TITLE:
The Equivalence between Special Relativity, Newtonian Physics and Quantum Mechanics in Complex Para-Space
AUTHORS:
Jerzy K. Filus
KEYWORDS:
Dramatic Simplification of Mathematical Apparatus, Special Relativity’s Hyperbolic versus Circular Versions, Some Equivalence of SR and Newton’s Theories, Algebra of Relativistic and the Corresponding Galilean Velocities
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.7,
July
21,
2025
ABSTRACT: Dramatic simplification of mathematical apparatus, special relativity’s hyperbolic versus circular versions, some equivalence of SR and Newton’s theories, algebra of relativistic and the corresponding Galilean velocities. Complex model, say
ℂ
3
, of para-space as alternative to the real M4 Minkowski space-time for both relativistic and classical mechanics was shortly introduced. As it turned out, the model and its theory have the power to bridge two and very likely also three physical theories in one common framework. The model, originally thought of as the model for special relativity only, exhibited the possibility also to model the classical Newtonian mechanics and, moreover, the two mechanics turned out to be equivalent in their algebraic and topological structures. As it follows from some additional analysis, placed in section 10, also the quantum mechanics (QM) may [hypothetically] be described as theory of the same
ℂ
3
complex domain. If so, the bridging power of the introduced complex model is striking even if QM does not seem to be just equivalent to the remaining two theories. As for the SR, in the new complex Euclidean framework, it is not only totally preserved but at many important issues extended. Thus, in the new model, several hardly understood facts from the real SR such as the universality of speed of light, the Lorentz contraction or twins’ paradox (especially the phenomena associated with the rocket’s changing direction) find clear explanation unknown in the real M4 version of this beautiful theory. The transition from the real to complex description (the only necessary prize) yields dramatic simplification of all the three theories which is specially striking in the case of quantum mechanics (provided that my hypothesis on QM will turn out to be the true one). Moreover, in this work, the algebraic structure of isomorphic vector spaces was imposed and analyzed on both: the set of all relativistic velocities and on the set of the corresponding to them “Galilean” velocities. In some association with that, the relativistic theory was closer analyzed. Namely, the two approaches: the hyperbolic (“classical” SR) with its four-vector formalism and Euclidean, where SR is modeled by the complex para-space
ℂ
3
, were analyzed and compared.