Ulf Klein
email: ulf.klein@jku.at

Quantizing Galilean spacetime -A reconstruction of Maxwell's equations in empty space

Keywords: Derivation of Maxwell's equations, Quantum of light, Photon hypothesis, Planck-Einstein relation, Particle-Field dichotomy, Generalized quantization MSC Classification: 70S99, 70H05, 81P05, 81S99, 83C50

As was recently shown, non-relativistic quantum theory can be derived by means of a projection method from a continuum of classical solutions for (massive) particles. In this paper we show that Maxwell's equations in empty space can be derived using the same method. In this case the starting point is a continuum of solutions of equations of motion for massless particles describing the structure of Galilean spacetime. As a result of the projection, the space-time structure itself is changed by the appearance of a new fundamental constant c with the dimension of a velocity. This maximum velocity c, derived here for massless particles, is analogous to the accuracy limit derived earlier for massive particles. The projection method can thus be interpreted as a generalized quantization. We suspect that all fundamental fields can be traced back to continuous sets of particle trajectories, and that in this sense the particle concept is more fundamental than the field concept.

Introduction

"..one is struck by the dualism which lies in the fact that the material point in Newton's sense and the field as continuum are used as elementary concepts side by side"

Albert Einstein [START_REF] Einstein | Autobiographical notes[END_REF] The disturbing dualism between the incompatible basic concepts particle and field has accompanied modern physics from the very beginning. If one wants to eliminate the particle concept in favor of the field concept, then one needs nonlinear field equations of unknown origin. The reverse way, namely the elimination of the field concept in favor of the particle concept, is much easier to accomplish, at least in a formal way. This elimination is actually an innocent looking standard technique of fluid mechanics known as Lagrangian to Eulerian transition [START_REF] Feldmeier | Theoretical Fluid Dynamics[END_REF][START_REF] Klein | From probabilistic mechanics to quantum theory[END_REF]. It is based on the existence of a continuum of solutions of first order ordinary differential equations which fill and thus generate the space under consideration [START_REF] Caratheodory | Calculus of Variations and Partial Differential Equations of the First Order, Part I[END_REF]; let us note for clarity that the Eulerian formulation is nothing but the standard formulation of physical fields.

A popular version of this deep particle-field duality is the particle-wave duality of quantum theory (QT). Einstein rejected this concept and considered QT to be a theory which is complete only with respect to statistical ensembles and not with respect to individual particles. From the point of view of this ensemble interpretation [START_REF] Ballentine | The statistical interpretation of quantum mechanics[END_REF], QT must be understood in terms of its relation to classical statistical ensembles. The general program of a "quantization", or better reconstruction of QT, then consists in the transformation of the basic equations of the theory of classical statistical ensembles into the basic equations and structures of QT. The first step of this transformation must be the Lagrangian to Eulerian transition.

This program has been realized in a series of papers of the present author which will be referred to as I [START_REF] Klein | From Koopman-von Neumann theory to quantum theory[END_REF], II [START_REF] Klein | From probabilistic mechanics to quantum theory[END_REF], III [START_REF] Klein | A reconstruction of quantum theory for nonspinning particles[END_REF], and IV [START_REF] Klein | A reconstruction of quantum theory for spinning particles[END_REF]. Reconstructing QT along these lines is very simple from a conceptual point of view. It requires, besides the transition to the Eulerian formulation, only two essential steps, namely a projection from phase space to configuration space and a linearization or randomization. Both steps can be motivated physically. A theory containing these two steps, regardless of the order, is referred to as "Hamilton-Liouville-Lie-Kolmogorov theory" (HLLK). This acronym is also used in this paper. As shown in I -IV, the success of the HLLK is remarkable -not only Schrödinger's equation but essentially all characteristic features of (single-particle) QT, like non-commuting observables, Born's rule, and even spin, have been derived.

The program underlying the HLLK has not yet been fully carried out. It is obviously incomplete with respect to the description of several particles and with respect to the extension to Minkowskian spacetime. One wonders, however, if this program cannot be completed in an even more fundamental sense. So far, we have studied only particles of non-zero mass. We have found, in IV, that such particles must be fermions with spin one-half. The second class of elementary particles of fundamental importance in nature are bosons, which have integer spin and are responsible for mediating interactions. The most important structureless particle of this class is the massless photon, which has spin 1, and mediates the electromagnetic interaction. The question arises whether the program of the HLLK can -in analogy to the situation with massive particles -also be used to derive the "quantum mechanical field equations of the photon", i.e. Maxwell's equations. In this work, which is the fifth in this series of papers, we show that this is indeed possible.

The ordinary QT of massive particles, derived in I to IV, describes the behavior of particles under a variety of external conditions (forces), each corresponding to a particular functional form of the Hamiltonian function. In contrast, in Maxwell's equations (in empty space) there is no quantity whose functional form could be varied; the form of the equations is fixed once and for all. In other words, Maxwell's equations describe only a single system. If one asks, what this single system might be, then the only possible (reasonable) answer seems to be: the empty space.

As a consequence, we need, as a starting point of the HLLK program, equations of motion for massless "particles" describing the structure of empty space. Such a differential equation, in which no inertial mass occurs, can be derived by means of a transformation from an inertial system to an arbitrarily moving coordinate system (see section 2). The result agrees with the equation of motion derived by Holland from Maxwell's equations [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF]. In Holland's paper, it is also mentioned that one can, in turn, derive the time-dependent part of Maxwell's equations from these equations of motion, using the standard quantization rules. Since Maxwell's equations are no longer Galilei-invariant, this means that, in the course of this standard quantization, the Galilei invariance is broken. In fact, the quantization process introduces, according to Holland, a new fundamental constant c and this implies a fundamental change from Galilean spacetime to Minkowskian spacetime [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF]. This is astonishing because physical constructions usually take place within a given spacetime. Holland's remarks provided a strong motivation for the author to try to realize the HLLK process for massless particles -despite this mismatch of symmetries which was irritating at first sight.

In this paper we follow the version of HLLK reported in I and II. The space generated in section 4 by means of the Lagrangian to Eulerian transition is the cotangent bundle of the configuration space R 3 × SO(3). We find that we have to exclude a certain subset of all possible trajectories from this process. This implies an additional condition, which leads later, in section 6.3, to the transversality conditions of the electromagnetic fields. It is this additional condition that allows for the first time a derivation of the complete set of Maxwell's equations in empty space. In section 5 the projection onto the configuration space R 3 × SO(3) is performed. This part represents the core of the HLLK-quantization. The projection creates a new fundamental constant c with the dimension of a velocity. Due to the additional rotational degrees of freedom, a further projection onto the Euclidean configuration space R 3 is required, which is carried out in section 6. The physical meaning of the system of equations derived in this way is discussed in section 7.

The last section 8 contains some reflections on the concepts of particles, fields, and quantization, as suggested by the success of the HLLK. We propose a new principle for describing the relation between a physical theory and a related "better" theory. This principle may be summarized as follows: "Get rid of idealizing assumptions"

Equations of motion for Galilean particles

Galilean spacetime is given by E 3 × R t , where E 3 is the Euclidean space equipped with an Euclidean distance function. A complete description of Galilean spacetime is given by its symmetry group, the 10-parameter Galilean group. Those elements of this group that are continuously connected with the identity can be expressed, in a Cartesian coordinate system, in the form

(x, t) → (Rx + gt + b, t + s) . ( 1
)
Here R is a 3 × 3 orthogonal matrix with determinant 1, x, g, b are threevectors, and s is a scalar parameter. The Galilei group, as Eq. ( 1) shows, has rotations, Galilei boosts, and space-and time translations as subgroups.

Our task is to find equations of motion whose solutions are as similar as possible to Eq. (1). We call the hypothetical particles that satisfy these equations Galilean particles. We introduce this name in order to emphasize the analogy with both the previous theory of particles with non-zero mass and also with the photons to be derived later. The above definition of the equations of motion is admittedly rather fuzzy; however, we will show that important physical results can be derived with their help.

Let us first consider Newton's equation of motion for a free particle ẍ = 0 and its general solution x(t) = v 0 t + x 0 . This solution already describes, in its dependence on the initial conditions, two of the above subgroups namely the Galilei boosts and the spatial translations. To account for the still missing rotations, one can tentatively use the defining equation ẋ = ω × x of the angular velocity ω. To recover the Galilei boosts and spatial translations, it is only necessary, to perform a derivation of this equation with respect to time. Considering only angular velocities constant in time (clearly, time-dependent angular velocities are not compatible with our similarity requirement) one obtains the equations ẍ = ω × ẋ, ω = 0, (2) which agree with the equations derived by Holland [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF]. These relations have a simple physical interpretation. Let us consider two coordinate systems, an inertial frame K, with basis vectors e i , and a rotating reference frame K , with basis vectors e i (t). Any position vector x(t) can be represented in either K or K . The relation between velocity v and acceleration a in K and the corresponding quantities v and a in K is given by

a = a + 2ω × v + ω × x + ω × (ω × x) (3) v = v + ω × x. (4)
If we set here v = a = 0 (and ω = 0), we obtain Eq. ( 2). The equation of motion [START_REF] Feldmeier | Theoretical Fluid Dynamics[END_REF] for Galilean particles thus simply states that for a point held fixed in K , the sum of the centripetal force and the centrifugal force vanishes. The general solution of (2) depends on the 9 integration constants ω, v 0 , x 0 . If the unit vector n = ω/ω parallel to ω and its magnitude ω = |ω| are introduced, the solution takes the form

x(t) = (n × v 0 ) 1 -cos ωt ω + (v 0 -v 0 • n n) sin ωt ω + v 0 • n n t + x 0 . ( 5
)
The generic solution, with v 0 neither parallel nor perpendicular to n, is shown in Fig. 1. It has the form of a spiral, i.e. a superposition of a circular motion with a Galilean boost. When v 0 is parallel to n, the circular motion disappears and only the Galilean boost survives, as shown in Fig. 2. We shall come back to this degenerate case in section 4 Finally, to investigate the similarity of the solutions (5) with the Galilean transformations (1), we use the relation

