ ∆ ∆ × n n n n x a j f t j f t n 1 exp 2 exp 2 (11)
This approximation is only true if modulations of an and fn are slower that the period of the sinusoid. This constraint is assumed to be verified. Then a linear transition is obtained between the imaginary part x 2 ,. and real part x 1 ,. of x n and x n+1 :

( ) ( ) ( ) ( ) π π π π + + ∆ - ∆ = ∆ ∆                     n n n n n n n n f t f t x x x x f t f t 1, 1 1, 2, 1 2, cos 2 sin 2 sin 2 cos 2 (13)
This formulation choice allows keeping the matrix Fn linear and time-independent if the signal is only amplitude modulated and not frequency modulated (the frequency fn keeps constant).

The instantaneous amplitude an is .

n ,n ,n a x x = + 2 2 1 2 (14)
As an and fn should be allowed to vary over time, it is proposed here to use the following non linear state space formulation : X n+1 =Φ(X n )+W n (15) where X n =(x 1,n x 2,n x 3,n ) T and W n , the process noise. A state variable x 3,n =2πf n ∆t was added to track the evolution of the instantaneous frequency fn. The transition between two time steps is composed by a sum of two parts: the stationary part and the evolutionary part. The stationary part links two successive points of a stationary sinusoid by Φ(.). Then Φ(.) is assumed as the non-linear transition function and is given by Φ(X n )=F n X n :

where

( ) ( ) ( ) ( ) ,n ,n n ,n ,n x x cos sin F x x sin cos   -   =       3 3 3 3 0 0 0 0 1 (16)
The two first components are related to the complex amplitude and are obtained by the previous linear relation.

The third component x 3,n+1 =x 3,n constrains the frequency not to change strongly between two time steps.

Up to now, the non-stationary behavior of the sinusoid was not modeled because it is not possible to express an exact equation for this evolution. We suppose that W n is a random variable whose probability law is Gaussian : W n =N(0,Q), where Q is its variance matrix. Then the variations of amplitude and frequency are allowed by random values of W n . In a first glance, it could seem strange to choose a random variable for an effect which is generally deterministic. For instance, the variation of frequency excitation of an engine is mainly deterministic. Anyway, this state space does not need to represent accurately the evolution of a sinusoid on a long period, but only step by step. Then on a short time scale, a random evolution of an and fn is enough to model a non-stationary sinusoid. For a signal composed of M observable modulated sinusoidal components, the size of the ROM is M and the size of the state function is 3M. and

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
+ - + - + - + -   -           =     -           n n n n n n M n M n M n M n x x x x F X x x x x 3, 3, 3, 3, 3 3 1 , 3 3 1 ,
+ - + - + -             =             n n n n M n M n M n x x x X x x x 1, 2, 3, 1 3 1 , 2 3 1 , 3 3 1 ,
This state space formulation is non linear: the state matrix F n (X n ) is varying over time and depends on the frequency modulation.

In reality, only the real part x 1,n of the analytic signal x n can be observed. Unlike the transition phase, the observation phase is completely linear

[ ] = + = + n n n n n n Z X V Z x V 1, 1 0 0 (18)
for one sinusoid and slightly more complex for M

sinusoids [ ] ( )

+ - = + = + + + n n n n n n M n Z X V Z x x V 1, 1 3 1 1 0 0 1 0 0 (19)
where V n is a noise observation random process.

Finally, a nonlinear discrete state space model has been derived to model the transition and observation of sinusoid components mixed with random processes ( ) ( )

+ + + + = Φ + = +    n n n n n n X X W Z H X V 1 1 1 1 (20)
where Φ(.) is the nonlinear transition function given by Φ(X n )=F(X n )X n and H(.) is the linear observation function given by :

H(X n+1 )=[1 0 0 … 1 0 0]X n+1 . ( 21
)

Series of Elementary Generic Models and bases : non oscillating systems

When the system is compose with first order ODEs or assumed as a sum of monotone responses, series of negative exponential are used. The continous state equation of the elementary generic model with control is

( ) ( ) ( ) ( ) 0 0 exp x t x aU t t aU δ = - - - + (22)
With :

x 0 , t 0 assumed as initial conditions, a and δ are constant to be identified. The DLSS of this equations is written as :

( )

1 e e 1 t t n n n x x U a δ δ -∆ -∆ + = + - (23)
With ∆t the time increment. The DNLSS of this equations is written as :

( )

1 e e 1 n n t t n n n n x x U a δ δ -∆ -∆ + = + - (24)
The complete SEGM is built as follows for the non linear state matrix and control matrix :

( ) ( ) 2, 2 3
1 , e e 0 0 0 0 0 0 1 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 n M n x t n n x t F X + - - ∆ - ∆           =               (25)
and

( ) ( ) ( ) ( ) 2, 2 3 1 , 3, 3,
e e 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n M n x t n n n x t n x B X x + - - ∆ - ∆   -           =     -           (26)
The observation equation is the same for oscillating an non oscillating systems. and the complete DNLSS is :

( ) ( ) ( ) 1 1 1 1 n n n n n n n n n n X X X W Z H X V F B U + + + + = = + + +    (27)

APPLICATIONS

Two kinds of applications are presented. The first one concerns oscillating systems with amplitude and frequency modulated signals tracked with EKF for CMS.

The second study is in the field of thermodynamic and is treated with UKF.

Oscillating systems and solutions of second order ODEs

Tracked signals are amplitude and frequency modulated.

The CMS is performed with SEKF based on SEGM. The expected ROM is composed with two EGM. The first observed component is around 22 Hz and the second around 50 Hz. For each component, the amplitude modulation is determined with the instantaneous amplitude:

n ,n ,n a x x = + 2 2 1 2 (28)
This amplitude could also be obtained by the complex analytic signal composed of the observed signal (real part) and its Hilbert transform (imaginary part). The frequency modulation is determined by the instantaneous frequency :

,n n x f π∆t = 3 2 (29)
For each tracked signal, spectral amplitudes show the main component and lateral modulated bands (Figure 1). The first tracked signal is frequency and amplitudemodulated (figure 1 and figure 2). The second tracked signal is only amplitude-modulated and not frequencymodulated.

Continuous State Systems

System identification has been based largely on discretetime models for a long time in the past, ignoring certain merits of the native continuous-time models.

In this section, we are interested in the procedure of determining directly a continuous-time model of a dynamical system from observed data. That means, the differential-equation model is valid whatever the command nature or sampling data considered. We will consider first linear time-invariant (LTI) continuous system. As a simple illustration of this stage, let us consider a first-order system described by a CLSS as follow:

{ } [ ]{ } [ ] { } [ ]{ } ( ) ( ) ( ) ( ) ( ) x t F x t B u t z t C x t = + = (30)
In particular:

{ } { } { } { } 1 1 1 2 2 2 1 11 12
21 22 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x t a b x t x t u t a b x t z t c c z t x t c c z t       = = +                 = =         (31)
where X(t) is the state vector, Y(t) the observation vector, U(t) the input vector, a1 and a2 invariant parameter system, b1 and b2 the parameters that locate the input vector to the state vector variables, and c11, c12, c21 and c22 the observation parameters.

The conversion of the Continuous system (30) into a recursive discrete system is performed using either EKF by means of exponential discretization technique [START_REF] Boyce | Elementary differential equations and boundary value problems[END_REF], (Annex A) or UKF using an implicit numerical integration method named Dormand-Prince method [START_REF] Dormand | A family of embedded runge-kutta formulae[END_REF], [START_REF] Mathews | Numerical methods using MATLAB[END_REF].

The resulting discrete-time Kalman model takes the following form:

1 1 1 1 1, 1 2, ( , , ) ( , )
- - - - -     = =             = =       k k k k k k k k k k k k k k x u x X 2, 11 , 12, 21, 22, 1 2 ∆
∆       =               EKF t a T a T k k k k k k k k k k k b b X x x e e c c c c a a (33) , 1 , 2, 1, 2, 1, 2, 11, 12, 21, 22,
  =   UKF k t k k k k k k k k k k X x x a a b b c c c c (34)

Numerical results

This section presents numerical simulation results aiming to evaluate the performance of the EKF and UKF methods regarding system identification based on data produced numerically by the reference model (30). The forcing term, u, is a square signal.

