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Remarks on the mathematical origin of wave mechanics and consequences for a quantum mechanics in a gravitational field

According to Schroedinger's ideas, classical dynamics of point particles should correspond to the"geometrical optics"limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the"geometrical optics"analogy leads to the correspondence between a classical Hamiltonian H and a"quantum"wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.


Introduction and statement of the main results
De Broglie found that the analogy between the variational principles which arise in geometrical optics and in Hamiltonian dynamics had its origin in a wave-particle duality valid for material particles as well as for electromagnetic radiation. Elaborating on de Broglie's ideas, Schrödinger emphasized that geometrical optics is merely an approximation to wave optics and, starting from the Hamilton-Jacobi equation of classical mechanics, he proposed his famous wave equation. For Schrödinger, the « matter waves » had to be generally more real than the particle aspect. Schrödinger's way to his equation was rather intuitive -although, in the application of this equation, he mastered advanced mathematics of his time. In today's quantum mechanics, the correspondence between a classical Hamiltonian and a « quantum » wave equation, as well as the rules of measurement, are introduced in a formal, axiomatic way (perhaps after having motivated this axiomatic by the examination of elementary examples). Moreover, the wave function is interpreted, not as representing a real wave, but instead as a tool to compute probabilities. While the quantum axiomatic works extremely well in the innumerable usual applications of quantum mechanics, it leads to ambiguities when one tries to extend it to a situation with curved space-timewhich is needed if one wishes to have a quantum mechanics in a gravitational field. Furthermore, the experimentally confirmed predictions of quantum mechanics, in situations of the Einstein-Podolsky-Rosen type, are difficult to interpret in the framework of physical realism. A realistic interpretation might be made easier if one would abandon the orthodox notion that the quantum objects are more likely to consist of point particles, and if one would come back, instead, to Schrödinger's concept according to which his extended waves describe an essential element of the physical reality of a quantum object. If, furthermore, one would partly relax the relativistic interdiction to pass beyond the velocity of light, the difficulties of interpretation might be even smaller [16]. The Lorentz-Poincaré version of special relativity gives support to this attitude, because it « desacralizes » the role of c. Due to relativistic mechanics, c remains an upper limit for the velocity of classical particles, but the fact that an ether is preserved in this version [12] allows to interpret the limiting velocity in an intuitive and less absolute way. If one thinks of material particles as of topologically structured flows in a fluid ether, the limit c applies to the organized flow as a whole, rather like the sound velocity [13], and does not forbid that parts of the flow may be superluminal -which is indeed the case in the slightly different context of purely electromagnetic particles [19].
The present work is an attempt to understand Schrödinger's wave mechanics in the light of the modern theory of (classical) waves, to give some (incomplete) support to the possibility of a realistic interpretation of his waves, and to draw conclusions for a possible extension of quantum mechanics to a situation with a gravitational field.
First, the correspondence between a linear wave equation and a classical Hamiltonian (algebraic in p) will be analysed in the framework of the theory of dispersion relations in a linear medium. It will be shown that this correspondence is fully justified under the assumption that the classical Hamiltonian should describe the trajectories corresponding to the « geometrical optics » limit of the considered wave equation. Thus, the wave equation deduced from a classical Hamiltonian is a priori no more « quantum » than is the d'Alembert equation for the classical electromagnetic potential. This correspondence is one-to-one also in the general case where the Hamiltonian contains mixed terms involving momentum p and position x (and time). This allows to use it also in the case with a gravitational field, the latter making the (extended) configuration space heterogeneous.
In the second part of the paper, it will be attempted to understand, from the point of view of wave mechanics, how eigenvalue problems do occur in quantum mechanics. In preparation for the gravitational context, it will be briefly recalled how stationary states are related to energy states, i.e. to eigenvalue problems for the spatial Hamiltonian operator. Then, the case of momentum states will be discussed in some detail. If one follows the principles of wave mechanics, the most essential property of a « pure momentum state » is that the group velocity should be spatially uniform. This property is true for a plane wave in the case of the usual Schrödinger equation, of course, but it is shared with a plenty of different wave functions. There are additional properties which should be satisfied by the wave function of a « pure momentum state » : it should obey the timedependent Schrödinger equation, and it also should satisfy the dispersion relation expressing the frequency as a function of the wave vector. Furthermore, if one wishes that such wave function may represent a single quantum object, it should be spatially limited in the minimum sense that the amplitude should vanish at infinity -a property obviously wrong for plane waves. It will be shown that these properties, taken together, lead to the unavoidable conclusion that: (i) pure momentum states can exist only in so far as the spatial variation of the force field is negligible, i.e., the potential must be linear in x; (ii) in that case, they do exist (thus for whatever time-dependence of the force field), and they are characterized by a wave which undergoes a mere translation with time, thus no deformation and, in particular, no spreading out at all; (iii) they must have a singularity, whose motion obeys Newton's second law.