r := R(n, φ)r = r -(r -r • n n) (1 -cos φ) + (n × r) sin φ, (6)
which describes the effect of a rotation R on a vector r when the angle of rotation φ and the axis n are used as parameters [START_REF] Goldstein | Classical Mechanics[END_REF]. With the help of ( 6) and the definitions r 0 = v 0 /ω, v = v 0 • n n, the solution (5) may be written in the form

x(t) = R(n, ωt -π/2)r 0 + v t + n × r 0 -n • r 0 n + x 0 . (7)
In this formula all types of transformations belonging to the Galilei group, with the exception of the time translations, occur. We have a rotation of the vector r 0 , with an axis n and an angle ωt -π/2, a Galilei boost, with the velocity v , and a spatial translation by n × r 0 -n • r 0 n + x 0 . However, the individual group parameters in [START_REF] Klein | A reconstruction of quantum theory for nonspinning particles[END_REF] are complicated functions of the initial conditions x 0 , v 0 , ω; in particular, the angle of rotation depends on time. Therefore [START_REF] Klein | A reconstruction of quantum theory for nonspinning particles[END_REF] is still very different from a real Galilei transformation, where all group parameters can be varied independently. As the solutions of (2) do not provide a complete description of of Galilean space we expect that the HLLK will only lead to physically meaningful results if additional restrictions are introduced.

Transition to Canonical Equations

In order to start the program of the HLLK, we must bring, as a first step, the equations of motion (2) of the Galilean particles into canonical form. For this purpose, the components of the angular velocity may be expressed by Euler angles, which allow to describe the instantaneous state of rotation of the particles. We follow here closely the procedure of Holland [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF][START_REF] Holland | The Quantum Theory of Motion[END_REF] using a standard notation for the Euler angles θ, φ, ψ [START_REF] Goldstein | Classical Mechanics[END_REF], which will alternatively be denoted by α k where α 1 = θ, α 2 = φ, α 3 = ψ.

The relation between the time derivatives αi of the Euler angles and the components of the angular velocity is given by αi

= A ik ω k , (8)
where the elements of the matrix A are periodic function of θ and φ,

A =   cos φ sin φ 0 -cot θ sin φ cot θ cos φ 1 csc θ sin φ -csc θ cos φ 0   . ( 9
)
Using the Euler angles, the original equations of motion (2) now take the form of two second order differential equations,

αl + A li ∂A -1 ik ∂α r αr αk = 0, ( 10
) ẍi = ijk A -1 js αs ẋk . (11)
Note that Eq. ( 10) only describes the constancy of ω over time and does not depend on x. The equations of motion [START_REF] Goldstein | Classical Mechanics[END_REF], [START_REF] Holland | The Quantum Theory of Motion[END_REF] of the Galilean particles have a somewhat unusual form which stems from the fact that each point of time t is assigned not only a position in R 3 but also a state of rotation, which is described by a point in the group space SO(3) of the rotation group. The 6dimensional configuration space R 3 ×SO(3) agrees with that of the rigid body, as already mentioned by Holland [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF][START_REF] Holland | The Quantum Theory of Motion[END_REF].

The equations ( 10), [START_REF] Holland | The Quantum Theory of Motion[END_REF] can easily be derived, by means of a variation with respect to x k and α k , from a Lagrangian L of the form

L(x, α, ẋ, α) = d 0 A -1 ik (α) ẋi αk , (12)
if one observes that the matrix elements of A fulfill the relations

∂A -1 kr ∂α s - ∂A -1 ks ∂α r A si = ijk A -1 jr . ( 13
)
Before we construct the canonical equations we note the following inconspicuous but important point. We introduced in Eq. ( 12) a constant d 0 as a prefactor. Neither its numerical value nor its dimension is determined by the form of the equations of motion. We can choose both freely, at least at this point. However, due to the projection onto the configuration space, to be performed later in section 5, a new fundamental constant, say c, will be generated in our theory. This seems to be a generic property of the HLLK (In the earlier case of massive particles studied in I-IV, Planck's quantum of action was generated in this way). Clearly, the choice of the constant d 0 is important because it determines this new constant c. In a sense, we are even able to derive the constant c, to the extent that we are able to justify our choice of d 0 sufficiently well. Of course, this derivation can not concern the numerical value of c, this value will be determined by nature alone. However, by means of a physically justified determination of the dimension of d 0 we can at least derive the dimension of c, which already gives us a crucial information about its physical meaning. A reasonable choice for the dimension of the Lagrangian L seems to be energy, the same dimension as in the theory of particles with non-zero mass. We have already chosen the designation (Galilean) "particles" for the objects, whose trajectories are determined by the solutions of Eq. ( 2). However, this was just a semantic assignment. If we now determine that d 0 has the dimension g cm (which gives L the dimension of an energy) then we make a formal determination of these objects as particles -which goes beyond the previous semantic assignment. In section 5.2 we will see that this choice of d 0 implies the dimension of a velocity for c.

The canonical equations may be constructed using the standard method. The momentum p k , canonically conjugate to x k , is given by

p i := ∂L ∂ ẋi = d 0 A -1 ik (α) αk . ( 14
)
Eq. ( 8) shows that p i = d 0 ω i . Since ωi = 0, the momentum p i is conserved. Thus, Galilean particles are subject to an unusual, strongly degenerate dynamics. The momentum π i canonically conjugate to α i is given by

π i := ∂L ∂ αi = d 0 A -1 ki (α) ẋk . ( 15
)
The momentum p k does not depend on ẋk but ony on αk ; an analogous behavior is shown by π k . Hamilton's function,

H(x, α, p, π) = d -1 0 A ki (α)p i π k (16)
does not depend on x, in agreement with the fact that p does not depend on time. The canonical equations are given by

ẋk = d -1 0 A ik (α)π i , αk = d -1 0 A ki (α)p i , ( 17
) ṗk = 0, πk = -d -1 0 ∂A ji (α) ∂α k p i π j . ( 18
)
The solutions of the ordinary differential equations ( 17), [START_REF] Wigner | Relativistic invariance and quantum phenomena[END_REF] are trajectories in a 12-dimensional phase space Ω [cotangent bundle of the rigid-body configuration space R 3 × SO(3)], which is spanned by the coordinates x, α, p, π.

Remarkably, the velocity ẋ is not proportional to the momentum p. Another interesting variable, defined by its components

m k = A ik (α)π i , (19)
takes the dimension of an angular momentum (if the parameter d 0 is defined as in section 3). The elements of the matrix A obey the relations

A rk ∂ r A ji - A ri ∂ r A jk = ikr A jr .
Using these identities it is easy to see that the m k fullfil also the Poisson bracket relations

{m i , m k } = ikl m l (20)
characterizing angular momentum components, as noted already by Holland [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF][START_REF] Holland | The Quantum Theory of Motion[END_REF]. Remarkably, according to the equations of motion [see the left member of ( 17)], the velocity v = ẋ agrees for arbitrary times with the angular momentum m, apart from a proportionality constant. The Hamiltonian function may consequently be written in the form H = v k p k . In order to get rid of the ugly prefactors d -1 0 the position variables x i may be replaced by the new variables

u i = d 0 x i , (21)
which have dimension g cm 2 . The new momentum variables, canonically conjugate to u i are then given by d -1 0 p i = ω i . Using the new variables u i , ω i the Lagrange function and the Hamilton function take the form

L(u, α, u, α) = A -1 ik (α) ui αk , (22) H(u, α, ω, π) = A ki (α)ω i π k , (23)
and the canonical equations are given by

uk = A ik (α)π i , αk = A ki (α)ω i , ( 24
) ωk = 0, πk = - ∂A ji (α) ∂α k ω i π j . ( 25
)
We will change freely between Eqs. ( 23)-( 25) and the original equations ( 16)- [START_REF] Wigner | Relativistic invariance and quantum phenomena[END_REF] where the important parameter d 0 appears explicitly.

The set of all solutions, together with a probability (or density) distribution at t = 0, defines a statistical (or particle) ensemble formulated by means of Lagrangian coordinates. The next step in the formulation of the HLLK is the transition to the usual (Eulerian) formulation, which describes physical properties by means of fields depending on the coordinates of a suitable space. In our case, this space is Ω.

Lagrangian to Eulerian transition

For the general considerations of this section, it is convenient to use the original equations of motion [START_REF] Good | Particle aspect of the electromagnetic field equations[END_REF], [START_REF] Wigner | Relativistic invariance and quantum phenomena[END_REF]. We write the solutions in the form

x k (t) = x k (t, x 0 , p 0 , α 0 , π 0 ), ( 26
)
p k (t) = p k (t, x 0 , p 0 , α 0 , π 0 ), (27)
α k (t) = α k (t, x 0 , p 0 , α 0 , π 0 ), (28)
π k (t) = π k (t, x 0 , p 0 , α 0 , π 0 ), (29)
where x 0 k , α 0 k , p 0 k , π 0 k are the initial values of the dynamic variables at t = 0. If these equations can be solved for x 0 k , α 0 k , p 0 k , π 0 k , then they define a flow in Ω, that is, there is a mapping from Ω onto itself at each instant of time t.

In order to discuss the physical meaning of this mapping it is convenient to abandon the special meaning of H and to substitute an arbitrary observable A(Q, P ) in place of H. We think of these observables, which depend on 2n phase space variables Q, P , as important physical quantities like energy, momentum or angular momentum (all defined by a fundamental symmetry of space-time). Each A(Q, P ) defines canonical equations, whose solutions (if invertibly) represent a group of transformations, each element of which maps the phase space onto itself. The observables A represent the classical equivalent of the associated Hermitian operators  in QT. The one-parameter group of transformations generated by A corresponds in QT to the one-parameter group of unitary operators generated by Â. These relations, basically known for a long time [START_REF] Sudarshan | Classical Dynamics: A Modern Perspective[END_REF] have recently been used by the author as an essential component in a reconstruction of QT, see II.