In the reference problem, (30) and (31), the terms [F], [B] and [C] are assumed known and {x} is determined through a numerical integration method implemented in Matlab (Ode23; Runge-Kutta method).

In the Figure 3 and4, we compare the response of the reference model (31) at points observation, Z, to the response of the identified system model (33), Z k , using Kalman filters, EKF and UKF, respectively,to evaluate its accuracy at five different times, 2 , 4 , 6 ,8 ,10 t s s s s s ≈ . Both EKF and UKF produce a good accuracy with almost equivalent convergence speed in that case, e.g. CLSS.

In the case of continuous non linear system, such as timedepend parameters a 1 and a 2 , only UKF is appropriate. To process it, we use the same model (30) and the UKF algorithm is unchanged in comparison with linear case. Here, the advantage of UKF as regards the implementation simplicity is highlighted with respect to the EKF. This latter methodology actually is not extended to the processing of nonlinear continuous systems due to the difficulty related to the analytical discrete model construction step starting for CNLSS.

CONCLUSION

Kalman Filters have been used successfully for compact model synthesis of oscillating and non oscillating systems.

The proposed method could be performed in line and with only few information about the complete model. The main challenge today is to work out an automatic method for initializing KF with robustness.

ANNEX A DETAILS OF FIRST ORDER STATE SYSTEM

The solution of Eq. 22.a on the time interval [tk tk+1] is:

Figure 1 :

 1 Figure 1 : Spectrums of two tracked signals. Symmetric lateral bands are due to modulation effects on spectrums

Figure 2 :

 2 Figure 2 : Time-frequency representation of the first tracked signal. Magnitude and frequency modulation are identified with a good accuracy. (Instantaneous frequency is identified with 4 significant digits).

Figure 3 :

 3 Figure 3: Comparison of response of reference model (black solid curve) to the response of identified system model at five times (starting points) using EKF; left, point observation 1, and right, point observation 2.

Figure 4 :

 4 Figure 4: Comparison of response of reference model (black solid curve) to the response of identified system model at five times using UKF; left, point observation 1, and right, point observation 2.

  Supposing u(t) constant over the sampling interval [tk tk+1], the discrete state space model is written as follows:


J P Pansart
email: jean-pierre.pansart@orange.fr

Electromagnetic field in 3 dimensionnal compact hyperbolic manifolds

Compact three dimensional spaces of constant curvature without border can be considered as cavities for free electromagnetic fields. For spacetimes of the form : t ⊗ staticspace , this note shows that the electromagnetic field spectrum is the same as the Laplacian eigenvalue spectrum of a scalar field in those spaces.

Introduction

An attempt to compute numerically the first Laplacian eigenvalues of a scalar field in 3-dimensional constant curvature compact hyperbolic manifolds without border, has been presented in [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF]. These geometries can be considered as cavities for free electromagnetic fields. The goal of the present note is to extend the results obtained in the scalar field case to the free electromagnetic field and to show that the eigenvalue spectra are the same. The space manifolds considered here, which are called M , are isometric to the quotient H n /Γ where H n is the n-dimensional constant curvature hyperbolic space (Appendix A), and Γ is a group of isometries of H n called the covering group in the following (in this note n = 3). For a complete study of constant curvature spaces see [START_REF] Wolf | Spaces of Constant Curvature[END_REF] . In this note the space-times in which the fields evolve are of the form : V = t ⊗ H 3 , V = t ⊗ M , where t is the time coordinate.

In the case of the scalar field there are, at least, two ways to compute the first Laplacian eigenvalues.

-Find a set of common eigenvectors of the Laplacian ∆ in H 3 and one generator of the covering group Γ, which will be named γ 0 , then expand the eigenvectors of the Laplacian in M on this set, and then impose the other periodicity constraints.

-Chose a Γ periodic test function in H 3 and use the Rayleigh theorem [START_REF] Chavel | Eigenvalues in Riemannian Geometry[END_REF] to find bounds on the eigenvalues. Another numerical method has been presented in [START_REF] Inoue | [END_REF]. The first method was chosen in [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF]. It could be directly applied to the case of 1 vector fields but this becomes complicated. The main part of this note shows that the Laplacian eigenvalue spectra of a scalar field and a free electromagnetic field are the same without calculating these eigenvalues. Once the eigenvalues are known it is easier to compute numerically the eigenvectors. The necessary elements are provided in appendices, but no attempt has been done to compute the eigenvectors explicitly. The identity of the spectra is shown in sections 2, 3 and 4. The technique used is to express scalar and vector fields as tensors built from spinors. Appendix A shows that, using cylindrical coordinates, the γ 0 periodicity condition are the same for scalar, vector or spinor fields.

The first section sets the notations and recalls very briefly some basic geometrical equations. Some technical calculations have been gathered in appendix B. Appendix C, although a little bit long for our purpose, discusses the advantages of considering the second order Dirac equation. At last, Appendix D provides solutions to the free electromagnetic field equations in V = t ⊗ H 3 which can be be used to build eigenvectors in M = H 3 /Γ , and discussed the existence of "plane wave" solutions .

1

Notations.

The space-time coordinates {x α } of a point x are labelled with Greek letters : α , β , γ ... , 0 ≤ α , β , γ, ... < n . The time coordinate is : x 0 . The vectors of the local natural frame are written : -→ e α , -→ e β , ... . When tensors are expressed with respect to local orthonormal frames they are labelled with Latin letters : a , b , c ... . The orthonormal local frame basis vectors are called :

-→ h a , and we set :

-→ h a = h α a -→ e α . The metric tensor is g αβ , and g αβ is its inverse. The determinant of the metric tensor is called g . The signature of the metric is : (+ ---) . In the case of local orthonormal frames, the metric tensor is written : η a b and its diagonal terms are : η aa = (+1 , -1 , -1 , -1) . Latin indices are lowered with η a b and raised with the inverse tensor η a b .

In the neighborhood of a given point, the local coordinates, with respect to the local orthonormal frame attached to this point, are given by the 1-forms : ω a = h a α dx α , which satisfy the structure equations :

dω a + ω a . b ∧ ω b = Σ a (1.
1) where : ω a . b = ω a . b γ dx γ are the connexion 1-forms and Σ a is the torsion 2-form. We shall also write:

ω a . b = ω a . b c ω c ↔ ω a . b c = ω a . b γ h γ c .
The connexion 1-forms are related to the connexion coefficients by :

ω a . b γ = Γ α . β γ h a α h β b + h a δ ∂ γ h δ b
The connexion coefficients are the sum of two terms :

Γ α . β γ = Γ α . β γ + S α . β γ
where the first term on the right is the Christoffel symbol and the second is the contorsion tensor. The contorsion is anti symmetric with respect to the two first indices : S α β γ + S β α γ = 0 . The torsion tensor is :

S α . β γ = 1 2 (Γ α . β γ -Γ α . γ β ) = 1 2 (S α . β γ -S α . γ β ) and inversely: S α . β γ = S α . β γ -S α β . γ -S α γ . β ; S α β . γ = g α δ S β δ γ The torsion 2-form is: Σ a = Σ a . b c ω b ∧ ω c = -h a α S α . β γ dx β ∧ dx γ We set : Γ a . b γ = Γ α . β γ h a α h β b + h a δ ∂ γ h δ b
The curvature 2-form is defined by :

Ω a . b = dω a . b + ω a . c ∧ ω c . b = R a . b c d ω c ∧ ω d 2 Outline.
In this note the electromagnetic field, which will be noted A α , is studied in static four dimensional space-times of the form :

V = t ⊗ H 3 , V = t ⊗ M .
In H 3 , elementary rigid motions, which are not symmetries, are either rotations around an axis or transvections (which generalize translations) along a geodesic called base geodesic. Each element of the covering group is the product of a rotation and a transvection having the same axis. Isotropic spaces are symmetric spaces and have no torsion. This applies to the spatial part only, but in the following calculations, the space -time torsion is set to S α .βγ = 0 . However, in appendices B and C torsion has been re-introduced to get more general relations.