The third part of the paper is devoted to the extension of wave mechanics in the presence of a gravitational field, as characterized by a curved space-time metric. It will be shown that, with a curved metric, Schrödinger's wave mechanics is inherently frame-dependent, hence a consistent wave mechanics in a gravitational field implies that a physically preferred reference frame must exist. This is, of course, incompatible with general relativity, but it is fully consistent with the « ether theory of gravitation » studied by the author. Thus, although one may propose gravitational extensions of quantum-mechanical wave equations without assuming any preferred reference frame, this seems rather formal from the viewpoint of Schrödinger's wave mechanics. In particular, it will be proved that a classical Hamiltonian exists for the motion of a test particle moving « freely » in a static gravitational field; that the wave-mechanical correspondence leads to unambiguously associate with this Hamiltonian a wave equation, which is thus the extension of the free Klein-Gordon equation; and that this wave equation, which also makes sense for a general gravitational field, is definitely a frame-dependent equation (in contrast to the obvious manifestly covariant extension of the free Klein-Gordon equation). The stationary states of this equation are defined: in the static case, they are solutions of an eigenvalue problem for a spatial operator. In the general case, one still may define stationary states, but they they do not seem to correspond to a spatial eigenvalue problem.
Moreover, it will be shown that the « wave-mechanically correct » extension of the free Klein-Gordon equation has the property that the associated energy-momentum tensor T does not satisfy T µ ν ; ν = 0, even in the static case. The latter equation is the dynamical equation in general relativity and other metric theories, and also, for the static case, in the preferred-frame theory studied by the author. Hence, it seems that, in the presence of a gravitational field, one has to make an unpleasant choice between wave mechanics and the most general form of classical dynamics.
In the Conclusion part, heuristic remarks about the concept of the « constitutive ether » are made. According to this concept, the quantum aspect of microphysics comes from the fact that the elementary objects are special kinds of flows in an imagined perfect fluid (or « micro-ether »), such as a vortex. On the other hand, gravitation would be a continuous variation, due to the accumulation of a great number of vortices, of the macroscopic pressure in the assumed substratum. This leads the author to conjecture that gravitation is not a quantized interaction, for it has the nature of a macroscopic correction: even though it affects matter down to the scale of elementary particles, the definition of the gravitational force is macroscopic. The source of the gravitational field is the classical energy-momentum tensor (or its energy component).

Dispersion equation and wave equation in a (classical) linear medium
As everyone knows, a general wave ψ may be defined as the product of a « smoothly varying » amplitude A(t, x) and an oscillating part φ(θ(t, x)), with φ a periodic scalar function of the dimensionless real variable θ, the latter or « phase » being itself a function of the time t and the position x. The « position » x belongs to an Ndimensional configuration space M, which may be the physical space (N = 3). In physics, φ is usually the complex exponential: φ(θ) = exp(iθ) (or its real part, cosθ ), but other « wave profiles » do also occur. The notion of a smoothly varying amplitude means that the wave structure may be recognized: to the very least, the relative amplitude variation δA/A should not exceed the order of δθ /T , with T the period of the wave profile φ. Now if we have a physical law for a given wave phenomenon, it will lead to a partial differential equation for the wave function ψ. Let us assume that ψ is a scalar and that this equation is linear: , is the relevant point of the product « time » × configuration space (extended configuration space R×M), and D µ ψ = ψ ,µ is the partial derivative. Quite often, the order of the equation is n = 2: Given the phase function θ(X ) of a general wave, one defines the wave covector K = ∇θ = (-ω, k), i.e. K µ = θ , µ . In the linear case, it is appropriate to consider a sinusoidal wave profile: φ(θ) = exp(iθ ), hence φ ,µ = iK µ φ. Let us consider a wave function ψ = A exp(iθ ), with A constant, which is an « elementary wave » at the point X considered, i.e., such that K µ , ν = 0 at point X. The necessary and sufficient condition for such wave function to obey Eq. (2.2) at point X is: the obvious generalization to Eq. (2.1) being (under the condition that all derivatives of the K µ 's, up to the order n -1, vanish at the point X considered):  [21]. However, one may mix odd and even derivatives, provided one accepts complex coefficients in the wave equation -as for Schrödinger's equation.
Now we state the essential result of this Section: the correspondence between the linear operator P and the function Π is one-to-one, in other words one may uniquely pass from the wave equation to the dispersion equation and conversely. This is undoubtedly true, since the data of either P or Π is equivalent to the data of the set of the scalar functions a 0 , a 1 µ , a 2 µν , and so on, of the point X in R×M. The important point is that this result is true also in the general case with variable coefficients, in which a 0 , a 1 µ , a 2 µν , ..., do depend on X. In the case with constant coefficients, the inverse correspondence, from (2.4) to (2.1), amounts to the substitution This remains true in the case with variable coefficients, provided one orders each monomial in the dispersion equation as in (2.3) or (2.4), i.e., « X before K ». This ordering is the natural ordering for a polynomial in K with coefficients that are functions of X. It corresponds to the ordering in Eq. (2.1) or (2.2), which also is the natural ordering for the general linear differential equation: the (variable) coefficient comes before the differentiation.

