The analogy of equation of rotation in complex plane with the Dirac equation , and its foundation

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.


Introduction
During the eighties of last century, R. Horodecki proposed Three Wave Hypothesis (TWH) where the particle is related to waves.This hypothesis implies that a massive particle is an intrinsically spatially as well as temporally extended non-linear wave phenomenon [1,2].The TWH is based on an assumption that, in a Lorentz frame where the particle is at rest it can be associated with an intrinsic nondispersive Compton wave.When the particle moves with velocity v (relative to the lab frame), it will be associated with the three waves: the superluminal de Broglie wave (of wavelength ), a subluminal dual wave (of wavelength ), and a transformed Compton wave (of wavelength ) [1,2].
Considering this system of waves in angular form and in single representation (instead of two dispersion relations of the de Broglie wave and dual wave) exhibits similarities with a system of two perpendicular circles [3,4].Thus, the combination of the three waves may form a classical kinematical bevel gear model.However, the main problem of TWH is lack of experimental evidence for the superluminal dual wave.

The position vector and the complex vector
A mathematical relationship between the position vector of a point in rolling circles system and a complex vector has been explained via a transformation under partial observation effect [5].The position vector of a point in this system is a real quantity.This project [5] propose a concept of partial observation to transfer the position vector to a complex function.The partial observation is based on inability to distinguish the complete system due to resolution limit.The work was not in quantum mechanics.

The aim of the work
In present work, we try to use the classical concepts mentioned above to formulate a model analogous to the relativistic quantum mechanics.Thus this approach does not base on the quantum postulates (wave function, and Dirac equation), but its goal is to go toward them.
At the end, we will demonstrate a comparison table to show the conventional relativistic quantum mechanics equations and the obtained equations by the present work.

Kinematical model
Based on 2012 work [5], the system of two rolling circles is shown in Figure 1.The circle of interest is guided circle of radius  � and the second circle is the guiding circle of radius where  � , are the radius of the circle, and the norm of the position vector of the circle centre (� � ) respectively.The angles , , and � are as shown in Figure 1.The guiding circle is of radius and its centre coordinates are (0,0).Let the ratio  �  ⁄ in Eq.( 1) is Point � is the point of contact between the two circles.For generality, let  � <  � .The ratio of the system is: Eq.1 can be formulated as a kinematical model: where, and � , � � , and � � represent the arc length made by point � , the angular velocity of point �, and the time variation of angle � .In terms of  and from Eq. ( 2),  can be written as: Eq. ( 5) gives full description of the location of the point.Then, from Eq. ( 5), the velocity equation of point � is:

Empiricist approach
To deal with the system as a physical object, testing (measuring) in a lab is an essential approach to prove its physical existence (positivism).In classical physics, the observable distinguishability is related to the optical resolution (Rayleigh criterion).The spatial resolution (�  ) is the minimum linear distance between two distinguishable points [6].
EPJ Web of Conferences 182, 02108 (2018) https://doi.org/10.1051/epjconf/201818202108ICNFP 2017 The system is fully observed (seen) when its dimensions are larger than the spatial resolution (�  ) and the angular frequencies (�  ) or: The system is said to be a classical physical system, and all its quantities are said to be physical and can be measured.The lab observer obeys these conditions.

Partial observation
The concept of partial observation and elimination has been proposed in Ref. [5].
Physically,  = � (elimination) suggests that no distinguishable spatial extent can be observed (empirical observation), in other words, the value of the parameter is less than the Rayleigh criterion (spatial resolution) [7].
Based on the resolution limit, the system is partially resolved under the condition and according to the ratio (4), it temporally resolved under the condition That implies � � �.Practically, elimination can be considered missing observation, indistinguishable data, etc.Thus, for partial observation  � = � � = � and the zero quantity is a practical approximation.
Inequalities (9, 10) describes the inability to resolve  � and � � (missing the data), whereas  � and � � can be resolved.Partial resolution refers to the inability to completely resolve of the kinematical system by using monochromatic light (�, �).�  is related to the wavelength so as �  is related to �.Then, the measured quantities are: and    � =  � , and The subscript � indicates resolved (measured) values.
When  � can not be detected, then the angle � can not be detected as well.Since � = � − �, then: Where the existence of angle � is related to the recognition of the guided circle of  � .

The system as seen by the Lab observer
The partial observation serves as a filter, see Figure 2.This filter makes a separation between two different worlds, the real full deterministic world (mathematical) and the physical world (observable) of the complex vector.What the lab observer (Figure 3) can see is not a point and not a sinusoidal wave in real space.For him/her there is a collapse of a real system to a complex mathematical form, which has no physical meaning.In such a case, the lab observer may use some of the quantum mechanics techniques (postulates): 1-Born's postulate(probabilistic interpretation of the wave function)

2-Correspondence principle (operators technique) 3-Von Neumann's postulate (wave function collapse)
The only possible utilisation of this complex function is by using quantum mechanics technique, like a statistical technique (Born rule), and the point location is probable.Note Figure3.

Figure 2.
The system and the lab observation.
The lab observer can recognise the observable parameters of the system ( �� , � �� ).According to Eq. ( 14) the lab observation can be considered as a combination of sinusoidal wave (probability wave) and point.With aid of the above postulates, the lab observer can recognise probable locations as in Figure 3.

Kinematic equations of the Lab observer
The complex vector can not be regarded as a physical expression.Thus, one may find in the statistical approach of the quantum mechanics a useful approach here as well.The concept of probable location of the point well be considered by the lab observer.
For lab observer who is assumed to be under the conditions of partial observation (Eqs.(13), and ( 14)), the equation of velocity Eq. ( 7) becomes (Appendix I): where  and � are coefficients related to the rotation of the system (Appendix I).Equation ( 15) is a complex velocity equation.The properties of the coefficients are shown in Appendix II.As shown in Figure 1 and Eq.( 1), � (��) is related to the rotation ((�� � ) of the position vector � of point �, and  (Eq.(5-A)) is related to the rotation direction (��  ) of  � .One can say that the cross product of the unit vectors is non-commutative: and that may be the physics behind the non-commutative Dirac coefficients.Thus for the lab observer, the definitions of  and � are changed to  and �.
With aid of the coefficients properties  and �.(same as those of relativistic quantum mechanics), the time differentiation of Eq. ( 15) is Eq. ( 17) looks like a complex acceleration equation for the complex velocity equation.

The seen system by the lab observer
Equations ( 13) and ( 14) are for point trajectory and sinusoidal wave in the complex plane.Thus, the lab observer with aid of quantum techniques may recognise the point and wave features together.The velocity � that appears in Eqs.(15, and 17), is (Eq.(2-A): The quantities with the prime symbol are the observed quantities by the lab observer.The point parameters are � � (the observable point velocity) and � �� whereas wave properties are  �� (Eq.( 18)) and ω  .
Duo to the equivalence of point and sinusoidal wave, then for the lab observer: The left-hand side is related to the point and the right-hand side is for the wave.Eq. ( 19) can be formulated as a ratio for that point wave combination: From Eq. (20) and Eq. ( 18) one can say that the velocity of the point is and where � �� phase velocity and is related to the phase of Eq. ( 14).

Comparisons
The similarity of the above-obtained equations' forms with the conventional equations of relativistic quantum mechanics and special relativity are shown in Table 1.

Figure 1 .
Figure 1.The real model.Rolling circles model.

Table 1 .
Comparisons of the equations of conventional quantum mechanics and special relativity with the analogical model forms.