Cosmology studies the evolution of universes whose unperturbed metric is spatially isotropic and of the form [START_REF] Landau | Classical Theory of Fields[END_REF] :

ds 2 = a 2 (η) dη 2 + γ ij dx i dx j
where η is called the conformal time and the indices i, j correspond to spatial coordinates. For these cosmological metrics, the equations of motion of scalar and vector fields show that the technique of variable separation can be used (separation of the conformal time from the spatial coordinates) if these fields are expanded on the eigenvectors of the 3-dimensional spatial Laplacian operator 3 ∆ . This justifies the form of the space-times V chosen above.

The most direct approach would be to proceed as in [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF], that is to say : find solutions of the electromagnetic field equations in V = t ⊗ H 3 which are also eigenvectors of one generator of Γ , and then impose the periodicity conditions corresponding to the other elements of the covering group. As in [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF], it is difficult to avoid using numerical method, which is not satisfactory. Appendix A shows why cylindrical coordinates are well suited to the search of solutions for vector and spinor fields.

Therefore we shall proceed as follows. Given a set of tensor fields satisfying some relations between them, it was shown in [START_REF]On the algebra of Dirac spinor densities: Factorization and inversion theorems[END_REF] that there exist a spinor field ψ such that they can be written as ψγ h ψ with : γ h = γ α1 γ α2 ... γ α h , where all the γ αi (1 ≤ i ≤ h) are different. In the following, we shall assume that scalar fields and vector fields can be written respectively (up to a multiplicative constant) as product of spinor fields of the form : ψϕ and ψγ α ϕ . One can show that interaction tensors of the form ψγ h ϕ satisfy also the relations (1.3) and (1.2) of [START_REF]On the algebra of Dirac spinor densities: Factorization and inversion theorems[END_REF] . In [START_REF]On the algebra of Dirac spinor densities: Factorization and inversion theorems[END_REF] , the construction does not require that the spinor ψ satisfies the Dirac equation. In section 3 and 4 it is assumed that ψ and ϕ satisfy the second order Dirac equation (Appendix C), and we shall look at the constraints on the spinors resulting from the scalar and electromagnetic field equations and show that such a construction is possible by comparing the number of constraints and the number of degrees of freedom.

If the scalar field S = ψϕ is an eigenvector of the Laplacian in H 3 the spinors ψ and ϕ must satisfy some constraints. This is shown in section 3. The vector field a α = ψγ α ϕ can not represent directly the electromagnetic field A α . A gauge change must be applied in order to satisfy the field equations and the Lorenz condition D α A α = 0. In section 4 we show that if : A α = a α + g αβ ∂ β f with f = µ ψϕ and if ψϕ is an eigenfunction of the 3-dimensional space Laplacian, then the electromagnetic field equations and the Lorenz condition can be satisfied and that they represent 4 constraints. In section 4 it is shown that the total number of constraints is less than the number of degrees of freedom, therefore it is possible to find ψ and ϕ such that the field equations are satisfied.

If the scalar field S (eigenvector of the spatial Laplacian) satisfies the periodicity conditions in H 3 for some frequencies (eigenvalues), then this must also be true for the spinors ψ and ϕ . Therefore this will also be true for the electromagnetic field A α , with the same frequencies, and conversely, showing that the scalar and the electromagnetic fields have the same spectra if M = H 3 /Γ .

3

The scalar field.

We first consider a free scalar field S whose Lagrangian is :

L = ∂ α S + g αβ ∂ β S -m 2 S + S . The Euler-Lagrange equations are : ∆S = 1 √ g ∂ α √ gg αβ ∂ β S = -m 2 S
We assume that S can be written as the product of two spinors : S = ψϕ (3.1) where ψ and ϕ satisfy the second order Dirac equation (C.16) with masses m 1 and m 2 respectively . Note that (3.1) is not the only scalar that one can build from two spinor fields. For instance we could have considered : S = aψ ϕ + bD α ψg αβ D β ϕ , where a, b are constant parameters, but this brings no simplification and no cancellation in the calculations.

Equations (B.4) and (C.16) give :

∆S = D α g αβ D β S = = R 2 -m 1 2 -m 2 2 ψϕ + 2g αβ D α ψD β ϕ (3.2)
If : ψ, ϕ ∼ e i ω t , S is independent of t . Then we chose : ψ ∼ e ± iωt ϕ ∼ e ∓iωt then :

3 ∆(ψ ϕ) = -4ω 2 ψϕ + m 1 2 + m 2 2 -R 2 ψ ϕ -2g αβ D α ψD β ϕ (3.
3) where 3 ∆ is the 3-dimensional Laplacian operator for M or H 3 .

We are interested by scalar fields which are eigenfunctions of the Laplacian on the manifold M : 3 ∆f = -λf , then one has the condition :

2g αβ D α ψD β ϕ ∼ ψϕ (3.4)
Remark : In [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF] and in many publications, the eigenvalue problem is written as :

3 ∆ f = -(1 + β 2 ) f which is convenient if spherical coordinates are used.
If M is a manifold without border, then for any function h (sufficiently well behaved) :

M ( 3 ∆f ) + hdV = -λ + M f + hdV = M ∂ α ( √ gg αβ ∂ β f + )hd 3 x = M ∂ α ( √ gg αβ ∂ β f + h)d 3 x -M g αβ ∂ β f + ∂ α h √ gd 3 x
The first integral is zero since M is assumed without border. If we set : h = Ct , then for λ = 0 , one has the constraint : M f dV = 0 . Is this condition compatible with the hypothesis (3.1) ? Let us consider : ψD 2 ϕ where D = γ α D α is defined in appendix C . Using (B. If : ψ ∈ D ± (see Appendix C) and ϕ ∈ D ∓ , then : M ψϕdV = 0 which is consistent with the above constraint.

4

The vector and electromagnetic fields.

The equation of motion of a vector field a α with mass m can be deduced from the Lagrangian :

L V = g αβ g µν D α a µ D β a ν -m 2 a µ a µ + L S
where L S represents source terms. There are other possible Lagrangians. The Euler-Lagrange equations are :

g αβ D α D β a µ -R µ .β a β = -m 2 a µ + J µ (4.
1) where J µ represents source terms. In the following we consider only free fields, then : J µ = 0 . The Lagrangian of the electromagnetic field A α is :

L A = -1 4 F αβ F αβ + L S where : F αβ = ∂ α A β -∂ β A α = D α A β -D β A α (
the last equality is true when there is no torsion). The corresponding equations of motion are :

g αβ D α D β A µ -R µ .β A β -g αµ D α D γ A γ = J µ (= 0) (4.
2) The gauge freedom is used to impose the Lorenz condition :

D γ A γ = 0 .
In the differential form formalism the Laplacian operator is defined as : ∆ ≃ dδ + δd where : δ is the adjoint of the differentiation operator d . Applying this definition to the 1-form ω = a α dx α gives :

-

∆a α = g γβ D γ D β a α -R α .β a β where : D β and R α
.β are the covariant derivative and the Ricci tensor computed with the Christoffel symbols only instead of the connexion coefficients, as if there were no torsion, although this expression is valid also with torsion. Note that : ∆d = d∆ and ∆δ = δ∆ , two properties which will be used later.