Group velocity and Hamiltonian motion
As mentioned above, we now need the preferred « projection » time t, with X = (t, x), and we seek to compute the frequency ω = −K 0 as a function of the « spatial » part k of K: ω = ω(k 1 , ..., k N ) = ω(k). When extracting the real roots of the polynomial equation Π X (K) = 0, considered as an equation for the unknown K 0 with the data k 1 , ..., k N , one may follow the different roots W 1 , ..., W n (at most n) as functions of X. It is assumed in the following that one such particular root has been identified, and we have thus: which is called « the » dispersion relation, it being kept in mind that several different such relations may in general be extracted from the unique dispersion equation [21]. Of course,W, which will be called the dispersion, is in general not a polynomial function of k at fixed X. Now let us assume that some wave function of the general form is such that the corresponding wave covector obeys exactly the dispersion relation (2.6) (we do not assume here that ψ is an exact solution of the wave equation (2.1), and indeed this is in general incompatible with our assumption). One defines the group velocity for this wave function as the « spatial » vector with components (Latin indices will be reserved for « spatial » components). This is the natural generalization [21] of the usual notion of group velocity in a uniform medium. From the definitions K µ = θ , µ and ω = − K 0 , and from the symmetry of the second derivatives of θ, one gets and since here ω(t, x) = W(k(t, x), x, t) by hypothese, Eq. (2.9) 1 may be rewritten as [21] ∂ This is a hyperbolic equation for the (spatial) wave covector k, showing that k propagates with the group velocity. It is immediate to verify that it can be put in the following characteristic form [21]: Thus, the motion of the wave vector in a general nonuniform medium is governed by a Hamiltonian dynamical system, the Hamiltonian being the dispersion W. To the author's limited knowledge of the literature on quantum mechanics, this crucial result has been overlooked there.

The quantum correspondence and its (non-)ambiguity
Let us consider a classical system of point particles with « position » x (in the configuration space M), its dynamics being given by a Hamiltonian system with Hamiltonian H and conjugate momentum p (a « spatial » covector): (2.12) Following de Broglie and Schrödinger, let us imagine that those microscopic objects, which we initially described as point particles, actually have a spatial structure made of some (unknown) waves; and that this classical Hamiltonian dynamics only describes the « skeleton » of the wave motion, in precisely the same way as geometrical optics describes the trajectories of « light rays », which constitute the skeleton of the light wave propagation pattern. Thus, we expect that the dynamics (2.12) should be that approximation of the wave motion which becomes exact at the « nil wave length » limit where, in the neighborhood of any point X in the extended configuration space, the wave may be considered as a plane wave, that is A ≈ Const. and δθ ≈ k.δ δx − ωδ t = K µ δ x µ (2.13) in Eq. (2.7); of course, condition (2.13) 2 means that the higher-order derivatives of θ can be neglected: K µ , ν ≈ 0, and so on. We seek for the linear wave equation ( , it should satisfy one and only one among the different possible dispersion relations (2.6), and we assume that it is indeed the case. Therefore, in the « geometrical optics » limit, the spatial wave covector k [associated with a wave function (2.7) obeying the wave equation (2.1)] obeys also the « Hamiltonian dynamics » (2.11): the only difference with a true Hamiltonian dynamics is that, in Eq. (2.11), a continuous distribution of Hamiltonian trajectories is involved. However, to account for the fact that (in its domain of relevance), the classical approximation (2.12) successfully considers point particles, we expect that, at least in some cases, the wave pattern is made of spatially concentrated « wave packets », in the exterior of which the amplitude is negligibly small. Obviously, we are led to admit that the Hamiltonian systems characterized respectively by the dispersion W (Eq. (2.11)) and by the classical Hamiltonian H (Eq. (2.12)) have exactly the same set of possible trajectories x = x(t).
This can be achieved by a canonical transformation (x, k, t, W ) → (x, p, t, H ) (thus with the « position » and time variables being left unchanged) only if k = p and W = H. But a canonical transformation means that the trajectories in the extended phase space, R×T*M, are the same: here we merely want that their projections in the extended configuration space R×M are the same. If we consider the Lagrangians Λ(x, x , t ) and L(x, x , t ) respectively associated with W and H by the Legendre transformation, the condition is that the extremals of the respective action integrals must be the same. The obvious, simplest way to ensure that this is true, is to impose that Λ and L are equal up to a multiplicative constant. We denote this constant by ... Thus, we are led to admit that L = Λ, whence follows that (2.14) and for the canonical conjugate momenta: The a priori interpretation of the correspondence (2.14) and (2.15)

Comments on the physical meaning
The usual correspondence (2.18), between a classical Hamiltonian and a « quantum » wave equation, may thus be arrived at by a seemingly new method, based on the dispersion relation in a linear medium. The interest one may find to this method is that it gives a rather rigorous basis to the essence of wave mechanics: the idea that classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation. It has been shown that this idea leads, modulo the study of classical dispersion relations [21], to the correspondence (2.18) in a natural and general way. The method is not limited to the case of a single particle: it works for any Hamiltonian system. Moreover, the method also indicates how to make the correspondence unambiguous in the case where the classical Hamiltonian contains mixed terms involving both the momentum p and the (configuration-)spacetime position X = (t, x). The rule to obtain the wave operator unambiguously is simple: just put the function of X as a multiplying coefficient before the monomial in p -as the dispersion equation has to be a polynomial in p at fixed X. This is important in the case with a gravitational field (part 4 of this paper), for a « curved spacetime » means that, for example, the kinetic energy becomes such mixed term (since it involves the metric tensor, that depends on X ).
