Submicroscopic Mechanics

Submicroscopic mechanics [1,2,3,4,5,6] describes the behaviour of the canonical particle in the real physical space constructed as the tessel-lattice of primary topological balls. The size of a cell in such mathematical lattice is identified with Planck's fundamental length $l_{\rm f} =\sqrt{\hbar G/c^3} \sim 10^{-35}$ m. The motion in the tessel-lattice is only deterministic, because a particle coming between the tessel-lattice's cells must interact with them and hence its path is traced. And at the same time we can calculate the particle's parameters (the kinetic energy, velocity, momentum, etc.) at any point of the particle's trajectory. The notion of the particle is exactly defined: it appears from an ordinary cell of the tessel-lattice when dimensional changes locally occur. In other words, the cell experiences fractal volumetric and surface deformations, which represent its mass and charge respectively.

A canonical particle is accompanied by its deformation coat in which oscillations of cells take place. The deformation coat can be simulated by a crystallite with the same radius $\lambda_{\rm Com} = h/(m c)$, i.e. the crystallite whose nodes are occupied by the same massive particles. The total mass of all these model particles is equal to the mass of the central particulate cell, $m_0$; these particles are found in a vibratory state, such that they can be described by the Lagrangian [3]

(1)
\begin{align} L=\frac 1{2}\sum_{\vec n, \beta}\mu_{\vec n}\dot\zeta^2_{\vec n \beta} -\frac 1{2} \sum_{\vec n,\beta\beta^{\prime}} \gamma_{\beta \beta^{\prime}} (\zeta_{\vec n\beta} - \zeta_{\vec n-\vec a, \beta})^2 \end{align}

where $\mu_{\vec n}$ is the mass of a particle located in the point $\vec n$ of the crystallite, $\zeta_{\vec n \beta}$, $(\beta =1,2,3)$ are three components of the shift of the particle from its equilibrium position $\vec n$, $\vec a$ and $\gamma_{\beta \beta^{\prime}}$ are the crystallite constant and the crystallite's force constant, respectively. The crystallite exists only in one excited state such that its unique mode is characterised by the vibrational energy $\hbar \omega_0 = m_0 c^2$. In this mode all particles vibrate in directions transferal to the vector of motion of the particle (along this vector vibrations are impossible owing to the migration of the crystallite as a whole along the mentioned vector).

The motion of a particulate cell accompanied by its deformation coat looks as follows: at each next movement, the particulate cell moves on the crystallite constant $a$, which is practically identical with the size of cell of the tessel-lattice (i.e. the Planck's fundamental length $l_{\rm f}$), the crystallite mode $\hbar \omega$ attacks the particle, knocking a fragment of its deformation out of it, or in physical terms, a fragment of mass $\delta m$.

The direction and the velocity of this elementary excitation called inerton poses the crystallite mode whose speed is identified with the velocity of light $c$; if $\upsilon$ is the velocity of the particle, then the direction and value of the $i$-inerton is found from the vector sum of velocities, such that the inerton velocity is $c_{\rm inert} = \sqrt{c^2 + \upsilon_i^2 }$. Ejected inertons must turn back to the particle, because otherwise the particle would loose its velocity (and also mass) will eventually stop. The number $N_{\rm inertons}$ of ejected inertons can be associated with a number of collisions of the particle with adjoining cells, which takes place in the section $\lambda /2$ where the velocity of the particle drops from the value $\upsilon$ to zero. For instance, in the case of an electron in the hydrogen atom $N_{\rm inertons} \approx \lambda /(2 l_{\rm f}) \sim 10^{25}$. Inertons ejected from the particle come back to it reflecting from the tessel-lattice. Returned inertons bring the velocity and mass back to the particle and, hence, they guide it in the next section $\lambda/2$ of the particle path. Such periodical motion can be described by equation

(2)
\begin{align} \mu d^2 r /dt^2 =- \gamma r . \end{align}

The maximal distance which the particle's inertons reach, the amplitude of the inerton cloud, is $r|_{\rm max} = \Lambda$.

Thus inertons periodically project out of the particle and then return. The motion of the particle and the inerton cloud enclosing it can be described by the Lagrangian (here it is simplified)

(3)
\begin{align} L= \frac 12 m {\dot x}^2 + \frac 12 \mu {\dot \chi}^2 - \frac {\pi}{T} \sqrt{m\mu} \ {\dot x} \chi \end{align}

where $T$ is the free run time of the particle between its collisions with the inerton cloud; then $1/T$ is the frequency of collisions.