We assume that one can write : a α = ψγ α ϕ where, as for the scalar field, ψ and ϕ satisfy the second order Dirac equation with masses m 1 and m 2 respectively. Then, using (B.4) and (C.16) , one has :

∆ a µ = R 2 -m 2 1 -m 2 2 ψγ µ ϕ -R µ .ν ψγ ν ϕ + 2g αβ D α ψ γ µ D β ϕ (4.
3) If one wants a α = ψγ α ϕ to represent the electromagnetic field, the Lorenz condition must be satisfied. Using the same calculation as for (B.7), we have :

D α a α = D α (ψγ α ϕ) = Dψ ϕ + ψDϕ If, as in the scalar case, ψ ∈ D ± and ϕ ∈ D ∓ then : D α a α = ∓i(m 1 + m 2 )ψϕ (4.
4) It is now assumed that the electromagnetic field A α is equivalent to the vector field a α up to a gauge change :

A α = a α + g αβ ∂ β f whence : D α A α = D α a α + ∆f = ∓i(m 1 + m 2 )ψϕ + ∆f If : f = µ ψϕ
where µ is a constant coefficient, the Lorenz condition looks like an eigenvalue condition. Let us assume that ∆(ψϕ) = λψϕ then the Lorenz condition is satisfied if : µλ = ±i(m 1 + m 2 ) , and the electromagnetic field is :

A α = ψγ α ϕ + g αβ µ∂ β (ψϕ) (4.
5) The free electromagnetic field equation is then such that : ∆a µ +∆(g µβ ∂ β f ) = 0 As seen above the Laplacian of a gradient field is equal to the gradient of the Laplacian, then using (4.3), the field equations are :

∆ A µ = R 2 -m 2 1 -m 2 2 ψγ µ ϕ -R µ .ν ψγ ν ϕ +2g αβ D α ψγ µ D β ϕ + g µβ µ∂ β (∆(ψ ϕ)) = J µ (4.6)
These equations represents 4 constraints. Are they independent and are they compatible with the Lorenz condition ? Since the divergence operator commutes with the Laplacian ( ∆δ = δ∆ ), we expect :

D µ J µ = D µ ∆A µ = ∆D µ A µ = 0 .
This is now checked directly. We first compute D µ J µ U where :

J U µ = g αβ D α ψγ µ D β ϕ . Using (B.7
) and (B.9) one has :

D µ J U µ = g αβ D T D α ψ D β ϕ + D α ψ D T D β ϕ D µ J U µ = 1 4 g αβ R cdγα ψγ dc γ γ D β ϕ + D β ψγ γ γ cd ϕ +g αβ D α DψD β ϕ + D α ψD β Dϕ With : γ cd γ e =
η de γ c -η ce γ d + γ cde and : γ e γ cd = η ce γ d -η de γ c + γ ecd and using the Bianchi identities of the first kind in the first bracket, it remains :

D µ J U µ = 1 2 R β c ψγ c D β ϕ + D β ψγ c ϕ + g αβ D α DψD β ϕ + D α ψD β Dϕ
Finally, with (B.3) :

D µ J U µ = 1 2 R β c D β (ψγ c ϕ) + g αβ D α DψD β ϕ + D α ψ D β
Dϕ where, according to our hypotheses : R µ .ν ∼ δ µ ν . Replacing this term in the divergence of (4.6) and using (4.4) gives :

D µ J µ = R 2 -m 2 1 -m 2 2 (∓)i(m 1 + m 2 )ψϕ +2g αβ D α DψD β ϕ + D α ψD β Dϕ + µD µ (g µβ ∂ β (∆(ψϕ)))
With ψ ∈ D ± and ϕ ∈ D ∓ and with :

D µ (g µβ ∂ β (∆(ψϕ))) = λD µ (g µβ ∂ β (ψϕ)) = λ∆(ψϕ) we obtain : D µ J µ = (∓)i(m 1 + m 2 ) R 2 -m 1 2 -m 2 2 ψϕ + 2g αβ D α ψD β ϕ -∆(ψϕ) which, with condition (3.2) gives : D µ J µ = 0 .
Therefore, the four equations (4.6) are compatible with the Lorenz condition and are in fact three independent equations only.

The constraints used are now recalled. The spinor fields ψ and ϕ are such that S = ψϕ , and are solutions of the Dirac equation. This is 1 + 4 x 2 constraints. Then the condition (3.4) is necessary if S is an eigenfunction of the spatial Laplacian. The vector (4.5) must satisfy the four equations (4.6) or three of them plus the Lorenz condition, that is four conditions. The total amounts to 14 conditions, the two spinors representing 16 real functions.

5 Appendix A. Coordinates and transport of local frames.

Hyperbolic spaces.

The hyperbolic n dimensional space H n is defined as the upper part of the sphere of radius |K| in the Minkowski space M n+1 . More precisely, if {x α } are Cartesian coordinates in M n+1 with origin O M , H n is the surface defined by: n-1 α=0

x α x α -x n x n = K where: K < 0 and : x n ≥ |K| . We set: R = |K| . The correspondance between the spherical coordinates (χ, θ, ϕ) of H 3 , where (θ, ϕ) are the usual polar angles with respect to some local orthonormal frame Oxyz and the coordinates of M 4 is :

x 0 = R shχc , c = cos(θ) , s = sin(θ) x 1 = R shχ s c ϕ ; c ϕ = cos(ϕ) , s ϕ = sin(ϕ) (A.1) x 2 = R shχ s s ϕ x 3 = R chχ χ ≥ 0 θ ∈ [0 , π] ϕ ∈ [0 , 2 π] Then the linear element of H 3 is: ds 2 = (dx 0 ) 2 + (dx 1 ) 2 + (dx 2 ) 2 -(dx 3 ) 2 = R 2 dχ 2 + sh 2 χ ( (dθ) 2 + s 2 (dϕ) 2 ) (A.
2) The coordinates (χ, θ, ϕ) are the Riemann normal (spherical) coordinates with origin at χ = 0 , which corresponds to the point (0, 0, 0, R) in M 4 . The curvature tensor is:

R α β γ δ = -1 R 2 (g α γ g β δ -g α δ g β γ )
, the Ricci tensor is :

R α β = -2 R 2 g α β ,
and the scalar curvature:

R H = -6 R 2 .
R is a scale factor, in the following it is set to 1.

H n , which is a space of constant curvature, is a symmetric space. A transvection in a symmetric space is an isometry which generalizes the notion of translation in Euclidean space. It is defined as the product of two successive symmetries with respect to two different points A and B . The geodesic going through these two points is invariant and is called the base geodesic. In H 3 one can perform a rotation around this base geodesic, it commutes with the transvection and the base geodesic is invariant. The base geodesic of a transvection γ is given by the intersection of the invariant plane, associated to the real eigenvalues of the SO(3, 1) element representing γ in M 4 , with H 3 .

The elements of the covering group Γ are screw motions. A screw motion is the product of a transvection by a rotation around the base geodesic of the transvection.

Since H 3 is isotropic, it is always possible to chose a particular generator of Γ such that its base geodesic defines the Oz axis of the spherical coordinates. This particular generator is named γ 0 in the present note.

The Laplacian operator acting on a form ω is defined as ∆ ≃ dδ + δd where δ is the adjoint of the differentiation operator d . ω of degree 0 corresponds to the case of a scalar field, and ω of degree 1 to the case of a vector field. The Laplacian operator commutes with all the isometry. It is then possible to find a common set of eigenvectors to ∆ and one of the generator of Γ (only one, because the generators of Γ do not, a priori, commute between themselves). The action of rotations around the Oz and Ox axis, and transvections along the Oz axis, on the vectors of local frames, has been described in ( [START_REF]A Clifford algebra gauge invariant Lagrangian for gravity. Part 2. Compatibility with General Relativity tests[END_REF] appendix A). If we call L the length of the transvection, which is twice the distance between A and B , a point p whose spherical coordinates are : (χ, θ, ϕ) is transformed, by a transvection along Oz , into a point q of coordinates : (χ q , θ q , ϕ q ) given by : ch(χ q ) = ch(χ) ch(L) + sh(χ) sh(L) c c q = (ch(χ) sh(L) + c sh(χ) ch(L))/ sh(χ q ) (A.3) ϕ q = ϕ where : c = cos(θ) (as in (A.1)) and c q = cos(θ q ) .

