having the form
4. Neorelativistic Laws of Conservation and Quantum Mechanics
From the condition of extremum of the functional (3.8) adjusted to four
cyclic coordinates t(t ), x0(t
), y0(t ), z0(t
) it particularly follows that all the four components Et,
Ex, Ey, Ez, of the
invariant energy vector
having the form
(4.1)
are unchanged in the given neorelativistic frame of reference 
in authonomic subspace 
of the whole space R7(gA,
gB), k = 6. So,
this is how the four laws of conservation look like, and here we have
(4.2)
While transition to another neorelativistic frame of reference in subspace 
,
which, actually, is a part of a tangent space R7(
),
k = 6 touching at the given magnitude of 
the psedoriemannean space V7(gA,
gB), k = 6 at its
curent point 
, the four
components (4.2) are transformed in 
in accordance with the Lorentz formulae. At the same time unit vectors 
are co-transformed in such a way that the invariant vector 
of type (4.1) remains unchanged, as an invariant contraction 
.
In the expression for the first constant 
,
appearing in (4.2), one may exclude the expression 
on the basic of the relationship 
in the form (3.7). (We mean that in every time t
it is 
). Instead of constant 
it is convenient to regard here another constant H relating
to 
through a relation 
.
(Of course, instead of funcion 
,
there exist a lot of anoher such, that 
,
when 
; why author takes
exactly this? Analogous question one can ask in case of formulae (3.7).
There exist also a lot of linear expressions of type 
;
why Lagrange takes exactly 
).
After a series of transformations one gets a following expresion for H:
(4.3)
where
(4.4)
Now it follows that the vector 
in space 
is obtained from
the artificial threecomponent “vector” 
,
considered in the given fixed frame of reference in subspace 
,
by the isomorphic mapping of negative unit vectors 
upon "our" usual unit vectors 
in space 
. Same way, the
vector 
is obtained as
a derivative 
, where 
is defined by (3.5). Here also is used isomorphic mapping of type 
.
The right part of the expression for H coincides by its appearance with the classical expression for the sum of kinetic energy T(t) and potential energy U(t), when instead of the invariant relativistic time t the absolute time t is used. This is why we can treat the quantity H as a neorelativistic Hamiltonian function for a closed system of two interacting material particles.
The magnitude H, exactly as in the classical case, does change
while its transition to a new frame of reference because of the change
of a "vector" 
. But the
expression for an internal Hamiltonian function Hin,t,
presented in the form
, (4.5)
is fully invariant, because it depends on the invariant vector 
and on the invariant magnitude dAB(t
) only.
From the first of the expressions (4.4) one can conclude that in the
given fixed frame of reference the direction of a canon velocity
of the centre of mass 
has the same unchanged direction as the canon impulse 
of the system. The direction of "vector" 
remains unchanged in the given frame of reference. But due to there existing
of a variable multiplier appearing to be an invariant function of t
, having the form 
, the
modules of the vector 
may periodically change its “length” (we write “lenghte” because vector 
in space 
has no length),
i.e. the "vector" 
in fixed
frame of reference will periodically and with some frequency w
either slightly increase or slightly decrease, because the magnitude of 
is very small when compared with unity.
If instead of the canon velocity 
one will consider a pseudovelocity 
then he will have to become convicted that the same impulse 
instead of formula (4.4) may be represented in the form
(4.6)
because 
. The quantity
M in (4.6) has the form
(4.7)
This M should be treated as a neorelativistic dynamic mass of
the isolated system of material particles considered in the given frame
of reference where the centre of masses has the pseudovelocity 
.
But the magnitude of 
,
having the form (4.7) with 
corresponds here to the invariant defective mass, which depends only on
the invariant internal energy 
of the whole system. If 
then the deffective mass 
tends to zero, and this is in agreement with mass annigilation.
If some so called elementary particle in reality is a system of two
or three “more elementary particles”, then this annigilation will correspond
to what in quantum mechanics is called "death of a particle", resulting
the action of the "death and birth" operator. Still, by any infinitesimally
small positive magnitude of e and by 
it may happen that the pseudovelocity 
would be as such as 
. Than
on the grounds of (4.7) we will get 
or even M will be more than M00 if 
.
That means that the particle will become "a newly born" one.
Furthermore, if one accepts that in the course of motion of the centre
of mass (of the above described particle-the-system) physically revealed
is not the unchanged pseudovelocity 
but just its variable canon velocity 
(i.e. the invariant time tis realy existing
one but t is, so to say, pseudotime) then this phenomena will explain
the reason of appearing of de'Broglie wave.
From the obvious condition for the modules 
of the energetic vector 
to be invariant and from the later expressions it follows that
. (4.8)
The expressions (4.8), (4.6) and (4.7) offer the sufficient generalisation
of the analogous expressions of STR valid, in fact, only for a single material
particle of mass m0. Particularly, in the analogous to
(4.8) expression, offered by STR, an unchanged mass of rest m0
is used instead of the offered by (4.8) defective mass 
,which can be changing in the course of possible interactions with another
"closed system" (considered as an “elementary” particle). Besides that
in the expression (4.8) may occure such situation when under certain conditions 
.
Then it will denote that the direction of the vector 
in the subspace 
will approach
an isotropic one. In this case four independent variations 
,
which are employed for varying the functional of action S, become
depending among themselves, because 
while
.
Then two independent laws of conservation (4.3) and (4.4) will be united
into one law of conservation, thus expressing "energy balance" in case
of "internal" or "external" energy radiation. This law (4.8) in case of 
by certain additional hypotheses may be given the form of Plank "law of
radiation", presented in quantum mechanics. These are only some of the
results making a "bridge" between neorelativism and quantum-mechanical
indeterminism.