Thus, a classical Hamiltonian (even in an implicit form such as Eq. (2.19)) gives unambiguously a linear wave equation for the more complex wave structure supposed to underly the approximate classical behaviour of a system of particles (the latter corresponding to the nil wave length limit). However, if one admits that it is a wave structure which is the fundamental behaviour, then one must expect, of course, that in some cases no classical Hamiltonian will be relevant, i.e., one must expect that the simple correspondence (2.18) is not always a sufficient tool to obtain a correct wave equation -an example is Dirac's equation.
If one admits that the quantum wave function ψ describes a physical wave, then one should be able to define a local energy density ρ for the wave, for which a local conservation should apply, involving also the force fields to which the quantum object is subjected (and which, conversely, the quantum object should in general influence). The « space » integral of ρ would be the total energy of the quantum object, which should be conserved if it is in a constant force field, and possibly also under more general conditions. For example, if the force field is electromagnetic, the energy of the field is conserved if the field lines of the Poynting vector field are closed, hence the possibility of « purely electromagnetic » quantum particles (see Vlasov [18] and Waite [19]). Thus, the energy density ρ would be substituted for the probability density of standard QM. Now, for the one-particle Schrödinger equation, there is the well-known local conservation equation, that involves only the wave function ψ, and which expresses the conservation of |ψ| 2 . Hence, one is tempted to assume ρ = |ψ| 2 in that case ... However, for other equations such as the Klein-Gordon equation, |ψ| 2 is not a conserved quantity, although the energy density of the Klein-Gordon field (the T 0 0 component of the associated energy-momentum tensor T) is well-defined, positive and conserved [6]. From the tentative point of view of a « mechanics of real microphysical waves », there does not seem to be any compelling reason to admit in general that the wave function ψ should be squareintegrable and that the integral of |ψ| 2 should be a conserved quantity -although there is indeed such reason for the usual one-particle Schrödinger equation. However, as long as the wave function is supposed to represent a single quantum particle (which is in principle impossible in the relativistic case, since the particle number is not conserved then), its spatial extension should be limited in some sense: for example, a function ψ such that |ψ| is constant in space should not be a correct wave function for a single particle.

Eigenvalues problems for energy states and for momentum states
The quantum-mechanical aspect in the correspondence between a Hamiltonian and a wave equation lies in the wave-particle duality: thus, a system of mass particles should be handled in general as a wave function defined on the extended configuration space and, conversely, a classical electromagnetic field obeying the d'Alembert equation should be treated, in the geometrical optics limit, as consisting of zero-rest-mass particles. However, the genuinely « quantum » (i.e., discrete) aspect of quantum mechanics is the role played by discrete eigenvalue spectra. It seems desirable that, as well as the above correspondence, also the eigenvalue problems should occur in a natural way: the best would be to avoid the a priori postulate that « the quantum observables are operators in a Hilbert space and the measurement operations lead to eigenvalues for these operators ». In this question, one should not limit oneself to wave equations that can be obtained by the correspondence discussed above.

Energy states and stationary waves
As it is well-known, the quantization of the energy levels appears naturally, in Schrödinger's wave mechanics, when one looks for stationary solutions of his equation. More generally, let H be a « quantum Hamiltonian », i.e., a linear operator acting on functions ψ defined on the extended configuration space R×M, and assume that it does not contain the time. 1 For a such « purely spatial » quantum Hamiltonian, one searches for stationary 1 In precise terms: H involves only « spatial » derivatives (e.g. in Eq.
A precise definition of stationarity may be that the time and « space » variables are separated: and that the time-dependence φ is a quasi-periodic function. Putting (3.2) in (3.1) gives hence the only possibility is that φ '/φ = Const ≡iω, whence with ω real in order that φ be quasi-periodic (and indeed periodic). Thus, the spatial part a is a solution of the eigenvalue problem (3.4) 2 , while the time part is sinusoidal with frequency ω = E/ , where E is the eigenvalue. Note that the elementary proof of this well-known result remains true in the case where the wave function ψ takes values that are not scalars, but elements of some vector space (e.g. a spinor space), provided one demands that the time-dependence, φ(t) in Eq. (3.2), is a scalar. Hence, it applies to Dirac's equation. Thus, as long as the wave equation is of the general Schrödinger type (3.1), one may dispense oneself from the a priori assumption that the energy levels are eigenvalues of the Hamiltonian operator, and replace it by the assumption that the energy levels correspond to stationary states. But what does an energy level mean? What is experimentally observed is a frequency spectrum (ω ' n ) of electromagnetic radiation, so one should postulate, more strictly, that any observed frequency ω ' of the emitted/absorbed radiation is the absolute difference between two eigenfrequencies: This is the Bohr relation, which was proposed in the framework of the light quanta hypothese, but it seems to be interpretable with a « real wave » picture, as Schrödinger suggested [14][15]. It then seems to indicate that there is some identity of nature between the « matter waves » and the electromagnetic waves, since the frequencies suffer no distortion. This would be understandable if all waves were in a common substratum or ether. However, the well-known difficulty for such a realistic interpretation is that the matter waves are in a configuration space.
