Abstract
We present the conjecture that the fundamental notions of physics, including such basic concepts as spin, charge(s), mass and spacetime, emerge from a mathematical theory having to do with fluctuations in the metrics of 2-manifolds. This involves looking at a set of 2-manifolds whose metrics depend on a global parameter \(t\), where each manifold is also associated with a set of six parameters whose values change continuously along \(t\) depending on the metrics and associated parameters of all the other manifolds. More intuitively, and specifically, what we do is define objects that generate oscillations on affine planes in \(R^3\), parameterized by the global parameter \(t\), where the plane associated with each object changes continuously along \(t\) depending on the oscillations that are generated by similar objects on other affine planes. We study some typical structures that arise when the oscillations’ amplitudes tend to zero, and show that the state of each such structure at any value of \(t\) can be associated with a real parameter \(0 \leqslant \beta < 1\), and that a special type of interaction with other similar structures exists which results in readjusting the value of \(\beta \). We go on to think of \(R^3\) at any value of \(t\) as corresponding to a hypersphere in \(R^4\) via the stereographic projection, where \(t\) is identified as the logarithm of hypersphere radius in the foliation of \(R^4\) in concentric hyperspheres. This leads to interpreting some curves arising from the structures we study as corresponding to certain \(S^2\) curves. We use both sides of that last correspondence to derive the free Dirac equation as the classical equation of motion of an abstract point particle, where our typical structures and their interactions are understood in connection with electrons interacting with an em radiation field, and the emergent description of any point in spacetime typically involves an infinite number of discrete values of \(t\). The Dirac equation is thus derived from the Frenet equations of our special family of curves, and the physical meaning is determined by the identification of physical and geometrical entities required by the derivation. We propose that the origin of the infinitesimal oscillations in our theory has to do with the axiomatization of the Euclidean plane and the construction of the real numbers. We briefly discuss the relevance of these ideas to the strong interaction and to gravity.
Highlights
Introduction and general outlineThe conjecture presented here is that the fundamental notions of physics, including such basic concepts as spin, charge(s), mass and spacetime, may emerge from a mathematical theory having to do with rotations in R4 and fluctuations in the metrics of 2-manifolds
I will try to show that the momentum terms in the Dirac equation emerge from the torsion terms in the second Frenet equation of our imagined momentary helix, and the mass term in the Dirac equation emerges from the curvature term in Frenet. This would be made possible by the fact that the curvature term becomes independent of β once we correctly identify the flow of physical time, and change variables
It seems that if we aim to identify a consistent system of relations between changes in β and the properties of external Elementary Metric Oscillator” (EMO), the interactions in which the external EMO shares the plane of a wheel, and its focal point is on the line determined by the effective focal point and the body focal point, at the right phase, are especially interesting
Summary
The conjecture presented here is that the fundamental notions of physics, including such basic concepts as spin, charge(s), mass and spacetime, may emerge from a mathematical theory having to do with rotations in R4 and fluctuations in the metrics of 2-manifolds. Note that the oscillation of EMO space relative to the wandering EMO space is determined individually for each EMO combination, so that by identifying the body plane with the momentary x, |cos(ωt)|y + sin(ωt)z plane we have effectively added a hidden (compact) dimension to our theory At this point we shall start discussing the possibility of identifying some of the variables involved with physical variables. I expect that quantum superposition would have to be understood via the vector addition of the velocities of the effective focal points of EMO combinations which are observed sharing a common body plane at a given physical time (but many different values of t) with their effective focal points effectively at the same point in physical space. I’ll finish the paper with a brief discussion of the possible origin of EMOs, and propose that EMOs may emerge naturally from a discussion of the construction of the real numbers and the geometric axiomatization of the Euclidean plane
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