In the present work we assume that analogous relations hold also for massless particles. Accordingly, A is identified with the Hamiltonian function [START_REF] Wigner | Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra[END_REF] of Galilean space-time. The variables Q and P are identified with the configuration space variables x, α and the corresponding canonically conjugate momenta p, π, respectively. We first assume that the mapping defined by ( 26)-(29), from Ω onto itself, is 1 : 1, but analyze this question more carefully at the end of this section.

The Lagrangian to Eulerian transition, which we perform now, is a straightforward generalization of the well-known method in fluid mechanics (where n = 3) to arbitrary n. The details were reported in II. From the canonical equations it follows that an arbitrary density ρ(x, α, p, π, t) obeys the Liouville equation

∂ρ ∂t + ∂ρ ∂x k ∂H ∂p k + ∂ρ ∂α k ∂H ∂π k - ∂ρ ∂p k ∂H ∂x k - ∂ρ ∂π k ∂H ∂α k = 0, (30)
During the reconstruction of QT in I-IV an analogous quantity ρ(x, p, t) was assigned the physical meaning of a probability density. In contrast to this we do not specify the physical meaning of the density ρ(x, α, p, π, t), at least at the moment. We assume further, as in I-IV, that the dynamical variable ρ(x, α, p, π, t) is not sufficient to perform the transition to a quantum theory. As a second variable we choose, again in analogy to I-IV, the variable S(x, α, p, π, t), defined by

S(x 0 , α 0 , p 0 , π 0 , t) = t du p k ∂H ∂p k + π k ∂H ∂π k -H . ( 31
)
The action variable S(x, α, p, π, t) obeys the following differential equation, which is referred to as action equation,

∂S ∂t + ∂S ∂x k ∂H ∂p k + ∂S ∂α k ∂H ∂π k - ∂S ∂p k ∂H ∂x k - ∂S ∂π k ∂H ∂α k = p k ∂H ∂p k + π k ∂H ∂π k -H. (32)
We can combine Eqs. (30), (32) into a single differential equation using the complex-valued field variable ψ(x, α, p, π, t), defined by

ψ = √ ρ exp ı w S . (33)
In this formula we have introduced a parameter w which has the dimension of an action and has no physical meaning at all (it is part of the definition of the quantity ψ and may take any numerical value). It follows immediately from (30) and (32) that ψ satisfies the homogeneous linear 'pre-Schrödinger' differential equation

w ı ∂ψ ∂t - ∂H ∂p k p k - w ı ∂ ∂x k ψ - ∂H ∂π k π k - w ı ∂ ∂α k ψ - w ı ∂H ∂x k ∂ ∂p k + ∂H ∂α k ∂ ∂π k ψ + Hψ = 0. ( 34
)
In this field-theoretic form of the equations of motion ( 17), [START_REF] Wigner | Relativistic invariance and quantum phenomena[END_REF], the projection onto the configuration space R 3 × SO(3) becomes particularly simple, as will be shown in the next section.

We now return to the question whether the mapping defined by the solutions of the equations of motion is 1 : 1. We use the equations of motion in the form (24),(25); the mapping has the form of Eqs. ( 26)-(29), with x and p replaced by u and ω. A point in Ω is denoted by x, its components are denoted by x µ , where (x µ ) = (u, ω, α, π) , µ = 1, .., 12. The initial values at time t = 0 are denoted by y, its components y µ are given by (y µ ) = u 0 , ω 0 , α 0 , π 0 . The mapping can then be written in the compact form x = f (t, y), and this mapping will be 1 : 1 if the determinant of the functional matrix

f =    ∂ 1 f 1 ∂ 2 f 1 • • • ∂ 12 f 1 . . . . . . . . . . . . ∂ 1 f 12 ∂ 2 f 12 • • • ∂ 12 f 12    (35)
(where ∂ µ denotes the derivative with respect to y µ ) is different from 0 everywhere in Ω. We are unable to perform a complete analysis of this determinant as the time-dependence of the Euler angles is very complicated [START_REF] Whittaker | A Treatise on the Analytical Dynamics of Particles and Rigid Bodies[END_REF], even for the simple case of constant angular velocity considered here. We will examine only the case where the velocity v is parallel to ω (this is the special case shown in Figure 2). This means v k := uk = rω k where r is a real number.

It is easy to see that in this special case the determinant of the functional matrix (35) vanishes.

To show this, we first assume that the parallelism between v k and ω k holds true only at t = 0. Thus, the initial values obey the relations v 0 k = rω 0 k . We know that ω is constant, thus ω k = ω 0 k in agreement with the left member of Eq. (25). Eq. ( 2) shows that all derivatives of v with respect to t vanish at t = 0. It follows that the parallelism v 0 k = rω 0 k at t = 0 implies the parallelism v k = rω k for arbitrary t. The left member of Eq. (24) implies A ik (α 0 )π 0 i = rω 0 k . Thus, the initial values of π i can be expressed by the initial values of α and ω using the relation π 0 l = rA -1 kl (α 0 )ω 0 k . As a consequene, the last three columns of the functional matrix (35) vanish, since nowhere in the solutions do the variables π 0 l appear any more. For the solutions obeying

v = rω, (36)
the Lagrangian to Eulerian transition cannot be carried out. These solutions must be excluded by an appropriate additional constraint. This will be done in section 6.3. An important part of HLLK is the projection from phase space Ω onto R 3 × SO(3). This step is similar to the usual quantization rule, in that the canonical momenta are replaced by operators acting on configuration space. This projection, which will be carried out in section 5, is, in the case of massless particles treated here, not sufficient to generate a physically meaningful field theory. In order to arrive at a field theory with independent variables t, x k a further reduction from R 3 × SO(3) to Euclidean configuration space R 3 must be performed. This step will be reported in section 6.

Projection to rigid-body configuration space

From now on, we use the original system of Eqs. ( 17), [START_REF] Wigner | Relativistic invariance and quantum phenomena[END_REF] for the variables x k , α k and the canonically conjugate momenta p k , π k . Our starting point is the linearized equation of motion (34) for ψ, with the Hamiltonian function [START_REF] Wigner | Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra[END_REF]. In Eq. (34) two constant parameters d 0 and w occur, which have -as part of the definition of the Lagrangian function and the dynamical variable ψ -no physical meaning at this time.

The projection

The projection onto R 3 × SO(3) is done exactly as in I, II. One replaces ψ(x, α, p, π, t) by an unknown function ψ(x, α, t) denoted for simplicity by the same symbol. Thus the third bracket in (34) disappears. To remove p k and π k from the operator expression, the only remaining option is to perform the following substitution by operators:

p k ⇒ pk := w ı ∂ ∂x k , π k ⇒ πk := w ı ∂ ∂α k . ( 37
)
This makes the first two brackets disappear and the Hamiltonian function H(x, α, p, π) becomes an operator Ĥ = H(x, α, p, π). So the well-known quantization recipe finds a natural explanation as a projection rule. We mention that the standard quantization recipe (without the α-degree of freedom) has also been used to derive Maxwell's equations from the relativistic particle equations of motion [START_REF] Raymer | The Maxwell wave function of the photon[END_REF]. The projection transforms Eq. (34) into the 'Schrödinger-like' differential equation

w ı ∂ ∂t + Ĥ ψ = 0, (38)
where, however, the exact form of the Hamiltonoperator Ĥ is not yet known, because the order of the noncommuting quantities A ki (α) and π has still to be fixed (there is no ordering problem between x and p because Ĥ does not depend on x). The standard method of symmetrization cannot be applied here, because of the non-euclidean configuration space R 3 × SO(3).

To clarify the ordering problem, we use the inner product

(ψ, φ) = Σ dσ ψ * (x, α, t)φ(x, α, t), (39)
where the abbreviations Σ = R 3 × SO(3), dσ = d 3 x d 3 α sin α 1 were used and the Euler angles vary according to

α 1 ∈ [0, π], α 2 ∈ [0, 2π] α 3 ∈ [0, 2π].
One can now show that the Hamilton operator

Ĥ = d -1 0 A ki (α)p i πk (40)
is hermitian with respect to the scalar product (39) as long as one restricts oneself to states in the space of 2π-periodic functions. The proof makes use of the relations

∂A ki ∂α k = cos α 1 sin α 1 A 1i . ( 41
)
and is otherwise similar to the standard case, though slightly longer. The reverse order of A ki (α) and πk can be excluded. As a consequence of the quantization, the angular momentum components m k , defined by [START_REF] Bialynicki-Birula | Photon wave function[END_REF], become operators

mk = A ik (α)π i = w ı A ik (α) ∂ ∂α i , (42)
The m k satisfy the commutation relations ([â, ô] = âô -ôâ)

[ mi , mk ] = - w ı ikl ml . (43)
In the proof of (43) the same identity is used as in the proof of the corresponding Poisson bracket relation [START_REF] Kiessling | On the quantum-mechanics of a single photon[END_REF]. Using the operators m, the Hamilton operator takes the form Ĥ = d -1 0 mk pk . It is then sometimes interpreted as helicity, a projection of angular momentum (or spin) onto momentum.

The velocity components v k become operators vk = d -1 0 mk due to quantization. When v is used, Ĥ takes the form Ĥ = vk pk , and thus becomes formally similar to the Hamilton operator of Dirac's theory in the limiting case of vanishing mass. Of course, in Dirac's theory, the velocity operator is given by c times α, where c is the speed of light, and α is a three-vector whose components are 4 × 4 matrices. In the present case, the velocity is still a differential operator, but we will perform, in section 6, a second projection which maps the velocity components to matrices.

A new fundamental constant

If we insert the definitions of pk and πk , and multiply with a factor w -1 , the Schrödinger-like equation (38) takes the form

1 ı ∂ ∂t -d -1 0 w A ki (α) ∂ ∂α k ∂ ∂x i ψ = 0. ( 44
)
Thus, the constants d 0 and w do no longer occur individually, but only in the combination d -1 0 w, which has the dimension of a velocity. While the quantities d 0 and w were originally introduced as purely mathematical definitions, now, after the projection, they can no longer be changed at will. Thus, a new natural constant d -1 0 w with the dimension of a velocity appears, which we denote by c. Experiments show that the constant c must be assigned the numerical value of the speed of light. As is well-known, the appearance c of has drastic consequences for the structure of spacetime.

During the quantization of massive particles, reported in I-IV, the parameter w was identified with Planck's quantum of action . This constant has the physical meaning of an accuracy limit for measurements, and is in this sense characteristic for QT. The absence of in (44) indicates that this equation should not be interpreted in the same probabilistic sense as Schrödinger's equation for massive particles. We discuss this point in more detail in section 7.

If we accept the independent existence of the constant c and identify the constant w with , the constant d 0 is given by

d 0 = c , ( 45
)
as the ratio of two fundamental constants of nature which characterize the transition from classical physics to QT and to special relativity, respectively. The constant d 0 itself does not occur in the field equation ( 44), but it occurs in the definition of the canonical momentum p k = d 0 ω k [see text following Eq. ( 14)]. Accordingly, the linear relationship cp k = ω k exists between the components p k of the canonical momentum of the Galilean particle and the angular velocity, or angular frequency, ω k . The corresponding relation between the absolute values p and ω is given by

cp = ω. ( 46
)
Equation ( 46) establishes a relation between particle properties and wave properties, as does the related equation p = h/λ between momentum p and wavelength λ. These famous relations, associated with the names of Einstein and De Broglie, were, remarkably, discovered very early in the development of QT [START_REF] Field | Relationship of quantum mechanics to classical electrodynamics and classical relativistic mechanics[END_REF]. We have found here, using the framework of HLLK, a new derivation of these relations.

Projection to Euclidean configuration space

We perform, as a first step of the projection to R 3 , a general expansion in the parameter space of the rotation group SO [START_REF] Klein | From probabilistic mechanics to quantum theory[END_REF]. A suitable set of basis functions is provided by the so-called Wigner D-matrices [START_REF] Wigner | Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra[END_REF]]

D l m,n (α) = e ımα2 d l m,n (cos α 1 )e ınα3 , ( 47
)
where the d l m,n are generalized associated Legendre functions. The D l m,n are the matrix elements of the rotation operator in a l, m (angular momentum) basis (l = 0, 1, 2, .., m = -l, .., +l). For each l, the (2l + 1) × (2l + 1) matrices D l m,n represent an irreducible representation of SO( 3). The homomorphism property may be expressed in the form

D l m,n (g 2 g 1 ) = l m =-l D l m,m (g 2 ) D l m ,n (g 1 ) , ( 48
)
where g 2 g 1 is the product of the rotations g 1 , g 2 ∈ SO(3). The D l m,n form an orthonormalized set of basis functions on SO(3),

D l m,n , D r p,q R = δ lr δ mp δ nq , (49)
with the inner product being defined by

(ϕ, χ) R = SO(3) d 3 α sin α 1 ϕ * (α)χ(α), (50)
Furthermore, one can show that this set is complete. Our wave function may consequently be written in the form

ψ(x, α, t) = ∞ l=0 l m=-l l n=-l F l m,n (x, t)D l m,n (α). ( 51
)
This expansion shows that ψ transforms under rotations according to a reducible representation, namely the direct sum of all irreducible representations. If we were to project Eq. ( 44) onto R 3 without any further constraints, we would have to introduce a variable with infinitely many components. This is of course not acceptable. We need a variable with a relatively small number of components which, more importantly, transforms according to an irreducible representation of the rotation group. Thus, the irreducibility postulate used so succesfully by Wigner plays a decisive role in the projection from R 3 × SO(3) to R 3 .

Reduction to l = 1

We choose the irreducible representation l = 1. As a basis in the corresponding subspace we use the three functions D 1 a,0 (α), a = -1, 0, 1; these transform according to the l = 1 representation, as shown by Eq. ( 48). In the following, we write D a instead of D 1 a,0 ommitting the indices 1, 0. Then the expansion (51) reduces to

ψ(x, α, t) = 1 a=-1 F a (x, t)D a (α), ( 52
)
where the three orthonormalized basis functions are given by

D -1 (α) = ı √ 3 4π e -ıα2 sin α 1 , ( 53
)
D 0 (α) = √ 3 2π √ 2 cos α 1 , ( 54
) D 1 (α) = ı √ 3 4π e ıα2 sin α 1 . ( 55
)
The projection to the three-dimensional subspace generates a theory in the Euclidean configuration space R 3 (a conventional field theory) for a wave function ψ(x, t) with the three components F -1 (x, t), F 0 (x, t), F 1 (x, t) (indices a, b running from -1 to 1). The Schrödinger-like differential equation (38) takes the form

ı ∂F a ∂t + d -1 0 pi M i a,b F b = 0, (56)
where the quantities M i a,b are the matrix elements of the operators mi in the l = 1 basis,

M i a,b = (D a , mi D b ) R , (57)
M 1 = √ 2   0 1 0 1 0 1 0 1 0   , M 2 = √ 2   0 ı 0 -ı 0 ı 0 -ı 0   , M 3 =   -1 0 0 0 0 0 0 0 1   . ( 58
)
Performing a necessary rearrangement of indices, the matrices M i coincide with the matrices J i derived by Holland from Maxwell's equations [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF]. The matrices M i fulfil the same commutation relations

M i , M k = - ı ikl M l , (59)
as the differential operators mk from which they were derived. The constant matrices M i are angular momentum operators (for l = 1), but their action consists only in a mixing of the three components of ψ. Thus, they do not describe an orbital angular momentum but an internal degree of freedom called spin (spin 1). Based on the present derivation, it is clear that the rotational motion associated with this internal degree of freedom must not be interpreted as rotation of individual massless particles (photons), but as collective rotational motion in a (statistical ?) ensemble of massless particles. The components vk of the velocity operator are transformed by the projection into the 3 × 3 matrices V k = d -1 0 M k . We note that both the velocity V k and the angular momentum M k describe internal degrees of freedom. Using the three-component wave function ψ, Eq. ( 56) may be written in the more compact form

ı ∂ ∂t + V i pi ψ = 0, (60)
which shows the formal similarity with the massless Dirac theory particularly clearly.

Transition to Cartesian basis

If we assume that the theory described by (60) makes physical sense, then it seems useful to replace the spherical coordinates with the more commonly used Cartesian coordinates. In concrete terms, this means that we have to look for a representation of the spin matrices that corresponds to the transformation behavior of real vector fields in R3. These are given by S i jk = ı ijk , or

S 1 = ı   0 0 0 0 0 1 0 -1 0   , S 2 = ı   0 0 -1 0 0 0 1 0 0   , S 3 = ı   0 1 0 -1 0 0 0 0 0   . ( 61
)
As for the prefactors ı , these were added in order to be able to use the classical result as part of QT; of course, the classical rotation operator itself does not depend on . The transition to the Cartesian basis is performed by means of the transformation

F a = U ai G i , S i nm = U -1 na M i ab U bm
, where indices a, b take values from -1, 0, 1, all other indices running from 1 to 3, and the unitary matrix U is given by

U = 1 √ 2   1 ı 0 0 0 √ 2 -1 ı 0   . ( 62
)
In terms of the new Cartesian field components G i and spin matrices S i the Schrödinger-like differential equation (56) takes the form

ı ∂G n ∂t + d -1 0 pi S i n,l G l = 0. ( 63
)
Denoting the components d -1 0 S i of the velocity operator with the same symbol V i as before, and using also the same symbol ψ for the complex quantity with components G 1 , G 2 , G 3 , Eq. (63) may also be written in the more compact form (60).

If we now introduce real vector fields E and B with components defined by G k = E k + ıB k , it follows immediately from (63) that these fields satisfy the two time-dependent electromagnetic field equations

∂ ∂r × B - 1 c ∂E ∂t = 0, (64)
∂ ∂r × E + 1 c ∂B ∂t = 0. ( 65
)
We have not yet used the constraint that states corresponding to Eq. ( 36) are not allowed. Next, we try to derive the two remaining Maxwell equations from this constraint.

Derivation of transversality conditions

The additional condition we have to take into account was formulated in section 4, in the context of the description of particle orbits in phase space. It stated that those orbits which fulfil the condition v = rω [see Eq. ( 36)] are to be excluded from consideration. If the relation p = d 0 ω [see text following Eq. ( 14)] is used, then the additional condition means that the velocity v must not be parallel to the direction of propagation n(p) = p/p, where p = p 2 1 + p 2 2 + p 2 3 . We have made three drastic changes to the original picture of the motion of (massless) particles in the course of the present development. In the first step, we moved from particle coordinates to space coordinates, replacing ordinary with partial differential equations. As a consequence, the concept of trajectories lost its meaning; it was replaced by the concept of continuous distributions in phase space. In the second step, we moved from phase space to rigid body configuration space. In this step the transition to QT was carried out -in the sense that observables in phase space were replaced by operators in configuration space. Remarkably, with this single step we left not only the realm of classical physics but also the realm of non-relativistic physics. In the third step the projection onto the Euclidean configuration space was performed. The number of degrees of freedom was further drastically reduced by replacing operators on SO(3) by 3 × 3 matrices.

Despite these major changes, the original condition (36) can easily be "translated" into the present version of the theory. Let us consider the operators of velocity V i and momentum pi , replacing the original quantities v i and p i . The eigenvalues of the components

V 1 = c ı   0 0 0 0 0 1 0 -1 0   , V 2 = c ı   0 0 -1 0 0 0 1 0 0   , V 3 = c ı   0 1 0 -1 0 0 0 0 0   . ( 66
)
of the velocity operator are 0, ±c. The particles that form the quantum mechanical counterpart to the original Galilean particles -we can call them photons -are thus either at rest or moving at the speed of light. This means that the photons that make up our ensemble may have unusual kinematics; for example, the velocity is always parallel to the spin as shown by the relation V i = d -1 0 S i . In order to find the possible values of V relative to the direction of propagation n(p), we solve the eigenvalue problem of the operator V i pi [START_REF] Good | Particle aspect of the electromagnetic field equations[END_REF]:

  E ıcp 3 -ıcp 2 -ıcp 3 E ıcp 1 ıcp 2 -ıcp 1 E     u 1 u 2 u 3   = 0. ( 67
)
We obtain three eigenvalues E a (a = 0, +, -) given by

E 0 = 0, E ± = ±cp, (68)
with the corresponding orthonormalized eigenvectors

u 0 (p) = 1 p   p 1 p 2 p 3   , u ± (p) = 1 2p 2 (p 2 1 + p 2 2 )   ±ıp 2 p -p 1 p 3 ∓ıp 1 p -p 2 p 3 p 2 1 + p 2 2   . ( 69
)
With the help of these solutions, one can form a complete basis of plane wave states φ a p,n (x, t) =

1 (2π ) 3/2 u a n (p)e ı (p k x k -E a (p)t) , (70)
and write the general solution of Maxwell's equations as a linear combination, with an amplitude A a (p), in the form

G n (x, t) = a G a n (x, t) = a d 3 p A a (p)φ a p,n (x, t). (71)
This decomposition will allow us to identify the part of the solution that has to be eliminated due to the original additional condition.

In the state with the eigenvalue E 0 = 0 the helicity is 0, i.e. the spin is perpendicular to the direction of propagation n(p). The associated eigenvector u 0 (p) describes a movement parallel to n(p), i.e. a longitudinal wave. In the states with E ± = ±cp the helicity is ±1, that is, the spin is parallel or antiparallel to n(p). The transition from the original particle picture to the present quantum picture led in a certain way to a simplification, insofar as only three directions of the original continuum of possible directions of velocity have now survived.

On the other hand, at first glance, a difficulty of intuitive interpretation arises in that the operators of spin and velocity are proportional to each other. The velocity operator therefore has the "wrong direction", so to say. However, the velocity is described by the eigenstate belonging to the operator of the velocity, and not by the operator itself. Therefore, the quantum mechanical state corresponding to the original particle state [v parallel to n(p)] is given by the longitudinal contribution G 0 n (x, t) associated with the eigenvalue E 0 = 0. To eliminate the term G 0 n (x, t) from Eq. ( 71) it is sufficient to postulate the validity of the relation

∂G n ∂x n = 0. ( 72
)
The longitudinal state is not compatible with this requirement due to the relation n(p) = u 0 (p), while the transversal states automatically fulfil this condition. After separating the real and imaginary parts, (72) leads to the transversality conditions

∂E ∂r = 0, ∂B ∂r = 0 (73)
for the real-valued physical fields E and B. The conditions (73) play a special role; as is well know, it is sufficient to ensure their validity at a single (initial) time. This special role is also visible in the formalism of second quantization. The absence of the longitudinal mode for massless particles can also be shown using Lorentz invariance; the same is true for the parallelism between the velocity operator and the spin operator [START_REF] Wigner | Relativistic invariance and quantum phenomena[END_REF]. It is satisfying that the HLLK provides an alternative and completely independent explanation for these facts.

Maxwell's equations, classical or quantum

We have already identified the system of Eqs. (64),(65),(73) with Maxwell's equations, but we may ask ourselves whether this identification is the only possible one within the framework of our theory.

In section 4 we introduced a density whose physical meaning and dimension was not specified. The formalism of HLLK yields the same field equations for all these densities. However, the physical meaning and dimension of the associated fields is determined by that of the densities; the dimension of the density coincides with the dimension of the square of the fields. Eqs. (64),(65) imply the conservation law 1 2c

∂ ∂t E 2 + B 2 + ∂ ∂r (B × E) = 0. ( 74
)
It follows that the density E 2 + B 2 integrated over whole space is a conserved quantity. If we interpret the fields E, B as electrodynamical fields, then this density is an energy density. In order to derive Maxwell's equations, one must therefore take a non-probabilistic quantity, namely an energy density, as the original density. This choice is consistent with the interpretation of Maxwell's equations as classical equations, where classical means the same as non-probabilistic. The HLLK thus provides the basis for a "second quantization" in the sense of the quantization of a classical field. As will be discussed in the next section, the process of HLLK can be thought of as a generalized quantization. If this terminology is accepted, the field quantization becomes a second quantization in the literal sense.

Regardless of the fact that there is no true probability density in Maxwell's equations, the 3-component quantity ψ introduced in section 6.2 (or a generalisation of it) is called the "wave function of the photon" by some authors [START_REF] Bialynicki-Birula | Photon wave function[END_REF]; this terminology has been criticised by other authors [START_REF] Kiessling | On the quantum-mechanics of a single photon[END_REF]. The question is not of paramount importance in view of the fact that numerous quantum-like structures can be found in Maxwell's equations, especially in the formulation of Eq. ( 60). These structures also became visible in the course of our derivation.

As already mentioned, the interpretation of the density, which is first introduced in the phase space, is completely arbitrary in the HLLK, and has no influence on the form of the final field equations in the configuration space. However, if a probability interpretation is assigned to the density, then one has to fulfil the additional condition that the integral of the density over the whole space must be equal to 1 (a similar operation exists also in the case of a particle density). For example, using the formalism introduced in I, II, we can also derive a classical Schrödinger equation where the square of the absolute value of the wave function has the meaning of a particle density. Conversely, instead of the classical Maxwell equations derived here, we can also derive a "quantum mechanical" equation, for a three-component complex quantity φ, with associated fields b and e. These fields satisfy Eqs. (64),(65),(73) and, in addition, must satisfy the probabilistic normalisation condition

d 3 x b 2 + e 2 = 1, (75)
with the integration extending over all space. The relation between this quantum-mechanical Maxwell wave function and the above classical wave function was found by Good [START_REF] Good | Particle aspect of the electromagnetic field equations[END_REF], [START_REF] Kiessling | On the quantum-mechanics of a single photon[END_REF]. He also showed that the expectation values of Maxwell's theory have the correct quantum mechanical form when using this wave function. These questions are relevant for the problem of second quantization, the introduction of an occupation-number formalism for a first-quantized theory, or the equivalent problem of quantizing a classical field [START_REF] Sebens | Electromagnetism as quantum physics[END_REF]. The additional complication that the relevant dynamical variables are not the fields but the potentials must also be taken into account [START_REF] Akhiezer | Quantum Electrodynamics[END_REF]. We do not want to discuss these questions, which are outside the scope of this paper, but mention that the HLLK provides a basis for both types of second quantization. We hope to return to these questions in a later paper, after the derivation of the Dirac equation.

Discussion

In I-IV Schrödinger's equation (and the entire formalism of QT as regards single particles) was derived with the help of the HLLK. In the present work, the complete set of Maxwell's equations was derived using the same method. It was thus possible to derive the two differential equations that are probably the most important in the field of fundamental physics with the help of this method.

These two equations are quite different in their physical meaning. We know from an enormous number of observations that the predictions of Schrödinger's equation are probabilistic in nature whereas the predictions of Maxwell's equations are deterministic, in the sense that they can be tested with the help of forces in single experiments. Schrödinger's equation thus belongs to QT, while the electromagnetic field is regarded as the prototype of a classical field par excellence; at least as long as one excludes experiments with very low intensity from consideration.

The HLLK was originally constructed to derive QT within the framework of a probabilistic world view. The fact that the same method can now be used to derive a deterministic (fundamental) system of classical field equations requires a reassessment of the significance of this method. One can define the HLLK as a generalized quantization method that starts from a continuum of particle trajectories, performs a projection onto the configuration space, and introduces a new natural constant in the course of this projection. In the case of QT, this new natural constant is , while in the case of Maxwell's equations it is the speed of light c.

This double success leads to the speculative question whether possibly all fundamental fields of physics can be defined on the basis of such a generalized quantization defined by the HLLK. It leads, in short, to the question of whether all fundamental fields are quantum fields in this sense. This would also mean that all fundamental fields of physics can be traced back to an associated particle system. With regard to the dichotomy mentioned at the beginning of this paper, it would mean that the particle concept is more fundamental than the field concept.

The natural constants generated in the course of the projection to the configuration space, and c, both have the physical meaning of a realistic limitation or -which is the same thing -the elimination of an unrealistic idealisation. Let us explain what we mean by that. Classical particle physics contains the unrealistic assumption that measurements with infinitely high accuracy are possible. The HLLK quantization that leads to Schrödinger theory eliminates this unrealistic assumption and creates an accuracy limit . As regards the massless case, let us consider the structure of Galilean space-time. Here we find the unrealistic assumption that infinitely large velocities are possible.

The HLLK quantization eliminates this unrealistic assumption by generating the Maxwell field -and with it a modified, relativistic space-time structure characterised by a maximum velocity c.

We find here a new ordering principle for the relation between different physical theories: Theory B is better than theory A if an inadmissible idealisation of theory A no longer occurs in theory B. In the present work, we have derived -following a path opened by Holland [START_REF] Holland | Hydrodynamic construction of the electrodynamic field[END_REF] -Minkowski space from Galilean space with the help of this ordering principle. We can now transfer classical mechanics (for particles with non-zero mass) into Minkowski space according to the well-known method. This relativistic mechanics still suffers from the inadmissible idealisation that measurements with infinitely high accuracy are possible. We expect that the elimination of this inadmissible idealisation with the help of the HLLK quantization will lead to the Dirac equation.

