Spaces of continuous dimension

Abstract

A space with dimension of continuum power is proposed, which is a natural generalization and development of the concepts of Euclidean and Hilbert spaces. The corresponding vector field is given by analogy with the counting case of a continuum of field functions. Differential operations are defined. The assumption of his physical reality shows the continuous quantum-mechanical Schrödinger equation. From the principle of correspondence to the finite-dimensional case, the derivatives are defined as fractional ones by means of interpolation (in order of differentiation) operators. A new kind of motion of matter is postulated - along the dimension of space (inter-dimensional oscillations), and on its basis quantum-mechanical determination of mass. The uncertainty principle is extended by the uncertainty factor of the "continuous number" of the coordinate.