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VIII. Discussion And Conclusions

Concluding this work, let us sketch its basic philosophy, starting this time from the point it reached in explaining what it proposed to explain. In the description of the gravitational field - or of every field for that matter - one has always to harmonize the omnipresence and permanence of the field (intrinsic attributes of the field concept) with the alleged contingency of events occurring in the field. At this point the usual Fourier analysis cannot help much, for the simple reason that the field is a uniform continuum: the time of evolution of apparatus is not the time of the field. We cannot say anymore “the apparatus selects a Fourier component of the field”. This was always a problem with the field measurements, and it occasionally turned into a big problem of the Theoretical Physics. A case in point is that of the infinite (self) energies leading to the renormalization procedure. It has been shown in the present work that, if it is to talk about harmonic oscillator structure in the case of a continuum field, we have to talk some other way. This way is precisely indicated.

Continuing with the philosophical development, one fundamental event, accessible to direct human perception and thus not involving the idea of measurement by interaction, is that of ‘particle in a point at an instant‘. To this event and its relationship with the field concept, the Science of Mechanics has dedicated a considerable analysis, mainly along the idea that the field determines characteristics of the trajectory of motion, this last one conceived as a succession of events. There is one historical case where the field is thought to determine the very characteristics of the particle, and that case comes out with the Mach principle referring to the (inertial) mass. Our contention in the present work is an improvement to the formulation of that principle. Indeed, as a result of the present work, we can add to this philosophical point our conclusion: the gravitational field is actually a kind of mediator between the cosmological (distant) interactions and the local system interactions. More specifically, the gravitational field is involved in a variation of the physical parameters of the harmonic oscillator. The time of this variation - which we like to call the cosmic time because it is a Stoler parameter related to the variation of intrinsic properties of the harmonic oscillator - is a solution of the Laplace equation, thus indicating a geographical dependence on a certain synchronization of the oscillators. This fact may have consequences, for instance, in our representation of a body interacting with the gravitational field. The local time - which is here the time of dynamics of the oscillator - is apparently independent of the cosmic time. This conclusion is, however, a little premature, inasmuch as the Israel-Wilson linear dependence between the amplitude of oscillator and gravitational field gives us a sense of incompleteness: according to Stoka Theorem we would expect an invariant cross ratio which is to be constructed here. This is apparently not the case, showing that the linear equation might be only a particular case of a more complex equation involving, possibly, what we have left aside in our present considerations: the transition electromagnetic fields.

Indeed, the electromagnetic field we are looking for in completing the image of measurement by departing from direct human perception must be a transition field, a perturbation going from one oscillator to another, more to the point, a reflection of the universality of the gravitational field as we defined it previously. In this work, however, we did not go into such details as the interaction properties of the oscillators. All we can say for now is that these interaction properties are in close connection with harmonic surfaces in space. A proper endeavor to describe the interaction here would require a reversal of the logic of initial general relativistic point of view. Namely we would have to submit the idea of particle, insofar as the point of space and the moment of time aspects of this concept are involved, to a logical analysis based on the field concept as omnipresent and permanent. To this task we reserve a future work.

The acceptance of universality as we understand it here came out with the second quantization, which is a kind of straightforward way to picture as accidental something that is actually permanent and omnipresent. The main idea is that the events necessarily require interaction for their very definition, and this definition amounts to the fact that a certain event does not occur if favorable field conditions are not met. It is these field conditions that can be labeled as contingent. In the specific case of gravitational field the situation remained always essentially classical. Indeed, it is hard to grasp as contingent something that seems to be the background of the everyday life. We have here as archetype the well-known anecdote with the Newton’s apple, showing how hard has the human spirit to struggle in order to escape from the chains of direct senses. It is nevertheless this background that we have to describe, starting evidently from the events composing the measurement process. Unfortunately, in describing the measurement process, we have to bring in all of the three characteristics of the field concept - omnipresence, permanence and universality - and Physics does not have a luxury of tools here beyond those considering contingent interaction as germane to the problem; which is why the physical thinking has sometimes been misled in its judgments. There is, however, an apparently new approach of the measurement of gravitational field trying to mix the permanence of this field with the contingency of the interaction involved in its measurement. This approach is the so-called continuous measurement one, and it suggests a way to physics of the gravitational field based on geometrical principles. The latest contention of this philosophy is that the motion can be viewed as a continuum measurement (Mensky 2002). Essentially, the continuous measurement problem has brought into light the idea of introducing a group theoretical approach to measurement, and the way to introduce it has been considered, in as much detail as possible, for the first time in (Mensky 1990). The general idea following from the present work in this respect is that a weight function necessary in the definition of the restricted path integrals, and usually introduced ad hoc, is just a natural thing to consider, for it has inscribed in it the apriori measure of the phase space as determined by the field mediating the measurements. The work gives also a recipe to construct weight function and probability density functions necessary for the continuous measurement problem and discusses their combination, thus issuing utilizable results. Based on these results, we venture to say that the Stoka Theorem allows building the general relation between the gravitational field and the electromagnetic field, whose first instance is Israel-Wilson compatibility condition, on a measurement basis. This invites to speculations about some obvious future developments. It becomes thus certain that as long as the transition electromagnetic field is explicitly involved the true characteristic of space is not flatness. In other words Israel - Wilson condition is just a particular case of a more general relation. In this respect, the present analysis allows one to say that the true nature of space is actually inscribed in the intrinsic properties of the matter, inasmuch as the universality of the gravitational field is involved. Indeed, if we are able to discover a SL (2, R) action accounting for the presence of the electromagnetic field as an interaction property, then we are in position to put this field in the correct universal form giving the general compatibility condition for the Ricci tensor of space. Finding this relation, a task for which we cannot see fit anything else but Stoka Theorem is, again, a matter for a future work.

Acknowledgment. If something positive has come out of the present analysis - and as far as the present author is concerned there is something positive for common knowledge - we made it plain that it is due to the Stoka Theorem. There may be, and certainly are, more modern geometrical ways to express the same things, however they surely do not allow for pure Physics to come so clearly into light. Being thankful for his life dedication to Geometry, the author would like to dedicate in return this modest contribution to Professor M. I. Stoka for his seventieth birthday. Hoping that it is a rightful return, the author also wishes to Professor Stoka many happy rightful returns of every one of his birthdays. May there be many of them too!

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