The solutions for the particle

(4)
\begin{align} {\dot x} = \upsilon_0 \cdot (1- |\sin(\pi t/T)|); \end{align}
(5)
\begin{align} x (t) = \upsilon_0 t + \lambda \cdot \{ (-1)^{[t/T]} \cos(\pi t/T) - (1+2[t/T]) \} \end{align}

show oscillations of the particle's parameters: the velocity periodically changes from $\upsilon$ to zero, $\upsilon_0 \rightarrow 0 \rightarrow \upsilon_0$ in each section $\lambda$ of the particle's path. Therefore, the section $\lambda$ is the spatial amplitude of the particle.

Analogously for the particle's inertons:

(6)
\begin{align} \chi = \frac {\Lambda}{\pi} |\sin(\pi t / T) |; \end{align}
(7)
\begin{align} \dot \chi = c (-1)^{[t/T]} \cos (\pi t / T), \end{align}

that is, the inerton cloud periodically leaves the particle and comes back and the parameter $\Lambda$ appears as the amplitude of oscillations of the inerton cloud.

The following relationships hold:

(8)
\begin{align} 1/T = \upsilon_0 / \lambda = c / \Lambda. \end{align}

Figures: Motion of the particle is associated with the ejection and reabsorption of its inerton cloud and shows an oscillation of its parameters; in particular, the velocity of the particle gradually decreases from the initial value $\underline{\upsilon_0}$ to zero and then increases again to $\underline{\upsilon_0}$ in each section $\underline{\lambda}$ of the particle's path.

With the use of the transformation

(9)
\begin{align} \dot \kappa = \dot \chi - \pi \sqrt {m / \mu} \ x / T \end{align}

we can obtain the Hamiltonian that describes the motion of a particle as to the centre of inertia of the system 'particle-inerton cloud':

(10)
\begin{align} H = \frac 12 \frac {p^2}{M} + \frac 12 M (2\pi /2T)^2 x^2 . \end{align}

However, this is the Hamiltonian of harmonic oscillator and hence such motion of the particle can be written in the form of the Hamilton-Jakobi equation for the shortened action $S_1$

(11)
\begin{align} \frac {1}{2m} (\frac {\partial S_1}{\partial x})^2 + \frac 12 m (2\pi /2T)^2 x^2 = E \end{align}

where $E$ is the energy of the moving particle. Introducing variables action-angle we obtain an increment of the action per cycle $T$:

(12)
\begin{align} \delta S_1 =\int p dX = E \cdot 2T. \end{align}

This equation can be rewritten though the frequency $\nu = 1/2T$. At the same time $1/T$ is the collision frequency of the particle with its inerton cloud. Taking into account that $E= m\upsilon^2 / 2$ we also can write (below $p_0 =m \upsilon_0$ is the initial momentum)

(13)
\begin{align} \delta S_1 = m \upsilon_0 \cdot \upsilon_0 T = p_0 \lambda. \end{align}

Identifying two left hand sides of equations (12) and (13), i.e. the increment of the action $\delta S_1$ per the period $T$, with Planck's constant $h$, we get two basic relationships of quantum mechanics

(14)
\begin{align} E=h\nu \ \ \ {\rm and } \ \ \ \lambda = h/p_0. \end{align}

These two de Broglie's relationships enable us to derive the Schrödinger equation (see de Broglie [7]). Thus the spatial amplitude $\lambda$, which has been introduced above, can be set equal to the de Broglie wavelength of a particle.

The availability of correlations $\Lambda =\lambda c/ \upsilon_0$ and $\lambda_{\rm Com} = h/(mc)$ and the de Broglie wavelength $\lambda =h/ (m \upsilon)$ allow us to deduce the very interesting relationship:

(15)
\begin{align} \Lambda = \lambda_{\rm Com} c^2 / \upsilon^2_0, \end{align}

which connects the amplitude of the inerton cloud $\Lambda$ with the size of the deformation coat (crystallite) $\lambda_{\rm Com}$.

From relationship (15) one can see that in the case of a small velocity of the particle, $\upsilon^2_0 / c^2 << 1$, the amplitude of the inerton cloud is significantly larger than the range of the deformation coat: $\Lambda >> \lambda_{\rm Com}$. The inerton cloud carries the kinetic energy of a particle and a detector will record the particle with the energy $E= m \upsilon^2 / 2$. Therefore, in this case for the description of such particle we have to use Schrödinger's formalism.