Fig. 1 Fig. 2

 12 Fig. 1 This is a generic trajectory for initial values v 0 neither parallel nor perpendicular to n

  Both associated eigenstates are perpendicular to n(p), and also perpendicular to each other [the relation u a (p), u b (p) = δ ab holds true for a inner product defined by (a, b) = k a * k b k ]. The movement therefore takes place in a plane perpendicular to n(p) and describes a transverse wave.


Wang Qudong
email: awangqudong@sjtu.edu.cn

Peng Jianguo
email: bpengjianguo@sjtu.edu.cn

Liu Manping
email: manping-liu@263.netd

Chen Yongjun
email: chenyongjun@sjtu.edu.cn

Ding Wenjiang

Michel Suéry
email: michel.suery@gpm2.inpg.fr

Jean Jacques Blandin

Wang Qudong1

Peng Jianguo1

Liu Manping1

Chen Yongjun1

Ding Wenjiang1

M Suéry2

J J Blandin2

Microstructure and mechanical properties of extruded AM50 + xCa magnesium alloys

Keywords: AM50 magnesium alloy, Calcium, Extrusion, Microstructure, Mechanical properties

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

Introduction

Magnesium alloy, as the lightest structural metal, has been widely used in automobile, electronics and aerospace industries [1] due to its desirable combination of properties, including low density, high specific strength and specific stiffness, superb damping and electromagnetic shielding capacities, excellent machinability and good castability [2] . However, the commonly used Mg alloys such as AZ91 alloy and AM60 alloy are unsuitable for use above 150-200˚C [3] . It was found that Ca addition greatly improves the elevated temperature properties of as-cast Mg-Al alloys because of the formation of the heat-resistant Al 2 Ca phase, which has a relatively high melting point [4] . Alloying with Ca is becoming more common in the development of cheap creep-resistant alloys, essentially replacing Mg 17 Al 12 with Al 2 Ca [4][5] . Besides, extrusion technique is effective to refine coarse microstructure and enhance mechanical properties of magnesium alloy. However, studies on extruded magnesium alloy are few and the studies on the effect of Ca on the microstructure and mechanical properties of extruded Mg-Al alloys are lack. In order to developed heat-resistant wrought magnesium alloys, AM50 magnesium alloys containing Ca were hot extruded and the

Material and Experimental Procedure

In the present experiments, commercial AM50 alloy was chosen as master alloy. The composition of the studied magnesium alloys containing different calcium is listed in Table 1. These alloys were melted under SF 6 +CO 2 atmosphere in electronic resistance furnace. The extruded process is: firstly, the as-cast ingots were held at 410˚C for 30 minutes, then the φ60mm ingots were hot extruded into bars of φ20mm at temperature 400˚C with a extrusion ratio of 1:9; secondly, the as-extruded φ20 mm bars was held at 300˚C for 30 minutes, then they were extruded into strips with section size 2×10mm at temperature 315˚C with an extrusion rate 1:16. The strips were machined into tensile specimens with a gage length 8mm and 2×2mm section. Microstructure was observed by optical microscopy and the tensile fractographs were observed by scanning electron microscopy ( PHILIPS SEM515).

Results and Discussion

Microstructure. The microstructure of as-extruded AM50 alloy consists of primary (Mg) solid solution and intermetallic Mg 17 Al 12 (Fig. 1 Tensile Properties. Fig. 2 shows the ultimate tensile strength of extruded AM50 alloys at different temperature. The results show that the tensile strength of AM50+xCa alloy is influenced by Ca content and experimental temperature. With the increasing temperature, tensile strength decreases. With increasing Ca content, tensile strength improves at room temperature, however, it reduces a little at 100˚C,150˚C and 200˚C. The effect of Ca content and temperature on elongation is shown in Fig. 3. The results show that the elongation decreased with the increasing Ca content. Besides, with increasing temperature, the elongation increases firstly, then decreases after temperature is higher than 150˚C. As can be seen, the extruded AM50+xCa alloys have the highest elongation at 150˚C.

c

  Al properties are investigated in this paper.

Fig. 1 .

 1 Fig.1.Optical microstructure of Extruded AM50+xCa alloys, AM50+xCa alloys, (a)AM50; (b)AM50+1Ca;(c)AM50+2Ca

Fig2.Fig. 4

 4 Fig2. Tensile strength of extruded AM50+xCaFig3.Elongation of extruded AM50+xCa AM50+xCa magnesium alloy magnesium alloys Tensile Fractograph. Fig.4(a-c) shows SEM images of tensile fractographs at room temperature.

Table 1

 1 Composition of the studied alloys (wt %)

		Al	Mn	Ca
	AM50	4.33	0.36	---
	AM50+1Ca	4.33	0.35	1.23
	AM50+2Ca	4.33	0.33	2.33

Acknowledgements

This work was supported by International Cooperation Found of Shanghai Science and Technology Committee (02SL002), P. R. China and the Regional council of Phone-Alpes, France.


Samir Sadok

Simon Leglaive

Renaud Séguier

A VECTOR QUANTIZED MASKED AUTOENCODER FOR SPEECH EMOTION RECOGNITION

Keywords: Self-supervised learning, masked autoencoder, vector-quantized variational autoencoder, speech emotion recognition

Recent years have seen remarkable progress in speech emotion recognition (SER), thanks to advances in deep learning techniques. However, the limited availability of labeled data remains a significant challenge in the field. Self-supervised learning has recently emerged as a promising solution to address this challenge. In this paper, we propose the vector quantized masked autoencoder for speech (VQ-MAE-S), a self-supervised model that is fine-tuned to recognize emotions from speech signals. The VQ-MAE-S model is based on a masked autoencoder (MAE) that operates in the discrete latent space of a vector quantized variational autoencoder. Experimental results show that the proposed VQ-MAE-S model, pre-trained on the VoxCeleb2 dataset and fine-tuned on emotional speech data, outperforms an MAE working on the raw spectrogram representation and other state-of-the-art methods in SER.

INTRODUCTION

Speech emotion recognition (SER) [START_REF] Ayadi | Survey on speech emotion recognition: Features, classification schemes, and databases[END_REF] is a research area focused on automatically identifying emotions from speech signals. With the growth of technology and the increasing use of speech-based interfaces, there is a growing demand for systems that can accurately recognize emotions in speech. In recent years, deep learning methods have played a major role in improving SER performance [START_REF] Berkehan | Speech emotion recognition: Emotional models, databases, features, preprocessing methods, supporting modalities, and classifiers[END_REF]. However, the scarcity and high cost of obtaining labeled speech data for emotion recognition pose a major difficulty. To address this challenge, researchers have shifted their focus towards self-supervised learning approaches [START_REF] Wang | A fine-tuned wav2vec 2.0/hubert benchmark for speech emotion recognition, speaker verification and spoken language understanding[END_REF][START_REF] Chen | Exploring wav2vec 2.0 fine-tuning for improved speech emotion recognition[END_REF]. In these approaches, models are pretrained on a self-supervised task, such as predicting masked tokens in speech signals, and then fine-tuned on a smaller set of labeled data for the SER task [START_REF] Gong | Ssast: Self-supervised audio spectrogram transformer[END_REF][START_REF] Pepino | Emotion recognition from speech using wav2vec 2.0 embeddings[END_REF]. This method has the advantage of scalability, as the self-supervised task can be trained on large amounts of unlabeled speech data, reducing the need for labeled data [START_REF] Liu | Audio self-supervised learning: A survey[END_REF][START_REF] Zhang | A survey on masked autoencoder for self-supervised learning in vision and beyond[END_REF]. Self-supervised training has been successfully applied in the field of SER, and has demonstrated promising results by allowing the model to learn useful representations of speech signals for emotion recognition, even in scenarios where labeled data is limited [START_REF] Pepino | Emotion recognition from speech using wav2vec 2.0 embeddings[END_REF][START_REF] Macary | On the use of self-supervised pre-trained acoustic and linguistic features for continuous speech emotion recognition[END_REF].

This article focuses on self-supervised SER with the masked autoencoder (MAE) approach [START_REF] He | Masked autoencoders are scalable vision learners[END_REF]. The MAE is an asymmetrical encoder-decoder architecture that relies on input masking [START_REF] He | Masked autoencoders are scalable vision learners[END_REF]. Originally developed in natural language processing (NLP) [START_REF] Devlin | Bert: Pre-training of deep bidirectional transformers for language understanding[END_REF], the MAE approach has also been applied to image analysis using vision transformers (ViT) [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF]. The MAE process involves dividing the input into non-overlapping patches, each represented by a token embedding. A large proportion of tokens are masked (usually 75% for image modeling and 15% for text modeling), and only the visible tokens are fed to the encoder. A lightweight decoder then reconstructs the image/text by combining the visible tokens from the encoder and learnable mask tokens. The cost function is applied only to the masked tokens. Recently, the MAE has been adapted for audio using 2D time-frequency representations such as the melspectrogram [START_REF] Gong | Ssast: Self-supervised audio spectrogram transformer[END_REF][START_REF] Baade | Mae-ast: Masked autoencoding audio spectrogram transformer[END_REF][START_REF] Xu | Masked autoencoders that listen[END_REF]. However, using L1 or L2 losses for reconstruction can result in a blurred image or a noisy audio signal. As He et al. suggest [START_REF] He | Masked autoencoders are scalable vision learners[END_REF], improving the quality of MAE predictions can potentially lead to better representations for downstream tasks. This paper introduces the vector quantized MAE for speech (VQ-MAE-S), a self-supervised model designed for emotion detection in speech signals. VQ-MAE-S is an adapted version of the Audio-MAE model proposed in [START_REF] Gong | Ssast: Self-supervised audio spectrogram transformer[END_REF][START_REF] Baade | Mae-ast: Masked autoencoding audio spectrogram transformer[END_REF][START_REF] Xu | Masked autoencoders that listen[END_REF]. Unlike Audio-MAE, VQ-MAE-S operates on the discrete latent representation of a vector-quantized variational autoencoder (VQ-VAE) [START_REF] Van Den | Neural discrete representation learning[END_REF] instead of the spectrogram representation. The pre-training of the model is performed on the VoxCeleb2 dataset [START_REF] Chung | Voxceleb2: Deep speaker recognition[END_REF], and fine-tuning is carried out on several standard emotional speech datasets. We conduct several experiments to study the impact of different model design choices (e.g., masking ratio, masking strategy, patch size, etc.). The experimental results demonstrate that the proposed VQ-MAE-S model yields improvements in SER performance compared to an MAE using raw spectrogram data and other state-of-the-art methods. The code and qualitative reconstruction results are available at https://samsad35.github.io/VQ-MAE-Speech/.

VECTOR QUANTIZED MASKED AUTOENCODER FOR SPEECH

This section presents the proposed self-supervised VQ-MAE-S model, which is represented in Figure 1. The model takes as input the power spectrogram of a speech signal, denoted by x ∈ R T ×D + , where T and D correspond to the time and frequency dimensions.

Vector quantized variational autoencoder

The proposed self-supervised approach builds upon the discrete latent representation of a VQ-VAE [START_REF] Van Den | Neural discrete representation learning[END_REF] assumed to be pre-trained and frozen (more details on the pre-training are given in Section 3.1.1). The VQ-VAE encoder is used to obtain a compressed and quantized representation xq ∈ Z T ×D of the input speech power spectrogram x ∈ R T ×D + . Each entry of xq corresponds to the index of a vector in the VQ-VAE codebook. The quantized representation xq keeps the aspect of a time-frequency representation, as the VQ-VAE is designed to be fully convolutional on the frequency axis and to process spectrogram frames independently. xq can thus be seen as a discrete and compressed representation of the speech power spectrogram x, where compression occurs along the frequency axis (D D). As illustrated in Figure 1 and discussed in the next subsections, the proposed self-supervised learning approach operates on this discrete and compressed representation before reconstruction with the VQ-VAE decoder.

Tokens creation, masking, and embedding

Discrete index tokens As illustrated in Figure 1, we build discrete index tokens from the quantized representation xq by dividing it into non-overlapping patches of size (t × d). This leads to nt = T t and n d = D d patches horizontally and vertically, respectively. The rep-

resentation xq ∈ Z (n t •t)×(n d •d) is then reshaped to Z (n t •n d )×(t•d)
and used as input to the masking process. Masking We apply masking to the (nt × n d ) time-frequency grid of discrete index tokens, each of which has a dimension of t • d. The masked tokens are then replaced with a trainable vector. We explore three different types of patch-based masking, where tokens are masked randomly along the time-frequency, time, or frequency dimensions. Additionally, we test the frame masking strategy that involves masking the entire discrete frames of xq (columns) instead of the patches. The effectiveness of these strategies is experimentally evaluated in the context of SER, including varying the ratio of masked tokens from 50% to 90%. Continuous embedding tokens The discrete index tokens correspond to the indices obtained through the quantization step of the pretrained VQ-VAE encoder. Before being input to the VQ-MAE-S encoder, these discrete tokens are replaced with trainable continuous embedding vectors taken from a codebook of dimension R k×e , where k is the number of codes in the codebook, and e is the dimension of each code. This is simply achieved by replacing the t • d indices of a discrete token by the corresponding t • d vectors of dimension e in the codebook. The codebook is initialized by the pretrained VQ-VAE codebook and it can be held frozen or fine-tuned. After this embedding process (represented by the orange block in Figure1), the discrete index tokens in Z (n t •n d )×(t•d) are transformed into continuous embedding tokens in R (n t •n d )×(t•d•e) . In the experiments, we will investigate the effect of varying the embedding size e on the SER performance.

VQ-MAE-S encoder, decoder, and loss function

Encoder The VQ-MAE-S encoder, similar to the ViT architecture [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF], consists of a single Transformer encoder [START_REF] Vaswani | Attention is all you need[END_REF]. This encoder is a stack of L residual blocks that includes a self-attention layer, a normalization layer, and a Multi-layer Perceptron (MLP) block. Trained position embeddings are added for each token [START_REF] Devlin | Bert: Pre-training of deep bidirectional transformers for language understanding[END_REF]. As in [START_REF] He | Masked autoencoders are scalable vision learners[END_REF], the encoder inputs are only the visible tokens; this is to learn a representation that relies on the context. In addition, we add a global trainable token [CLS] as in [START_REF] Devlin | Bert: Pre-training of deep bidirectional transformers for language understanding[END_REF], which will be used for SER tasks. Since most of the tokens are masked and only the visible tokens are fed to the encoder, this resolves the quadratic complexity issue inherent in transformer models with respect to the sequence length [START_REF] He | Masked autoencoders are scalable vision learners[END_REF]. As a result, the number of attention blocks in the encoder can be increased without experiencing computational inefficiencies. Decoder The VQ-MAE-S decoder takes in both visible and masked tokens along with an additional position embedding. It consists of attention blocks similar to the encoder, but with fewer consecutive blocks (L ) compared to the encoder (L > L ). The decoder also includes a linear layer at the end that maps to the size of the VQ-VAE codebook. The output of this linear layer corresponds to the logits of the discrete index tokens. After applying a max operation, we obtain a reconstruction x rec q of the indices xq that were provided by the VQ-VAE encoder. Loss function To train the VQ-MAE-S model, we minimize the cross-entropy loss applied only to the masked discrete index tokens:

LVQ-MAE-S = cross-entropy xq(ΩM), x rec q (ΩM) , (1)
where xq(ΩM) and x rec q (ΩM) represent the masked tokens in xq and their reconstructions, respectively.

EXPERIMENTS

3.1. Pre-training, fine-tuning, and evaluation

Pre-training of VQ-MAE-Speech

To pre-train VQ-MAE-S, we used the VoxCeleb2 dataset [START_REF] Chung | Voxceleb2: Deep speaker recognition[END_REF], which provides an extensive collection of audio speech data from opensource media, including speech segments corrupted by various realworld noises. We restricted our use of the dataset to a subset of around 1000 hours of audio-visual speech, including 2170 speakers.

The VQ-VAE model was trained on the same portion of the VoxCeleb2 dataset using short-time Fourier transform (STFT) power spectrograms (x). The STFT is computed using a 64-ms Hann window (1024 samples) and a 70% overlap, resulting in sequences of D = 513-dimensional discrete Fourier coefficients. The VQ-VAE architecture is symmetrical with respect to the encoder and the decoder, with three 1D convolution (encoder) or transposed convolution (decoder) layers on the frequency axis and a residual convolution layer. The model processes each frame independently with no time dependency. For each speech power spectrogram frame of size to increase the use of the different codes in the codebook, as in [START_REF] Yu | Vector-quantized image modeling with improved vqgan[END_REF]. We evaluate the impact of different encoder architectures on the performance of VQ-MAE-S. The architecture is denoted as VQ-MAE-S-n, where n refers to the number of attention blocks in the encoder. The decoder is fixed at four attention blocks. Each selfattention layer in these blocks is subdivided into four heads. The model is trained using the AdamW optimizer [START_REF] Loshchilov | Decoupled weight decay regularization[END_REF] with a cosine scheduler to adjust the learning rate, with a 100-epoch warm-up period. The parameters of the optimizer, similar to [START_REF] He | Masked autoencoders are scalable vision learners[END_REF], are β2 = 0.9, β2 = 0.95, and weight decay= 0.05. The base learning rate follows the linear scaling rule [START_REF] Goyal | Accurate, large minibatch sgd: Training imagenet in 1 hour[END_REF] lr = (base lr = 1e -3) × (batchsize = 512)/256. We distributed the pre-training of VQ-MAE-S on 4 NVIDIA HGX A100. As in [START_REF] Xu | Masked autoencoders that listen[END_REF], no data augmentation is performed. For more information on the architecture, please refer to our publicly available implementation.

SER fine-tuning details

Only the encoder of the VQ-MAE-S model is fine-tuned for SER. We propose two approaches: The first one uses the global token [CLS] as input to a single linear layer, followed by a max operation to perform emotion classification. The second approach, referred to as Query2Emo and inspired by [START_REF] Liu | Query2label: A simple transformer way to multi-label classification[END_REF], involves cross-attention between all sequence tokens as value/key and the emotion classes represented by trainable embedding as query. Query2Emo has a single attention block for both the encoder and decoder. For these two approaches, we use the AdamW optimizer [START_REF] Loshchilov | Decoupled weight decay regularization[END_REF] with a cosine scheduler to adjust the learning rate and with a 40-epoch warm-up period. The parameters of the optimizer, similar to [START_REF] He | Masked autoencoders are scalable vision learners[END_REF], are β2 = 0.9, β2 = 0.95, and weight decay = 0.05. The base learning rate is 1e-4. For the loss function, we adopt the asymmetric loss [START_REF] Ridnik | Asymmetric loss for multi-label classification[END_REF] adapted for single labels instead of the cross entropy as it yields better results.

Emotional databases for fine-tuning and evaluation

We fine-tune and evaluate the proposed approaches on four emotional speech audio databases. [START_REF] Steven | The ryerson audiovisual database of emotional speech and song (ravdess): A dynamic, multimodal set of facial and vocal expressions in north american english[END_REF]: This English database consists of 1440 audio files recorded by 24 professional actors and labeled with eight different emotional expressions (neutral, calm, happy, sad, angry, fearful, disgust, surprised). RAVDESS-Song [START_REF] Steven | The ryerson audiovisual database of emotional speech and song (ravdess): A dynamic, multimodal set of facial and vocal expressions in north american english[END_REF]: Same as the RAVDESS-Speech database, but utterances are sung a capella. This database contains a total of 1012 audio files recorded by 23 actors and labeled with six emotions (neutral, calm, happy, sad, angry, and fearful). IEMOCAP [START_REF] Busso | Iemocap: Interactive emotional dyadic motion capture database[END_REF]: This database comprises approximately 12 hours of audio, annotated with several emotions, but only four emotions (neutral, happy, angry, and disgusted) have been retained to ensure a balanced distribution. It consists of dyadic sessions in which actors participate in improvisations or scripted scenarios. EMODB [START_REF] Burkhardt | A database of german emotional speech[END_REF]: The German EMODB database consists of 535 utterances spoken by ten professional speakers. It includes seven emotions (anger, boredom, anxiety, happiness, sadness, disgust, and neutral).