Depending on the problem it could be more convenient to use cylindrical coordinates as described in [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF]. The axis of these cylindrical coordinates is chosen to be the same as the polar axis of the spherical coordinates. These coordinates are named : z which is defined as the distance between the origin of the coordinates and the orthogonal projection of a point on the geodesic Oz , ρ the radius equal to the distance between the point and its projection , ϕ the cylindrical angle, the same angle as for spherical coordinates.

The correspondance between the cylindrical coordinates of H 3 and those of the Minkowski space M 4 is given by : x = (shρ c ϕ , shρ s ϕ , chρ shz, chρ chz) , where : c ϕ and s ϕ are defined above in (A.1) . The metric is :

ds 2 = dρ 2 + sh 2 (ρ)(dϕ) 2 + ch 2 (ρ)(dz
    x ′0 x ′1 x ′2 x ′3     =     c α -s α 0 0 s α c α 0 0 0 0 chL shL 0 0 shL chL         x 0 x 1 x 2 x 3    
where :

c α = cos(α) , s α = sin(α) .
The vectors e α of the natural local frames are given in M 4 by : e 1 = (chρ c ϕ , chρ s ϕ , shρ shz, shρ chz) e 2 = (-shρ s ϕ , shρ c ϕ , 0, 0) e 3 = (0, 0, chρ chz, chρ shz)

It is now easy to find how such screw motions along the Oz axis act. Using cylindrical coordinates, a screw motion along the Oz axis transforms the local frame basis vectors as :

γ 0 (h a (x)) = h a (γ 0 x) (A.4)
In the case of the spherical coordinates, the transported frame would be equivalent to the local frame up to a rotation since rotations and transvections are isometries of H n . The advantage of using cylindrical coordinates is the following. A vector field can be written as : V (x) = V a (x)h a (x). If this field is periodic under the action of γ 0 , only the components have to be periodic, there is no additional rotation to take into account. These components can be expanded on fonctions of the form :

V a ∼ e i(νϕ+kz) with the constraint : να + kL = 2πm where m is a relative integer, as for the scalar case ( [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF]).

Likewise, spinors are defined with respect to (space-time) local orthonormal frames, therefore, using cylindrical coordinates, γ 0 periodicity conditions apply only to the spinor components.

Spherical spaces.

The spherical n dimensional space S n is defined as the sphere of radius R in the Euclidean space E n+1 . For S 3 in E 4 :

x 0 = R sin χ c x 1 = R sin χ s c ϕ x 2 = R sin χ s s ϕ x 3 = R cos χ (A.5) χ , θ ∈ [0 , π] ϕ ∈ [0 , 2
π] Then the linear element of S 3 is:

ds 2 = (dx 0 ) 2 + (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 = R 2 dχ 2 + sin 2 χ ( (dθ) 2 + s 2 (dϕ)
2 ) (A.6) As above R is set to 1. With the same notations as in the hyperbolic case, a point of coordinates : (χ, θ, ϕ) is transformed , by a transvection along Oz , into a point whose coordinates are given by : cos(χ q ) = cos(χ) cos(L) -sin(χ) sin(L) c c q = (cos(χ) sin(L) + c sin(χ) cos(L))/ sin(χ q ) (A.7)

ϕ q = ϕ
The correspondance between the cylindrical coordinates of S 3 , which are defined as in the hyperbolic case , and those of the Euclidean space E 4 is given by : x = (s ρ c ϕ , s ρ s ϕ , c ρ s z , c ρ c z ) , where : c ϕ and s ϕ are defined in (A.1) , c ρ = cos(ρ) , s ρ = sin(ρ) , c z = cos(z) , s z = sin(z) . The metric is :

ds 2 = dρ 2 + s 2 ρ (dϕ) 2 + c 2 ρ (dz) 2
In E 4 , a screw motion whose base geodesic is Oz has the form :

    x ′0 x ′1 x ′2 x ′3     =     c α -s α 0 0 s α c α 0 0 0 0 c L s L 0 0 -s L c L         x 0 x 1 x 2 x 3    
where : c L = cos(L) , s L = sin(L) . The vectors of the natural local frames are given in E 4 by : e 1 = (c ρ c ϕ , c ρ s ϕ , -s ρ s z , -s ρ c z ) e 2 = (-s ρ s ϕ , s ρ c ϕ , 0 , 0) e 3 = (0 , 0 , c ρ c z , -c ρ s z ) which can be used to show that (A.4) is also true in the spherical case.

6 Appendix B. Tensors built from spinors, derivatives and Laplacian.

The calculations of this appendix are performed in the more general case of spaces with torsion and with coupling to gauge fields W α , although this additional environment will not be used in the main section of this note. We consider tensors of the form : ψ γ h T x ϕ where : γ h = γ α1 γ α2 ... γ α h , in which all the γ αi (1 ≤ i ≤ h) are different (appear only once) , and where T x belongs to a representation of the Lie algebra of the gauge group . We set :

D α = ∂ α + Γ α + W α -S α (B.1) where : Γ α = Γ c d α γ c γ d 4
, S α = S γ . αγ and : W α = W x α T x . W x α is the gauge field . It is chosen real, and for a unitary group with real parameters :

T x + = -T x . The tensor derivative is : D β ψ γ h T x ϕ = ∂ β ψ γ h T x ϕ + Γ γp . ϕp β ψ γ γ1 ... γ ϕp ... γ γ h T x ϕ D β ψ γ h T x ϕ = D β ψ γ h T x ϕ + ψ γ h T x D β ϕ +Γ γp . ϕp β ψ γ γ1 ... γ ϕp ... γ γ h T x ϕ + 1 4 Γ c d β ψ γ c γ d , γ h T x ϕ + 2 S β ψ γ h T x ϕ -W y β ψ γ h (T y + T x + T x T y ) ϕ (B.
2) In the following discussion: {h} means a fixed set of indices {α 1 , ... , α h } all different, as above.

If : c, d ∈ {h} , or if : c, d / ∈ {h} then : γ c γ d , γ h = 0 . If : c = α j ∈ {h} , d / ∈ {h} → γ c γ d , γ h = -2 η c c γ α1 ... γ αj-1 γ d γ αj+1 ... γ α h . If : d = α j ∈ {h} , c / ∈ {h} → γ c γ d , γ h = 2 η d d γ α1 ... γ αj-1 γ c γ αj+1 ... γ α h .
Recalling that spinors are defines with respect to orthonormal frames, and therefore that : Γ c d α = -Γ d c α these contributions cancel with the third term of the right member of (B.2), and it remains :

D β ψ γ h T x ϕ + C z . x y W y β ψ γ h T z ϕ = D β ψ γ h T x ϕ + ψ γ h T x D β ϕ + 2 S γ . β γ ψ γ h T x ϕ (B.
3) where : C z . x y are the structure constant of the gauge group : [T x , T y ] = C z . x y T z .

The relation (B.3) is very general, in the rest of this note, we shall not consider any more gauge fields, then (B.3) will be used with : T x = I . The Laplacian of the tensor field ψ γ h ϕ involves the term : D α g αβ D β ψ γ h ϕ which can be written (if there is no gauge field):

D α g αβ D β ψ γ h ϕ = g α β D α D β ψ -Γ γ . β α D γ ψ γ h ϕ +ψ γ h g αβ D α D β ϕ -Γ γ . β α D γ ϕ (B.4) +2g αβ D α ψ γ h D β ϕ + 2D α S α ψ γ h ϕ , (W α = 0)
Let us consider the current :

J µ = g α β ψ α γ µ ϕ β (B.5
) where : ψ α = D α ψ , ϕ β = D β ϕ and let us calculate its divergence :

div(J) = 1 √ g ∂ µ ( √ g J µ ) .
Using the identity :

∂ µ γ µ = [γ µ , Γ µ ] - ∂µ √ g √ g γ µ -2 S µ γ µ (B.6) one obtains : div(J) = ∂ µ g α β ψ α γ µ ϕ β + g αβ Dψ α ϕ β + ψ α Dϕ β where : D = γ α (∂ α + Γ α + W α -S α ) .
Finally, with :

∂ µ g αβ = -Γ α . δ µ g δβ -Γ β . δ µ g δα div(J) = g αβ D T ψ α ϕ β + ψ α D T ϕ β (B.
7) where we have set :

D T ψ α = Dψ α -Γ γ
. α µ γ µ ψ γ . Now, in (B.7), we would like to replace D T ψ α by D α Dψ :

D T ψ α = γ β ([D β , D α ] ψ + D α D β ψ) -γ β Γ γ . α β D γ ψ
Using the relation (C.15) for the commutator and the identity :

∂ α γ β = γ β , Γ α -Γ β . γ α γ γ (B.8) we obtain : D T D α ψ = D α Dψ + γ β 1 4 R cdβα γ cd + G βα + ∂ α S β -∂ β S α ψ +2γ β S γ . β α D γ ψ (B.9)
7 Appendix C. The second order Dirac equation.

