The work is an attempt to transfer a structure from Euclidean plane (pure geometrical) under the physical observation limit (resolving power) to a physical space (observable space). The transformation from the mathematical space to physical space passes through the observation condition. The mathematical modelling is adopted. The project is based on two stapes: (1) Looking for a simple mathematical model satisfies the definition of Euclidian plane; (2)That model is examined against three observation resolution conditions (resolved, unresolved and partially resolved). The simplest mechanical model satisfies the definition of Euclidian plane is a planetary gear. The interesting examination of the mechanical model is that is under partial resolution. That examination shows analogous equation for Euler’s formula. The derived complex formula contains the resolved (observable) quantities of the mechanical system and satisfies the linear wave equation. The interpretation of this complex formula is: it is a function related to the position vector of a point in the small wheel of the partially resolved planetary gear system. The function is in terms of the observable quantities only. The work shows the possibility of transformation from real to complex space. The work is purely classical but the result of the partial resolution shows a function similar to the Quantum mechanics wave function.
The work is an attempt to transfer a structure from Euclidean plane (pure geometrical) under the physical observation limit (resolving power) to a physical space (observable space). The transformation from the mathematical space to physical space passes through the observation condition. The mathematical modelling is adopted. The project is based on two stapes: (1) Looking for a simple mathematical model satisfies the definition of Euclidian plane; (2)That model is examined against three observation resolution conditions (resolved, unresolved and partially resolved). The simplest mechanical model satisfies the definition of Euclidian plane is a planetary gear. The interesting examination of the mechanical model is that is under partial resolution. That examination shows analogous equation for Euler’s formula. The derived complex formula contains the resolved (observable) quantities of the mechanical system and satisfies the linear wave equation. The interpretation of this complex formula is: it is a function related to the position vector of a point in the small wheel of the partially resolved planetary gear system. The function is in terms of the observable quantities only. The work shows the possibility of transformation from real to complex space. The work is purely classical but the result of the partial resolution shows a function similar to the Quantum mechanics wave function.
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.
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.
Based on de Broglie’s wave hypothesis and the covariant ether, the Three Wave Hypothesis (TWH) has been proposed and developed in the last century. In 2007, the author found that the TWH may be attributed to a kinematical classical system of two perpendicular rolling circles. In 2012, the author showed that the position vector of a point in a model of two rolling circles in plane can be transformed to a complex vector under a proposed effect of partial observation. In the present project, this concept of transformation is developed to be a lab observation concept. Under this transformation of the lab observer, it is found that velocity equation of the motion of the point is transformed to an equation analogising the relativistic quantum mechanics equation (Dirac equation). Many other analogies has been found, and are listed in a comparison table. The analogy tries to explain the entanglement within the scope of the transformation. These analogies may suggest that both quantum mechanics and special relativity are emergent, both of them are unified, and of the same origin. The similarities suggest analogies and propose questions of interpretation for the standard quantum theory, without any possible causal claims.
With approaching quantum/noncommutative models for the deep microscopic spacetime in mind, and inspired by our recent picture of the (projective) Hilbert space as the model of physical space behind basic quantum mechanics, we reformulate here the WWGM formalism starting from the canonical coherent states and taking wavefunctions as expansion coefficients in terms of this basis. This provides us with a transparent and coherent story of simple quantum dynamics where both the wavefunctions for the pure states and operators acting on them arise from the single space/algebra, which exactly includes the WWGM observable algebra. Altogether, putting the emphasis on building our theory out of the underlying relativity symmetry -- the centrally extended Galilean symmetry in the case at hand -- allows one to naturally derive both a kinematical and a dynamical description of a quantum particle, which moreover recovers the corresponding classical picture (understood in terms of the Koopman-von Neumann formalism) in the appropriate (relativity symmetry contraction) limit. Our formulation here is the most natural framework directly connecting all of the relevant mathematical notions and we hope it may help a general physicist better visualize and appreciate the noncommutative-geometric perspective behind quantum physics. It also helps to inspire and illustrate our perspective on looking at quantum mechanics and quantum physics in general in direct connection to the notion of quantum (deformed) relativity symmetries and the corresponding quantum/noncommutative models of spacetime as various levels of approximations all the way down to the Newtonian.
The aim of the paper is to seek and identify mathematical models designated to understand better the quantum scale phenomena, models which could permit to link them with the real world phenomena. This shall be made by analyzing the common features exhibited by both, normal scale and quantum scale phenomena and then, by creating mathematical models that are describing both, the quantum and the real world. One main idea of the paper, on which the involved models are built is that observed physical quantities such as position and momentum of a quantum particle, but also the laws of their transformation (of that physical quantities) from one state into another such as for example the probability wave function described by the Schrödinger’s equation, are signals transmitted out of the quantum world into the real physical world by the a certain real, not yet detected wave function, wave function, underlying the visible measurable reality and which is describing this more complex reality underlying the observable one, and unavailable not only to our limited senses but also not available, even though using the new discovered technologies. The two observed physical quantities, position and momentum, are describing a quantum system, that is to say the quantum system can be represented either as a wave function of the position of the particle or as a wave function of the momentum of the particle, this two different functions being linked one to another by means of the Fourier Transform. The paper is furthermore describing the way in which the theory of Signal Processing, a powerful and useful tool, is to be used in order to model both, quantum and large scale phenomena.