Canonical Particle

A canonical particle is a particle that can be described in the framework of quantum mechanical formalism. Elementary particles can be viewed as canonical particles (electron, positron, proton, neutron, etc.) but isolated atoms would also fall under this definition. A canonical particle has such quantum characteristics as the de Broglie's wavelength, the $\psi$-wave function, spin, etc. Its energy, momentum and the moment of momentum are determined throuh the Schrödinger and Dirac quantum equations. In quantum mechanics a canonical particle is a wave-particle and its characteristics are bound by de Broglie relationships: $E = h \nu$ and $\lambda = h/p$.

It is known from experiments on the scattering of light by these particles that canonical particles possess a radius of some hardness, which has been referred to as the Compton wavelength, $\lambda_{\rm Com} = h/(mc)$, where $m$ is the particle's mass and $c$ is the velocity of light.

Why we speak about some radius of hardness? Because this directly follows from the theoretical consideration of collisions of quanta of light with electrons (see, e.g. the description of the experiment in the well-known remarkable book by Born [1]).

Indeed, the equation characterising the energy conservation law and two appropriate equations for the momentum conservation law have the form

(1)
\begin{align} h\nu + m_0 c^2 = h\nu^\prime +mc^2, \end{align}
(2)
\begin{align} \frac{h \nu}{c}=\frac{h \nu^\prime}{c} \cos \phi + m \upsilon {\kern 1pt} \cos \alpha, \end{align}
(3)
\begin{align} 0=\frac{h\nu^\prime}{c} \sin\phi - m\upsilon {\kern 1pt} \sin\alpha. \end{align}

Here, $\nu$ is the frequency of quantum of light, $m_0$ and $m=m_0/\sqrt{1-\upsilon^2/c^2}$ are the electron's rest mass and the total mass, respectively; $\phi$ is the angle of deviation of the quantum of light after the scattering with the electron and $\alpha$ is the angle of deviation of the electron after collision with the quantum of light.

Figure 1: Diagram of momenta in the Compton's effect.

The final expression, Compton's formula for the change of the wavelength of the quantum of light due to the collision with the electron, can easily be derived from equations (1)-(3):

(4)
\begin{align} \Delta \lambda = \lambda^{\prime} - \lambda = c {\kern 1pt}\Big( \frac {1}{\nu^{\prime}} - \frac{1}{\nu} \Big) = (1- \cos \phi) \frac{h}{m_0 c}, \end{align}

where $h/(m_0 c) = \lambda_{\rm Com}$ is the Compton wavelength of the electron. This length, $\lambda_{\rm Com}$, is not formal. We can rewrite equations (1)-(3) in the form, which explicitly shows how spatial intervals characterising the scattering objects behave,

(5)
\begin{align} \frac{1}{\lambda} +\frac{1}{\lambda_{\rm Com}} = \frac{1}{\lambda^{\prime}} + \frac{1}{\lambda_{\rm Com}} \frac{1}{\sqrt{1-\upsilon^2/c^2}}, \end{align}
(6)
\begin{align} \frac{1}{\lambda} = \frac{1}{\lambda^{\prime}} {\kern 1pt} \cos \phi + \frac{1}{\lambda_{\rm Com}} \frac{\upsilon}{ c}{\kern 1pt}\cos \alpha, \end{align}
(7)
\begin{align} 0= \frac{1}{\lambda^{\prime}} {\kern 1pt} \sin \phi - \frac{1}{\lambda_{\rm Com}} \frac{\upsilon}{c}{\kern 1pt}\sin \alpha. \end{align}

Equations (5)-(7) show that the quantum of light, which is described by the wavelength $\lambda$, is scattered by an object that has the characteristic length, Compton's wavelength $\lambda_{\rm Com}$; and then the length $\lambda_{\rm Com}$ influences the $\lambda$ such, the latter is changed. Hence, the object's wavelength $\lambda_{\rm Com}$ is much more rigid, than wavelength $\lambda$ of the running quantum of light. Eqs. (5) to (7) result in the same Compton's expression (4).

Thus, canonical particles possess an actual radius of hardness, which is determined by the Compton's expression $\lambda_{\rm Com} = h/(m_0 c)$. Notwithstanding this, in orthodox quantum mechanics the size of a canonical particle does not play a part in the theory. In high energy physics a particle is regarded as pointlike.

At the same time in submicroscopic mechanics, which starts from the size of an elementary cell of physical space, i.e. the tessel-lattice, the real size of a canonical particle plays an important role. Since the size of cell in the tessel-lattice is identified with the Planck fundamental length $l_{\rm f} =\sqrt{\hbar G/c^3} \sim 10^{-35}$ m, it is quite reasonable to suggest this scale to be the size of the kernel of an elementary particle (recall that in agreement with the theory of real physical space an elementary particle appears directly from a cell of the tessel-lattice owing to fractal changes in its volume and surface).

It is known from solid state physics that the emergence in the crystal lattice of a foreign particle or an ion/atom leads to a deformation of the crystal lattice in the vicinity of the particle, which is called a deformation coat.

Because of that, in the tessel-lattice, a deformation coat is also formed around a created elementary particle [2,3]. Hence it is logical to assume that the created particle distorts the tessel-lattice up to the radius $\lambda_{\rm Com}$, which manifests itself in experiments on the scattering of light [1]. For instance, for the electron $\lambda_{\rm Com, {\kern 2pt}e} \sim 10^{-12}$ m and for the proton $\lambda_{\rm Com, {\kern 2pt}p} \sim 10^{-15}$ m. In the deformation coat, cells of the tessel-lattice are shifted from their equilibrium position to the particulate cell (i.e. the kernel of the particle); the importance of this deformation gradually decreases as the boundary of the coat is reached. Behind the deformation coat an unmovable particle does not express itself; here the tessel-lattice remains in the degenerate state. Clearly in line with the conservation law, the deformation of the particulate ball has to be compensated by the surrounding distortion of the tessel-lattice (i.e. the deformation coat). Hence the initial volume of a spatial locality which includes the particle with its deformation coat, remains the same as that of the degenerate tessel-lattice.

Figure 2: the particle - moving to the right - pulls its deformation coat with it and emits a cloud of inertons

A moving particulate cell also pulls its deformation coat with it. The state of the coat migrates together with the particle through a relay mechanism: in each point of the particle's path it induces the deformation coat onto the surrounding cells. At the backside of the particle the tessel-lattice restores its degenerate state.

The tessel-lattice has its dynamics that makes itself evident in the trajectory of motion of the particle. In particular, the particle's original characteristic is the value of the appropriate deformation, i.e. the mass $m$; the deformation coat's original characteristic is the value of shift of cells from their equilibrium positions, i.e. the tension of cells $\xi$. Consequently, in a dynamic system we should anticipate the oscillation between mass and tension, between the parameters $m$ and $\xi$. The motion of an inerton, an elementary excitation of the tessel-lattice, induces the periodical transformation of the parameters $m$ and $\xi$. The behaviour of a moving particle that emits and reabsorbs its inertons and the interaction of the moving particle with its deformation coat is the subject of study of submicroscopic mechanics.

### Bibliography

[1] M. Born, Atomic physics, Blackie and Son Limited, London-Glasgow, 1963; Supplement 10. The Compton effect (Russian translation: Mir, Moscow, 1965; p. 389).

[2] 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).

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

page revision: 25, last edited: 23 Oct 2009 18:14