Let ψ be a spinor. The second order Dirac equation is obtained by applying to it twice the Dirac operator. This formulation has some advantages and has been studied in [START_REF] Petroni | Second order wave equation for spin fields[END_REF] . This appendix summarizes some of its properties without entering into details. Consider the Lagrangian :

L = γ α D α ψ γ β D β ψ -m 2 ψψ (C.1
) where : γ α = h α a γ a , and the Dirac matrices γ a are defined with respect to an orthonormal frame (in 4-dimensional Minkowski space). In the following we assume that the γ a matrices are such that : γ 0 (γ a ) + γ 0 = γ a and : γ 0+ = γ 0 . One can find a description of Clifford algebra and their representations in [10]. The operator : D α = ∂ α + Γ α + W α -S α has been introduced in (B.1) . The sum : D = γ α D α is the the Dirac operator.

The Euler-Lagrange equations give :

D 2 ψ = h α a γ a D α h β b γ b D β = -m 2 ψ (C.2)
From the Lagrangian we deduces the energy-momentum tensor :

T a α = Dψ γ a D α ψ + D α ψ γ a Dψ -Lh a α (C.
3) the spin tensor :

S αcd = 1 4 Dψ γ α γ c γ d ψ -ψ γ c γ d γ α Dψ (C.4
) and the current associated to the gauge invariance of the electromagnetic field A α (W α = ieA α ) :

J α = i Dψγ α ψ -ψγ α Dψ = -ψγ α iDψ + h.c.
(C.5) where h.c. means : Hermitic conjugate. Using (B.7) one checks directly the conservation law :

1 √ g ∂ α ( √ gJ α ) = 0
We define :

γ 5 = γ 0 γ 1 γ 2 γ 3 (C.6) which satisfies : γ 5 2 = -1 , γ 5 + = -γ 5 (C.7)
The sets of solutions of the type : iDψ = ±mψ are called respectively :

D ± . If ψ ∈ D ± then : iγ 5 ψ ∈ D ∓ .
Now we consider the charge conjugation operation and proceed as in the case of the linear Dirac equation by taking the complex conjugate of equation (C.2) in order to reverse the sign of the electromagnetic charge, and set : ψ c = Cψ * where C is a matrix operator. The charge conjugate spinor ψ c is required to satisfy also the Dirac equation (C.2) with opposite charge, which leads to the condition :

Cγ a * C -1 = ±γ a (C.8) The sign ambiguity comes from the fact that the matrices γ a are in even number in equation (C.2). Applying (C.8) to itself gives :

C(Cγ a * C -1 ) * C -1 = (CC * )γ a (CC * ) -1 = γ a therefore : CC * = kI ,
where k is a complex number, since it commutes with all the Dirac matrices. Then :

C = k(C * ) -1 ⇒ C = k((kC * -1 ) * ) -1 = (k/k * )C ,
therefore k is real. At last, imposing that (ψ c ) c equals ψ up to a phase implies k = ±1 . One can add a last contraint which is that the probability (see farther) is conserved, that is :

ψ + c ψ c = ψ + ψ whatever ψ is : (ψ + ψ) * = ψ c + ψ c * = ψ c + ψ c + = ψ c + ψ c = (Cψ * ) + Cψ * = (ψ + (C + C) * ψ) *
γ a R = Xγ a D X -1 .
Then the condition (C.8) is still true if C above is replaced by : XCX -1 * . Now we look at the charge conjugation effect on the Energy-Momentum tensor (C.3) and on the current (C.5) . We have :

C ± (γ a (∂ β + Γ β + ieA β -S β )ψ) * = ±γ a (∂ β + Γ β -ieA β -S β )ψ c where, in this relation : ψ c = C ± ψ * . As a consequence : C ± (Dψ) * = ±Dψ c and : J α c = -ψ c γ α iDψ c + h.c. = -C ± ψ * γ α iDψ c + h.c. = -ψ * + C ± + γ 0 iγ α (±)C ± (Dψ) * + h.c. = -(±) ψ * + γ 0 * i(γ α Dψ) * + h.c. = (±) (ψγ α iDψ) * + h.c. = (∓)J α (C.10
) With the solution (C.9) the electromagnetic current is inverted as it should be.

Let us calculate the first term of the Energy-momentum tensor (C.3) for the charge conjugate spinor :

Dψ c γ a D α ψ c = (±C ± Dψ * ) + γ 0 (±)C ± (γ a D α ψ) * = (Dψ) + * C -1 ± γ 0 C ± (γ a D α ψ) * = ±(Dψ) + * γ 0 * (γ a D α ψ) * = ± Dψγ a D α ψ * = ± Dψγ a D α ψ
+ the same treatment applies to L , and finally :

T a α (ψ c ) = ±T a α (ψ) (C.
11) The solution C = C + keeps the energy positivity and reverse the electromagnetic current as desired.

As seen above, if ψ ∈ D ± then :

ψ c = C + ψ * ∈ D ∓ and J α (ψ c ) = ±2mψγ α ψ = -J α (ψ)
. The 4-vector ψγ α ψ is usually used to define the probability density current. Although it is invariant under charge conjugation, therefore keeping the positivity of the probability density, it is not a conserved current within the second order equation formalism, because, applying (B.7) gives :

D α ψγ α ψ = Dψψ + ψDψ ( however, if ψ ∈ D ± , D α ψγ α ψ = 0 ).
The solution has been given in [START_REF] Petroni | Second order wave equation for spin fields[END_REF] , for solutions of (C.2), by defining the probability current :

J α P = 1 2 ψγ α ψ + 1 m 2 Dψγ α Dψ (C.
12) which satisfies the conservation law : 1 √ g ∂ α ( √ gJ α P ) = 0 . Applying the charge conjugation operation we obtain : [START_REF] Strichartz | Explicit Solutions of Maxwell's Equations on a Space of Constant Curvature[END_REF]) and if : ψ ∈ D ± the probability current is as usual :

J α P (ψ c ) = J α P (ψ) (C.
J α P = ψγ α ψ .
The current (C.12) can be obtained by transforming the second degree equation (C.2) into a first degree system. For that purpose we set : γ α D α ψ = imϕ Then equation (C.2) can be put in the form :

γ α D α ψ ϕ = im 0 1 1 0 ψ ϕ
The matrix in the right member is the Pauli matrix σ x . This equation is re-written :

(σ x ⊗ γ α )D α ψ ϕ = im ψ ϕ
Therefore equation (C.2) is equivalent to the usual linear Dirac equation in (n + 2) dimensional Minkowski space with spinors not depending on the coordinates x n , x n+1 . For the extension to higher dimensions of the standard 4-dimensional γ a matrices see for instance [START_REF]A Clifford algebra gauge invariant Lagrangian for gravity[END_REF] . Then the 6-dimensional vector Ψγ α n+2 Ψ , where : Ψ = ψ ϕ and : γ α n+2 are the Dirac matrices in 6 dimensions, gives the current (C.12) for α < 4 . Its conservation law can be checked directly, and can be considered as a consequence of the conservation law of the 6-dimensional current.

Remark : As seen above, in the second order formalism, the solution ψ c = C + ψ * conserves the energy positivity, inverts the sign of the electromagnetic current, and keeps the positivity of the probability current. Applying the charge conjugation operation does not change the coupling to the connection, then an antiparticle is coupled to the gravitation field as a particle.

