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Disregarding Frequency: the Spherical Inversion

Notice that the alleged physical condition of limitation of velocities to values smaller than the speed of light in vacuum is here what one sometimes sees termed in the literature as “theoretical folklore”. Indeed, this condition is basically a consequence of the relativistic mass transformation formula, whereby if the velocity of body approaches the speed of light, the mass associated by an observer at rest goes necessarily to infinity. It is only human to think that something strange, unphysical, happens there, so we have to limit the considerations to speeds smaller than that of light. It is this way that the limitation of velocities came up as a physical condition. Useless to say, the modern theoretical physics found methods to deal with superluminal velocities. This fact compels us to somehow formally frame the initial statement that was the basis of de Broglie’s association between group of waves and particle, in order to delineate it from unsecured additions. One way to do this, worthy of consideration inasmuch as it suggests a fruitful approach of the problem, is indeed to formally accept as physical all the velocities smaller than that of light, and analyze the consequences. This fact can easily be accomplished if we seek a three-dimensional extension of the theory.

Notice that the core of the association wave-particle is the idea of frequency as coming directly from Planck’s quantization through the concept of quantum. While the original Planck’s considerations were referring to blackbody radiation for which the notion of frequency makes perfect sense, de Broglie made an unsecured extension of the area of application of that notion. That should have made him uneasy, to the extent that he felt necessary to further elaborate on the subject. However, as his later works prove de Broglie was exclusively dealing with the frequency: he struggled to clarify the nature of the frequency he introduced, and the works cited by M. Lochak in his Note [3] are to be taken along that idea. As the history proves, the nature of that frequency is still unclear. It is just natural then, to explore alternatives under the broad idea of the association wave-particle.

The first reaction to this seems to present itself naturally: try to avoid the notion of frequency. This can be done easily by exploiting a relation between velocities as suggested when using the frequency itself. More to the point, the relation between phase and group velocities as given in the equation (11) above is the core of the whole construction: mathematically it represents an inversion which, while facilitated by using the frequency, can be described in general by dispensing with this concept. In case one seeks an extension of this relation for the 3D vectors, the most natural and direct one is the inversion of velocity vectors with respect to the sphere of speeds of light in vacuum. In order to do this it is necessary to look at equation (11) as representing a relation between magnitudes of the vectors representing the group and phase velocities. Then the relation between the two vectors can be written as

which is, indeed, the expression of a geometrical inversion of the velocity vectors with respect to the sphere

representing the possible speeds of light in vacuum. Thus if v is a vector inside the sphere (13) u is necessarily a vector outside that sphere, and vice versa. This way we have a clear distinction between the velocities smaller than that of light and the velocities greater than that of light. If we deny physical reality to these last ones, then they do not represent physical entities: in de Broglie’s language they represent just phase waves. In this formalism, having very much in common with Synge’s approach to de Broglie’s waves [7], a superluminal theory cannot therefore refer but to phase waves, some kind of mathematical beings, having no physical reality.

By equation (12) the correspondence between phase waves and particles is one-to-one: there is only one phase wave associated to a particle, and vice versa. And there is no contradiction in this. The idea of group of waves enters the stage here as related to indecision of the reference frame. Indeed, the equation (12) should be valid in a stationary reference frame. In a reference frame moving with the velocity w, the associated wave will have the speed u’ where

Applying the inversion formula to this wave velocity, gives the particle velocity

The very same goes for phase waves; one can write

thereby associating an ensemble of waves to a phase wave, in one-to-one correspondence with the ensemble of reference frames.

A little digression is now in order. First of all, the correspondence between particles and waves is here one-to-one and, as already noticed, there is no contradiction for this association, inasmuch as the exterior of the light sphere in velocity space is exclusively reserved for “unphysical” phase waves. There is, however, a subtle issue here. One can see that the idea of group of waves, which is the key in energetical description of particles, may become itself unphysical if we do not accept that the classical vector composition applies to velocities greater than that of light. Even at a superficial consideration, one can see the trouble boiling here. We need therefore to answer at least to a question like: has this purely geometrical speculation any bearing with some physical facts? Well, if one forgets about Relativity, it certainly has. Indeed, let us take one of the equations (15) or (16), and calculate the magnitude of the vector from its left hand side. We have, for instance,

From the first one of these equations we find, by expanding to the first order in s, the formula

where θ is the angle between the direction of the motion of reference frame and that of the velocity v. When v and v’ represent speeds of light in a transparent medium, which is plainly a physical situation, the equation (18) gives the Fresnel dragging. The second expression in equation (17) has been written here only in order to compare this inversion formula with the formula used by Lorentz in deducing the equation of the Fresnel dragging from his theorem of corresponding states [8]. This fact may give some physical credentials to the inversion formula as long as it refers to vectors characterizing the speed of waves. After all, the article [2] reporting on a possible physical meaning of de Broglie’s frequency, refers to experiments with particles (electrons) in crystals, the counterpart of historical experiments with light. However, in the case of particles the extension of the physical meaning of the inverse of velocity bears a marked statistical character. Come to think of it: the frequency, being it classical or de Broglie frequency, has always a statistical character, passing unnoticed most of the times. This fact suggests insisting on some statistical issues related to frequency that were plainly indicated by de Broglie’s later works on the subject.

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