In deriving the conclusion (3.4) that a stationary wave has a sinusoidal time-dependence and that its amplitude a(x) is the solution of an eigenvalue problem, no restriction is imposed on the spatial part of the wave equation, but the time part is assumed to be the r.h.s. of Eq. , is in general a matter of convenience. As a matter of fact, the K-G equation may be rewritten in the Schrödinger form (3.1), by introducing a 2-component wave function [9]. However, it is more expedient here to stay with the scalar form (3.6).

Momentum states: a generalization of plane waves
In QM, it is admitted that a (single) quantum particle has a well-defined momentum p, if and only if its wave function (in the physical space, which coincides, in the case of a single particle, with the configuration space) is that of a plane wave. Moreover, only the spatial dependence of the wave function is envisaged, so that the « wave function » (at a given time) is A exp(i k.x). Thus, the first wave function encountered in QM has constant modulus and, in particular, it is not square-integrable, although for the case of one quantum « particle » obeying Schrödinger's equation, this is a necessary requirement (cf. the end of § 2.4). Actually, it is easy to show that a plane wave cannot obey the time-dependent Schrödinger equation unless the potential V is a constant number (thus unless the « particle » is in a free motion); this will be proved in passing below. Perhaps one may ask if the wave function of a moving particle could be defined as a function of time also, and if its spatial extension could be limited in some sense.
Thus, we consider a wave function of the general form (2.7), and we ask under which particular conditions it could be taken to represent a quantum object having, at any given time t, a well-defined momentum p(t). In agreement with the notion that the velocity of the particle is related to the group velocity of the associated wave (Eq. (2.11)), we impose a first requirement: (i) the group velocity C must be a function of t only. This seems to be a natural condition. However, C depends on the dispersion W (Eq. (2.8)). The latter is found by the correspondence (2.14), in the particular case that the quantum wave equation can be deduced from a classical Hamiltonian by the correspondence (2.18). In the general case, the dispersion equation is found directly from the wave equation by the transition from Eq. (2.1) to Eq. (2.4), thus by the substitution D µ → i K µ , provided the wave equation can be put in a form where it is the same for each component of the wave function; then, several dispersions (whence several C vectors) may be defined by extraction (see § (2.2)). In any case, the definition of C depends on the wave equation. Hence, the natural condition (i) implies that the very definition of a « pure momentum state » should depend on the quantum wave equation. Let us assume that this is the one-particle Schrödinger equation, associated with the non-relativistic Hamiltonian (2.17). The group velocity is then (3.10) so that, in the case of the usual Schrödinger equation, condition (i) is equivalent to the requirement that (i)' the spatial wave covector k is a function of t only 4 . Thus ∇ x θ = k(t), which is equivalent to say that the phase θ has the form θ(t, x) = k(t ).x − f (t ). (3.11) Note that, under condition (i)', the frequency is not necessarily uniform: Obviously, condition (3.11) is fulfilled by a plenty of possible wave functions, since no restriction is imposed on the amplitude A. But since we recognized that our definition of a momentum state depends on the wave equation, we have one more reason to impose on the wave function ψ the condition that (ii) ψ should obey the wave equation -which is anyway a necessary requirement 5 . Furthermore, we assume from now on that the amplitude A, as well as the phase θ, is real. This is the necessary condition under which the writing of the complex function ψ in the form (2.7) is (practically) unambiguous: otherwise, the relation between ψ and its phase θ would be totally ambiguous. (The restriction that A also should be real does not play a role in the study of dispersion relations, since one then considers constant amplitudes.) After an easy algebra, the time-dependent Schrödinger equation for the wave function (2.7), with the phase (3.11), is then found equivalent to the two real equations: potential A: in that case, one finds C = [k -(q/c)A]/m (consistently with the fact that, for a classical particle governed by the relevant Hamiltonian, P -(q/c)A = m x is the « true » momentum, where P is the canonically conjugate momentum, to which applies the relation P = k ). Hence, k = k(t ) is then incompatible with C = C(t ) unless A = A(t) (i.e., no magnetic field). 5 In the case where a magnetic field is present, the expression of the group velocity involves A that depends on the gauge condition (note 4). Therefore, one is really enforced to impose the wave equation: if one changes the gauge condition, the wave function, hence also the wave covector k, will then transform according to a simple rule [11], and it is easy to verify that the group velocity is then independent of the gauge.