Equation (C.2

) is now transformed in order to match equation (B.4) :

D 2 ψ = h α a γ a D α h β b γ b D β ψ = h α a γ a h β b γ b D α D β ψ + h α a γ a D α , h β b γ b D β ψ = -m 2 ψ
The commutator reduces to :

D α , h β b γ b = (∂ α + Γ α + W α -S α ) h β b γ b -h β b γ b D α = γ b ∂ α h β b + 1 4 Γ cdα γ c γ d , γ b h β b
and finally :

D α , h β b γ b = -γ b h γ b Γ β .γα
Writing :

γ α γ β = h α a γ a h β b γ b = h α a h β b γ a γ b = h α a h β b (η ab + γ ab ) = g αβ + γ
αβ , the second order equation becomes :

g αβ D α D β ψ -Γ γ .βα D γ ψ + 1 2 γ αβ [D α D β ] ψ +γ αβ S γ .αβ D γ ψ = -m 2 ψ (C.14) where the commutator is : [D α , D β ] = 1 4 R cdαβ γ cd + G αβ + (∂ β S α -∂ α S β ) (C.15) and : G αβ = ∂ α W β -∂ β W α + [W α , W β ]
The second term of (C.14) implies a term : R cdab γ ab γ cd where :

γ a γ b γ c γ d = γ abcd + η ab γ cd -η ac γ bd + η ad γ bc + η bc γ ad -η bd γ ac +η cd γ ab + η ab η cd -η ac η bd + η ad η bc

If the torsion is null, then : R cdab γ abcd = 0 by the Bianchi identities of the first kind. Using the fact that the Ricci tensor is symmetric if the torsion is null, the second order Dirac equation is :

g αβ D α D β ψ -Γ γ .βα D γ ψ -1 4 Rψ + 1 2 G αβ γ αβ ψ = -m 2 ψ (C.16) (S α
.βγ = 0) where R is the scalar curvature.

8 Appendix D. Solutions to the free electromagnetic field equations in t ⊗ H 3 .

This appendix looks for solutions to the electromagnetic field equations in V = t ⊗ H 3 which can be used as a basis to expand eigenvectors . A subclass of them can be interpreted as the equivalent of the plane wave solutions in Minkowski space , in the neighborhood of one geodesic . For local plane wave solutions see also [START_REF] Strichartz | Explicit Solutions of Maxwell's Equations on a Space of Constant Curvature[END_REF] . We shall use cylindrical coordinates (see Appendix A and [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF]). The space-time metric used is :

ds 2 = dρ 2 + sh 2 (ρ)dϕ 2 + ch 2 (ρ)dz 2 -dt 2
The symmetry axis of the cylindrical coordinates (Oz axis) is chosen to be the base geodesic of one generator element of the covering group Γ called γ 0 . In [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF] the scalar functions which are eigenvectors of the Laplacian and γ 0 at the same time are of the form : ϕ µν ∼ I µ,ν (ρ) exp(i(µ z z + νϕ) (D.1) These functions are invariant with respect to γ 0 if : µ z L + να = 2πm where m is a relative integer , L is the length of the transvection and α is the rotation angle around the base geodesic.

The coordinates of a vector are written on the local natural base as :

-→ v = v ρ-→ e ρ + v ϕ-→ e ϕ + v z -→ e z It will be convenient to use orthonormal local frames defined with respect to the natural frames by :

-→ h a = h α a -→ e α and write : -→ v = v a -→ h a , v α = h α a v a but in order to avoid any confusion between the component names and to which local frame they correspond, we rename them :

-→ v = u a -→ h a .
With the above metric we have :

u 1 = v a=1 = v α=1 , u 2 = v a=2 = shρ v α=2 , u 3 = v a=3 = chρ v α=3 .
All the Christoffel symbols having an index equal to 0 are null and the other are equal to those of the space metric alone. The non zero Christoffel symbols are :

Γ 1 22 = Γ 1 33 = -shρ chρ Γ 2 12 = chρ/shρ Γ 3 13 = shρ/chρ (D.
2) The non zero coefficients of the first structure equations are :

ω 1 = dρ , ω 2 = shρ dϕ , ω 3 = chρ dz (D.3) ω 2
. 1 = chρ dϕ , ω 3 . 1 = shρ dz which satisfy (1.1).

The following equations will be used :

D 2 v a = D β (g βγ D γ v) a = g βγ ∂ β (D γ v) a + ω a . bβ D γ v b -Γ δ . γβ D δ v a (D.4)
where :

(D γ v) a = ∂ γ v a + ω a .bγ v b D α v α = 1 shρ chρ ∂ ρ (shρ chρ u 1 ) + 1 shρ ∂ ϕ (u 2 ) + 1 chρ ∂ z (u 3 ) + ∂ 0 (v 0 ) (D.5)
The electromagnetic field equations are (4.1) :

∆A β ≡ D α (g αγ D γ A β ) -R β α A α = J β (D.6
) here : J β = 0 since we consider free fields , with the Lorenz condition : D α A α = 0 . Using the cylindrical coordinates, (D.4) becomes :

(D 2 v) a=1 = ∆(v a=1 ) -2 chρ sh 2 ρ ∂ ϕ u 2 -2 shρ ch 2 ρ ∂ z u 3 -2 + 1 sh 2 ρ ch 2 ρ u 1 (D.7a)
where : ∆(v a=1 ) means the Laplacian operator applied the the component :

u 1 = v a=1 as if it were a scalar function. (D 2 v) a=2 = ∆(v a=2 ) + 2 chρ sh 2 ρ ∂ ϕ u 1 -ch 2 ρ sh 2 ρ u 2 (D.7b) (D 2 v) a=3 = ∆(v a=3 ) + 2shρ ch 2 ρ ∂ z u 1 -sh 2 ρ ch 2 ρ u 3 (D.7c) (∆v) α=0 = 3 ∆(v α=0 ) -∂ 00 v 0
(D.7d) where : 3 ∆ is the spatial Laplacian. Now, by analogy with the plane wave solutions of the euclidean space, we consider waves propagating along the Oz axis, and try solutions of the form :

A 1 = A 1 (ρ) cos(νϕ + τ )e i(ωt-kz) , A 2 = -A 2 (
ρ) sin(νϕ + τ )e i(ωt-kz)

A 3 = A 3 (ρ) cos(νϕ + τ )e i(ωt-kz) (D.8) where τ is a phase and, a priori, k = k(ρ). In this appendix ω has not exactly the same meaning as in section 3 (it is up to a factor 2) . Since the Lorenz condition does not fix completely the gauge, we chose : A 0 = 0 .

In the euclidean case, the equivalent of equation (D.7c) is homogeneous and A 3 = 0 is a solution. Setting A 1 = A 2 gives solutions of the form :

A 2 = J ν-1 ( √ ω 2 -k 2 ρ)
, where J ν-1 is the cylindrical Bessel function of index ν -1 . When ω = |k| and ν = 1 , one obtains the standard plane wave solution propagating along the Oz axis with electric field aligned with the Ox axis if τ = 0 and aligned with the Oy axis if τ = -π/2 . With the hypothesis (D.8) the Lorenz condition (D.5) is :

1 shρ chρ ∂ ρ (shρ chρ A 1 ) -iA 1 ∂ ρ k z -ν shρ A 2 -ik chρ A 3 = 0 (D.9) and the z dependence implies ∂ ρ k = 0 . For H 3 : R β α = -2δ β α → R b a = -2δ b a
for the spatial components of the Ricci tensor (R = 1) . Using expressions (D.7) in (D.6) gives the following equations :

1 shρ chρ ∂ ρ shρ chρ ∂ ρ A 1 -1 + 1 sh 2 ρ ch 2 ρ + ν 2 sh 2 ρ A 1 -k 2 ch 2 ρ A 1 (D.10a) +2ik shρ ch 2 ρ A 3 + 2ν chρ sh 2 ρ A 2 = -(1 + ω 2 )A 1 1 shρ chρ ∂ ρ shρ chρ ∂ ρ A 2 -ν 2 + 1 sh 2 ρ A 2 -k 2 ch 2 ρ A 2 +2ν chρ sh 2 ρ A 1 = -(1 + ω 2 )A 2 (D.10b) 1 shρ chρ ∂ ρ shρ chρ ∂ ρ A 3 + 1 ch 2 ρ -ν 2 sh 2 ρ A 3 -k 2 ch 2 ρ A 3 -2ik shρ ch 2 ρ A 1 = -(1 + ω 2 )A 3 (D.10c)
There are 3 unknown functions and 4 equations including the Lorenz condition (D.9) . First consider equation (D.10b). If we had : chρ A 1 = A 2 , it would become homogeneous :

1 shρchρ ∂ ρ shρchρ∂ ρ A 2 -(ν-1) 2 sh 2 ρ A 2 -k 2 ch 2 ρ A 2 = -(1 + ω 2 )A 2
which is the defining equation of the scalar field Laplacian eigenfunction radial part (equation ( 14) in [JPP0] ), then : A 2 ∼ I k,ν-1 (ρ) would be a solution.

