ProtoQuant.com

IV. A Matter of Principle in the Classical Framework

The above treatment of the particle dynamics, leads to some important consequences: first of all, if we admit that the particle exchanges energy with its environment, and this exchange is taken into consideration by a 2-form like (4), the general principles of the Invariantive Mechanics lead us directly to an extended particle model, as envisioned by Nonlinear Electrodynamics [5]. In view of equation (10), the mechanical properties of this particle model may not be independent of the electrodynamic properties, inasmuch as they are formally expressed in the very same way. In other words, while the Theoretical Electrodynamics may be a universal language for the physics of interactions, the electromagnetism may be just a particular instance of the interactions in question. This fact is reflected here by the existence of an unknown metric tensor which is responsible for the stresses induced in the interaction process (Eq. (13)). Such a treatment of the particle dynamics, points toward an invariantive formalism that might unify Mechanics, Electrodynamics and Thermodynamics. Finally, we observe that conceiving the mass this way turns out to be equivalent with an extended Mach Principle, extended in the sense that dynamics is not exclusively determined by what we know as inertia.

It is worth taking time for a little elaboration on this type of Mechanics. The equation (2) indicates still another supporting characteristic of the theory. Namely, from a very classical (i.e non relativistic) point of view, we have two differential 1-forms characterizing a particle

(18)

Then the equation (2) for these forms is equivalent to the first equation from (3). The development of Modern Physics shows that, in general, the momentum and velocity are correlated not by a linear relation, but by something more complicated. This observation has been brought about by Dirac [6], and it is at the base of his theory of constraints. Now, we are not able to find that relationship between momentum and velocity, but we may use (2) for the differential forms (18), to show that, in general, a first step in writing dynamics of a particle is to accept that

(19)

where Ω₀ is a physical 2-form of energy. If this is zero, then we obtain the classical result of proportionality of momentum and velocity. Otherwise, the relationship between the two vectors is more involved, a fact already proved by the development of Physics. To bring just a simple example, consider the Aharonov-Bohm effect [11]. Let us assume that Ω₀ does represent electromagnetic fields. Accounting for their existence would, of course, mean updating the definition of the momentum form, as always has been the case. First of all, equation (19) implies the following update:

(20)

and the connection of Mechanics with Magnetism for instance, would then read: the momentum 1-form (20) is null (i.e. the momentum can be purely mechanically defined) whenever and wherever the 2-form of magnetic flux is zero (i.e. there is no magnetic field) and reciprocally. Then, using an appropriate theorem of Cartan, the magnetic flux 2-form, B say, can be represented as

(21)

where a ≡ A_μdx^μ is a conveniently chosen 1-form. Now, if the magnetic field is existent but its flux is zero over some regions of Space, the equation (21) shows that over those regions we must have

(22)

where q is a scalar. Thus, the momentum must be updated to account for the existence of the magnetic field, by adding a vector that proves to be the classical potential vector. Needles to say, this procedure can be iterated as many times as we feel necessary, but then the bottom line is that the Aharonov-Bohm effect can be described as a classical topologic effect: as long as we have at our disposal the classic-quantum correspondence rules, the fact that it is also a quantum effect emerges quite naturally. This view of the relation experiment-theory may have an impact upon our association wave-particle.

Pages: 1 2 3 4 5 6 7