Hence, we may wish to impose the additional requirement that (iii) the dispersion relation is satisfied: (3.15) and this requirement implies that Eq. (3.13) is equivalent to the Laplace equation: However, Eq. (3.15), in which we recall that p = p(t) ≡ k (t), implies also that V is a linear function of x. Indeed, since ω is given by Eq. (3.12), we deduce from (3.15) that Thus, considering a uniform force field F(t), let us study such a pure momentum state precisely. Given the initial momentum p 0 , its time evolution is completely determined by Newton's second law (3.18). Due to Eq. (3.10) with k = k(t), the momentum may really be considered as uniformly distributed in the wave. The dispersion relation (3.15), i.e., « E = ω », determines the frequency ω. Note that it depends on space and time. The group velocity being known, Eq. (3.14) shows that the amplitude A undergoes a translation with the uniform velocity C(t), i.e., A simply follows the spatially uniform motion. In particular, A is determined at any time if it is known initially. The spatial variation of A is determined by the Laplace equation (3.16). Here, we wish that (a) A is defined in the whole space (except perhaps at a point singularity) and harmonic, and (b) A tends towards zero as r ≡ |x|→ ∞ (this is a minimum requirement of spatial limitation). First, these conditions imply that there is a ball |x| ≤ r 1 such that A is not bounded in this ball, since otherwise it would be harmonic except perhaps at one point, and bounded in the whole space, hence [7, p. 305] it would be harmonic in the whole space; being harmonic and bounded in the whole space, it would be a constant, hence zero by condition (b). Hence, A is not harmonic at one point of the ball |x| ≤ r 1 (since otherwise it would be continuous, hence bounded, in this ball), which we choose as the origin. Thus, A has a point singularity at x = 0. Now if we impose the subsidiary condition that A is not more singular than 1/r, i.e., rA is bounded as r → 0, then we get by the same theorem [7, p. 305] that there is a constant R such that A -R/r is a harmonic function φ on the whole space. But φ is bounded on the whole space since it is bounded in every bounded domain and since, as A and 1/r, it tends towards zero as r → ∞. Hence, φ is a constant a, which must be zero. Thus, conditions (a) and (b) imply that A = R/r with R an arbitrary constant and r ≡ |x| after the appropriate choice of the origin x = 0, unless one would accept that A be more singular than 1/r. Condition (b) gives A ≈ R/r as r → ∞, independently of the latter condition [7, p. 314]. Unfortunately, R/r is not squareintegrable. Note that, if we replace condition (b) by the less severe condition that (b)' A is bounded as r → ∞, just the same arguments lead to A = α + R/r. Now let us consider, for R and a > 0, the function From a physical point of view, this continuous function is undistinguishable from R/r if α is small enough, and it has a finite square integral. Of course, A does not satisfy ∆A = 0 on the sphere r = R/α, but A vanishes from that sphere and, in traversing the sphere, its derivatives undergo an arbitrarily small discontinuity if α is small enough. We note finally that different amplitudes functions A may be found if one accepts that A has, for instance, a one-dimensional singularity (such as a circle, as for a torus vortex). However, as long as one demands that A vanish at infinity, a singularity is unavoidable since A must be a harmonic function. 6

Comments on the physical interpretation of the « momentum states »
Thus, in a uniform force field -∇V = F(t), the Schrödinger equation admits the wave functions

20)
as square-integrable pure momentum states (see the proposed definition after Eq. (3.18)). The wave covector k(t) and the function f (t) are determined (up to their arbitrary initial value) by the potential V(x, t); in particular, the momentum p(t) = k (t) evolves according to Newton's second law (3.18).
The arbitrary numbers R and α determine the constant « spatial extension » of a such wave function, which may be taken to be R' = R/α. As outlined above, α should be considered as a small « cut-off ». Such wave functions may indeed appear to be more correct realizations of a « pure momentum state » than a plane wave, because, in contrast to plane waves: (i) they obey Schrödinger's equation (except at the singularity x = a(t) and on the « cancellation sphere » |x − a(t)| = R/α), and (ii) they are square-integrable in space.
In replacing A R α by R/r, one gets momentum states that satisfy Schrödinger's equation everywhere except at the singularity and that vanish at spatial infinity, though without having a finite square integral. The fact that the evolution of the momentum is governed by Newton's second law seems to make the transition from classical mechanics to wave mechanics quite transparent: classical mechanics would envisage as a point particle what is in fact a spatially extended wave. And since the « pure momentum state » remains so only in a uniform force field, classical mechanics 6  The answer is, obviously, that the latter applies (as a consequence of the QM postulates regarding the « collapse of the wave function ») to the outputs of measurements of momentum and position. Whereas the present result applies to the objective description of a « quantum particle » (wave function) in a spatially uniform force field, thus excluding a necessarily perturbating action of measurement. This result is hence compatible with the axioms of QM if they are taken in the operational sense, and it has no immediate consequence on the possible predictions that one can make in QM. We also note that, since the spatial extension of a « pure momentum » wave function is arbitrary, one still may say in a restricted sense that « the particle has not a welldefined position ». (This is no surprise if one considers that there is indeed no point particle but an extended wave!) Another one might consider, on the contrary, that « the position of the particle » is perfectly defined at any time, as the position a(t) of the singularity! The present result may contribute to give hope that one may go beyond QM towards a theory based on Schrödinger's notion of « real waves ». In any case, it indicates that QM as it is does not preclude Schrödinger's viewpoint (which appears essentially compatible with de Broglie's), at least at the stage of the scalar Schrödinger equation that does not integrate the notion of the quantum spin.
Finally, if one tries to interpret the wave function ψ as a real wave, one may venture to guess which physical quantity could wave as ψ. If one dares to think of a mechanical wave in a fluid « micro-ether », it certainly could not be the displacement -due to the singularity. Similarly, it seems to be excluded that ψ could be the pressure, because the micro-ether should be barotropic (Romani [13], Arminjon [1]), so an infinite amplitude would also occur for the density oscillation. Instead, the real part of ψ could be perhaps a (any?) component of the velocity u of the flow, or instead, possibly, of grad u. This suggests again -but this time directly from the quantum-mechanical framework -the idea (tracing back at least to Kelvin, with e.g. Romani [13] and Winterberg [22][23] as recent defenders) that the elementary particles could be ideal vortices in a perfectly fluid « micro-ether ».

