**SPФ symmetry** is combined three-component symmetry of
similarity of objects, phenomena and processes at different scale levels of
matter. With its help in the Theory of
Infinite Hierarchical Nesting of Matter the similarity of matter levels is described
and the invariance of action of the physical laws at these levels is proved.
With the help of SPФ symmetry the meaning of the scale dimension and scale relativity is clarified.

The theorem of the SPФ symmetry
was proved by Sergey Fedosin in 1999. ^{[1]}

In the combined SPФ
transformation the invariance of physical laws is revealed as the result of
transition from one scale level of matter to other levels of matter. To move
from one level of matter to another using SPФ it is necessary simultaneously to
make the transformation of the speeds S, the transformation of sizes (scales) P
and the transformation of masses Ф. The values S, P and Ф are fixed by the
corresponding coefficients of similarity between the levels, where the
transition is made. The examples of similarity coefficients are given in the
articles: similarity of matter levels, quantization of parameters of cosmic systems,
hydrogen system. Since the objects of
lower level of matter are part of the object of higher scale level of matter,
it allows us on the basis of physical laws and equations of the matter state to deduce the relations between the
coefficients of similarity S, P and Ф. ^{[1]}

After substituting in the
Lagrangian, which determines the laws of motion of a physical system, the new
variables, taking into account the SPФ transformations, the Lagrangian does not
change its form. This means that the physical laws are not changed during
transitions between different levels of matter and the corresponding phenomena
occur in the similar way. In particular, at each level of matter, we can
introduce its own Dirac constant as the characteristic angular momentum ( spin)
and the quantum of action of typical objects and also write its own Heisenberg
uncertainty principle. Another example is the stellar constants corresponding
to the level of stars.

If in the system of physical
units CGS we make similarity transformation for masses, sizes and speeds, it
turns out that in the Newton's law of gravitational
attraction we need to transform not only masses and sizes, but also the
constant of gravitation. At the same time in the system of physical units CGS
the vacuum permittivity is equal to 1 and transformation applies only to the
forces, charges and sizes. This means that in the transition to the atomic
systems the ordinary gravitation is replaced by the strong gravitation and a new constant
appears – strong gravitational constant.
The strong gravitation is responsible for the integrity of the elementary
particles, including nucleons, and in the gravitational
model of strong interaction it is an integral part of the strong
interaction. ^{[2]} The transition to the atomic
systems is accompanied by a significant increase not only of the gravitation but
also of the electromagnetic fields acting near the elementary particles. The
value of the vacuum permittivity does not change in the CGS, or in any other
system of physical units.

According to the substantial neutron model and the substantial proton model, the equation of
the state of the nucleon matter is similar to the equation of the state of neutron stars matter. The dependences of the mass on the
radius are also similar. SPФ symmetry allows us to understand the dependences,
arising between the mass and the electric charge of the proton, to justify the model of quark quasiparticles, to approach
the essence of gravitational and electromagnetic forces in the framework of the
Le Sage's theory of gravitation.

Based on the postulate on
equality of the kinetic energy flux and the fluxes of gravitational (in the
strong gravitation field) and electromagnetic energies in the electron matter formulated by Sergey Fedosin, quantization of the
energy levels and the angular momentum of the electron during its rotation in
the atom is derived. The similar idea is used with respect to the Solar system,
showing the probable cause of the discrete planetary orbits.^{[3]}
The law of redistribution of the energy fluxes formulated by Fedosin allows us
to find the stationary state of rotation of the nucleon and the neutron star
similar to it, to connect many other phenomena in the microworld and macroworld.
In particular, it is assumed that the equilibrium of nucleons in the atomic
nucleus is due to the equality of the forces and energies associated with the
attraction from the strong gravitation and the repulsion from the gravitational torsion field (the given
forces are the main components of the nuclear forces in the gravitational model
of strong interaction).

The combined three-component
symmetry is also the CPT symmetry connecting the properties of particles and antiparticles
with each other.^{[4]} There are works in which the SPФ
symmetry is confirmed.^{[5]} ^{[6]}

As in SPФ
symmetry, in scale
relativity by Laurent Nottale
the fundamental laws of physics cannot involve scales themselves because their
values are arbitrary choices. But this scale relativity is connected mostly
with geometry of space-time in order explaining of physical properties of
particles including their mass and charge. Such approach is close to attempts
of general relativity to explain gravitational force with the help of metric
tensor and space-time curvature. In contrast, the scale relativity of scale dimension is
other example of relativity which extends the special relativity to five
dimensions of space-time.

1.
^{1.0} ^{1.1} Fedosin S.G. Fizika i filosofiia podobiia:
ot preonov do metagalaktik, Perm, (1999-06-09) 544 pp. ISBN
5-8131-0012-1.

2. Sergey Fedosin, The physical theories and infinite hierarchical nesting of
matter, Volume 1, LAP LAMBERT Academic Publishing,
pages: 580, ISBN-13: 978-3-659-57301-9.

3. Comments to the book: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages. ISBN 978-5-9901951-1-0. (in Russian).

4.
Griffiths,
David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons,
Inc. ISBN 0-471-60386-4.

5.
Recami E. Multi-verses, Micro-universes and
Elementary Particles (Hadrons). arXiv:physics/0505149v123,
May 2005.

6.
R. L.
Oldershaw. Discrete Scale
Relativity. Astrophysics and Space Science, Vol. 311, No. 4, pgs. 431-433,
October 2007.

- Similarity
of matter levels
- Quantization
of parameters of cosmic systems
- Discreteness
of stellar parameters
- Hydrogen
system
- Stellar
constants
- Stellar
Dirac constant
- Stellar
Planck constant
- Strong
gravitation
- Gravitational model of strong interaction
- Model of
quark quasiparticles
- Substantial
electron model
- Substantial
neutron model
- Substantial
proton model