We assume that :

A 1 = f (ρ)/chρ (D.11) The Lorenz condition (D.9) is then : ikA 3 = ∂ ρ f + chρ shρ f -ν chρ shρ A 2 (D.
12) which can be used in (D.10a). Combining with (D.10b) one obtains : Each single ν mode is an eigenvector common to the Laplacian and γ 0 if : kL + να = 2πm where m is a relative integer, for a given value of β = ω , and can be used to find solutions to the free electromagnetic field equations in compact hyperbolic manifolds as expansion over these modes.

A 2 ± f ∼ I k,ν∓1 (
In Euclidean space, the case ν = 1 corresponds to plane waves propagating in the Oz direction with linear polarization. With the gauge A 0 = 0 , the electric field is co-linear to -→ A which is then contained in the constant phase surfaces z = Ct , in other words : A 3 = 0 . In H 3 this is impossible, because if : A 3 = 0 , then equation (D.7c) implies : ∂ z u 1 = 0 , and ∂ z (D.7a) would give ∂ z u 2 = 0 . This is not a propagating wave. Therefore, the vector -→ A can not be tangent to the constant phase surface z = Ct everywhere.

Using the recurrence relations for the cylindrical function derivatives [START_REF]Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces[END_REF], one gets if ν = 1 :

T 3 1,-1 = (k 2 -1-β 2 )
2

Re I ik+1,1 + kIm I ik+1,1 with :

4 Im I ik+1,1 = -k shρ chρ I k,2 and :

T 3 1,1 = 4Re I ik+1,1 = 4 shρ chρ I k,0 -(k 2 -1-β 2 ) 8 I k,2
The choice : k 2 -1 -β 2 = 0 and : d 1 = 0 gives : A 2 = -c 1 I k,0 sin(ϕ + τ 1 )e i(ωt-kz) A 1 = c 1 chρ I k,0 cos(ϕ + τ 1 )e i(ωt-kz) (D.15) -∂ z A 3 = ikA 3 = -c 1 k 2 shρ 4chρ I k,2 cos(ϕ + τ 1 )e i(ωt-kz) For ρ → 0 , ikA 3 ∼ O(sh 3 ρ) which means that in the vicinity of the Oz axis the vector -→ A , and therefore the electric field, are tangent to constant phase planes. It can be checked that near the Oz axis the magnetic field is also tangent to these constant phase plane. In conclusion, the solution (D.14) looks like an Euclidean plane wave if : ν = 1 in the neighborhood of the geodesic Oz .

In order to get more insight, we can look at the behavior of solution (D.15) when ρ → ∞. The cylindrical radial functions I k,ν → Q ν cos(βρ+ κ ν )/ √ shρchρ where Q ν and κ ν are respectively an amplitude and a phase. A 1 becomes negligible with respect to A 2 and A 3 . The potential vector is perpendicular to the radial vector -→ h 1 and the solution looks like a elliptical polarisation in the radial direction. This can be compared with the solution obtained in spherical coordinates. The electromagnetic potential is expanded on the orthonormal set of vector spherical harmonics [START_REF] Kristensson | Spherical vector waves[END_REF] :

-→ A = f 1 (χ) -→ V 1 (θ, ϕ) + f 2 (χ) -→ V 2 (θ, ϕ) + f 3 (χ)
-→ V 3 (θ, ϕ) where :

-→ V 1 (θ, ϕ) = Y m l (θ, ϕ) -→ h 1 (here -→ h 1 is the radial vector attached to the spherical coordinates), and -→ V 2 (θ, ϕ) and -→ V 3 (θ, ϕ) are the other vector spherical harmonics. For each pair (l, m) the field equations are satisfied by (up to a constant factor) : When : χ → ∞ the radial function behaves as : φ l β ∼ cos(βχ + κ l )/shχ , then f 1 becomes negligible compared with f 2 , f 3 , and the electromagnetic field is orthogonal to the radius direction, as in the cylindrical case. This is consistent since the plane z = Ct is asymptotically tangent to the cone of summit O and angle : cos(θ) = thz .

f 1 = φ l β /shχ , f 2 = 1 √ l(l+1)shχ

3 ) : ψD 2 ϕ

 32 = ψγ β D β Dϕ = D β ψγ β Dϕ -D β ψγ β Dϕ The first term of the right member is a divergence, therefore, if ∂M = 0 : ψD 2 ϕdV = -DψDϕdV . If ϕ is a solution of the second order Dirac equation : DψDϕdV = m 2 2 ψϕdV

) 2 A

 2 rotation by an angle α around the Oz axis changes only the azimuthal angle : ϕ → ϕ + α. A transvection along the Oz axis of length L = 2 AB changes only the z coordinate : z → z + L. The action of a screw motion, along the Oz axis, is given in M 4 coordinates by :

  and therefore : C + C = I . In the Dirac representation of the γ a matrices : C = γ y is a solution of equation (C.8) with the minus sign. We name it C -. C = iγ 5 C - (C.9) is a solution of (C.8), with plus sign. It is named C + . Both solution satisfy the relation C + C = I . If : ψ ∈ D ± then : ψ c = C -ψ * ∈ D ± and : ψ c = C + ψ * ∈ D ∓ . The Dirac representation has been chosen to find an explicit expression of C . For any other representation R equivalent to the Dirac representation, the γ a matrices satisfy an equivalence relation of the type :

D. 13 ) 3 .

 133 If ν = 0 , the only possibility is :f ∼ I k,1 .This solution satisfies equations (D.10a), (D.10b) and the Lorenz condition defines A It remains to check the compatibility with equation (D.10c). The Laplacian operator (see appendix A and section 4) ∆ = dδ + δd commutes with δ , then if the Lorenz condition is satisfied one has : δ∆ ω = ∆δ ω = 0 which shows that the equations (D.10) are not independent, and that (D.10c) is automatically satisfied. This can be checked directly. After some algebra we obtain : ∂ z (D.10c) = 0 , and since (D.10c) ∼ e -ikz this means that (D.10c)=0 .The equations of the electromagnetic field with the Lorenz condition are satisfied by the hypothesis (D.8) and the solution (D.11), (D.13). The general solution for a field propagating along the Oz axis is :A 2 = ν [c ν I k, ν-1 + d ν I k, ν+1 ] (-sin(νϕ + τ ν ))e i(ωt-kz) (D.14a) A 1 = 1 chρ ν [c ν I k, ν-1 -d ν I k, ν+1 ] cos(νϕ + τ ν )e i(ωt-kz) (D.14b) -∂ z A 3 = ν c ν T 3 ν,-1 -d ν T 3 ν,1 cos(νϕ + τ ν )e i(ωt-kz) (D.14c) where, with ε = ±1 : T 3 ν,ε = ∂ ρ I k, ν+ε + chρ shρ I k, ν+ε (1 + εν)

∂ χ (sh 2 χf 1

 1