General framework and the question of the reference frame and the reference time
We adopt the general framework of theories in which gravitation changes the flat Galilean spacetime metric into a curved metric γ γ. The latter is the physical metric, in the sense that, for example, the proper time defined along a trajectory as the integral of dτ = (γ µν dx µ dx ν ) 1/2 /c, (4.1) is assumed to be the time that may be measured by a physical clock moving along this trajectory. As it has been emphasized previously [3], this assumption of a « physical » pseudo-Riemannian metric does not imply per se that the relativity principle (principle of manifest covariance) must remain true in the presence of gravitation. In any such theory, including general relativity, the Poincaré-Einstein synchronization convention can be applied only to infinitesimally distant clocks [10]. At first sight, this seems to reinforce the relativity of simultaneity, because clocks may then be synchronized, in general, only along an open line in space. However, there exists (always locally i.e. in an open neighborhood of any space-time point, and indeed in the space-time as a whole, unless a « pathological » space-time such as Gödel's is considered), a « globally synchronized » reference frame (GSRF), i.e., such that the γ 0i components are zero in some adapted local coordinates (x µ ). (« Adapted » coordinates are ones in which any observer bound to the reference frame has constant space coordinates (x i ).) The Poincaré-Einstein synchronization extends to the whole of a GSRF and defines, in this frame, a privileged family of time coordinates, that exchange by space-independent transformation of time: x' 0 = φ(x 0 ), under which the « γ 0i = 0 » condition is stable [3]. Now we wish to extend Schrödinger's wave mechanics to the case with a gravitational field. To this end, we just need to make sure that the tools which are used there remain at our disposal with a curved space-time metric γ γ. As noted in Section (2.1), the correspondence, at any given point X of the extended configuration space, between the wave equation (here of order n ≤ 2) and the dispersion equation demands that one restricts oneself to infinitesimally linear coordinate changes; and, for that purpose considered alone, the class of locally geodesic coordinate systems appears naturally. However, a clear space/time separation, as it occurs in Newtonian physics, has to be used so as to define the group velocity and the associated Hamiltonian system (2.11) (cf. Section (2.2)). Finally, the latter system is assumed to give the same set of trajectories as a certain classical Hamiltonian (2.12) (cf. Section (2.3)).
Therefore, we a priori need to introduce a preferred time coordinate (up to a scale factor) and hence to restrict ourselves to some subclass of coordinate systems, two of which should exchange by a purely spatial and infinitesimally linear transformation: restricting the discussion to a « one-particle » system, the admitted transformations are given by (In a subsequent step, one may examine the covariance of the obtained wave equation under a larger group of transformations. However, the need to fix the time coordinate will be further justified below: see after Eq. (4.4).) Furthermore, a classical (though « relativistic ») Hamiltonian system (2.12) must be found. 8 The first step is hence to ask: in the curved space-time of some theory of gravitation, can one select in a natural way a group of transformations of the form (4.2)? Note that such selection includes the choice of a particular reference frame. In view of the outmost importance taken by the notion of a global simultaneity in Newtonian physics (which is the concrete arena for Hamiltonian systems), it indeed seems natural to require that the preferred time t is a globally synchronized time, which means that a GSRF should be considered. However, there is a continuous class of GSRF 's, that undergo a « pure strain » (i.e., no « rotation ») with respect to each other, and one would have to ask if one may then reexpress the wave mechanics 8 Although it indeed must be relativistic in the sense of being consistent with a pseudo-Riemannian space-time, with the mass increase with velocity, etc., the system (2.12) must be expressed with such preferred time coordinate. That is, H in Eq. (2.12) is a Hamiltonian with the space, not the space-time, as the configuration space (in the case of one particle); and with a true time coordinate, not the proper time τ of Eq. (4.1), as the time t, for the simple reason that τ is defined only once the line in space-time is known -whereas, in a « true » Hamiltonian system (2.12), the time t is defined independently of the position in the configuration space! covariantly from one to another. In what follows, one GSRF will be considered. The coordinate changes internal to such frame are [3] x' 0 = φ (x 0 ), The further restriction to locally geodesic coordinate systems [γ µν,ρ (X) = 0] is then impossible in general (no LGCS is in general bound to the given GSRF, in other words the local observer of the given frame is in general not in a « free fall »). One class of coordinate systems, stable under infinitesimally linear spatial changes, and compatible with (4.3), appears naturally: those systems whose space coordinates are locally geodesic at X for the spatial metric g in the given GSRF: g ij, k (X ) = 0 (i, j, k = 1, 2, 3 Thus, the fixing of the time coordinate is physically imposed. Actually, the author has proposed a preferred-frame, scalar theory of gravitation, according to which gravitation breaks the Lorentz invariance (see Refs. 1, 2, 4). The preferred frame (« macro-ether » or simply « ether ») of this theory is, by hypothese, a GSRF: a global simultaneity is defined in terms of the « absolute time » T, which is that measured by an observer bound to the ether and far enough from massive bodies. Hence, the answer to the above question is that, in a preferred-frame theory of gravitation as the one just evocated, one group (4.2) appears naturally, namely those coordinate systems whose time component is x 0 = αT with T the absolute time, and whose spatial part verifies (4.4); and that it is not the case for theories starting from the relativity postulate (or here the principle of « manifest covariance »).

