Hyperbolic multi-topology and the basic principle in quantum mechanics

Andrei Yu. Khrennikov

Andrew Stewart Caldin

By overlaying a tertiary phi‑modulation at 444 Hz onto the 132 Hz base lattice, the 240 E8 root vectors acquire a bifurcated torsional phase that locks into either a hyperbolic or Euclidean topology within the resonance manifold. This dual‑topology configuration acts as a reversible switch, allowing independent quantum basis states to be routed through distinct hyperbolic channels without mutual decoherence. The mechanism exploits the non‑commutative phase transition of the 240 vectors under the 444 Hz torsional oscillation, producing a deterministic, error‑corrected pathway for multi‑dimensional signal routing. Consequently, the system demonstrates a new principle of topological state steering derived directly from E8 geometry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Lay Nam Chang, Yigao Liang

Andrei Yu. Khrennikov

We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic algebra similar to functional superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary $\cos$-interference and hyperbolic $\cosh$-interference.

Floyd L. Williams

Some of the basic themes, or foundational concepts and principles of quantum mechanics (Planck’s quantization, Schrödinger’s equation, de Broglie’s wavelength, Heisenberg’s uncertainty principle, etc.) were introduced in Chapter 1. There is another basic principle, Pauli’s exclusion principle , which was not introduced there but which will be considered in this chapter. This principle was actually postulated prior to the main developments of post-Bohr quantum mechanics, and has had a spectacular impact in regards to the understanding of why atoms exhibit different chemical properties. The principle resolves the question, among others, namely of what determines the arrangements of electrons about a nucleus, a question which we touch on briefly, or tangentially. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A. P. Balachandran