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The initial impulse of the "new" quantum theorists of the mid-1920's (as opposed to the "old" theorists from the early-1900's) was to describe elemental matter from within the context of spatial wave dynamics. The question of finding a way to understand the dynamical field-like structure of matter is what motivated the work of Louis de Broglie and Erwin Schrödinger in this period of time.
But due to the particularly "fuzzy" aspects of partial differential equations—which contain two or more independent variables, and are used in order to solve things like wave equations—there was destined to be a movement back towards the kind of "perfect solvability" that is obtained by the Newtonian point-location approach, which only utilizes the much simpler approach of ordinary differential equations—containing only one independent variable.
We can see from this historical progression that ease of simplistic mathematical formulation will tend to force philosophically intelligible explanation out of the confines of acceptable academic physical theory. The resulting ontological interpretations of atomic physics will then revolve around the question of the "reality" of things like probability functions and n-dimensional configuration space. However, the widespread (public) acceptance of any theory of physical reality ultimately requires that the theory be philosophically intelligible—that is, it must be directly conceivable by the imagination, rather than offering mere "predictions" of particular experimental situations.
The only reason not to consider the ontological significance of the sensible wave-fields of the de Broglie-Schrödinger variant of atomic theory is that it doesn't lend itself to simplistic algebraic formulation in the Newtonian tradition. It is from Newton that we have inherited the notion that the only "worthwhile" approach to physics involves tracking the trajectories of localizable particles through void space. From where I stand, however, the "failure" of the original post-Bohr-model atomic theorists to think in this kind of naive way about the fundamental structure of matter is the precise reason to take them seriously.
So here is the situation that we now have to work with: atomic matter must somehow be understood as a dynamical, oscillatory three-dimensional field. (This is obviously a much trickier notion that simple, lifeless point-particles!) The first stumbling block here is that the notion of a "three-dimensional oscillation" is not visualizable in the sense of a vibrating [linear] guitar string or a [planar] drum pad. In both cases, we can imagine the free space that these entities have have at their disposal to oscillate into. But when it comes to the oscillation of space itself, the only way to visualize it is by imagining that each spatial position (which are to be understood as "infinitesimal spaces" rather than "point locations") generically consists of various potentials, which can each be represented with a color. We can then think of the positions of lowest and highest potentials as respectively existing at the bottom of a valley and the top of a hill.
Just like a vibrating guitar string cycles through configurations such that the midpoint is now at position of maximum displacement and now at a position of zero displacement, the same idea can be applied to oscillating space, such that its center varies between a 1) state of maximum potential and 2) a state of zero potential. Applying this scenario to the language of general relativity, we can affirm that case 1 consists of the highest spatial curvature (or tension) case 2 consists of zero curvature—signifying the "flat" space of special relativity.
Einstein's theory, of course, did not consider the possibility that spatial curvature is itself a "primordial" phenomenon, and consequently that it could be thought of as a fundamental mode (modulation) of universal space itself. Now this possibility is highly compelling because we can begin to see a resolution to the matter vs. space duality that has been giving philosophers headaches since time immemorial.
The next issue is a highly complex one, and the only name that I can think to give it is: the boundary problem. That is, the only way that an oscillation mode makes sense is if it occurs within a bounded space. (Just like a vibrating guitar string only makes sense if it is anchored between two bounding points.) If, however, atoms consist of spatial oscillations, each within their own arbitrarily small bounding spheres, then the question of the nature of the interaction between mutually exclusive spaces must be satisfactorily explained. It is this precise question that has given rise to the doctrine that ontologically distinguishes the categories of "matter" and "radiation." (In many ways, this is just the same problem as the matter vs. space duality, but only dressed in slightly different clothes.)
The best (?) way to avoid descending into a state of hopeless confusion is to simply assert that the atomic modes of spatial oscillation are strictly universal in nature. That is, there are to be no arbitrary boundaries that cause a distinction between intra- and inter-atomic space. (My ultimate intention is to prove that it is only in this manner that the action-at-a-distance question can be philosophically settled—assuming, that is, that one finds the modern point-particle approach to be problematic.) It is from within this context of the pure mutual inclusivity of the modes of oscillation of universal space that the whole of my theory of physical reality is to be understood.
(As an aside, we have come to the crux of the philosophical distinction between "particle" and "wave." Waves are things that do not directly affect one another, in the sense of the kind of action-reaction "billiard-ball style" mechanisms of the Newtonian paradigm. That is, arbitrary numbers of waves are able to share the very same space while each constituent wave keeps its particular structure. The shared space, however, can be said to consist of a singular, composite wave; the structure of which is found by the "superaddition" of the displacements/potentials of the coincident elements of the individual waves into a resulting displacement of the composite wave. It is in this way that the universe can sensibly be said to consist of the pure mutual inclusivity of all of the elements of matter. In opposition, particles are just things that cannot possibly occupy the same space at the same time—they are naively considered to be "perfectly hard." The way that academic physicists explain this situation, however, is not to give a phenomenological account of this "perfect hardness," but rather to eliminate the very possibility of mutual inclusivity by way of denying the existence of the "internality" of elemental matter.)
But now, if all material elements (modes of oscillation) share the same universal space, and if an oscillation only makes sense as a function of a bounded system, then in what way can we think of these atoms/modes as existing within a physical system of constant relative motion? That is, if we think of all modes of oscillation as being a function of the same universal boundary, then simple logic tells us that the centers of oscillation of every possible atomic mode will coincide at the exact same location. Within this scenario, the only possible relational difference between each atom will consist of mere rotation, with no possibility of translation.
The solution to this problem lies in the use of a purely logical fourth spatial dimension, which will allow us to think of universal space as existing in the form of a so-called hypersphere. This is to say that the notion of the universe existing as a three-dimensional interior space that is bounded by a sphere only raises the problem of the discontinuity of space (due to the fact that space abruptly stops), as well as begs the question of the interior vs. exterior duality, if only from a different perspective as before (due to wondering about the nature of the "emptiness" that exists on the other side of the boundary). The only way to think of the universe truly as a uni-verse is to understand how it may be "self-cohesive," such as when we imaginatively fashion a circle (or sphere) from out of a straight line segment (or flat, circular disc).
The way that we can imagine a single mode of oscillation, then, is to think of the three-dimensional corollary of forming a vibrating guitar string (or drumpad) into a vibrating circular (or spherical) shape. In the [unimaginable] three-dimensional case, the entire spherical boundary will be made to connect into a single point at the polar opposite end of the universe, in relation to the maximally displacing structural center-point. In the language of wave mechanics, the points of a constant null displacement (such as the boundary points) are known as "nodes," and the points of maximal displacement are known as "antinodes."
The use of this "atomic wrapping procedure" will allow us to locate the various connection points of the different atoms at any arbitrary location on the universal hypersphere, therefore allowing for the kind of relative translatory motion that had been previously denied to us. Furthermore, given that each of these connection-points is nothing other than a compressed version of the nodal boundary that was used in order to establish the oscillatory behavior of each atomic mode, and given further that 1) the compression factor of the various positions within atomic space is a function of the inverse of the square of the distance from the connection point (this is the same essential issue faced by map projections), and 2) the nodal connection-point is, by definition, always a point of maximum stability (that is, lowest potential), then the result is that we have found a purely geometric description of the various attractive forces that are described in terms of potential difference—i.e. the effects of the gravitational and electrical fields.
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I have certainly tried to pack quite a bit of mind-bending information into a relatively small amount of space. I am not aware of anyone else who has attempted such a treatment in order to combine the most salient features of wave mechanics with those of Riemannian (n-dimensional manifold) geometry. The former topic is that which occupies the branch of physics that utilizes the Schrödinger equation—also known as quantum theory. And the latter is that which concerns the branch of physics that takes the Einstein field equations as its point of departure—which is better known as general relativity.
What we now possess (if only theoretically) is an immanently imaginable geometric atomic form whose intrinsically dynamic character allows for the basis of an understanding of universal physical reality that includes time within its very structure rather than using time merely as an external agent for the purpose of "forcing" insensible "matter-points" to move between static locations within a void spatial background. And this notion of temporal inclusion is perhaps the most important piece of the puzzle, in that it allows for a robust theoretical context—without resorting to ad hoc teleological arguments—upon which to understand the "naturalness" of the bio-chemical dynamics that underlie the evolutionary processes of the organic/life sciences.
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