Comments on special relativity and on the covariance of Hamilton's equations
(i) In the case of a flat space-time metric (special relativity), a particular class of GSRF 's does appear naturally, namely that of the (global) inertial frames (those frames which are defined by a Galilean coordinate system). And in any inertial frame, an even smaller group than (4.2) occurs naturally: that of the global Cartesian coordinates (combined with the change of the time scale a). In that case, one thus naturally builds one wave mechanics per inertial frame, and the question is then: what happens in the transition from one inertial frame to another? It seems that, as far as only the wave equation is concerned, nothing happens: if a wave equation is deduced, by the quantum correspondence (2.18), from a Lorentzcovariant expression of the Hamiltonian, H = H(p,x, t), then this wave equation itself is Lorentzinvariant, of course.
(ii) At first sight, it seems that the a priori restriction to coordinate changes defined by (4.2) is not imposed by the Hamiltonian dynamics in itself, but instead by the correspondence between the Hamiltonian and the wave equation. Indeed, a Hamiltonian system like (2.12) remains in the same form even after the most general canonical transformation of the extended phase space (although the Hamiltonian is changed if the transformation depends on time), and these transformations make a very large class. In such transformations, the position x, momentum p, and time t variables are considered on just the same footing: the only condition is that the 1-form p.dx -H dt is invariant up to an exact 1-form dS [5, p. 239]. However, the covariance of the Hamiltonian system (2.12) under canonical transformations is not in the same sense as (and indeed it is a weaker statement than), for instance, the covariance of the geodesic equation of motion for a free particle in a gravitational field (in general relativity and other « metric theories »). To be precise, consider a merely spatial transformation: This may be seen as a canonical transformation with generating function By definition, the canonical variables transform indeed as it is therefore a covector only for space transformations that are infinitesimally linear at the point considered, again. In other words, Eq. (2.12) 1 is covariant only if the variation of ∂ x i /∂ x' j along the trajectory is « forgotten » -in contrast to the geodesic equation, in the transformation of which this variation is accounted for, as one easily checks. This difficulty comes from the fact that, in Hamilton's equations, (dp i /dt ) is indeed not a true covector but instead the projection, on the dp i part of the coordinate system (dx i , dp i ) on T(T*M) (x,p) (deduced from the data of the coordinate system x i on M), of a tangent vector ξ ξ∈ T(T*M) (x,p) . Only in the case where M is vector space E is there a coordinate-independent decomposition of ξ ξ into a vector dx/dt and a covector dp/dt (due to the fact that, in that case, T(T*M) (x,p) is canonically identified to E×E*). This canonical identification (implying an intrinsic decomposition for x) is defined through the intermediate of the class of the linear coordinate systems (the linear bijections of E onto R N ) (see Dieudonné [8,). Thus, the a priori restriction to transformations all having the form (4.2) may also be seen as resulting from the need to fix the time coordinate (as it is fixed in physics that furnishes the classical Hamiltonian and that demands a precise form of the wave equation), plus the need that Hamilton's equations be truly covariant under the allowed coordinate transformations.

Klein-Gordon equation in a gravitational field
As an illustrative test of the foregoing framework, let us investigate the possible extension of the free Klein-Gordon equation (3.6) to the case where a gravitational field is present. Thus, we consider a massive test particle subjected only to the gravitation. We first have to ask whether a Hamiltonian H does govern the motion of a such « free » test particle in a gravitational field. This feature may depend on the law of motion in the theory of gravitation. Precisely, it may depend on whether the test particle follows a geodesic line of metric γ γ (as in « metric theories ») or, instead, its motion is defined by an extension of Newton's second law with a gravity acceleration field g, generally incompatible with geodesic motion (as in the « ether theory of gravitation » (ETG) studied by the author). The conclusion of § 4.1, that the necessary restriction on the coordinate systems (Eq. (4.2)) may be fulfilled in a natural way only in such preferred-frame theory, gives some justification for starting the present discussion with the investigated ETG. However, the result that will be obtained, and that applies to a static gravitational field, is also true in metric theories since, in the case of a static field, the equation of motion in the ETG is the geodesic equation, thus the same as in metric theories [2][3].
In the ETG, the energy of the particle in the gravitational field is defined by where β ≡ √γ 00 , γ v ≡ 1/√(1−v 2 /c 2 ), the velocity vector v (in the preferred frame) and its modulus v being evaluated in terms of the local physical standards of space and time: (4.11) The energy defined for metric theories by Landau & Lifchitz [10], turns out to have the same general expression (4.10). It is conserved if and only if the field is constant: this is true for the ETG as well as for metric theories [3]. It is hence natural to try e as a Hamiltonian H(p,x,T), with a canonical momentum p that is yet to be determined. The Lagrangian associated with H is defined by Legendre transformation: