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Is it possible to derive a mathematical physical form for Euler’s formula in the reality of complex harmonic oscillator?

Mohammed I. Sanduk

Published Nov 8, 2022• 0 citations

Abstract

Abstract Euler’s formula is a pure mathematical formula. In spite of its nonphysical nature, the form of this formula has a genuine fundamental role in quantum mechanics, but with no physical reason for this role. Philosophically, Heisenberg has proposed Kantian ideas in physics like “nature in itself”, unobservable nature, and “nature as appears”. He proposed a possibility of mathematical formulation for a system in nature in itself. In the present work, a mathematical model of rolling circles in real space has been proposed as a model for a system in nature in itself. Observation process acts as a transformation of the system to a model in observable nature. Partial observation transforms the real position vector of a point in that system to a form similar to the Euler’s formula. That causes the appearance of the imaginary unit. Interestingly, the transformed real kinematical forms exhibit similarity with the forms of relativistic quantum mechanics.

DOI: 10.21203/rs.3.rs-2237855/v1

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Abstract Euler’s formula is a pure mathematical formula. In spite of its nonphysical nature, the form of this formula has a genuine fundamental role in quantum mechanics, but with no physical reason for this role. Philosophically, Heisenberg has proposed Kantian ideas in physics like “nature in itself”, unobservable nature, and “nature as appears”. He proposed a possibility of mathematical formulation for a system in nature in itself. In the present work, a mathematical model of rolling circles in real space has been proposed as a model for a system in nature in itself. Observation process acts as a transformation of the system to a model in observable nature. Partial observation transforms the real position vector of a point in that system to a form similar to the Euler’s formula. That causes the appearance of the imaginary unit. Interestingly, the transformed real kinematical forms exhibit similarity with the forms of relativistic quantum mechanics.

Abstract Euler’s formula is a pure mathematical formula. In spite of its nonphysical nature, the form of this formula has a genuine fundamental role in quantum mechanics, but with no physical reason for this role. Philosophically, Heisenberg has proposed Kantian ideas in physics like “nature in itself”, unobservable nature, and “nature as appears”. He proposed a possibility of mathematical formulation for a system in nature in itself. In the present work, a mathematical model of rolling circles in real space has been proposed as a model for a system in nature in itself. Observation process acts as a transformation of the system to a model in observable nature. Partial observation transforms the real position vector of a point in that system to a form similar to the Euler’s formula. That causes the appearance of the imaginary unit. Interestingly, the transformed real kinematical forms exhibit similarity with the forms of relativistic quantum mechanics.

A solution to the part and whole problem is presented in this paper by using a complex mathematical representation that permits to define the Holon concept as a Basic Unit System that remains itself in spite of complex operations such as integration and derivation. This can be done because of the remarkable isomorphic property of Euler Relation. We can then define a domain independent both of the observer and the object, as within it, the object is embedded. We will then be able to have a Quantum Mechanics solution without the "observer drawback", as Karl R. Popper tried to find all his life but from the philosophical point of view and which was Einstein main concern about QM too. A unit that has always similar or identical structure or form, despite even complex operations such as integration and derivation, is the ideal unit for the new sciences of complexity or just the systems sciences too, where structure or form, wholeness, organization, and complexity are main requirements. A table for validating the results obtained is presented in case of the pendulum formula.

Mathematically, the imaginary unit 𝑖 has long been regarded as an algebraic generator extending the real numbers to the complex field. Yet a century after the birth of quantum mechanics (1925–2025), the answer to its oldest question is now clear: quantum mechanics is not merely a mathematical structure—it is a physical reality, and all quantities appearing in its non-commutative relations are physical. Therefore, 𝑖 is not a mere algebraic extension but the universal operator demanded by the geometric completeness of non-commutative reality between position and momentum. The imaginary unit is a physical reality. The canonical relation [𝑥, 𝑝] = 𝑖ℏ is not merely algebraic—it is a geometric law, showing that the complex structure of quantum mechanics arises inevitably from physical reality itself. It stands without contradiction, without complexity. This is nature. In the (𝑥, 𝑝) plane, 𝑖 represents a 90° rotation and ℏ quantizes the curvature of that rotation. Complex numbers thus arise as the natural geometric language of non-commutative physics, providing a geometric resolution to the century-old question of physical completeness first raised by Einstein, Podolsky and Rosen (1935).

A solution to the part and whole problem is presented in this paper by using a complex mathematical representation that permits to define the Holon concept as a unit that remains itself in spite of complex operations such as integration and derivation. This can be done because of the remarkable isomorphic property of Euler Relation. We can then define a domain independent of the observer and the object where the object is embedded. We will then be able to have a Quantum Mechanics solution without the "observer drawback", as Karl R. Popper tried to find all his life but from the philosophical point of view and which was Einstein main concern about QM. A unit that has always similar or identical structure or form, despite even complex operations such as integration and derivation, is the ideal unit for the new sciences of complexity or just the systems sciences where structure or form, wholeness, organization, and complexity are main requirements. A table for validating the results obtained is presented in case of the pendulum formula.

Complex numbers are broadly used in physics, normally as a calculation tool that makes things easier due to Euler's formula. In the end, it is only the real component that has physical meaning or the two parts (real and imaginary) are treated separately as real quantities. However, the situation seems to be different in quantum mechanics, since the imaginary unit i appears explicitly in its fundamental equations. From a learning perspective, this can create some challenges to newcomers. In this article, four conceptually different justifications for the use/need of complex numbers in quantum mechanics are presented and some pedagogical implications are discussed.

The Three Wave Hypothesis (TWH) has been proposed by Horodecki in 1981. Sanduk attributed TWH to a classical kinematical model of two rolling circles in 2007. In a previous project in 2012, it is shown that the position vector of a point in a system of two rolling circles can be transformed to a complex vector under the effect of partial observation. The present work tries to develop this concept of transformation. Under this transformation, it is found that the kinematical equations of the motion of point can be transformed to equations analogise the relativistic quantum mechanics equations. Many analogies have been found and are listed in a comparison table. These analogies may sagest that both of the quantum mechanics and the special relativity are emergent, and are of the same origin.