RAVDESS-Speech

For a fair comparison with the literature, we perform 5-fold cross-validation by separating the speakers' identity between the fine-tuning phase and the evaluation phase.

Performance on speech emotion recognition

Table 1 highlights the importance of pre-training and fine-tuning the VQ-MAE-S-12 model for SER. Pre-training significantly improves the SER performance, with the accuracy increasing from 28.6% to 76.7% on the RAVDESS-Speech dataset. Fine-tuning the encoder is also crucial, as freezing the encoder results in a 19.1% accuracy loss. We also compared our approach with SpecMAE-12, an MAE model that directly uses speech power spectrogram patches and aims to reconstruct masked patches using L2 loss. Our approach outperforms SpecMAE-12 by 24.5% in terms of accuracy, which clearly shows the benefit of working on the discrete representation of the VQ-VAE instead of the raw spectrogram representation for training the MAE.

Table 5 compares the SER performance (accuracy and F1-score) of the proposed VQ-MAE-S model (with classification from the [CLS] token), its improved version VQ-MAE-S + Query2Emo, the SpecMAE-12 baseline, and three state-of-the-art methods: SSAST [START_REF] Gong | Ssast: Self-supervised audio spectrogram transformer[END_REF], MAE-AST [START_REF] Baade | Mae-ast: Masked autoencoding audio spectrogram transformer[END_REF], and a supervised self-attention-based approach [START_REF] Chumachenko | Self-attention fusion for audiovisual emotion recognition with incomplete data[END_REF] on the four evaluation databases. Two configurations of masking are considered: random frame masking (Frame) and ran- We conducted several experiments to assess the importance of several hyperparameters of the proposed VQ-MAE-S model, which are presented and discussed in the following paragraphs. Impact of the masking strategy The choice of the masking strategy in MAE-based self-supervised models can have a significant impact on the performance of auxiliary tasks [START_REF] He | Masked autoencoders are scalable vision learners[END_REF]. As shown in Table 2, which compares the performance of emotion recognition across four masking types discussed in Section 2.2, random frame-based masking outperforms random patch-based masking. In particular, the frame-based approach achieves performance gains of 15.1%, 12.3%, and 4.1% compared to random patch-based masking on the time, frequency, and time-frequency axes, respectively. These results are consistent with those reported in prior studies [START_REF] Gong | Ssast: Self-supervised audio spectrogram transformer[END_REF][START_REF] Baade | Mae-ast: Masked autoencoding audio spectrogram transformer[END_REF], and are highlighted in Table 5 for the four evaluated databases. Impact of the masking ratio Figure 2 shows the relationship between masking ratio (x-axis) and SER accuracy (y-axis) for the patch (blue) and frame (red) masking using VQ-MAE-S. The results indicate that patch masking achieves maximum accuracy at around 80% masking ratio, after which performance declines. The trend can be attributed to the complexity of the task, where higher masking ratios improve representation learning and SER accuracy. In contrast, the performance of frame-based masking remains relatively stable. However, pushing the complexity too high (e.g., 90%) leads to a loss of relevant contextual information and decreased SER accuracy. Impact of the encoder depth The impact of the encoder depth on SER performance is presented in Table 4, where the number of atten-tion blocks is varied. The results show that, to some extent, increasing the number of blocks in the encoder leads to improved performance. When the number of blocks becomes very high (in this case, VQ-MAE-S-20), there is no further improvement, and performance actually decreases (-2.4% accuracy) compared to VQ-MAE-S-16. Impact of the continuous embedding token size The impact of the continuous embedding token size on the SER performance on the RAVDESS-Speech dataset is shown in Table 3, by varying the dimension (e) of the codes in the dictionary. Although we did not conduct an extensive study on this parameter, we tested three exponentially increasing token sizes (160, 320, 640). It is observed that using a size of 320 leads to an improvement in SER performance compared to sizes 160 and 640, with an increase of +5.8% accuracy and +1.3% accuracy, respectively. Impact of the discrete index token size Figure 3 illustrates the impact of the dimensions of the discrete index tokens (t and d) on SER performance on the RAVDESS-Speech dataset. d represents the discrete token size on the frequency axis, while t represents the discrete token size on the time axis. Our study reveals that both t and d significantly affect SER performance. Therefore, it is important to carefully select the values of (t, d). Based on our experiment, we suggest fixing them to (t = 10, d = 4).

CONCLUSION

VQ-MAE-S is a novel approach that uses a pre-trained VQ-VAE model to adapt the MAE for speech representation learning, using discrete tokens obtained from the quantization step of the VQ-VAE. Our method outperforms other MAE methods based on spectrograms or mel-spectrograms for SER on several standard datasets. Several experiments also highlighted the impact of VQ-MAE-S hyperparameters on SER performance. For future work, we plan to investigate other masking strategies to further improve SER performance, and we aim to combine the MAE with contrastive methods to improve the learned audio representation and achieve even better SER performance.

Fig. 1 :

 1 Fig. 1: VQ-MAE-S model structure.

Fig. 2 :

 2 Fig. 2: Impact of the masking ratio for VQ-MAE-S-12.

Fig. 3 :

 3 Fig. 3: Impact of the discrete index token size (t, d) on SER.

Table 1 :

 1 Performance of VQ-MAE-S-12 on RAVDESS-Speech. Masking method: Random time-frequency masking (Patch-tf ), Ratio: 80%, freeze refers to the freezing of the encoder of VQ-MAE-S.

	Method	Pre-train freeze Acc. (%)
	VQ-MAE-S-12	26.8
	VQ-MAE-S-12	57.6
	VQ-MAE-S-12	76.7
	SpecMAE-12	52.2

Table 2 :

 2 Performance of VQ-MAE-S-12 on RAVDESS-Speech for different masking strategies with a ratio of 80%.

	Masking method	Acc. (%) f1-score (%)
	Random frame masking (Frame)	80.8	80.5
	Random frequency masking (Patch-f )	65.7	65.0
	Random time masking (Patch-t)	68.5	68.7
	Random time-frequency masking (Patch-tf )	76.7	75.9

Table 3 :

 3 Performance

	of VQ-MAE-S-12 on RAVDESS-Speech
	for different continuous embedding sizes. Masking method:
	Random time-frequency masking (Patch-tf ); Ratio: 80%
	Token dim.	Parm. (M) Acc. (%) f1-score (%)
	(t • d • (e = 4))=160	≈5	70.9	70.1
	(t • d • (e = 8))=320	≈20	76.7	75.9
	(t • d • (e = 16))=640	≈80	75.4	75.3

Table 4 :

 4 Performance

	of VQ-MAE-S on RAVDESS-Speech for differ-
	ent encoder depths. Masking method: Random time-frequency mask-
	ing (Patch-tf ); Ratio: 80%.
	Encoder Depth Parm. (M) Acc. (%) f1-score (%)
	VQ-MAE-S-6	≈12	65.8	65.3
	VQ-MAE-S-12	≈20	76.7	75.9
	VQ-MAE-S-16	≈25	79.2	79.4
	VQ-MAE-S-20	≈30	76.8	76.2

D = 513, the VQ-VAE compresses it into a discrete latent vector (a row of xq) of size D = 64. The VQ-VAE codebook contains k = 256 codes of dimension e = 8. Such a low dimension is chosen

Table 5 :

 5 Overall results (accuracy (%) and f1-score (%)) on the four evaluation databases.

	DATASET	RAVDESS-Speech	RAVDESS-Song	IEMOCAP	EMODB
	Metrics	Accuracy f1-score Accuracy f1-score Accuracy f1-score Accuracy f1-score
	Self-attention audio [26]	58.3	-	-	-	-	-	-	-
	SSAST [5] (Patch-tf )	-	-	-	-	54.3	-	-	-
	MAE-AST [13] (Patch-tf )	-	-	-	-	58.6	-	-	-
	SpecMAE-12 (Patch-tf )	52.2	52.0	54.5	53.9	46.7	45.9	57.2	57.0
	VQ-MAE-S-12 (Patch-tf )	76.7	75.9	84.0	84.0	61.9	61.2	85.7	85.8
	VQ-MAE-S-12 (Patch-tf ) + Query2Emo	78.2	77.5	83.7	83.4	63.1	62.5	88.4	88.3
	VQ-MAE-S-12 (Frame)	80.8	80.5	84.2	84.3	65.2	64.9	87.0	87.0
	VQ-MAE-S-12 (Frame) + Query2Emo	84.1	84.4	85.8	85.7	66.4	65.8	90.2	89.1
	dom time-frequency patch masking (Patch-tf ). The results indicate
	that the proposed models outperform all other methods across all
	databases. For random time-frequency patch masking (Patch-tf ),
	VQ-MAE-S achieves 15.2% better accuracy than SpecMAE, 7.6%
	better accuracy than SSAST, and 3.3% better accuracy than MAE-
	AST on the IEMOCAP dataset. The accuracy improvement over the
	supervised method on the RAVDESS-Speech database is of 18.4%.
	Query2Emo also contributes to the SER performance, with a gain of
	1.5%, 1.2%, and 2.7% compared to VQ-MAE-S (Patch-tf ) alone on
	RAVDESS-Speech, IEMOCAP, and EMODB, respectively.