Igor Bayak
Dynamic flows on a sphere
The question of a materialistic interpretation of the principle of least action and quantum mechanics is studied. Going beyond the framework of classical mechanics, we moved from representing a particle as a material point in physical space (i.e., a geometric point with its own mass) to representing a particle as a singularity of the vector field of velocities of matter particles, closed by their streamlines in a double ring on a 3-dimensional sphere, which has its own angular velocity in the 4-dimensional Euclidean space of the evolving 3-sphere $\mathbb{R}^{4}= r\ast S^{3}$, where the radius of the sphere $r$ and the evolutionary parameter $\tau$ are related by the dependence $\tau = \log(r)$. In other words, from now on the inertial manifold of a dynamical system of 4-dimensional space, formed by limit cycles closing on the 3-sphere, is the prototype of a material point, and since absolute time is a function of the polar coordinate of space $\mathbb{R}^{4}$, the dynamical system represents a vector field of the velocity of matter in evolution. Moreover, the interpretation of the streamlines of the vector field of 4-dimensional space (corresponding to logarithmic spirals in $\mathbb{R}^{4}$) as all possible trajectories of free motion of limit cycles leads to the necessity of extending 3-dimensional Euclidean space to a cylindrical manifold $\mathbb{R}^{3}\times S^{1}$, onto which the Minkowski space is wound. This allowed us to interpret the principle of least action as the statement that a particle moves along a path that minimizes the total number of revolutions of the singularity ring. At the same time, compactifying Minkowski space into the product $S^{3}\times S^{1}$ allows us to interpret quantum mechanics as the problem of a random walk of the singularity ring on Minkowski space. However, choosing an evolving 3-sphere as the metaphysical extension of space is a simplification, and to account for all physical phenomena (including gauge connections), it is necessary to study an evolving hypersphere of 8-dimensional space with a neutral metric that fits onto the 7-sphere of 8-dimensional Euclidean space.