When the velocity of a particle is close to the velocity of light, $\upsilon_0 \sim c$, the amplitude of the inerton cloud comes very close to the range of the deformation coat, $\Lambda \sim \lambda_{\rm Com}$. But the deformation coat together with the kernel (particulate cell) is specified by the total energy of the canonical particle and the detector will record the particle just with this energy, $E = m_0 c^2 / \sqrt {1- \upsilon_0^2 /c^2}$. Because of that, for the description of the particle in this situation we have to use the Dirac formalism.

The analysis above shows that it is the deformation coat that causes a peculiar phase transition from the Schrödinger formalism to the Dirac formalism when the particle's velocity $\upsilon$ approaches the speed of light $c$. Moreover, in addition the particle features also an inner motion (asymmetrical pulsations), which is mapped on the formalism of quantum mechanics as the particle's spin.

The inerton cloud is expanded up to the distance $\lambda /2$ along the particle path and occupies a band width $2 \Lambda$ in transversal directions. The formalism of quantum mechanics does not take the reality of the inerton cloud into consideration but fills a range around the particle with an abstract wave $\psi$-function.

The results stated in this article enable us to reveal the true physical interpretation of the wave function $\psi$ as the particle's field of inertia.

Then the expression the "material wave" acquires a real sense, because now behind this term we see not an abstract probability $| \psi (\vec r) |$, but the material field of inertia of the particle and inertons become carriers of this field.

Such an interpretation of the physical nature of the wave $\psi$-function completely satisfies those conditions that Louis de Broglie laid down, namely: There should be another solution for the Schrödinger equation and the wave function should have a true causal physical meaning and not statistical.

It is interesting to note that this allegedly abstract function $\psi$ was directly observed in experiment [8] (so it is not so abstract!). The researchers put in the title of their paper: "Looking at electronic wave functions…". The inerton field has been detected in our experiments.

Furthermore, submicroscopic mechanics is a starting point for an understanding and the derivation of Newton's law of universal gravitation, the problem of quantum gravity and the nuclear forces.

### Bibliography

[1] V. Krasnoholovets, D. Ivanovsky, Motion of a particle and the vacuum, Physics Essays 6, no. 4, pp. 554-563 (1993) (also http://arXiv.org/abs/quant-ph/9910023).

[2] V. Krasnoholovets, Motion of a relativistic particle and the vacuum, Physics Essays 10, no. 3, pp. 407-416 (1997) (also http://arXiv.org/abs/quant-ph/9903077).

[3] V. Krasnoholovets, On the nature of spin, inertia and gravity of a moving canonical particle, Indian Journal of Theoretical Physics 48, no. 2, pp. 97-132 (2000) (also http://arXiv.org/abs/quant-ph/0103110).

[4] V. Krasnoholovets, Space structure and quantum mechanics, Spacetime & Substance 1, no. 4, 172-175 (2000) (also arxiv.org e-print archive http://arxiv.org/abs/quant-ph/0106106).

[5] V. Krasnoholovets, Submicroscopic deterministic quantum mechanics, International Journal of Computing Anticipatory Systems 11, pp. 164-179 (2002) (also http://arxiv.org/abs/quant-ph/0109012).

[6] V. Krasnoholovets, Gravitation as deduced from submicroscopic quantum mechanics, http://arXiv.org/abs/hep-th/0205196.

[7] L. de Broglie, Les incertitudes d'Heisenberg et l'interpretation probabiliste de la mechanique ondulatoire (Gauthier-Villars, Bordas, Paris, 1982), ch. 2, sect. 4. Russian translation: Соотношения неопределенностей Гейзенберга и вероятностная интерпретация волновой механики /Heisenberg’s uncertainty relations and the probabilistic interpretation of wave mechanics/ (Mir, Moscow, 1986), pp. 50-52.

[8] G. Briner, Ph. Hofmann, M. Doering, H. P. Rust, A. M. Bradshaw, L. Petersen, Ph. Sprunger, E. Laegsgaard, F. Besenbacher and E. W. Plummer, Looking at electronic wave functions on metal surfaces, Europhysics News 28, 148-152 (1997).

page revision: 12, last edited: 11 Jan 2009 18:00