This article is outside the theme I’m carrying out in the blog. I decided to publish it anyway because the topic seems interesting to me to divulge. It is known that all vector fields that depend exclusively on the radial coordinate (are independent of the angle) admit a potential and the work done by the field in a movement from any point to any other point in space is independent of the type of path chosen for this move. Moreover, if the variability of the field is not only radial but, in particular, it is due to the inverse square of the radius, the field becomes conservative(1).
Some simple examples.
Imagine a cloud of millions of midges buzzing in the air. Imagine drawing a closed surface of any shape that encloses a portion of the cloud of midges. Whether the cloud of flies moves to a certain direction, or whether it remains on average in the same position of space, the number of flies entering the closed surface equals the number of flies coming out of the surface. This concept is valid only if the flies are not disturbed in some way in their buzzing or if midges are not born or die in the volume enclosed by the surface. The first case corresponds to creating a potential difference. E.g. if inside the surface we introduce a little bird the flies fly away radially and in this case the flies will all be outgoing. Or again, if inside the surface we inject a poisonous gas that kills the midges, the number of flies that cross the surface in the unit of time (the algebraic sum of those entering more than those leaving) will equal the number of midges that die in the unity of time. These are intuitive and reasonable concepts. The only small effort is to think of vector fields as if they were fluttering flies.
The differential and integral calculus of vector fields(2) is a very elegant way of dealing with these topics so fascinating that it risks diverting our concentration to the purely mathematical aspects, sometimes making us lose sight of the physical meaning of the phenomena(3).
Given these premises, we can state in extreme synthesis that field theory depends on the type of field and on the modalities of diffusion in space moment by moment: both these entities (vectorial fields and spaces) are susceptible of extension and abstraction. For example, the zero-dimensional space (the point) can be extended to one-dimensional (straight), extensible to two and three-dimensional (Cartesian coordinates in the plane and in space)(4). Mathematically it is possible to increase the number of coordinates (Hilbert spaces) until they become infinite (Banach spaces). Extradimensional extensions are also possible, as in the space-time case where, in addition to adding a “contaminated” temporal coordinate from the spatial variables, the spatial coordinates are also “contaminated” by the temporal coordinate as the Lorenz trasformations allows to make. Incidentally, with the extension to the space-time dimension the electrostatic field becomes electromagnetic and electrodynamic(5) and the gravitational field becomes relativistic. This extension/abstraction process allows us to imagine a possible description of the fields in “spaces” of a completely different nature from those usually treated by field theory.
Even for the vectorial fields it is possible to imagine a process of abstraction. E.g. why not think of a “thought” field, a “love” field, an “information” field? Obviously their domains would be particular “spaces” such as the “space of thought”, the “space of love”, the “information space”. At this point, in analogy to what happens in other fields, on could imagine the validity of principles such as the overlapping effects principle, the “conservation of thought”, etc. With these hypotheses it would be possible to apply field theory also to other “fields” not studied by physics such as those mentioned above. My hypothesis is that in this way it becomes possible to move from a qualitative description (philosophical, sociological, psychological, etc.) to a quantitative description (physical, scientific) of entities such as thought, love, information, etc. The “power of thought”, the “color” of love, “the energy of love, etc. they would become measurable quantities. In analogy to what is done for the vectorial fields studied by physics, e.g. for the “space of thought” it can be imagined initially homogeneous and isotropic and we can imagine a radial or even inverse quadratic radial transmission. Starting from these assumptions, the results would be analyzed, rectifying these hypoteses on the basis of the inconsistencies found until a congruent treatment was reached.
(1) As well known two examples of conservative fields are the electrostatic field and the gravitational field. In the electrostatic field, measurements have been carried out showing that the variability from the inverse square of the distance is valid up to distances of the order of 10-15 meters (nuclear distances). For shorter distances it seems that Coulomb’s law is less; in particular, the experimental results indicate values about 10 times weaker than those calculated with the Coulomb law. One of the hypotheses is that electrons and protons are not point charges but widespread charges. See also E. Williams, J.Faller, H. Hill, New Experimental Test of Coulomb’s Law: A Laboratory Upper Limit on Photon Rest Mass, in Physical Review Letters, vol. 26, No. 12, 1971, pp. 721-724, Bibcode: 1971PhRvL..26..721W, DOI: 10.1103 / PhyRevLett.26.721.
(2) Field theory makes use of spatial and vectorial derivatives (gradient, divergence, rotor) and second derivates (laplacian) and is based on the rules of vector calculus and on the Stokes and Gauss theorems. In the space-time extension, the Laplacian becomes the Dalembertian.
(3) In a subsequent article we will go into detail on the dipolar electrostatic field of the water molecule and on the VSEPR theory (Valence Shell Electron Pair Repulsion).
(4) The bi / three-dimensional treatment is independent of the type of coordinates chosen, whether they are Cartesian, polar, cylindrical, geodetic or other in the case, for example, of use of non-Euclidean geometries.
(5) In electrostatic/magnetostatic the Maxwell equations are separable (two equations relating only to electrostatics and two to magnetostatic only).